(The problem came to me via Aardvark (vark.com), which is a service where you can send questions to random people, and random people send you questions. Aardvark tries to match up questions with people who may know the answer. I find it interesting to see the questions I receive.)

The brute force solution

If 1 appears a times, 2 appears b times, and 3, 4, and 5 appear c, d, and e times, then the total number of possibilities is 12!/(a!b!c!d!e!) - i.e. the multinomial coeffient.

Then we just need to sum across all the different combinations of a through e. a has to be at least 1, and can be at most 8 (in order to leave room for the other 4 digits). Then b has to be at least 1 and at most 9-a, and so on. Finally, e is whatever is left over.

The Python code is straightforward, noting that range(1, 9) counts from 1 to 8:

from math import factorial
total = 0
for a in range(1, 8+1):
  for b in range(1, 9-a+1):
    for c in range(1, 10-a-b+1):
      for d in range(1, 11-a-b-c+1):
        e = 12-a-b-c-d
        total += (factorial(12) / factorial(a) / factorial(b)
          / factorial(c) / factorial(d) / factorial(e))
print total

(def factorial (n)
  (if (<= n 1) 1
     (* n (factorial (- n 1)))))

(let total 0
  (for a 1 8
    (for b 1 (- 9 a)
      (for c 1 (- 10 a b)
        (for d 1 (- 11 a b c)
          (let e (- 12 a b c d)
            (++ total (/ (factorial 12) (factorial a) (factorial b)
              (factorial c) (factorial d) (factorial e))))))))
  total)

The generalized brute force solution

We can improve the solution by writing code to generate the vectors of arbitrary digits and length. In Python, we can use yield, which magically gives us the vectors one at a time. The vecs function recursively generates the vectors that sum to <= t. The solve routine adds the last entry in the vector to make the sum exactly equal the length. My multinomial function is perhaps overly fancy, using reduce to compute the factorials of all the denominator terms and multiply them together in one go.

from math import factorial

# Generate vectors of digits of length n with sum <= t, and minimum digit 1.
def vecs(n, t):
  if n <= 0:
    yield []
  else:
    for vec in vecs(n-1, t-1):
      for last in range(1, t - sum(vec) + 1):
        yield vec + [last]

def multinomial(n, ks):
  return factorial(n) / reduce(lambda prod, k: prod * factorial(k), ks, 1)
  
# Find number of sequences of digits 1 to n, of length len,
# with each digit appearing at least once
def solve(n, len):
  total = 0
  for vec0 in vecs(n-1, len-1):
    # Make the vector of length n summing exactly to len
    vec = vec0 + [len - sum(vec0)]
    total += multinomial(len, vec)
  return total

In Arc, doing yield would need some crazy continuation code, so I just accumulate the list of vectors with accum and then process it. The code is otherwise similar to the Python code:

(def vecs (n t)
  (if (<= n 0) '(())
    (accum accfn
      (each vec (vecs (- n 1) (- t 1))
       (for last 1 (- t (reduce + vec))
          (accfn (cons last vec)))))))

(def multinomial (n ks)
  (/ (factorial n) (reduce * (map factorial ks))))

(def solve (n len)
  (let total 0
    (each vec0 (vecs (- n 1) (- len 1))
        (++ total (multinomial len
                    (cons (- len (reduce + vec0)) vec0))))
    total))

The dynamic programming solution

The above shows that S(5, 12) = 5 * (S(5, 11) + S(4, 11)). In general, we have the nice recurrence S(n, len) = n * (S(n, len-1) + S(n-1, len-1)). Also, if you only have a single digit, there's only one solution, so S(1, len) = 1. And if you have more digits than length to put them in, there are clearly no solutions, so S(n, len) = 0 if n > len. We can code this up in a nice recursive solution with memoization:

memo = {}
def solve1(digits, length):
  if memo.has_key((digits, length)):
    return memo[(digits, length)]
  if digits > length:
    result = 0
  elif digits == 1:
    result = 1
  else:
    result = digits * (solve1(digits-1, length-1) + solve1(digits, length-1))
  memo[(digits, length)] = result
  return result

Note that the actual number of different function evaluations is pretty small, but the same ones keep getting evaluated over and over exponentially often. The traditional way of fixing this is to use dynamic programming. In this approach, the structure of the algorithm is examined and each value is calculated just one, using previously-calculated values. In our case, we can evaluate S(1, 0), S(2, 0), S(3, 0), and so on. Then evaluate S(2, 1), S(2, 2), S(2, 3) using those values. Then evaluate S(3, 2), S(3, 3), and so on. Note that each evaluation only uses values that have already been calculated. For an introduction to dynamic programming, see Dynamic Programming Zoo Tour.

Rather than carefully looping over the sub-problems in the right order to implement a dynamic programming solution, it is much easier to just use memoization. The trick here is once a subproblem is solved, store the answer. If we need the answer again, just use the stored answer rather than recomputing it. Memoization is a huge performance win. Without memoization solve1(10, 30) takes about 15 seconds, and with it, the solution is almost immediate. (solve1 10 35) takes about 80 seconds.

The Arc implementation is nice, because defmemo creates a function that is automatically memoized.

(defmemo solve1 (digits length)
  (if (> digits length) 0
      (is digits 1) 1
      (* digits (+ (solve1 (- digits 1) (- length 1))
                   (solve1 digits (- length 1))))))

A faster dynamic programming solution

How many sequences are missing 1 digit? S(4, 12) is the number of sequences containing at least one of 1 through 4, and by definition missing the digit 5. Of course, we need to consider sequences missing 4, 3, 2, or 1 as well. Likewise, there are S(4, 12) of each of these, so 5 * S(4, 12) sequences missing exactly one digit.

Next, sequences missing exactly 2 of the digits 1 through 5. S(3, 12) is the number of sequences containing at least one of 1 through 3, so they are missing 4 and 5. But we must also consider sequences missing 1 and 2, 1 and 3, and so on. There are 5 choose 2 pairs of digits that can be missing. Thus in total, (5 choose 2) * S(3, 12) sequences are missing exactly 2 of the digits.

Likewise, the number of sequences missing exactly 3 digits is (5 choose 3) * S(2, 12). The number of sequences missing exactly 4 digits is (5 choose 4) * S(1, 12). And obviously there are no sequences missing 5 digits.

Thus, we get the recurrence S(5, 12) = 5^12 - 5*S(4, 12) - 10*S(3, 12) - 10*S(2, 12) - 5*S(1, 12)

In general, we get the recurrence:

The same dynamic programming techniques and memoization can be applied to this. Note that the subproblems all have the same length, so there are many fewer subproblems than in the previous case. That is, instead of a 2-dimensional grid of subproblems, the subproblems are all in a single row.

In Python, the solution is:

memo = {}
def solve2(digits, length):
  if memo.has_key((digits, length)):
    return memo[(digits, length)]
  if digits > length:
    result = 0
  elif digits == 1:
    result = 1
  else:
    result = digits**length
    for i in range(1, digits):
      result -= sol2(i, length)*choose(digits, i)
  memo[(digits, length)] = result
  return result

(def choose (m n)
  (/ (factorial m) (factorial n) (factorial (- m n))))

(defmemo solve2 (digits length)
  (if (> digits length) 0
      (is digits 1) 1
      (- (expt digits length)
         (reduce +
           (map [* (choose digits _) (solve2 _ length)]
             (range 1 (- digits 1)))))))

The mathematical exponential generating function solution

Discussion

The Python and Arc code was pretty similar. Python's yield construct was convenient. Arc's automatic memoization was also handy. On the whole, both languages were about equally powerful in solving these problems.

Acknowledgement: equations created through CodeCogs' equation editor

My IPv6 explorations are roughly driven by Hurricane Electric's IPv6 Certification levels. To get from "Newbie" to "Enthusiast" you have to have an IPv6 web server. I expected I could just use the web server you're viewing right now (provided through Pair), but Pair (like most web providers) doesn't support IPv6. So I figured I'd just set up my own simple web server.

To make things more interesting, I decided to set up the server on my Sheevaplug, a very small plug-based computer running Linux. (See my previous articles about Arc on the Sheevaplug, and Arduino with the Sheevaplug.) The things I describe will work just as well on a standard Linux or Windows box, though.

Setting up the SixXS IPv6 tunnel on the Sheevaplug was much easier than setting it up on Windows; I just followed the directions. One gotcha: if your clock is wrong, aiccu will silently fail - check /var/log/syslog.

IPv6 with the Python web server

import socket
from BaseHTTPServer import HTTPServer
from SimpleHTTPServer import SimpleHTTPRequestHandler

class MyHandler(SimpleHTTPRequestHandler):
  def do_GET(self):
    if self.path == '/ip':
      self.send_response(200)
      self.send_header('Content-type', 'text/html')
      self.end_headers()
      self.wfile.write('Your IP address is %s' % self.client_address[0])
      return
    else:
      return SimpleHTTPRequestHandler.do_GET(self)

class HTTPServerV6(HTTPServer):
  address_family = socket.AF_INET6

def main():
  server = HTTPServerV6(('::', 80), MyHandler)
  server.serve_forever()

if __name__ == '__main__':
  main()

The key change to support IPv6 is subclassing HTTPServer and setting the address_family to IPv6. The other key change is starting the server with the name '::', which is the IPv6 equivalent of '', and binds to no specific address. Similar changes work with related Python classes such as SocketServer.

IPv6 with Arc's web server

I simply start a server thread on port 80, and then define a simple web operation on the home page (represented as || for obscure reasons).

arc> (thread (serve 80))
arc> (defop || req (pr "Welcome to Arc!  Your IP is " (req 'ip)))

Unfortunately, when I tried running the same Arc code on the Sheevaplug or another Linux box, it refused to work at all with IPv6. The problem turned out to be misconfiguration in the compilation of Racket (the new name for mzscheme). To get IPv6 working, you can recompile Racket following the instructions here. After recompiling, I was able to get Arc working on my Sheevaplug with IPv6. Eventually this fix will get into the official builds.

Note that implementing web pages in Arc requires much less boilerplate than in Python. This isn't too surprising, given that the primary application of Arc is web serving.

Putting the server's IPv6 address into DNS

I had the domain name nlanguages.com available, and wanted to point it at my IPv6 web server. To do this, I went to the DNS Zone File Editor for my nameserver (godaddy.com in my case). I created one AAAA entry for host @ to point to my IPv6 address, and a second AAAA entry for host ipv6 to point to my IPv6 address. The @ provides an entry for my top-level domain (nlanguages.com), and the second provides an entry for ipv6.nlanguages.com. I then tested the DNS entries with a nslookup query, asking for the AAAA record: nslookup -q=aaaa nlanguages.com

The result is that http://ipv6.nlanguages.com will access my IPv6 server (if you have IPv6 access.)

Conclusion

The exponential generating function solution

For our specific problem, we want to exclude solutions where a digit appears 0 times, but we'll stick with the generalized approach of excluding solutions where a digit appears exactly p times, and then we'll set p to 0 later. We'll let d be the range of each digit (e.g. if the digits are 1 through 5, then d=1). And n will be the total number of digits in the sequence (e.g. n=12 in our original problem).

First, we create the generating function for a single digit, say 1. If the digit appears zero times, we represent it by term x^0. If it appears 1 time, we have x^1. If it appears twice, we have x^2/2!. (We divide by 2! because there are 2! equivalent ways the digit can appear twice.) Likewise, if it appears three times, we represent it by x^3/3!. We end up with x^0 + x^1 + x^2/2! + x^3/3! + x^4/4! + ... Now for the crazy part. This is the series expansion of e^x, so we replace this with e^x. Finally we subtract x^p/p!, which is the "forbidden" case. We end up with the generating function e^x - x^p/p!.

If we have d choices for each digit, we multiply together d copies of the generating function, which is of course raising it to the power of d:

To get our solution, if we want to find how many solutions are n digits long, we simply extract the coefficient of x^n/n! from the above generating function, and that gives us the value we want:

Next step: how do we evaluate the above? First, let's expand the right half using the standard binomial formula:

Now we'll take a quick detour through some generating function facts, which I will present without proof.

Important generating function facts

Unless n=0. Then you're looking for the coefficient of x^n in 1. This is simply 1 if n=0, and 0 otherwise.

Or unless n<0. Then the result is 0.

Second important generating function fact: if you're looking for the coefficient of, say, x^9 in an expression multiplied by x^3, then you can look instead for the coefficient of x^6 in the expression, if you use the appropriate scaling:

Solving one term of the summation

If n<kp, then the coefficient is 0.

If d != k, then from the second important fact, the value is:

If k = d, then the exponential drops out and the value is 0 if n != kp (i.e. n != dp), and otherwise:

Putting the summation together

where limit = min(d-1, n/p) (or d-1 if p=0). This ensures that n-kp >=0.

On the other hand, if n = dp, then we need to handle the k=d term.

Note that if n=dp, then n-kp>0, so we don't need to add an extra upper limit on k.

Solving the original problem

Thus, the number of sequences of length 12 consisting of the digits 1 through 5, with each digit appearing at least once, is given by n=12 and d=5:

Happily, this is the same result we got in Part I.

Implementing this in Python

from math import factorial

def solve(d, n, p):
  total = 0
  for k in range(0, d):
    if n-k*p < 0: break
    total += (pow(-1, k) * choose(d, k) *
              factorial(n) / factorial(n-k*p) *
              pow(d-k, n-k*p) / pow(factorial(p), k))
  if n == d*p:
    total += pow(-1, d) * factorial(n) / pow(factorial(p), d)
  return total

import itertools

def bruteforce(d, n, p):
  total = 0
  for seq in itertools.product(range(0, d), repeat=n):
    ok = 1
    for i in range(0, d):
      if len([x for x in seq if x==i]) == p:
        ok = 0
        break
    total += ok
  return total

Conclusion

The Space Shuttle has three independent hydraulic systems to operate engine valves, actuators, landing gear, and so forth during launch and landing. The hydraulic pumps are powered by three Auxiliary Power Units (or APUs), which are hydrazine-powered turbines. Each APU is 88 pounds and produces 135 horsepower (which is about the same horsepower as a Honda Accord).

Hydrazine is a highly-toxic rocket fuel; when exposed to a catalyst, it energetically decomposes into hot gases at 1700°F. It is convenient for applications such as the APU, since it doesn't need oxygen, and the decomposition can be easily started and stopped.

(Click on the image for tons of detailed information.)

Apparently the fuel heaters in APU 1 are not working. Since the hydrazine fuel will freeze at 34°F, each APU has redundant heaters to keep the system above 45°F. Since the heaters are redundant, the Space Shuttle would still be able to operate with the current problem, but would not be able to handle another failure. If the second fuel heater failed, then the fuel would freeze and the APU would not be able to work. Since there are three APUs, even this failure would not be a major problem. But still, you wouldn't want to take off with the heater not working, because losing hydraulic pressure would be a very bad thing.

According to articles, the fuel heater problem is due to a lack of power from the Aft Cabin Load Control Assembly, a switchbox that powers a heater circuit for Auxiliary Power Unit 1. There are three Aft Load Controller Assemblies, as well as many other Load Controller Assemblies. (Sources are inconsistent about whether it is called a Control vs Controller.)

A complex Electrical Power System provides power to all parts of the Shuttle. Three fuel cells (10 kW each) generate 28-volt direct current. The fuel cells feed three main DC power buses, as well as powering AC inverters to feed three AC buses with three-phase, 117-volt, 400-hertz AC power. From the fuel cell, power goes to a Distribution Assembly (DA), then to the aft Power Controller Assemblies, and then to the Aft Load Controller Assemblies.

The Load Control Assemblies contain solid-state switching devices for loads up to 20 amps, and relays for loads up to 135 amps. These switching devices are internally fused.

Reportedly there is a short or other electrical fault in the Aft Load Controller Assembly 2, which is causing the APU heater to fail to operate. The fuel is being drained from Endeavour so technicians can access the assembly and resolve the problem. If I'm interpreting everything correctly, it seems like they'll need to replace one of the internal fuses in the Load Control Assembly.

Complexity and the Space Shuttle

The water spray boiler doesn't have anything to do with the current launch delay, other than being part of the APU, but it provides an interesting example of the complexity of the systems involved. To summarize the complexity along just this one path, the engines require hydraulic pressure, which requires APUs to power the hydraulic pumps, which require a lubricating oil system, which requires a complex boiler system, which requires a own control and monitoring system. And this is just one small sub-path! I'm ignoring equally complex systems such as the APU injector cooling system (more water and pressurized nitrogen), or the APU fuel pump (which for instance, has a catch bottle in case its seals leak, a drain port if the catch bottle overflows, and associated monitoring system).

Conclusion

The images and much of the information above is from the 1988 Shuttle Reference Manual at shuttlepresskit.com. This manual goes into extreme detail and is very interesting (if you find this sort of thing interesting). I should probably make it clear that this posting is based on what I've read; I have no connection with the space program.

P.S. I've found extensive details on the LCA and launch issues are available at nasaspaceflight.com, e.g. Endeavour receives her new LCA – Blown driver examined.

The talk inspired me to read The Art of Computer Programming, Volume 4A: Combinatorial Algorithms. As he described in the talk, he gives a reward check (for 1 hexadecimal dollar) to anyone who finds an error in one of his books, so I set myself the goal of finding an error in the book while on vacation.

After a lot of study, I thought I'd found an error in a diagram, but then I realized I was confused. Next, I found a blatant error in an appendix, but that error had already been discovered and was listed in the errata. Finally I found an error on page 574 that I hoped hadn't been found yet and sent it off to Professor Knuth.

I was delighted to receive my reward check today, which Wikipedia calls "among computerdom's most prized trophies". That's a big exaggeration but nonetheless, I'm happy to get one. Note that the check is from the fictional Bank of San Seriffe to avoid check fraud problems from all the images of his checks on the web.

As for the book itself, it's interesting even if you're not after a reward, although very challenging to read. Volume 4a describes combinatorial algorithms, including more ways of computing permutations and combinations than you can shake a stick at. It also has an extensive discussion of BDDs and ZDDs, which are newish data structures for representing sets. The section on bitwise tricks and techniques is interesting if you like HAKMEM-style tricks such as reversing the bits in an integer through a short sequence of incomprehensible operations.

I have to admit that trying to find an error in a book is a strange vacation goal, but I'm glad I succeeded and Knuth's book taught me some interesting algorithms in the process.

P.S. I was very surprised to see this article on the front page of Hacker News. To answer the questions there and below, the error I found was in volume 4a page 574 (as the memo line on the check shows). The solution to exercise 67 on that page says a particular circuit uses 6 ANDN gates, which I thought should be NAND gates. It gets a bit more complicated because Knuth was referring to the ANDN op code for MMIX, but there was still a mistake with and-not versus not-and. (The other error I noticed was n choose k on page 824, but I checked the errata and it had already been found.)

For those who want technical details, I hacked together the following quick sketch to generate the interesting patterns you can see on the oscilloscope:

void setup()
{
  pinMode(4, OUTPUT);
  pinMode(5, OUTPUT);
}

int state = 1;
int count1 = 0;
int state2 = 1;
int count12 = 0;
int max1 = 20;
int max2 = 200;
int t = 0;

void loop() {

  if (count1-- <= 0) {
    state = 1-state;
    digitalWrite(5, state);
    count1 = max1;
  }
  if (count12-- <= 0) {
    state2 = 1-state2;
    digitalWrite(4, state2);
    count12 = max2;

    if (t++ > 20000) {
      max2 -= 1;
      if (max2 < 1) {
        max2 = 500;
      }   
      t = 0;
    }
  }
}

I should point out the outputs generated this way are rather unstable because many thing can interrupt the timing loop. The above code is provided just in case your are curious. I don't recommend using this approach for anything real; using the PWM timers would yield much cleaner results.

Anyone have other ideas for easy ways to generate cool oscilloscope patterns with an Arduino?

I decided to analyze the game mathematically. I made the simplifying assumption that each team had 50-50 odds of winning each point. I found the analysis interesting, and it turns out to have close ties to Pascal's Triangle, so I'm posting it here in case anyone else is interested.

Volleyball games are scored using the rally point system, which means that one team gets a point on every serve. (Back in the olden days, volleyball used side-out scoring, which meant that only the serving team could get a point. Fortunately, rally point scoring is more mathematically tractable. Rally point scoring also keeps the game advancing faster.) The winner of each match is the best out of three sets (a set is the same as a game). In the league I was watching, the winner of a game is the first team to get 25 points and be ahead by at least 2. (Except if a third tiebreaker game is needed, it only goes to 15 points instead of 25.)

A few cases are easy to analyze mathematically. If we assume each team has a 50-50 chance of scoring each point and the score is tied, each team obviously has a 50% chance of winning the game. (With side-out scoring, it makes a difference which team is serving, but for rally point scoring we avoid that complication.) The second obvious case is if a team has 25 points and the other team has 23 or fewer points, the first team has 100% chance of winning (since they already won).

I will use the notation P(m,n) for the chance of the first team wining if the score is m to n. From above, P(n, n) = 50%, P(25, n) = 100% for n <= 23, and P(m, 25) = 0% for m <= 23.

The chance of winning in other cases can be calculated from the assumption that a team has a 50% chance of winning the point, and a 50% chance of losing: the chance of winning is the average of these two circumstances. Mathematically, we get the simple recurrence:

For instance, if the score is 25-24, if the first team scores, they win. If the second team scores, then the score is tied. In the first (winning) case, the first team wins 100%, and in the second (tied) case, the first team wins 50%. Thus, on average they will win 75% of the time from a 25-24 lead. That is, P(25, 24) = 75%, and by symmetry P(24, 25) = 25%. (Surprisingly, these are the only scores where the requirement to win by 2 points changes the odds.)

Likewise, if the score is 24-23, half the time the first team will score a point and win, and half the time the second team will score a point and tie. So the first team has 1/2 * 100% + 1/2 * 50% = 75% chance of winning, and P(24, 23) = 75%.

More interesting is if the score is 24-22, half the time the first team will score a point and win, and half the time the second team will score, making the score 24-23. We know from above that the first team has a 75% chance of winning from 24-23, so P(24, 22) = 1/2 * 100% + 1/2 * 75% = 87.5%.

We can use the recurrence to work backwards and find the probability of winning from any score. The following table shows the probability of winning for each score. The first team has the score on the left, and the second team has the score on the top.

Table with odds of winning when the score is m to n

Any particular chance of winning can be easily read from the table. For instance, if the score is 15-7, look where row 15 and column 7 meet, and you'll find that the first team has a 94% chance of winning. (This is P(15, 7) in my notation.)

The table illustrates several interesting characteristics of scores. The odds fall away from 50% pretty rapidly as you move away from the diagonal (i.e. away from a tied score). Points matter a lot more near the end of the game, though: you've only got a 1% chance of winning from an 18-24 position, while being six points behind at the beginning (0-6) still gives you an 18% chance of winning. However, a big deficit is almost insurmountable - if you're behind 0-15, you have less than a 1% chance of catching up and winning. (Note that 0% and 100% in the table are not exactly 0% and 100%, because there's always some chance to win or lose.)

Note that each score is the average of the score below and the score to the right - these are the cases where the first team gets the point and the second team gets the point. This corresponds directly to the equation above.

The table could be extended arbitrarily far if neither team gets a two point lead, but those cases are not particularly interesting.

Generating the score table with dynamic programming

(def scorePercent (s1 s2 max)
  (if (and (>= s1 max) (>= s1 (+ s2 2))) 100.
      (and (>= s2 max) (>= s2 (+ s1 2))) 0
      (is s1 s2) 50.
      (/ (+ (scorePercent s1 (+ s2 1) max)
            (scorePercent (+ s1 1) s2 max)) 2)))

arc> (scorePercent 24 22 25)
87.5
arc> (scorePercent 20 22 25)
22.65625

There are a couple ways to improve performance. The hard way of solving the dynamic programming problem without this exponential blowup is to carefully determine an order in which each value can be calculated exactly once by working backwards, until you end up with the desired value. For instance, if the values are calculated going up the columns from right to left, each value can be computed immediately from two values that have already been computed, until we end up efficiently computing the whole table in approximately 25*25 steps. This requires careful coding to step through the table in the right order and to save each result as it is calculated. It's not too hard, but there's a much easier way.

The easy way of solving the problem is with memoization - when an intermediate value is calculated, remember its value in case you need it again, instead of calculating it over and over. With memoization, we can compute the results in any order we want, and automatically each result will only be computed once.

In Arc, memoization can be implemented simply by defining a function with defmemo, which will automatically memoize the results of the function evaluation:

(defmemo scorePercent (s1 s2 max)
  (if (and (>= s1 max) (>= s1 (+ s2 2))) 100.
      (and (>= s2 max) (>= s2 (+ s1 2))) 0
      (is s1 s2) 50.
      (/ (+ (scorePercent s1 (+ s2 1) max)
            (scorePercent (+ s1 1) s2 max)) 2)))

The above function generates a single entry in the table. To generate the full table in HTML with colored cells, I used a simple loop and Arc's HTML generating operations. If you're interested in Arc programming, the full code can be downloaded here.

Mathematical analysis

Pascal's Triangle

The game probabilities come from the triangle of partial sums of binomial coefficients, which is a lesser-known sequence that is easily derived from Pascal's Triangle. This sequence, T(n, k) is formed by summing the first k elements in the corresponding row of Pascal's Triangle. That is, the first element is the first element in the same row of Pascal's triangle, the second is the sum of the first two elements in Pascal's triangle, the third is the sum of the first three, etc.

T - the partial row sums of Pascal's Triangle

With function T defined above, the volleyball winning probabilities are given simply by:

For example, P(23,20) = T(6, 4)/2^6 = 89.0625%, which matches the table.

Intuitively, it makes sense that the probabilities are related to Pascal's Triangle, because each entry in Pascal's Triangle is the sum of the two values above, while each probability entry is the average of the value above and the value to the right in the table. Because taking the average divides by 2 in each step, an exponent of 2 appears in the denominator. The equation can be proved straightfowardly by induction.

The importance of a point

Table with importance of the next point when the score is m to n

The values in the table make intuitive sense. If one team is winning by a lot, one more point doesn't make much difference. But if the scores are close, then each point counts. Each point counts a lot more near the end of the game than at the beginning. The first point only makes an 11% difference in the odds of winning, while the if the score is 23-23, the point makes a 50% difference (75% chance of winning if you get the point vs 25% if you miss the point). This table is sort of a derivative of the first table, showing where the values are changing most rapidly.

The importance of a point as defined above closely matches the behavior of the spectators. If the score is very close at the end of the game, the audience becomes much more animated compared to earlier in the game.

The "importance" is mathematically simpler than the probability of winning derived earlier. If the current score is 25-a, 25-b, then the importance is given by the simple equation:

This can proved straighforwardly from the equation for P(x, y). For example, if the score is 18-12, the importance is C(7+13-2, 6) / 2^(7+13-2) = 18564 / 262144 = 7.08%.

Conclusions

However, I think the model is still useful, since it provides some quantitative answers to the original questions, and confirms some intuitions. In addition, the mathematics turned out to be more interesting than I was expecting, with the surprising connection to Pascal's Triangle.

You may have seen the amazing "Inner Life of a Cell" video, which has spectacular animations of the activities inside a cell as a whilte blood cell responding to inflammation. (There's also a longer narrated version at the BioVisions website.)

Cells are very crowded

Molecules move very very fast

Because cells are so crowded, molecules can't get very far without colliding with something. In fact, a molecule will collide with something billions of times a second and bounce off in a different direction. Because of this, molecules are doing a random walk through the cell and diffusing all around. A small molecule can get from one side of a cell to the other in 1/5 of a second.

As a result of all this random motion, a typical enzyme can collide with something to react with 500,000 times every second. Watching the video, you might wonder how the different pieces just happen to move to the right place. In reality, they are covering so much ground in the cell so fast that they will be in the "right place" very frequently just by chance.

In addition, a typical protein is tumbling around, a million times per second. Imagine proteins crammed together, each rotating at 60 million RPM, with molecules slamming into them billions of times a second. This is what's going on inside a cell.

I'm not blaming the makers of "Inner Life of a Cell" for slowing down the action in a cell. If the video were totally realistic, you wouldn't see anything, since the action would be too fast to even see a blur. But keeping the real speed of the cell in mind can clear up a lot of things, such as how molecules find their way around.

The incredible speed and density of cells also helps explain why it's so difficult to simulate what's happening inside a cell. Even with a supercomputer, there's way too much going on inside a cell to simulate it without major simplifications. Even simulating a single ribosome is a huge computational challenge.

Molecular motors sprint, not walk

It's remarkable enough that cells contain these mechanical walkers, but I recently learned that they aren't plodding along, but actually sprint at 100 steps per second. If you watch the video again, imagine it sped up to that rate.

Cells are powered by electric motors spinning at 40,000 rpm

Mitochondria also provide a fascinating look at just how fast things are inside cells. You may know that mitochondria are the power plants of cells; they take in food molecules, process it through the famous citric acid cycle, and then use oxygen to extract more energy, which is provided to the rest of the cell through molecules of ATP, the cell's "energy currency".

The same Harvard group that created "Inner Life of a Cell" also created a two minute sequel, "Powering the Cell: Mitochondria", which shows mitochondria in action. Around the 1:10 mark, the video shows the rotating ATP synthase enzymes creating glowing ATP molecules.

Watching the leisurely turning enzymes illustrates one of the amazingly complex mechanisms in a cell. But what really surprised me was to learn that in real life, these enzymes spin at up to 700 revolutions per second, which is faster than a jet engine. As I said earlier, cells are really, really fast.

If you're interested in more about this mechanical motor, you'll probably enjoy PDB's molecule of the month article.

Conclusions

The BioVisions videos are very interesting, and I highly recommend watching them. (I also found Molecular Biology of the Cell very interesting and readable, but as it is a 1200+ page text, I don't imagine many people would read it unless they had to. But if you're still reading this article, maybe you're one of those people.)

Previous techniques to display fractional-exponent fractals force the multi-valued complex root to have a single value, which distorts the "true" fractal. By computing all the possible root values in parallel, I determine the "true" form of the fractal.

The following image shows the multi-branch fractal for z^2.5+c. Click on the image (or any of the other images) for a full-size version.

The problem with square roots

The disadvantage of making the square root function discontinuous is the resulting fractatals will have discontinuities. In addition, the appearance of the fractal will change if the arbitrary cut is made in a different location. Thus, in a sense, if you generate a fractal based on z^2.5+c, you're not seeing the real fractal, but artifacts based on arbitrary decisions.

Multi-valued complex functions can be expressed as Riemann surfaces. Instead of being defined on the complex plane, the function is defined on a surface which locally looks like the complex plane, but can have more structure. For instance, the following illustration shows the Riemann surface for the complex square root. Note that for each point (except 0), there are two values for the square root.

How do you compute a complex root?

Because I'm using square roots instead of arbitrary powers (for now), I can use a simpler complex square root formula (details at Wikipedia). The following code takes a complex number x + i * y, and returns the primary square root x1 + i * y1. Note that the negative -(x1 + i * y1) is the other square root.

public static void csqrt(double x, double y, ref double x1, ref double y1)
{
  double m = x * x + y * y;
  double r = Math.Sqrt(m);
  x1 = Math.Sqrt((r + x) / 2.0);
  if (y > 0) {
    y1 = Math.Sqrt((r - x) / 2.0);
  } else {
    y1 = -Math.Sqrt((r - x) / 2.0);
  }
}

Generating the real fractal, with all the branches

To generate the multi-branch fractal, we can test each point to see if any branch converges. However, the result is more interesting if we count how many of the branches converge for each pixel. The result can be anywhere between all of them (in the middle of the fractal) to none of them (outside the fractal).

To decide if a point diverges, I use the standard escape-time technique of checking if the magnitude of z exceeds a bound. If z exceeds the bound, I know the point diverges. If z doesn't exceed the bound by the end of the iterations, I assume it doesn't diverge. This is not necessarily true, which is why the accuracy of a fractal increases as the number of iterations increases. I test for divergence with a bound of magnitude^2 > 4; the exact value of the bound doesn't make much difference as long as it is large enough to guarantee divergence.

The following code shows how the number of convergent branches c is computed recursively. Note that (z25x, z25y) is one of the values of z^2.5, and (-z25x, -z25y) is the other. The key to the multi-branch fractal is that both paths are explored, rather than just one. For a particular pixel, eval(x, y, x, y, 0) is called and the the result is displayed with a suitable colormap.

int eval(double zx, double zy, double cx, double cy, int n)
{
  if (n == max)
  {
    return 1;
  }
  // zsquared is z^2, zroot is sqrt(z), z25 is z^2.5
  double zsquaredx = zx * zx - zy * zy;
  double zsquaredy = 2 * zx * zy;
  double zrootx = 0, zrooty = 0;
  CMath.csqrt(zx, zy, ref zrootx, ref zrooty);
  double z25x = zsquaredx * zrootx - zsquaredy * zrooty;
  double z25y = zsquaredx * zrooty + zsquaredy * zrootx;

  int c = 0;
  // Use the first root
  double newx1 =  z25x + cx;
  double newy1 =  z25y + cy;
  if (newx1 * newx1 + newy1 * newy1 < 4)
  {
    c += eval(newx1, newy1, cx, cy, n + 1);
  }

  // Use the second root
  double newx2 = -z25x + cx;
  double newy2 = -z25y + cy;
  if (newx2 * newx2 + newy2 * newy2 < 4)
  {
    c += eval(newx2, newy2, cx, cy, n + 1);
  }
  return c;
}

The multi-branch fractal is exponentially slow compared to regular escape time fractals. At each iteration, there are two choices of square root, with the consequence that we evaluate 2^n values at each pixel, rather than n with a normal escape-time fractal. Unfortunately, this makes computation very expensive. For a regular escape-time fractal, you might use an iteration depth of hundreds for each pixel. But for the multi-branch fractal, it gets very slow if you go above about 12 iterations.

The above algorithm provides detail of the "inside" of the multi-branch fractal. Note that there is a central region where every branch converges. This isn't too surprising, since if c is sufficiently small, z^2.5 will converge with either branch. Outside this region is a complex area where points just barely converge on some branches, and flipping the branch anywhere will make the point diverge. The eight "snowflake" buds are what I find most interesting; these are regions that diverge for almost all branches, but converge for the "right" branches.

The resulting fractal is obviously symmetric when reflected in the y axis or when rotated by 60 degrees. The proofs are straightforward . In comparison, the regular z^2.5+c fractal is not rotationally symmetric because of the effect of branch cuts.

I believe the multi-branch fractal is connected (unlike the regular z^2.5+c fractal), but I don't have a proof. Interestingly, the fractal has some holes (i.e. is not simply connected). I believe these happen where different branches overlap in such a way that they happen to leave gaps, but on a particular branch (whatever that means) the fractal does not have holes.

The best way to see the holes is to look at the "outside" of the fractal. Instead of counting how many branches converge, the code can be easily modified to determine the maximum number of iterations it takes for a branch to diverge. This is similar to the standard escape-time fractal algorithm with level sets approaching the fractal (except of course, it uses multiple branches). In the image below, you can see dark blue spots inside the fractal near the "snowflakes" that look like image noise. These are actually areas that are outside the fractal with complex structure.

Stepping through iteration-by-iteration

Snowflakes and the Monkey's Paw

Edge detection

Comparison with the "regular" escape-time fractal

Below is the same region of the fractal, displayed using the multi-branch algorithm. Note that there is much more detail provided by the multi-branch algorithm. Also note that the stable self-similar region looks very much like the overall multi-branch fractcal.

The regular fractal is actually a subset of the multi-branch fractal, since each computation in the single-branch fractal will be done in one of the paths of the multi-branch fractal. In the image below, the regular fractal has been overlayed on the multi-branch fractal. Note that the regular fractal exactly falls onto the multi-branch fractal, but is missing most of the branchess. Clicking on the image below will show an animation flipping between the regular fractal and the multi-branch fractal.

One surprising thing is how different the regular and multi-branch fractals look in general. The regular fractal looks much more "Mandelbrot-like" with its sequences of bulbs. I expect that the multi-branch fractal also has a similar structure, but hidden by the overlapping branches.

Another way of seeing how the fractals are related is to overlay the regular fractal with the edge map of the multi-branch fractal. In the following image, both fractals are rendered to a depth of 5 iterations. The regular fractal is displayed in cyan on top of the edges of the multi-branch fractal. Note that the edges match exactly, which is expected from the mathematics. Note also that the regular fractal jumps from curve to curve as it hits the branch cuts, rather than following a single curve. Also note how much of the multi-branch struture is missed by the regular fractal. (The regular fractal is very "blobby" because the iteration count is so low. A higher iteration could would make it too hard to see the edges.)

What's next?

A more interesting exploration is looking at the fractal in three dimensions. In particular, I want to examine the Riemann surface structure in more detail. I think separating out the sheets of the surface will expose much more of the fractal structure, which gets hidden when all the sheets are projected together. I tried to compute the Riemann surface for just two iterations of a similar function using Mathematica but the result is almost incomprehensible:

I think the orbit behavior of individual starting points is a key to understanding this fractal. For instance, which starting points yield a 1-cycle, 2-cycle, and so forth. It's hard to define these cycles exactly, because of the multi-value nature of the fractal. A value can converge to a fixed point on one branch, but not another.

Another mathematical feature that I think is key to understanding the fractal is points where the value goes to zero. The function has many more zeros than I expected, and they are concentrated at "interesting" points of the fractal. The zeros are where the Riemann surfaces come together, and also the points where the boundary forms "loops".

I've been exploring different ways of displaying cycles and zeros, and hope to post images soon, but this post is already very long, so I'll leave those for Part II.

Related work

I started investigating multi-branch fractional exponents many years ago but computers weren't powerful enough at the time, so my investigation didn't get very far. My negative integer exponents were easier to compute and I wrote a paper about those fractals: ``An Investigation of z -> 1/z^n+c,'' Computers & Graphics, 17(5), Sep. 1993, pp 603-607.

Joshua Sasmor has done an extensive investigation of non-integer exponents in his PhD thesis "Fatou, Julia and Mandelbrot Sets for Functions with Non-Integer Exponent" and in the paper "Fractals for functions with rational exponent", Computers and Graphics 28(4). Also of interest is his presentation Julia Set and Branch Cuts which shows the Julia set for z^2.5 - 1/2 + i/10 using inverse iteration instead of escape time. I suspect that his inverse iteration Julia set algorithm yields results similar to applying my multi-branch algorithm to Julia sets. I haven't explored this yet, but if true, it would provide more evidence that the multi-branch algorithm gives the "real" structure of the fratals.

There are several interesting videos that show the evolution of the generalized Mandelbrot set as the exponent changes. A few examples are Mandelbrot Set from 1 to 100 with zoom, Cut along negative X axis, and Cut along negative Y axis. It is interesting to compare the last two, to see how different the results are if the branch cut is placed in a different location.

Conclusion

Looking at the page, I couldn't figure out how CSS and JavaScript could perform the effects on the page - the way the icons moved around, the angles of the icons and the page, or the way the page blurred and appeared. Using Inspect Element in the browser showed a whole bunch of complex divs, but didn't give much clue as to how it works.

I set out to understand in detail how the page works. In the process, I learned a lot of interesting JavaScript and CSS techniques, and I'll share them with you in this article.

So how does the page work?

Generating totally new content from the existing page

The following image shows the part of the page that is generated by the above HTML snippet. You can see a clear mapping from the headings and links in the source above to the text and icons in the image below, but this is all dynamically synthesized. That is, the JavaScript goes through the existing page and for each heading, list item, link, etc, it generates a bunch of entirely new elements. The source above is not rendered at all.

Note that the text in the HTML is displayed below the icons. In addition, the images are not explicitly specified in the HTML, but come from the id attributes, which I explain in more detail below. I also explain later how the class attributes work.

Most web pages use CSS to style the HTML content of the page with the desired page formatting and layout. Worrydream.com uses a very different approach where the existing page contents is input used to generate entirely new page content. This is one of the most interesting techniques of worrydream.com. It's even more impressive that the "template" content renders nicely as a fallback mechanism for Internet Explorer and other unsupported browsers.

The page implements many different JavaScript objects

The Site object is the key object that does most of the work. The main logic in Site scans through the divs in the original document and creates a SiteSection and SiteSectionTitleSet for each div. SiteSection in turn extracts the h2 and ul tags from the original page, creates a SiteStripSegment for each one, and lays them out on the page into a collection of SiteStrips. The Site object also creates many relatively minor components: SiteBackground for the page background, SiteContactSet for the sharing links, SiteDoodle for thie images at the top, SiteHomeButton, shadows, SitePageArrowRegion for the left and right buttons, and the custom scrollbars SiteXScroller and SiteYScroller.

The page format explodes dramatically when an icon is clicked

The SitePageSet becomes visible after an icon is clicked. The SitePageSet manages the blurred page images, most of which extend off to the left and right of the visible screen. To initialize it, SiteSection added a SitePage to the SitePageSet for each element that it processed.

The ContentContainer holds the actual page contents when an icon is clicked. When I first looked at the site, I figured "Oh, the page you click on is loaded into an iframe." It turns out the site is considerably more complex. Simple pages are loaded into an iframe but there is special logic do display Vimeo, movies, images, or embedded HTML.

Pre-computed thumbnails and page images

Similarly, all the page icons are stored in a thumbnail directory. Interestingly, the page images and icon images do not explicitly appear in the source, but are dynamically created. The id attribute on each link is used to generate the URLs. For instance a link with id="ScrollTabs" corresponds to images named ScrollTabs.jpg in the PageImages and ThumbnailImages directories.

The two lessons are: first, sometimes it's better to use an offline brute-force solution such as precomputing blurred images for every page, rather than trying to do it dynamically. And second, you can use a naming system to generate image links, rather than hardcoding them.

Details and techniques used by the site

BVLayer library

Each layer is implemented as a div, normally with another BVLayer div as a parent. This explains why looking at the page structure with Inspect Element just shows a huge number of divs.

Most of the JavaScript objects in worrydream.com are subclasses of BVLayer, with additional functionality implemented in the subclass. For some objects, this additional functionality is just a background image or an event handler, while other objects may be extremely complex, for instance Site, which hold the top-level logic for the page, or SiteSection, which has the layout engine for the page.

The library is complex - almost 1000 lines of JavaScript, and provides many functions. It provides a system to transform layers by moving or rotating them, as well as implementing animation. It includes a framework to handle mouse and touchpad events and provides browser-independent abstractions.

The touch events API

Worrydream.com makes heavy use of touch events. The BVLayer library builds an event system on top of mouse and touch events to detect movements, taps, double taps, and to implement touchable regions. This allows the site to support complex interations both on desktop and touch devices.

Web fonts

The web fonts feature allows a website to download desired fonts, rather than being limited to the standard browser fonts. A simple explanation of @font-face is here or you can read the official W3C CSS Fonts document. Web fonts can be obtained from a variety of sources, such as Google web fonts, Typekit, or Fonts.com.

MooTools

The worrydream.com website makes heavy use of MooTools, but builds complex layers of abstraction on top of it.

More information on MooTools is available at MooTools.net, Wikipedia, the MooTorial tutorial site, or books such as MooTools 1.2 Beginner's Guide.

Object parameters come from the CSS class names

<li class="pageColor-0c0c0c pageWidth-900 pageHeight-970 injectContent-1">

The worrydream.com code uses an interesting properties system (see Site.mergePropertiesFromElement) to turn the "classnames" into object properties. This function parses each hyphen-separated entry in the class attribute, so the above would generate the properties {pageColor: "0c0c0c", pageWidth: 900, pageHeight: 970, injectContent: 1}. These properties are then used to control the new elements that get created on the page.

The properties system provides a few additional features. For instance, it supports string, integer, and percent types - scalePercent-68 turns into {scale: 0.68}. It also allows properties to be inherited from other elements.

This class-based property system is a clever way to pass arbitrary parameters into the page-generation system, while causing these parameters to be ignored by browsers that are rendering the original page. There are over 30 different properties used to change the rendering style for particular sections (filmEdges-1 to give the top strip film-style edges), specify special content (vimeo-23839605 to load a Vimeo video), provide special behavior (magicSubtitle-1 to enable the Easter Egg), specify dimensions (pageWidth-960), and many other functions.

Special support for Ajax URLs

This style of URL is a standard technique for dynamic pages, allowing the use of the back arrow, bookmarks, and sharing links. Normally, if you change a page's URL via JavaScript, the entire page will reload, which is generally undesirable for a dynamic site. However, everything after the pound sign is a URL fragment identifier, which can be modified without reloads. Dynamic pages take advantage of this - they can update the fragment identifier in the URL to reflect page state, without triggering a disruptive page load. The second aspect of fragments in dynamic pages is if the anchor is changed (either by the user including the anchor in a URL or by the back arrow), the JavaScript must update the page to display the "right" content for that anchor.

But what is the exclamation point doing in there? This enables web crawlers to crawl the content, using an Ajax crawling standard. This allows Web crawlers to get the HTML page contents without needing to execute the JavaScript.

This Ajax crawling technique is important for any site that dynamically generates pages with JavaScript and wants the pages to get rendered properly by web crawlers.

CSS transformations

Most modern browsers support 2D transforms through CSS, allowing an element to translate(), rotate(), scale(), skew(), or be transformed through an arbitrary matrix(). (spec) This can be done easily in CSS, for example:

-webkit-transform: rotate(5deg);

Animation through CSS transitions

Transitions can be implemented by setting a duration:

-webkit-transition-duration: 1s;

Hardware Acceleration

Attention to detail

Another hidden feature is the "Easter Egg" that is activated when clicking on the "purveyor of impossible dreams" subtitle at the top of the page. Its implementation is a significant amount of code, but I'll leave the details as a surprise.

The scrollbars at the top and right of the page are not standard browser scrollbars, but custom-implemented scrollbars with their own styling and logic.

The Twitter, RSS, and email icons at the top of the page are not simply icons, but SiteContactSet and SiteContact classes with their own logic, as well as separate implementations for the bottom of the page.

The page background is not just a simple background, but a set of SiteBackground classes implemented from BVLayer. The shadows around the edges of the page are also implemented through BVLayer.

The page contains complex logic to lay out the icons according to the page dimensions and redo the layout if the browser is resized.

My conclusion is that a site like worrydream.com isn't made by simply adding some JavaScript functions to a page, but by implementing every aspect of it with careful attention to the details. I hate to imagine how much time it must have taken to implement the site.

Conclusions

By examining the worrydream.com site in detail, I learned a lot about how to build a complex site out of JavaScript and take advantage of CSS3 functions. I hope that you have also learned some interesting techniques by reading this article.

Instead of a conventional linear power supply, Holt built one like those used in oscilloscopes. It switched the power on and off not sixty times per second, but thousands of times; this allowed it to store the power for far less time, and thus throw off less heat. "That switching power supply was as revolutionary as the Apple II logic board was," Jobs later said. "Rod doesn't get a lot of credit for this in the history books but he should. Every computer now uses switching power supplies, and they all rip off Rod Holt's design."

The history of switching power supplies turns out to be pretty interesting. While most people view the power supply as a boring metal box, there's actually a lot of technological development behind it. There was, in fact, a revolution in power supplies in the late 1960s through the mid 1970s as switching power supplies took over from simple but inefficient linear power supplies, but this was a few years before the Apple II came out in 1977. The credit for this revolution should go to advances in semiconductor technology, specifically improvements in switching transistors, and then innovative ICs to control switching power supplies.[2]

Some background on power supplies

A typical linear power supply uses a bulky power transformer to convert the 120V AC into a low AC voltage, converts this to low voltage DC with a diode bridge, and then uses a linear regulator to drop the voltage to the desired level. The linear regulator is an inexpensive easy-to-use transistor-based component that turns the excess voltage into waste heat to produce a stable output. Linear power supplies are almost trivial to design and build.[6] One big disadvantage however, is they typically waste about 50-65% of the power as heat,[7] often requiring large metal heat sinks or fans to get rid of the heat. The second disadvantage is they are large and heavy. On the plus side, the components (other than the transformer) in linear power supplies only need to handle low voltages and the output is very stable and noise-free.

A switching power supply works on a very different principle: rapidly turning the power on and off, rather than turning excess power into heat. In a switching power supply, the AC line input is converted to high-voltage DC, and then the power supply switches the DC on and off thousands of times a second, carefully controlling the time of the switching so the output voltage averages out to the desired value. Theoretically, no power gets wasted, although in practice the efficiency will be 80%-90%. Switching power supplies are much more efficient, give off much less heat, and are much smaller and lighter than linear power supplies. The main disadvantage of a switching power supply is it is considerably more complex than a linear power supply and much harder to design.[8] In addition, it is much more demanding on the components, requiring transistors that can efficiently switch on and off at high speed under high power. The switches, inductors, and capacitors in a switching power supply can be arranged in several different arrangements (or topologies), with names such as Buck, Boost, Flyback, Forward, Push-Pull, Half Wave, and Full-Wave.[9]

History of switching power supplies to 1977

The computer industry started using switching power supplies in the late 1960s and they steadily grew in popularity. Examples include the PDP-11/20 minicomputer in 1969,[20] the Honeywell H316R in 1970,[21] and Hewlett-Packard's 2100A minicomputer in 1971.[22][23] By 1971, companies using switching regulators "read like a 'Who's Who' of the computer industry: IBM, Honeywell, Univac, DEC, Burroughs, and RCA, to name a few."[21] In 1974, HP used a switching power supply for the 21MX minicomputer,[24] Data General for the Nova 2/4,[25] Texas Instruments for the 960B,[26] and Interdata for their minicomputers.[27] In 1975, HP used an off-line switching power supply in the HP2640A display terminal,[28] Matsushita for their traffic control minicomputer,[29] and IBM for its typewriter-like Selectric Composer[29] and for the IBM 5100 portable computer.[30] By 1976, Data General was using switching supplies for half their systems, Hitachi and Ferranti were using them,[29] Hewlett-Packard's 9825A Desktop Computer[31] and 9815A Calculator[32] used them, and the decsystem 20[33] used a large switching power supply. By 1976, switching power supplies were showing up in living rooms, powering color television receivers.[34][35]

Switching power supplies also became popular products for power supply manufacturers starting in the late 1960s. In 1967, RO Associates introduced the first 20Khz switching power supply product,[36] which they claim was also the first switching power supply to be commercially successful.[37] NEMIC started developing standardized switching power supplies in Japan in 1970.[38] By 1972, most power supply manufacturers were offering switching power supplies or were about to offer them.[5][39][40][41][42] HP sold a line of 300W switching power supplies in 1973,[43] and a compact 500W switching power supply[44] and a 110W fanless switching power supply[45] in 1975. By 1975, switching power supplies were 8% of the power supply market and growing rapidly, driven by improved components and the desire for smaller power supplies for products such as microcomputers.[46]

Switching power supplies were featured in electronics magazines of this era, both in advertisements and articles. Electronic Design recommended switching power supplies in 1964 for better efficiency.[47] The October 1971 cover of Electronics World featured a 500W switching power supply and an article "The Switching Regulator Power Supply". A long article about power supplies in Computer Design in 1972 discussed switching power supplies in detail and the increasing use of switching power supplies in computers, although it mentions some companies were still skeptical about switching power supplies.[5] In 1973, Electronic Engineering featured a detailed article "Switching power supplies: why and how".[42] In 1976, the cover of Electronic Design[48] was titled "Suddenly it's easier to switch" describing the new switching power supply controller ICs, Electronics ran a long article on switching power supplies,[29] Powertec ran two-page ads on the advantages of their switching power supplies with the catchphrase "The big switch is to switchers",[49] and Byte magazine announced Boschert's switching power supplies for microcomputers.[50]

A key developer of switching power supplies was Robert Boschert, who quit his job and started building power supplies on his kitchen table in 1970.[51] He focused on simplifying switching power supplies to make them cost-competitive with linear power supplies, and by 1974 he was producing low-cost power supplies in volume for printers,[51][52] which was followed by a a low-cost 80W switching power supply in 1976.[50] By 1977 Boschert Inc had grown to a 650-person company[51] that producing power supplies for satellites and the F-14 fighter aircraft,[53] followed by power supplies for companies such as HP[54] and Sun. People often think of the present as a unique time for technology startups, but Boschert illustrates that kitchen-table startups were happening even 40 years ago.

The advance of the switching power supply during the 1970s was largely driven by new components.[55] The voltage rating of switching transistors was often the limiting factor,[5] so the introduction of high voltage, high speed, high power transistors at a low cost in the late 1960s and early 1970s greatly increased the popularity of switching power supplies.[5][6][21][16] Transistor technology moved so fast that a 500W commercial power supply featured on the cover of Electronics World in 1971 couldn't have been built with the transistors of just 18 months earlier.[21] Once power transistors could handle hundreds of volts, power supplies could eliminate the heavy 60 Hz power transformer and run "off-line" directly from line voltage. Faster transistor switching speeds allowed more efficient and much smaller power supplies. The introduction of integrated circuits to control switching power supplies in 1976 is widely viewed as ushering in the age of switching power supplies by drastically simplifying them.[10][56]

By the early 1970s, it was clear that a revolution was taking place. Power supply manufacturer Walt Hirschberg claimed in 1973 that "The revolution in power supply design now under way will not be complete until the 60-Hz transformer has been almost entirely replaced."[57] In 1977, an influential power supply book said that "switching regulators were viewed as in the process of revolutionizing the power supply industry".[58]

The Apple II and its power supply

Steve Jobs said that every computer now rips off Rod Holt's revolutionary design.[1] But is this design revolutionary? Was it ripped off by every other computer?

As illustrated above, switching power supplies were in use by many computers by the time the Apple II was released. The design is not particularly revolutionary, as similar simple off-line flyback converters were being sold by Boschert[50][60] and other companies. In the long term, building the control circuitry out of discrete components as Apple did was a dead-end technology, since the future of switching power supplies was in PWM controller ICs.[2] It's surprising Apple continued using discrete oscillators in power supplies even through the Macintosh Classic, since IC controllers were introduced in 1975.[48] Apple did switch to IC controllers, for instance in the Performa[61] and iMacs.[62]

The power supply that Rod Holt designed for Apple was innovative enough to get a patent,[63] so I examined the patent in detail to see if there were any less-obvious revolutionary features. The patent describes two mechanisms to protect the power supply against faults. The first (claim 1) is a mechanism to safely start the oscillator through an AC input. The second mechanism (claim 8) returns excess energy from the transformer to the power source (especially if there is no load) through a clamp winding on the transformer and a diode.

The AC start mechanism was not used by the Apple II,[59] but was used by the Apple II Plus,[64] Apple III,[65] Lisa,[66] Macintosh,[67] and Mac 128K through Classic.[68] I could not find any non-Apple power supplies that use this mechanism,[69] except for a 1978 TV power supply,[70] and it became obsoleted by IC controllers, so this mechanism seems to have had no impact on computer power supply design.

The second mechanism in Holt's patent, the clamp winding and diode to return power in a flyback converter, was used in a variety of power supplies until the mid-1980s and then disappeared. Some examples are the Boschert OL25 power supply (1978),[60] Apple III (1980),[65] Apple's power supply documentation (1982),[59] Tandy hard drive (1982),[71] Tandy 2000 (1983),[72][73] Apple Lisa (1983),[66] Apple Macintosh (1984),[67] Commodore Model B128 (1984),[74] Tandy 6000 (1985),[75] and Mac Plus (1986) to Mac Classic (1990).[68] This flyback clamp winding seems to have been popular with Motorola in the 1980s, appearing in the MC34060 controller IC datasheet,[76] a 1983 designer's guide[77] (where the winding was described as common but optional), and a 1984 application note.[78]

Is this flyback clamp winding the innovation of Holt's that other companies ripped off? I thought so, until I found a 1976 power supply book that described this winding in detail,[35] which ruined my narrative. (Also note that forward converters (as opposed to flyback converters) had used this clamp winding dating back to 1956,[79][80][81] so applying it to a flyback converter doesn't seem like a huge leap in any case.)

One puzzling aspect of power supply discussion in the book Steve Jobs[1] is the statement that the Apple II's power supply is "like those used in oscilloscopes", since oscilloscopes are just one small use for switching power supplies. This statement apparently arose because Holt had previously designed a switching power supply for oscilloscopes,[82] but there's no other connection between Apple's power supply and oscilloscope power supplies.

The biggest impact of the Apple II on the power supply industry was on Astec - the Hong Kong company that manufactured the power supply. Before the Apple II came out, Astec was a little-known manufacturer, selling switching DC-DC inverters. But by 1982, Astec had become the world's top switching-powers-supply manufacturer, almost entirely based on Apple's business, and kept the top spot for a number of years.[83][84] In 1999, Astec was acquired by Emerson,[85] which is currently the second-largest power supply company after Delta Electronics.[86]

A little-known fact about the Apple II power supply is that it was originally assembled by middle-class California housewives as piecework.[83] As demand grew, however, power supply construction was transferred to Astec, even though it cost $7 a unit more. Astec was building 30,000 Apple power supplies monthly by 1983.[83]

Power supplies post-Apple

I will compare the IBM 5150 PC power supply with the Apple II power supply in detail to show their commonalities and differences. They are both off-line flyback power supplies with multiple outputs, but that's about all they have in common. Even though the PC power supply uses an IC controller and the Apple II uses discrete components, the PC power supply uses approximately twice as many components as the Apple II power supply. While the Apple II power supply uses a variable frequency oscillator built out of transistors, the PC power supply uses a fixed-frequency PWM oscillator provided by the NE5560 controller IC. The PC uses optoisolators to provide voltage feedback to the controller, while the Apple II uses a small transformer. The Apple II drives the power transistor directly, while the PC uses a drive transformer. The PC checks all four power outputs against lower and upper voltage limits to make sure the power is good, and shuts down the controller if any voltages are out of spec. The Apple II instead uses a SCR crowbar across the 12V output if that voltage is too high. While the PC flyback transformer has a single primary winding, the Apple II uses an extra primary clamp winding to return power, as well as an another primary winding for feedback. The PC provides linear regulation on the 12V and -5V supplies, while the Apple II doesn't. The PC uses a fan, while the Apple II famously doesn't. It's clear that the IBM 5150 power supply does not "rip off" the Apple II power supply design, as they have almost nothing in common. And later power supply designs became even more different.

The IBM PC AT power supply became a de facto standard for computer power supplies. In 1995, Intel introduced the ATX motherboard specification,[88] and the ATX power supply (along with variants) has become the standard for desktop computer power supplies, with components and designs often targeted specifically at the ATX market.[89]

Computer power systems became more complicated with the introduction of the voltage regulator module (VRM) in 1995 for the Pentium Pro, which required lower voltage at higher current than the power supply could provide directly. To supply this power, Intel introduced the VRM - a DC-DC switching regulator installed next to the processor that reduces the 12 volts from the power supply to the low voltage used by the processor.[90] (If you overclock your computer, it is the VRM that lets you raise the voltage.) In addition, graphics cards can have their own VRM to power a high-performance graphics chip. A fast processor can require 130 watts from the VRM. Comparing this to the half watt of power used by the Apple II's 6502 processor[91] shows the huge growth in power consumption by modern processors. A modern processor chip alone can use more than twice the power of the whole IBM 5150 PC or three times that of the Apple II.

The amazing growth of the computer industry has caused the power consumption of computers to be a cause for environmental concern, resulting in initiatives and regulations to make power supplies more efficient.[92] In the US, Energy Star and 80 PLUS certification[93] pushes manufacturers to manufacture more efficient "green" power supplies. These power supplies squeeze out more efficiency through a variety of techniques: more efficient standby power, more efficient startup circuits, resonant circuits (also known as soft-switching and ZCT or ZVT) that reduce power losses in the switching transistors by ensuring that no power is flowing through them when they turn off, and "active clamp" circuits to replace switching diodes with more efficient transistor circuits.[94] Improvements in MOSFET transistor and high-voltage silicon rectifier technology in the past decade has also led to efficiency improvements.[92]

Power supplies can use the AC line power more efficiently through the technique of power factor correction (PFC).[95] Active power factor correction adds another switching circuit in front of the main power supply circuit. A special PFC controller IC switches this at a frequency of up to 250kHZ, carefully extracting a smooth amount of power from the power supply to produce high-voltage DC, which is then fed into a regular switching power supply circuit.[13][96] PFC also illustrates how power supplies have turned into a commodity with razor-thin margins, where a dollar is a lot of money. Active power factor correction is considered a feature of high-end power supplies, but its actual cost is only about $1.50.[97]

Many different controller chips, designs, and topologies have been used for IBM PC power supplies over the years, both to support different power levels, and to take advantage of new technologies.[98] Controller chips such as the NE5560 and SG3524 were popular in early IBM PCs.[99] The TL494 chip became very popular in a half-bridge configuration,[99] the most popular design in the 1990s.[100] The UC3842 series was also popular for forward converter configurations.[99] The push for higher efficiency has made double forward converters more popular,[101] and power factor correction (PFC) has made the CM6800 controller very popular,[102] since the one chip controls both circuits. Recently, forward converters that generate only 12V have become more common, using DC-DC converters to produce very stable 3.3V and 5V outputs.[94] More detailed information on modern power supplies is available from many sources.[103][104][98][105]

Modern computers contain a surprising collection of switching power supplies and regulators. A modern power supply can contain a switching PFC circuit, a switching flyback power supply for standby power, a switching forward converter to generate 12 volts, a switching DC-DC converter to generate 5 volts, and a switching DC-DC converter to generate 3.3 volts,[94] so the ATX power supply can be considered five different switching power supplies in one box. In addition, the motherboard has a switching VRM regulator to power the processor, and the graphics card has another VRM, for a total of seven switching supplies in a typical desktop computer.

The technology of switching power supplies continues to advance. One development is digital control and digital power management.[106] Instead of using analog control circuits, digital controller chips digitize the control inputs and use software algorithms to control the outputs. Thus, designing the power supply controller becomes as much a matter of programming as of hardware design. Digital power management lets power supplies communicate with the rest of the system for higher efficiency and logging. While these digital technologies are largely used for servers now, I expect they will trickle down to desktop computers eventually.

To summarize, the original IBM 5150 PC power supply was different in almost every way from the Apple II power supply, except both were flyback power supplies. More recent power supplies don't even have that in common with the Apple II. It's absurd to claim that power supplies are ripping off Apple's design.

Famous switching power supply designers

The irony in Steve Jobs' comment that Rod Holt doesn't get a lot of credit is that Rod Holt's work is described in dozens of books and articles about Apple, from Revenge of the Nerds in 1982[109] to 2011's best-selling Steve Jobs biography, which makes Rod Holt easily the most famous power supply designer ever.

Conclusion

If you're interested in power supplies, you might also like my article Tiny, cheap, and dangerous: Inside a (fake) iPhone charger.

Notes and references

[1] Steve Jobs, Walter Isaacson, 2011. Rod Holt's power supply design for the Apple II is discussed on page 74. Note that the description of a switching power supply in this book is rather garbled.

[2] PWM: From a Single Chip To a Giant Industry, Gene Heftman, Power Electronics Technology, pp48-53, Oct 2005.

[3] Preliminary Site Planning: Cray-1 Computer (1975) The Cray-1 used two 200 HP (150KW) motor-generator units to convert 250A 460V input AC into regulated 208V, 400Hz power; each motor-generator was approximately 3900 pounds. The 208V 400Hz power was fed into 36 separate power supplies that used twelve-phase transformers but no internal regulators. These power supplies famously formed the 12 benches around the Cray computer. Photographs of the Cray power components are in Cray-1 S Series Hardware Reference Manual (1981). This high-frequency motor-generator setup may seem strange, but the IBM 370 used a similar setup, see Announcing: IBM System/370 Model 145.

[4] Many larger computers used ferroresonant transformers for regulation. For instance, the power supply for the IBM 1401 computer used a 1250W ferroresonant regulator, see Reference Manual, 1401 Data Processing System (1961), p13. The HP 3000 Series 64/68/70 also used ferroresonant transformers, see Series 64/68/70 Computer Installation Manual (1986), p2-3. DEC used ferroresonant and linear power supplies almost exclusively in the early 1970s, including for the PDP-8/A (picture in "Power-supply choice looms large in sophisticated designs", Electronics, Oct 1976, volume 49, p111).

[5] "Power Supplies for Computers and Peripherals", Computer Design, July 1972, pp 55-65. This long article on power supplies has a lot of discussion of switching power supplies. It describes the buck (series), boost (shunt), push-pull (inverter), and full bridge topologies. The article says that the voltage rating of the switching transistor is the limiting parameter in many applications, but "high voltage, high speed transistors are increasingly available at low cost - and important factor in the more widespread use of switching regulator supplies." It concludes that "Availability of high voltage, high power, switching transistors at moderate prices is providing extra impetus to the use of high efficiency switching regular [sic] supplies. Substantial increase in usage is expected this year."

The article also says, "One of the more controversial topics is the continuing debate on the value of switching type supplies for computer applications, as contrasted with conventional series transistor regulators." This is echoed by some of the vendor comments. One skeptic was Elexon Power Systems, which "does not regard switching regulators as 'the answer.' They plan to disclose an entirely new power supply approach in the near future." Another was Modular Power Inc, who "recommend against switching regulators except when small size, light weight, and high efficiency are primary considerations, as in portable and airborne equipment." Sola Basic Industries said "their engineers are highly skeptical of the long term reliability of switching regulators in practical mass production designs, and predict transistor failure problems."

The "vendor comments" section of the article provides insight into the technology of the power supply industry in 1972: Hewlett Packard "specifies that a major influence today is the ready availability of high speed, high current, low cost transistors accelerated by the current trend toward switching type regulators. The company makes extensive use of switching in a full range of high power designs." Lambda Electronics "makes extensive use of switching regulators for outputs above about 100 W" which are designed to avoid fan cooling. Analog Devices offered precision supplies that use switching techniques for high efficiency. RO Associates "considers growth of switching power supplies to be a major change in the power supply design area". They offered miniature 20-kHz supplies, and low cost 60-kHz supplies. Sola Basic Industries "predict that minicomputer manufacturers will be using more transformerless switching regulators in 1972, for high efficiency and reduced size and weight." Trio Laboratories "indicates that computer and peripheral manufacturers are turning to switching types because pricing is now more competitive and applications are requiring reduced size."

[6] Practical switching power supply design, Marty Brown, 1990, p17.

[7] See the comment section for a detailed discussion of linear power supply efficiency.

[8] Power Supply Cookbook, Marty Brown, 2001. Page 5 discusses the relative development time for different power supply technologies, with a linear regulator taking 1 week of total development time, while a PWM switching regulator takes 8 person-months.

[9] A summary of different topologies is in SMPS overview and Power Supply Topologies. Details are in Microchip AN 1114: SMPS Topologies and Topologies for switched mode power supplies

[10] Lifetime Achievement Award Recipient Robert Mammano, Power Electronics Technology, Sep 2005, pp 48-51. This article describes the Silicon General SG1524 (1975) as the IC that ushered in the age of switching regulators and switch-mode power supplies.

[11] IBM Customer Engineering Reference Manual: 736 Power Supply, 741 Power Supply, 746 Power Distribution Unit (1958), page 60-17. The power supply for the 704 computer consists of three refrigerator-sized cabinets full of vacuum tubes, fuses, relays, mechanical timers, and transformers, using 90.8KVA of power. It used multiple regulation techniques including saturable-reactor transformers and thermistor-based reference voltage. The DC outputs were regulated by a 60 Hz thyratron switching mechanism. Thyratrons are switching vacuum tubes that control the output voltage (much like TRIACs in a common dimmer switch). This can be considered a switching power supply (see Power supplies, switching regulators, inverters, and converters, Irving Gottlieb, pp 186-188).

[12] In their ads, Pioneer Magnetics claims to have designed their first switching power supply in 1958. For instance, see Electronic Design, V27, p216.

[13] Unity power factor power supply, patent 4677366. Pioneer Magnetics filed this patent in 1986 on active power factor correction. See also Pioneer Magnetics' Why PFC? page.

[14] An early switching power supply was described in "A Transistor Power Converter-Amplifier", D. A. Paynter, General Electric Co., Solid-State Circuits Conference, 1959, p90-91. Also see related 1960 patent 3067378, Transistor Converter.

[15] Nondissipative DC to DC Regulator Converter Study, Goddard Space Flight Center, 1964. This survey of transistorized DC-DC converters shows about 20 different switching designs known in the early 1960s. The flyback converter is notably absent. Many other NASA reports on power converters from this time period are available from NASA Technical Reports Server.

[16] A detailed history of switching power supplies appears in S.J. Watkins' M.Phil. thesis Automatic testing of switched-mode power supplies, in the chapter History and Development of Switched-Mode Power Supplies Pre 1987.

[17] Switching Power Supply Development History, TDK Power Electronics World. This provides a very brief history of switching power supplies. TDK also has a surprisingly detailed discussion of switching power supplies in comic form: TDK Power Electronics World.

[18] "Satellite power supply has variable pulse width", Electronics, Feb 1962, p47-49. This article by Elliot Josephson of Lockheed describes a constant-frequency PWM DC-DC converter for satellites. See also patent 3219907 Power Conversion Device.

[19] The Spacecraft Power Supply System, Telstar, 1963. The Telstar satellite obtained power from solar cells, storing the power in NiCad batteries. Efficiency was critical for the satellite, so a DC-to-DC switching voltage regulator was used, with a buck converter converting the variable battery voltage into stable -16 V DC at up to 32 watts at up to 92% efficiency. Because the satellite needed a wide range of voltages, up to 1770 volts for the RF amplifier, additional converters were used. The regulated DC was inverted to AC, fed into transformers, and rectified, to produce the required voltages.

[20] Some PDP models, such as the PDP-11/20 used the H720 power supply (see PDP handbook, 1969). This power supply is described in detail in H720 Power Supply and Mounting Box Manual (1970). The 25 pound power supply uses a power transformer to generate 25V DC, and then switching switching regulators (buck converter) to generate 230W of regulated +5 and -15 volts. Because transistors of the era couldn't handle high voltages, the DC voltage had to be reduced to 25 volts by a large power transformer.

[21] "The Switching Regulator Power Supply", Electronics World v86 October 1971, p43-47. This long article on switching power supplies was featured on the cover of Electronics World. The article is worth looking up, if only for the picture of the F-111 aircraft's switching power supply, which looks so complex that I'd almost expect it to land the plane. The switching power supplies discussed in this article combine a switching DC-DC inverter with a transformer for isolation with a separate buck or boost switching regulator. As a result, the article claims that switching power supplies will always be more expensive than linear power supplies because of the two stages. Modern power supplies, however, combine both stages. The article discusses a variety of power supplies including the 250W switching power supply used by the Honeywell H316R. The article says that the switching regulator power supply had come of age because of new advances in high-speed and high-power transistors. The cover shows a 500W switching power supply that according to the article could not have been built with the transistors available just a year and a half earlier.

[22] A Bantam Power Supply for a Minicomputer, Hewlett-Packard Journal, October 1971. Circuit details in High Efficiency Power Supply patent 3,852,655. This is a 492W off-line power supply using inverters followed by 20V switching regulators.

[23] The HP2100A was introduced in 1971 with a switching power supply (see HP2100A Main Specifications). It is claimed to have the first switching power supply in a minicomputer 25 Years of Real-Time Computing, but the PDP-11/20 was earlier.

[24] A Computer Power System for Severe Operating Conditions, p21, Hewlett-Packard Journal, Oct 1974. The 21MX minicomputer used a 300 W, off-line, switching preregulator to generate regulated 160V DC which was fed into switching dc-to-dc converters.

[25] Data General Technical Manual Nova 2, 1974. The Nova 2/4 used switching regulator to generate 5V and 15V, while the larger 2/10 used a constant voltage transformer. The manual says, "At the higher current losses associated with a computer, the losses [from linear regulators] may become excessive, and for this reason the switching regulator is often used, as in the NOVA 2/4."

[26] Model 960B/980B Computer Maintenance Model: Power Supply The power supply for the Texas Instruments 960B minicomputer used a switching regulator for the 150W 5V supply and linear regulators for the other voltages. The switching regulator consists of two parallel buck converters running at 60kHz, and using 2N5302 NPN switching transistors (introduced in 1969). Because the transistors have a 60V maximum rating, the power supply uses a transformer to drop the voltage to 35V that is fed into the regulator.

[27] M49-024 and M49-026 Switching Regulated Power Supply Instruction Manual, Interdata, 1974. These off-line half-bridge power supplies provided 120W or 250W and were used by the Interdata minicomputers. The switching oscillator used 555 and 556 timer chips.

[28] 2640A Power Supply, Hewlett-Packard Journal, June 1975, p 15. "A switching mode power supply was chosen because of the efficiency and space requirement." Also Data Terminal Technical Information. Another point of interest is its case molded of structural foam (p23) which is very similar to the Apple II's foam-molded plastic case (see page 73 of Steve Jobs), and a couple years earlier.

[29] "Power-supply choice looms large in sophisticated designs", Electronics, Oct 1976, volume 49. p107-114. This long article discusses power supplies including switching power supplies in detail. Note that the Selectric Composer is very different from the popular Selectric typewriter.

[30] IBM 5100 Portable Computer Maintenance Information Manual. The IBM 5100 was a 50-pound portable computer that used BASIC and APL, and included a monitor and tape drive. The power supply is described on page 4-61 as a small, high power, high frequency transistor switching regulator supply that provides 5V, -5V, 8.5V, 12V, and -12V.

[31] The HP 9825A Desktop Computer from 1976 used a switching regulator for the 5V power supply. It also used a foam-molded case, predating the Apple II's; see 98925A Product Design, Hewlett-Packard Journal, June 1976, p5.

[32] Mid-Range Calculator Delivers More Power at Lower Cost, Hewlett-Packard Journal, June 1976 discusses the 5V switching power supply used by the 9815A calculator.

[33] DEC's H7420 power supply is described in decsystem 20 Power Supply System Description (1976). It holds 5 switching regulators to provide multiple voltages, and provides about 700 W. The power supply uses a large transformer to reduce the line voltage to 25V DC, which is passed to the individual switching regulators, which use a buck topology to obtain the desired voltage (+5, -5, +15, or +20).

The decsystem 20 minicomputer was a large system, consisting of three refrigerator-sized cabinets. It took a hefty 21.6 KW of three-phase input power, which is regulated by a combination of switching and linear regulators. It contained seven H7420 power supplies and about 33 individual switching regulator units, as well as a linear regulator for the CPU that used -12V DC at 490A.

[34] Switching power supplies for television receivers seemed to gain momentum around 1975-1976. Philips introduced the TDA2640 for television switched-mode power supplies in 1975. Philips published a book, Switched-mode power supplies in TV receivers in 1976. One drawback of the increasing use of switched-mode power supplies in TVs was they caused interference with amateur radio, as discussed by Wireless World, v82, p52, 1976.

[35] "Electronic Power Control and Digital Techniques", Texas Instruments, 1976. This book discusses switching power supplies in detail.

Chapter IV "Inverter/Converter Systems" describes a simple 120W flyback power supply using a SCR-driven BUY70B power transistor. Of note, this circuit uses an additional primary winding with diode to return unused energy to the source.

Chapter V "Switching Mode Power Supplies" describes the construction of a 5V 800W switching power supply based on an off-line switching shunt regulator followed by a DC-DC converter. It also describes a fairly simple multiple-output flyback power supply controlled by a SN76549, designed for a large screen color television.

[36] Power Electronics Milestones, Power Sources Manufacturers Association.

[37] In 1967 RO Associates introduced the first successful switching power supply product, the 20-kHz, 50-W switching power supply, the Model 210. (See "RO first into switching supplies", Electronic Business, Volume 9, 1983, p36.) They claimed to be the leaders in switching power supplies by 1976. Their 1969 patent 3564384 "High Efficiency Power Supply Apparatus" describes a half-bridge switching power supply that is surprisingly similar to the ATX power supplies popular in the 1990s, except mag amp circuits controlled the PWM rather than the ubiquitous TL494 controller IC.

[38] Nippon Electronic Memory Industry Co (NEMIC, which ended up part of TDK-Lambda) started developing standardized switching power supplies in 1970. History of TDK-Lambda Corporation.

[39] "I forecast that the majority of companies, after several false starts in the power-supplies field will be offering, by the end of 1972, ranges of switching power supplies with acceptable specifications and RFI limits.", page 46, Electronic Engineering, Volume 44, 1972.

[40] Power supply manufacturer Coutant built a power supply called the Minic using "a relatively new switching regulator technique". Instrument practice for process control and automation, Volume 25, p471, 1971.

[41] "Switching power supplies reach the marketplace", p71, Electronics & Power, February 1972. The first "transformerless" switching power supply reached the UK market in 1972, APT's SSU1050, which was an adjustable 500W switching power supply using a half-bridge topology. This 70-pound power supply was considered lightweight compared to linear power supplies.

[42] This article explains switching power supplies in depth, describing the advantages of off-line power supplies. It describes the half-bridge MG5-20 miniature switching power supply built by Advance Electronics. The article says, "The widespread application of microelectronic devices accentuated the sheer bulk of conventional power supplies. Switching converters have now become viable and offer appreciable savings in volume and weight." "Switching power supplies: why and how", Malcolm Burchall, Technical Director,Power Supplies Division, Advance Electronics Ltd. Electronic Engineering, Volume 45, Sept 1973, p73-75.

[43] High-Efficiency Modular Power Supplies Using Switching Regulators, Hewlett-Packard Journal, December 1973, p 15-20. The 62600 series provides 300W using an off-line, half-bridge topology switching power supply. The key was the introduction of 400V, 5A transistors with sub-microsecond switching times. "A complete 300 W switching regulated supply is scarcely larger than just the power transformer of an equivalent series-regulated supply, and it weighs less - 14.5 lbs vs the transformer's 18 lbs."

[44] A High-Current Power Supply for Systems That Use 5 Volt IC Logic Extensively, Hewlett-Packard Journal, April 1975, p 14-19. The 62605M 500W switching power supply for OEMs at 1/3 the size and 1/5 the weight of linear supplies. Uses an off-line half bridge topology.

[45] Modular Power Supplies: Models 63005C and 63315D: This 110W 5V power supply used an off-line forward converter topology, and was convection cooled without a fan.

[46] "The penetration of switching supplies in the US power-supply market will grow from 8% in 1975 to 19% by 1980. This increasing penetration corresponds to the worldwide trend and represents a very high growth rate." Several reasons were given for this predicted growth, including "the availability of better components, reduced overall cost, [...] and the advent of smaller products (such as microcomputers) that make smaller power units desirable." Electronics, Volume 49. 1976. Page 112, sidebar "What of the future?"

[47] Seymour Levine, "Switching Power Supply Regulators For Greater Efficiency." Electronic Design, June 22. 1964. This article describes how switching regulators can increase efficiency from less than 40 percent to more than 90 percent with substantial savings in size, weight, and cost.

[48] The cover of Electronic Design 13, June 21, 1976 says, "Suddenly it's easier to switch. Switching power supplies can be designed with 20 to 50 fewer discrete components than before. A single IC performs all of the control functions required for a push-pull output design. The IC is called a regulating pulse-width modulator. To see if you would rather switch, turn to page 125." Page 125 has an article, "Control a switching power supply with a single LSI circuit" that describes the SG1524 and TL497 switching power supply ICs.

[49] In 1976, Powertec was running two-page ads describing the advantages of switching power supplies, titled "The big switch is to switchers". These ads described the benefits of the power supplies: with twice the efficiency, they gave off 1/9 the heat. They had 1/4 the size and weight. The provided improved reliability, worked under brownout conditions, and could handle much longer power interruptions. Powertec sold a line of switching power supplies up to 800W. They suggested switching power supplies for add-on memory systems, computer mainframes, telephone systems, display consoles, desktop instruments, and data acquisition systems. Pages 130-131, Electronics v49, 1976.

[50] Byte magazine, p100 June 1976 announced the new Boschert OL80 switching power supply providing 80 watts in a two-pound power supply, compared to 16 pounds for a less-powerful linear power supply. It was also advertised in Microcomputer Digest, Feb 1976, p12.

[51] Robert Boschert: A Man Of Many Hats Changes The World Of Power Supplies: he started selling switching power supplies in 1974, focusing on making switching power supplies simple and low cost. The heading claims "Robert Boschert invented the switching power supply", which must be an error by the editor. The article more reasonably claims Boschert invented low-cost, volume-usage switching-mode power supplies. He produced a low-cost switching power supply in volume in 1974.

[52] The Diablo Systems HyTerm Communications Terminal Model 1610/1620 Maintenance Manual shows a 1976 Boschert push-pull power supply and a 1979 LH Research half-bridge power supply.

[53] Boschert's F-14 and satellite experience was touted in ads in Electronic Design, V25, 1977, which also mentioned volume production for Diablo and Qume.

[54] An unusual switching power supply was used by the HP 1000 A600 computer (see Engineering and Reference Documentation) (1983). The 440W power supply provided standard 5V, 12V, and -12V outputs, but also a 25kHz 39V AC output which was used to distribute power to other cards within the system, where it was regulated. The Boschert-designed off-line push-pull power supply used a custom HP IC, somewhat like a TL494.

[55] Multiple 450v switching transistor product lines were introduced in 1971 to support off-line switching power supplies, such as the SVT450 series, the 40850 to 4085 from RCA, and the 700V SVT7000 series.

[56] PWM: From a Single Chip To a Giant Industry, Power Electronics Technology, Oct 2005. This article describes the history of the power supply control IC, from the SG1524 in 1975 to a multi-billion dollar industry.

[57] "The revolution in power supply design now under way will not be complete until the 60-Hz transformer has been almost entirely replaced.", Walter Hirschberg, ACDC Electronics Inc, CA. "New components spark power supply revolution", p49, Canadian Electronics Engineering, v 17, 1973.

[58] Switching and linear power supply, power converter design, Pressman 1977 "Switching regulators - which are in the process of revolutionizing the power supply industry because of their low internal losses, small size, and weight and costs competitive with conventional series-pass or linear power supplies."

[59] Multiple Apple power supplies are documented in Apple Products Information Pkg: Astec Power Supplies (1982). The Apple II Astec AA11040 power supply is a simple discrete-component flyback power supply with multiple outputs. It uses a 2SC1358 switching transistor. The 5V output is compared against a Zener diode and control feedback and isolated through a transformer with two primary windings and one secondary. It uses the flyback diode clamp winding.

The AA11040-B (1980) has substantial modifications to the feedback and control circuitry. It uses a 2SC1875 switching transistor and a TL431 voltage reference. The AA11040-B was apparently used for the Apple II+ and Apple IIe (see hardwaresecrets.com forum). The silk screen on the power supply PCB says it is covered by patent 4323961, which turns out to be "Free-running flyback DC power supply" by Elliot Josephson and assigned to Astec. The schematic in this patent is basically a slightly simplified AA11040-B. The feedback isolation transformer has one primary and two secondary windings, the reverse of the AA11040. This patent is also printed on the Osborne 1 power supply board (see Osborne 1 teardown), which also uses the 2SC1875.

The Apple III Astec AA11190 uses the flyback diode clamp winding, but not Holt AC startup circuit. It uses a 2SC1358 switching transistor; the feedback / control circuitry is very similar to the AA11040-B. The Apple III Profile disk drive power supply AA11770 used the flyback diode clamp winding, a 2SC1875 switching transistor; again, the feedback / control circuitry is very similar to the AA11040-B. The AA11771 is similar, but adds another TL431 for the AC ON output.

Interestingly in this document Apple reprints ten pages of HP's "DC Power Supply Handbook" (1978 version used by Apple) to provide background on switching power supplies.

[60] Flyback converters: solid-state solution to low-cost switching power supply, Electronics, Dec 1978. This article by Robert Boschert describes the Boschert OL25 power supply, which is a very simple discrete-component 25W flyback power supply providing 4 outputs. It includes the flyback diode clamp winding. It uses a TL430 voltage reference and optoisolator for feedback from the 5V output. It uses a MJE13004 switching transistor.

[61] The Macintosh Performa 6320 used the AS3842 SMPS controller IC, as can be seen in this picture. The AS3842 is Astec's version of the UC3842 current-mode controller, which was very popular for forward converters.

[62] Power supply details for the iMac are difficult to find, and there are different power supplies in use, but from piecing together various sources, the iMac G5 appears to use the TDA4863 PFC controller, five 20N60C3 MOS power transistors, SG3845 PWM controller, TL431 voltage references, and power supervision by a WT7515 and LM339. A TOP245 5-pin integrated switcher is also used, probably for the standby power.

[63] DC Power Supply, #4130862 which was filed in February 1978 and issued in December 1978. The power supply in the patent has some significant differences from the Apple II power supply built by Astec. Much of the control logic is on the primary side in the patent and the secondary side in the actual power supply. Also, the feedback coupling is optical in the patent and uses a transformer in the power supply. The Apple II power supply doesn't use the AC feedback described in the patent.

[64] A detailed discussion of the Apple II Plus power supply is at applefritter.com. The description erroneously calls the power supply a forward converter topology, but it is a flyback topology. Inconveniently, this discussion doesn't match the Apple II Plus power supply schematics I've found. Notable differences: the schematic uses a transformer to provide feedback, while the discussion uses an optoisolator. Also, the power supply in the discussion uses the AC input to start the transistor oscillation, while the schematic does not.

[65] Apple III (1982). This Apple III power supply (050-0057-A) is almost totally different from the Apple III AA11190 power supply. This is a discrete-component flyback power supply with an MJ8503 switching transistor driven by a SCR, the flyback clamp winding, and 4 outputs. It uses the Holt AC startup circuit. The switching feedback monitors the -5V output with a 741 op amp and is connected via a transformer. It uses a linear regulator on the -5V output.

[66] Apple Lisa (1983). Another discrete-component flyback power supply, but considerably more complex than the Apple II, with features such as standby power, remote turn-on via a triac, and a +33V output. It uses a MJ8505 NPN power transistor driven by a SCR for switching. It uses the Holt AC startup circuit. The switching feedback monitors the +5V sense (compared to the linearly-regulated -5V output) and is connected via a transformer.

[67] Macintosh power supply. This flyback power supply uses the diode clamp winding and the Holt AC startup circuit. It uses a 2SC2335 switching transistor controlled by a discrete oscillator. The switching feedback monitors the +12V output using Zener diodes and a LM324 op amp and is connected through an optoisolator.

[68] Mac 128K schematic, Mac Plus discussion. This flyback power supply uses the diode clamp winding and Holt AC startup circuit. It uses a 2SC2810 switching transistor controlled by discrete components. The switching feedback monitors the 12V output and is connected via an optoisolator. Interestingly, this document claims that the power supply was notoriously prone to failure because it didn't use a fan. The Mac Classic power supply appears to be identical.

[69] TEAM ST-230WHF 230 watt switch mode power supply. This schematic is the only non-Apple computer power supply I have found that feeds raw AC into the drive circuit (see R2), but I'm sure this is just a drawing error. R2 should connect to the output of the diode bridge, not the input. Compare with R3 in an almost-identical drive circuit in this ATX power supply.

[70] Microprocessors and Microcomputers and Switching Mode Power Supplies, Bryan Norris, Texas Instruments, McGraw-Hill Company, 1978. This book describes switching power supplies for televisions that use the AC signal to start oscillation.

[71] Tandy hard drive power supply (Astec AA11101) . This 180W flyback power supply uses the diode clamp winding. It uses a 2SC1325A switching transistor. The oscillator uses discrete components. Feedback from the 5V rail is compared against a TL431 voltage reference, and feedback uses a transformer for isolation.

[72] Tandy 2000 power supply (1983). This 95W flyback power supply uses the MC34060 controller IC, a MJE12005 switching transistor, and has the flyback clamp winding. It uses a MC3425 to monitor the voltage, has a linear regulator for the -12V output, and provides feedback based on the 5V output compared with a TL431 reference, feed through an optoisolator. The 12V output uses a mag amp regulator.

[73] The Art of Electronics has a detailed discussion of the Tandy 2000 power supply (p 362).

[74] Commodore Model B128. This flyback power supply uses the diode clamp winding. It uses a MJE8501 switching transistor controlled by discrete components, and the switching feedback monitors the 5V output using a TL430 reference and an isolation transformer. The 12V and -12V outputs use linear regulators.

[75] Tandy 6000 (Astec AA11082). This 140W flyback power supply uses the diode clamp winding. The circuit is a rather complex discrete circuit, since it uses a boost circuit described in Astec patent 4326244, also by Elliot Josephson. It uses a 2SC1325A switching transistor. It has a slightly unusual 24V output. One 12V output is linearly regulated by a LM317, and the -12V output controlled by a a MC7912 linear regulator, but the other 12V output does not have additional regulation. Feedback is from the 5V output, using a TL431 voltage source and an isolation transformer. There's a nice photo of the power supply here.

[76] MC34060 controller IC (1982) documentation.

[77] The Designer's Guide for Switching Power Supply Circuits and Components, The Switchmode Guide, Motorola Semiconductors Inc., Pub. No. SG79, 1983. R J. Haver. For the flyback converter, the clamp winding is described as optional, but "usually present to allow energy stored in the leakage reactance to return safely to the line instead of avalanching the switching transistor."

[78] "Insuring Reliable Performance from Power MOSFETs", Motorola application note 929, (1984) shows a flyback power supply using the MC34060 with the clamp winding and diode. This can be downloaded from datasheets.org.uk.

[79] For more information on forward converters, see The History of the Forward Converter, Switching Power Magazine, vol.1, no.1, pp. 20-22, Jul. 2000.

[80] An early switching converter with an diode clamp winding was patented in 1956 by Philips, patent 2,920,259 Direct Current Converter.

[81] Another patent showing an energy-return winding with diode is Hewlett-Packard's 1967 patent 3,313,998 Switching-regulator power supply having energy return circuit

[83] "Switching supplies grow in the bellies of computers", Electronic Business, volume 9, June 1983, p120-126. This article describes the business side of switching power supplies in detail. While Astec was the top switching power supply manufacturer, Lambda was the top AC-DC power supply manufacturer because it sold large amounts of both linear and switching power supplies.

[84] "Standards: A switch in time for supplies", Electronic Business Today, vol 11, p74, 1985. This article states that Astec is the world's leading merchant maker of power supplies and the leader in switching power supplies. Astec grew almost solely on supplying power supplies to Apple. This article also names the "big 5" power supply companies as ACDC, Astec, Boschert, Lambda, and Power One.

[85] Astec Becomes Wholly-Owned Subsidiary of Emerson Electric, Business Wire, April 7, 1999.

[86] An industry report on the top power supply companies as of 2011 is Power Electronics Industry News, v 189, March 2011, Micro-Tech Consultants. Also, Power-Supply Industry Continues March to Consolidation, Power Electronics Technology, May 2007 discusses various consolidations.

[87] The SAMS photofact documentation for the IBM 5150 provides a detailed schematic of the power supply.

[88] Wikipedia provides an overview of the ATX standard. The official ATX specification is at formfactors.org.

[89] ON Semiconductor has ATX power supply Reference Designs, as does Fairchild. Some ICs designed specifically for ATX applications are SG6105 Power Supply Supervisor + Regulator + PWM, NCP1910 High Performance Combo Controller for ATX Power Supplies, ISL6506 Multiple Linear Power Controller with ACPI Control Interfaces, and SPX1580 Ultra Low Dropout Voltage Regulator.

[90] Intel introduced the recommendation of a switching DC-DC converter next to the processor in Intel AP-523 Pentium Pro Processor Power Distribution Guidelines, which provides detailed specifications for a voltage regulator module (VRM). Details of a sample VRM are in Fueling the megaprocessor - A DC/DC converter design review featuring the UC3886 and UC3910. More recent VMR specifications are in Intel Voltage Regulator Module (VRM) and Enterprise Voltage Regulator-Down (EVRD) 11 Design Guidelines (2009).

[91] The R650X and R651X microprocessors datasheet specifies typical power dissipation of 500mW.

[92] Power Conversion Technologies for Computer, Networking, and Telecom Power Systems - Past, Present, and Future, M. M. Jovanovic, Delta Power Electronics Laboratory, International Power Conversion & Drive Conference (IPCDC) St. Petersburg, Russia, June 8-9, 2011.

[93] The 80 Plus program is explained at 80 PLUS Certified Power Supplies and Manufacturers, which describes the various levels of 80 PLUS: Bronze, Silver, Gold, Platinum, and Titanium. The base level requires efficiency of at least 80% under various loads, and the higher levels require increasingly high efficiencies. The first 80 PLUS power supplies came out in 2005.

[94] A few random examples of power supplies that first generate just 12V, and use DC-DC converters to generate 5V and 3.3V outputs from this: ON Semiconductor's High-Efficiency 255 W ATX Power Supply Reference Design (80 Plus Silver), NZXT HALE82 power supply review, SilverStone Nightjar power supply review.

[95] Power supplies only use part of the electricity fed through the power lines; this gives them a bad "power factor" which wastes energy and increases load on the lower lines. You might expect that this problem arises because switching power supplies turn on and off rapidly. However, the bad power factor actually comes from the initial AC-DC rectification, which uses only the peaks of the input AC voltage.

[96] Power Factor Correction (PFC) Basics, Application Note 42047, Fairchild Semiconductor, 2004.

[97] Right Sizing and Designing Efficient Power Supplies states that active PFC adds about $1.50 to the cost of a 400W power supply, active clamp adds 75 cents, and synchronous rectification adds 75 cents.

[98] Many sources of power supply schematics are available on the web. Some are andysm, danyk.wz.cz, and smps.us. A couple sites that provide downloads of power supply schematics are eserviceinfo.com and elektrotany.com.

[99] See the SMPS FAQ for information on typical PC power supply design. The sections "Bob's description" and "Steve's comments" discuss typical 200W PC power supplies, using a TL494 IC and half-bridge design.

[100] A 1991 thesis states that the TL494 was still used in the majority of PC switched mode power supplies (as of 1991). The development of a 100 KHZ switched-mode power supply (1991). Cape Technikon Theses & Dissertations. Paper 138.

[101] Introduction to a two-transistor forward topology for 80 PLUS efficient power supplies, EE Times, 2007.

[102] hardwaresecrets.com states that the CM6800 is the most popular PFC/PWM controller. It is a replacement for the ML4800 and ML4824. The CM6802 is a "greener" controller in the same family.

[103] Anatomy of Switching Power Supplies, Gabriel Torres, Hardware Secrets, 2006. This tutorial describes in great detail the operation and internals of PC power supplies, with copious pictures of real power supply internals. If you want to know exactly what each capacitor and transistor in a power supply does, read this article.

[104] ON Semiconductor's Inside the power supply presentation provides a detailed, somewhat mathematical guide to how modern power supplies work.

[105] SWITCHMODE Power Supply Reference Manual, ON Semiconductor. This manual includes a great deal of information on power supplies, topologies, and many sample implementations.

[106] Some links on digital power management are Designers debate merits of digital power management, EE Times, Dec 2006. Global Digital Power Management ICs Market to Reach $1.0 Billion by 2017. TI UCD9248 Digital PWM System Controller. Free ac/dc digital-power reference design has universal input and PFC, EDN, April 2009.

[107] Rudy Severns, Lifetime Achievement Award Winner, Power Electronics Technology, Sept 2008, p40-43.

[108] Where Have All the Gurus Gone?, Power Electronics Technology, 2007. This article discusses the contributions of many power supply innovators including Sol Gindoff, Dick Weise, Walt Hirschberg, Robert Okada, Robert Boschert, Steve Goldman, Allen Rosenstein, Wally Hersom, Phil Koetsch, Jag Chopra, Wally Hersom, Patrizio Vinciarelli, and Marty Schlecht.

[109] The story of Holt's development of the Apple II power supply first appeared in Paul Ciotti's article Revenge of the Nerds (unrelated to the movie) in California magazine, in 1982.

Detailed explanation

Isolation and safety

USB charging protocols

Advances in AC power adapters

Conclusions

Notes and references

How it works

In more detail, the AC line power is first converted to high voltage DC[1] by a diode bridge. The DC is switched on and off by a transistor controlled by a power supply controller IC. The chopped DC is fed into a flyback[2] transformer which converts it into low voltage AC. Finally, this AC is converted into DC which is filtered to obtain smooth power free of interference, and this power is output through the USB jack. A feedback circuit measures the output voltage and sends a signal to the controller IC, which adjusts the switching frequency to obtain the desired voltage.

The side view above shows some of the larger components. The charger consists of two circuit boards, slightly under one inch square each.[3] The top board is the primary, which has the high voltage circuitry, and the bottom board, the secondary, has the low voltage output circuitry. The input AC first passes through a fusible resistor (striped), which will break the circuit if there is a catastrophic overload. The input AC is converted to high-voltage DC, which is smoothed by the two large electrolytic capacitors (black with white text and stripe) and the inductor (green).

Next, the high voltage DC is chopped at high frequency by a MOSFT switching transistor, which is the large three-pinned component in the upper left. (The second transistor clamps voltage spikes, as will be explained below.) The chopped DC goes to the flyback transformer (yellow, barely visible behind the transistors), which has low voltage output wires going to the secondary board below. (These wires were cut during disassembly.) The secondary board converts the low voltage from the transformer to DC, filters it, and then feeds it out through the USB connector (the silver rectangle in the lower left). The gray ribbon cable (just barely visible on the lower right under the capacitor) provides feedback from the secondary board to the controller IC to keep the voltage regulated.

The picture above shows the flyback transformer (yellow) more clearly, above the USB jack. The large blue component is a special "Y" capacitor[4] to reduce interference. The controller IC is visible above the transformer on the top of the primary board.[5]

The circuit in detail

The primary

The primary circuit board is packed with surface mounted components on both sides. The inner side (diagram above) holds the large components while the outer side (diagram below) has the controller IC. (The large components were removed in the diagrams, and are indicated in italics.) Input power is connected to the corners of the board, goes through the 10&ohm; fusible resistor, and is rectified to DC by the four diodes. Two R-C snubber circuits absorb EMI interference created by the bridge.[6] The DC is filtered by the two large electrolytic capacitors and the inductor, producing 125-340V DC. Note the thickness of the circuit board traces connecting these capacitors and other high-current components compared to the thin control traces.

The power supply is controlled by an 8-pin STMicrosystems L6565 quasi-resonant SMPS controller chip.[7] The controller IC drives the MOSFET switching transistor which chops the high voltage DC and feeds it into the primary winding of the flyback transformer. The controller IC takes a variety of inputs (secondary voltage feedback, input DC voltage, transformer primary current, and transformer demagnetization sensing) and adjusts the switching frequency and timing to control the output voltage through complex internal circuitry. The current sense resistors let the IC know how much current is flowing through the primary, which controls when the transistor should be turned off.

The second switching transistor, along with some capacitors and diodes, is part of a resonant clamp circuit that absorbs voltage spikes on the transformer. This unusual and innovative circuit is patented by Flextronics.[8][9]

The controller IC needs DC power to run; this is provided by an auxiliary power circuit consisting of a separate auxiliary winding on the transformer, a diode, and filter capacitors. Since the controller IC needs to be powered up before the transformer can start generating power, you might wonder how this chicken-and-egg problem gets solved. The solution is the high-voltage DC is dropped to a low level through startup power resistors to provide the initial power to the IC until the transformer starts up. The auxiliary winding is also used by the IC to sense transformer demagnitization, which indicates when to turn on the switching transistor.[7]

The secondary

The USB output also has specific resistances connected to the data pins to indicate to the iPhone how much current the charger can supply, through a proprietary Apple protocol.[10] An iPhone displays the message "Charging is not supported with this accessory" if the charger has the wrong resistances here.

The secondary board contains a standard switching power supply feedback circuit that monitors the output voltage with a TL431 regulator and provides feedback to the controller IC through the optocoupler. A second feedback circuit shuts down the charger for protection if the charger overheats or the output voltage is too high.[11] A ribbon cable provides this feedback to the primary board.

Isolation

You might expect the primary board to have the dangerous voltages and the secondary board to have the safe voltages, but the secondary board consists of two areas: the hazardous area connected to the primary board, and the low-voltage area. The isolation boundary between these areas is about 6mm in the Apple charger and can be seen in the above diagram. This isolation boundary ensures that dangerous voltages cannot reach the output.

There are three types of components that cross the isolation boundary, and they must be specially designed for safety. The key component is the transformer, which provides a way for electrical power to reach the output without a direct electrical connection. Internally, the transformer is extensively insulated, as will be shown below. The second component type is the optocouplers, which send the feedback signal from the secondary to the primary. Internally, the optocoupler contains a LED and a photo-transistor, so the two sides are connected only by light, not by an electrical circuit. (Note the silicone insulation on the secondary side of the optocouplers to provide extra safety.) Finally, the Y capacitor is a special type of capacitor[4] that lets EMI (electromagnetic interference) escape between the high-voltage primary and the low-voltage secondary.

The above picture shows some of the isolation techniques. The secondary board (left) has the blue Y capacitor. Note the lack of components in the middle of the secondary board, forming an isolation boundary. The components on the right of the secondary board are connected to the primary board by the gray ribbon cable so they are at potentially high voltages. The other connection between the boards is the pair of wires from the flyback transformer (yellow) delivering the output power to the secondary board; these were cut to separate the boards.

Schematic

These circuits are very small

Transformer teardown

The transformer measures roughly 1/2" by 1/2" by 1/3". Inside, the transformer has three windings: a high voltage primary input winding, a low voltage auxiliary winding to provide power to the control circuits, and a high-current low voltage output winding. The output winding is connected to the black and white wires coming out of the transformer, while the other windings are connected to the pins attached to the bottom of the transformer.

The outside of the transformer has a couple layers of insulating tape. The second line appears to start with "FLEX", for Flextronics. Two grounded strands of wire are wrapped around the outside of the transformer to provide shielding.

After removing the shielding and the tape, the two halfs of the ferrite core can be removed from the windings. Ferrite is a rather brittle ceramic material, so the core broke during removal. The core surrounds the windings and contains the magnetic fields. Each core piece is roughly 6mm x 11mm x 4mm; this style of core is known as EQ. The circular center section is very slightly shorter than the ends, creating a small air gap when the core pieces are put together. This 0.28mm air gap stores the magnetic energy for the flyback transformer.

Underneath the next two layers of tape is a 17-turn winding of thin varnished wire, which I think is another shield winding to return stray interference to ground.

Underneath the shield and another two layers of tape is the 6-turn secondary output winding that is connected to the black and white wires. Note that this winding is heavy-gauge wire, since it is feeding the 1A output. Also note that the winding is triple-insulated, which is a UL safety requirement to ensure that the high voltage primary remains isolated from the output. This is one place where cheap chargers cheat - they just use regular wire instead of triple-insulated, and also skimp on the tape. The result is there's not much protecting you from high voltage if there's an insulation flaw or power surge.

Under the next double layer of tape is the 11-turn heavy gauge primary power winding, that powers the controller IC. Since this winding is on the primary side, it doesn't need to be triple insulated. It's just insulated with a thin layer of varnish.

Under the final double layer of tape is the primary input winding, which is 4 layers of approximately 23 turns each. This winding receives the high voltage input. Since the current is very low, the wire can be very thin. Because the primary has about 15 times as many turns as the secondary winding, the secondary voltage will be 1/15 the primary voltage, but 15 times the current. Thus, the transformer converts the high voltage input to low voltage, high current output.

The final picture shows all the components of the transformer; left to right shows the layers from the outside tape to the innermost winding and bobbin.

Apple's huge profit margins

Apple's 2008 charger safety recall

In 2008, Apple recalled the iPhone chargers due to a defect that the AC prongs could fall off the charger and get stuck in an outlet.[15] The faulty chargers had the prongs attached with what was described as little more than glue and "wishful thinking".[15] Apple replaced the chargers with a redesigned model indicated by the green dot marking shown above (which counterfeit chargers inevitably imitate).

I decided to see what safety improvements Apple made in the replacement charger, and compare with other similar chargers. I tried pulling out the prongs of the Apple charger, a Samsung charger, and a counterfeit charger. The counterfeit prongs came out with a tug with pliers, as there's basically nothing anchoring them but friction. The Samsung prongs took a lot of pulling and twisting with pliers, since they have little metal tabs holding them in place, but eventually they came out.

When I moved on to the Apple charger, the prongs didn't budge, even with my hardest pulling with pliers, so I got out the Dremel and ground through the case to find out what was holding the prongs. They have large metal flanges embedded in the plastic of the case, so there's no way a prong can come loose short of the destruction of the charger. The photo shows the Apple plug (note the thickness of plastic removed from the right half), the prong from the counterfeit charger held in only by friction, and the Samsung prong held in by small but sturdy metal tabs.

I'm impressed with the effort Apple put into making the charger more safe after the recall. They didn't just improve the prongs slightly to make them more secure; clearly someone was told to do whatever it takes to make sure there's absolutely no way the prongs could possibly come loose again under any circumstances.

What makes Apple's iPhone charger special

The input AC is filtered thorugh a tiny ferrite ring on the plastic case (see photo below). The diode bridge output is filtered by two large capacitors and an inductor. Two other R-C snubbers filter the diode bridge, which I've only seen elsewhere in audio power supplies to prevent 60Hz hum;[6] perhaps this enhances the iTunes listening experience. Other chargers I disassembled don't use a ferrite ring and usually only a single filter capacitor. The primary circuit board has a grounded metal shield over the high-frequency components (see photo), which I haven't seen elsewhere. The transformer includes a shield winding to absorb EMI. The output circuit uses three capacitors including two relatively expensive tantalum ones[14] and an inductor for filtering, when many supplies just use one capacitor. The Y capacitor is usually omitted from other designs. The resonant clamp circuit is highly innovative.[9]

Apple's design provides extra safety in a few ways that were discussed earlier: the super-strong AC prongs, and the complex over-temperature / over-voltage shutdown circuit. Apple's isolation distance between primary and secondary appears to go beyond the regulations.

Conclusions

If you're interested in power supplies, please take a look at my other articles: tiny, cheap, dangerous: Inside a (fake) iPhone charger, where I disassemble a $2.79 iPhone charger and discover that it violates many safety rules; don't buy one of these. Also take a look at Apple didn't revolutionize power supplies; new transistors did which examines the history of switching power supplies. To see Apple's adapter disassembled, check out videos created by scourtheearth and Ladyada. Finally, if you have an interesting charger lying around that you don't want, send it to me and maybe I'll write up a detailed teardown of it.

Also see comments on Hacker News.

Notes and references

[2] The Apple power supply uses a flyback design, where the transformer operates "backwards" from how you might expect. When a voltage pulse is sent into the transformer, the output diode blocks the output so there is no output - instead a magnetic field builds up. When the voltage input stops, the magnetic field collapses causing voltage output from the transformer. Flyback power supplies are very common for low-wattage power supplies.

[3] The primary board measures about 22.5mm by 20.0mm, while the secondary board is about 22.2mm by 20.2mm. [4] For more information on X and Y capacitors, see Kemet's presentation and Designing low leakage current power supplies.

[5] For clarity, some insulation was removed before taking the pictures in this article. The Y capacitor was covered with black heat shrink tubing, there was tape around the side of the circuit, the fusible resistor was covered with black heat shrink tubing, and there was a black insulating cover over the USB connector.

[6] Snubber circuits can be used to reduce 60 Hz hum generated by the diode bridge in audio power supplies. A detailed reference on R-C snubbers for audio power supply diodes is Calculating Optimum Snubbers, and a sample design is An Audio Amplifier Power Supply Design.

[7] The power supply is controlled by the L6565 quasi-resonant SMPS (switched-mode power supply) controller chip (datasheet). (To be sure, the chip could be something else, but the circuit exactly matches the L6565 and no other chip I examined.)

To improve efficiency and reduce interference, the chip uses a technique known as quasi-resonance, which was first developed in the 1980s. The output circuit is designed so when the power is switched off, the transformer voltage will oscillate. When the voltage hits zero, the transistor switches back on. This is known as Zero Voltage Switching because the transistor is switched when there is essentially no voltage across it, minimizing wasted power and interference during switching. The circuit remains on for a variable time (depending on the power required), and then switches back off, repeating the process. (See Exploring quasi-resonant converters for power supplies for more information.)

One interesting consequence of quasi-resonance is the switching frequency varies depending on the load (with 70kHz as a typical value). Early power supplies such as the Apple II power supply used simple variable-frequency circuits to regulate the power. But in the 1980s, these circuits were replaced by controller ICs that switched at a fixed frequency, but varied the width of the pulses (known as PWM). Now, advanced controller ICs have gone back to variable frequency controls. But in addition, super-cheap knockoff power supplies use variable frequency circuits almost identical to the Apple II. So both high-end and low-end chargers are now back to variable frequency.

It took me a long time to realize that the "FLEX01" marking on the controller IC indicates Flextronics, and the X on the chip was from the Flextronics logo: . I assume the chip has these markings because it is being manufactured for Flextronics. The "EB936" marking on the chip could be Flextronics' own part number, or a date code.

[8] I thought Flextronics was just an electronics assembler and I was surprised to learn that Flextronics does a lot of innovative development and has literally thousands of patents. I think Flextronics should get more credit for their designs. (Note that Flextronics is a different company than Foxconn, which manufactures iPads and iPhones and has the controversy over working conditions).

The picture above is from Flextronics Patent 7,978,489: Integrated Power Converters describes an adapter that looks just like the iPhone charger. The patent itself is a grab bag of 63 assorted claims (spring contacts, EMI shields, thermal potting material), most of which are not actually relevant to the iPhone charger.

[9] Flextronics Patent 7,924,578: Two Terminals Quasi Resonant Tank Circuit describes the resonance circuit used in the iPhone charger, which is shown in the following diagram. Transistor Q2 drives the transformer. Transistor Q1 is the clamp transistor, which directs the voltage spike from the transformer into resonance capacitor C13. The innovative part of this circuit is that Q1 doesn't need special drive circuitry like other active clamp circuits; it is self-powered via the capacitors and diodes. Most charger power supplies, by contrast, use a simple resistor-capacitor-diode clamp which dissipates the energy in the resistor.[18]

Later Flextronics patents extend the resonance circuit with even more diodes and capacitors: see patents 7,830,676, 7,760,519, and 8,000,112

[10] Apple indicates the charger type through a proprietary technique of resistances on the USB D+ and D- pins. For details on USB charging protocols, see my earlier references.

[11] One puzzling feature of the Apple charger is the second feedback circuit monitoring the temperature and output voltage. This circuit on the secondary board consists of a thermistor, a second 431 regulator, and a few other components to monitor the temperature and voltage. The output is connected through a second optocoupler to more circuitry on other side of the secondary board. Two transistors are wired in a SCR-like crowbar latch that will short out the auxiliary power and also shut down the controller IC. This circuit seems excessively complex for this task, especially since many controller ICs have this functionality built in. I could be misunderstanding this circuit, because it seems that Apple unnecessarily took up space and expensive components (maybe 25 cents worth) implementing this feature in such a complex way.

[12] Note the mysterious "For use with information technology equipment" on the outside of the charger. This indicates that the charger is covered by the safety standard UL 60950-1, which specifies the various isolation distances required. For a brief overview of isolation distances, see i-Spec Circuit Separation and some of my earlier references.

[13] Some notes on the components used: On the primary board, the JS4 package is two diodes in a single package. The input diodes labeled 1JLGE9 are 1J 600V 1A diodes. The switching transistors are 1HNK60 600V 1A N-channel MOSFETs. The values of many of the resistors and capacitors are indicated through standard SMD three-digit markings (two digits and then a power of ten, giving ohms or picofarads).

On the secondary board, the "330 j90" capacitor is a Sanyo POSCAP tantalum polymer 300mF 6.3V capacitor (j indicates 6.3V and 90 is a date code). 1R5 indicates a 1.5uH inductor. GB9 is a AS431I low cathode current adjustable precision shunt regulator, and 431 is a regular TL431 regulator. SCD34 is a 3A 40V schottky rectifier. YCW is an unidentified NPN transistor and GYW is an unidentified PNP transistor. The Y capacitor labeled "MC B221K X1 400V Y1 250V" is a 220nF Y capacitor. The "107A" capacitor is a 100 µF 10V tantalum capacitor (A indicates 10V). The optocouplers are PS2801-1. (All these component identifications should be considered tentative, along with the schematic.)

[14] In order to get a rough idea of how much the components in the charger cost, I looked up the prices of some components on octopart.com. These prices are the best prices I could find after a brief search, in quantities of 1000, attempting to match the parts accurately. I have to assume Apple's prices are considerably better than these prices.

A few notes. Flyback transformers are generally custom and prices are all over the place, so I don't have much confidence in that price. I think the POSCAP price is high because I was looking for the exact manufacturer, but tantalum capacitors are fairly expensive in general. It's surprising how cheap SMD resistors and capacitors are: a fraction of a penny.

[15] Apple's safety recall of chargers was announced in 2008. Blog reports showed that the prongs on the charger were attached only by 1/8" of metal and some glue. Apple Recalls iPhone 3G Power Adapters in Wired provides more details.

[16] Low-quality chargers interfere with touchscreens, and this is described in detail in Noise Wars: Projected capacitance strikes back. (Customers also report touchscreen problems from cheap chargers on Amazon and other sites.)

[17] There are many industry designs for USB AC/DC converters in the 5W range. Sample designs are available from iWatt, Fairchild, STMicroelectronics, Texas Instruments, ON Semiconductor, and Maxim.

[18] When a diode or transistor switches, it creates a voltage spike, which can be controlled by a snubber or clamp circuit. For a lot of information on snubbers and clamps, see Passive Lossless Snubbers for High Frequency PWM Conversion and Switchmode Power Supply Reference Manual.

The above picture shows the twelve chargers I analyzed.[2] The charger in the upper-left is the cube-shaped Apple iPhone charger. Next is an oblong Samsung adapter and a cube Samsung adapter. The Apple iPad power adapter is substantially larger[3] than the iPhone charger but provides twice the power. The HP TouchPad power charger has an unusual cylindrical shape. Next is a counterfeit iPhone charger, which appears identical to the real thing but only costs a couple dollars. In the upper right, the Monoprice iPhone charger has a 30-pin dock connector, not USB. The colorful orange charger is a counterfeit of the Apple UK iPhone charger. Next is a counterfeit iPad charger that looks just like the real one. The Belkin power adapter is oval shaped. The KMS power supply provides four USB ports. The final charger is a Motorola Charger.

Summary of ratings

	Model	Overall rating
Apple iPhone	Apple A1265
Samsung oblong	Samsung travel adapter ETA0U60JBE
Samsung cube	Samsung travel adapter ETA0U80JBE
Apple iPad	Apple 10W USB Power Adapter A1357
HP TouchPad	Hewlett Packard LPS AC/DC Adaptor P/N 157-10157-00
Counterfeit iPhone	Fake Apple A1265 "Designed by California"
Monoprice	Monoprice Switching Mode Power Supply MIPTC1A
Counterfeit UK	Fake Apple A1299
Counterfeit iPad	Fake Apple 10W USB Power Adapter A1357
Belkin	Belkin UTC001
KMS	KMS-AC09
Motorola	Motorola AC Power Supply DC4050US0301

Inside a charger

Internally a charger is an amazingly compact switching power supply that efficiently converts line AC into 5 volt DC output. The input AC is first converted to high-voltage DC. The DC is chopped up tens of thousands of times a second and fed into a tiny flyback transformer. The output of the transformer is converted to low-voltage DC, filtered, and provided as the 5 volt output through the USB port. A feedback mechanism regulates the chopping frequency to keep the output voltage stable. Name-brand chargers use a specialized control IC to run the charger, while cheap chargers cut corners by replacing the IC with a cheap, low-quality feedback circuit.[4]

A poor design can suffer several problems. If the output voltage is not filtered well, there will be noise and spikes due to the high-frequency switching. At extreme levels this could damage your phone, but the most common symptom is the touchscreen doesn't work while the charger is plugged in.[1] A second problem is the output voltage can be affected by the AC input, causing 120 Hz "ripple".[5] Third, the charger is supposed to provide a constant voltage. A poor design can cause the voltage to sag as the load increases. Your phone will take longer to charge if the charger doesn't provide enough power. Finally, USB chargers are not all interchangeable; the wrong type of charger may not work with your device.[6]

Counterfeits

I've taken apart several counterfeit chargers and readers have sent me photos of others. Surprisingly, the counterfeit chargers I've examined all use different circuitry internally. If you get a counterfeit, it could be worse or better than what I've seen.

How do you tell if a charger is counterfeit? The fakes are very similar; it's hard for me to tell, even after studying many chargers. There's a video on how to distinguish real and fake chargers through subtle differences. You can also weigh the charger (if you have an accurate scale), and compare with the weights I give above. The easiest way to get a genuine Apple charger is fork over $29 to an Apple store. If you buy a $2 "Original Genuine Apple" charger on eBay shipped from China, I can guarantee it's counterfeit. On the other hand, I've succeeded in buying genuine used chargers from US resellers for a moderate price on eBay, but you're taking a chance.

The following picture shows a counterfeit charger that burned up. The safety issues with counterfeits are not just theoretical; when hundreds of volts short out, the results can be spectacular.

Indicated charger type

There are many different charger types, but only a few are used in the chargers I examined. A USB charger that follows the standard is known as a "dedicated USB charger". However, some manufacturers (such as Apple, Sony, and HP) don't follow the USB standard but implement their own proprietary charger types. Apple has separate charger types for 1 amp (iPhone) and 2 amp (iPad) chargers. HP has a special type for the HP TouchPad.

The point is that USB chargers are not interchangeable, and devices may not work if the charger type doesn't match what the device expects. The table below shows the type of charger, the current that the label claims the charger provides, the current it actually provides, and the charger type it indicates to the device.

The types of the counterfeit chargers are a mess, as they advertise one power level, actually supply a different power level, and have the charger type for a third level. For example, the counterfeit iPhone charger is advertised as supplying 1 amp, but has the 2A charger type, so an iPad will expect 2 amps but not obtain enough power. On the other hand, the counterfeit iPad charger claims to supply 2 amps, but really only supplies 1 amp and has a 1A type.

Efficiency

I also measured efficiency of the chargers under load.[11] One of the benefits of switching power supplies over simpler linear supplies is they are much more efficient at converting the input power to output. The chargers I measured all did pretty well, with 63% to 80% efficiency. The HP charger was the winner here.

	Vampire	milliwatts	Efficiency	Percent
Apple iPhone		195		74
Samsung oblong		19		76
Samsung cube		86		77
Apple iPad		62		78
HP TouchPad		91		80
Counterfeit iPhone		375		63
Monoprice		78		72
Counterfeit UK		103		63
Counterfeit iPad		95		66
Belkin		234		66
KMS		179		69
Motorola		59		75

The chargers up close

Apple iPhone and counterfeit

The above photo shows a real iPhone charger (left) and a counterfeit (right); the two chargers are almost identical, down to the green dot. If you look closely, the genuine one says "Designed by Apple in California", while the counterfeit has the puzzling text "Designed by California". The counterfeit also removed the "Apple Japan" text below the plug. I've seen another counterfeit that says "Designed by Abble" (not Apple). I assume the word "Apple" is removed for legal or trademark reasons, since the word "Apple" is often (but not always) missing from counterfeits.

Samsung oblong

I call this charger the Samsung oblong charger, to distinguish it from the Samsung cube charger.

Samsung cube

The Samsung cube charger is shaped very similarly to the Apple iPhone charger. Internally, however, it turns out to be entirely different.

Apple iPad and counterfeit

The photo above shows a real iPad charger (left) and a counterfeit (right). The counterfeit has almost identical text, but without "Designed by Apple in California. Assembled in China", "Listed" under UL, and the manufacturer "Foxlink". Inexplicably this sanitization left "TM and © 2010 Apple Inc".

The above photo shows a real iPad charger on the left and a fake iPad charger on the right, with the plug removed. The most visible difference is the real charger has a round metal grounding post, while the fake has plastic. (The US plug isn't grounded, but in other countries the lack of ground in the counterfeit could pose a safety hazard.)

HP TouchPad

The HP TouchPad charger has a very unusual cylindrical shape, which is striking if perhaps not practical. The charger twists apart, allowing the plug to be replaced for different countries. (It took me weeks to discover this feature.)

Monoprice

The Monoprice charger isn't a USB charger, but instead has a 30-pin iPhone dock connector attached. It is a relatively large charger.

Counterfeit UK

This charger is a counterfeit of the Apple UK iPhone charger. They've removed Apple from the text, but left Emerson Network Power, which I'm sure is not the actual manufacturer. The genuine Apple UK charger can be distinguished by a serial number inside the USB connector.

Belkin

The Belkin charger eschews the minimal design styling of most chargers, with a roughly oval cross section, curves and ribs, and a cover over the USB port.

KMS

The KMS charger is unusual in providing 4 USB ports. It also gives off a blue glow while in use. The plug can be removed and replaced for use in different countries, similar to the iPad and HP TouchPad chargers. I couldn't find any UL safety approval on this charger, but I did find a report of one catching fire.

Motorola

The Motorola charger has the lowest listed power output, 850mA. The back of it has a holographic sticker (like a credit card), which may ward off counterfeiters, even though it's unlikely for anyone to counterfeit this charger. I wonder though why Apple doesn't use holograms or other anti-counterfeiting techniques, given the large number of counterfeit Apple chargers being sold.

Delivery of advertised power

The counterfeit chargers perform extremely poorly, putting out a fraction of the expected power. Charging your device with one of these chargers will be a slow, frustrating experience. In particular, the counterfeit UK charger only produces a third of the expected power. Although the label claims the charger works on 100-240 volts, it's clearly not designed to work on US power.

The iPad is a surprise, putting out less power than expected. Despite being nominally a 10 watt charger, the label says it provides 5.1V and 2.1A, which works out to 10.7 watts. However, the maximum power I measured is 10.1 watts (4.4 volts at 2.3 amps, as shown in the Power section below). Since the measured power is slightly less than advertised, it only gets four bolts.

	Rating	Label	Watts from label	Measured watts
Apple iPhone		5V 1A	5.0	6.0
Samsung oblong		5V 0.7A	3.5	4.0
Samsung cube		5V 1A	5.0	5.5
Apple iPad		5.1V 2.1A	10.7	10.1
HP TouchPad		5.3V 2.0A	10.6	11.4
Counterfeit iPhone		5V 1A	5.0	2.7
Monoprice		5V 1A	5.0	5.7
Counterfeit UK		5V 1A	5.0	1.7
Counterfeit iPad		5.1V 2.1A	10.7	5.9
Belkin		5V 1A	5.0	5.6
KMS		5V 2.1A	10.5	10.9
Motorola		5.1V .85A	4.3	4.3

Power quality

	Spikes	Noise	Ripple
Apple iPhone
Samsung oblong
Samsung cube
Apple iPad
HP TouchPad
Counterfeit iPhone
Monoprice
Counterfeit UK
Counterfeit iPad
Belkin
KMS
Motorola

The following oscilloscope traces show the output signal (yellow) and frequency spectrum (orange). The left images provide high-frequency information on the output voltage. The right images show the low-frequency information on the output voltage.[13]

The desired voltage graph is a flat, thin yellow line indicating totally smooth power. However, some factors mess this up. First, any ripple from the power line will show up as 5 sinusoidal peaks in the first (high-frequency) yellow line. High-frequency noise will widen the yellow line. Voltage spikes will appear as vertical spikes in the yellow line.

The plots also show the frequency spectrum in orange, from 0 at the left to 230 kHz at the right. The desired graph would have the orange spectrum near the bottom of the screen. Thus, the power quality exponentially gets worse as the orange line gets higher. The left (high frequency) spectrum generally shows noise at the switching frequency of the charger (and harmonics). The right (low frequency) spectrum typically shows spikes at multiples of 120 Hz, caused by ripple from the 60 Hz power.[5]

Apple iPhone

The ripple is clearly visible as the waves in the yellow trace on the left and as the spikes (at 120 Hz and 240 Hz) in the orange trace on the right.

The iPhone charger performs extremely well at filtering out spikes and noise, the best of the chargers I measured. Apart from the 120 Hz spikes, the noise spectrum (orange) is flat and very low. The power quality is so good, I checked the results several times to make sure I wasn't missing something.

Samsung oblong

The Samsung charger's output has a lot more noise than the iPhone charger. This is visible in the thickness and jaggedness of the yellow output curves. The orange frequency spectrum on the left shows large peaks at harmonics of the switching frequency. The 120 Hz spike on the right is a bit lower than the iPhone charger, so the ripple filtering is a bit better.

Samsung cube

The Samsung cube charger shows some noise in the output (yellow). The frequency spectrum shows wide peaks at multiples of the the switching frequency, about 90kHz. There's some ripple.

Apple iPad

The iPad charger almost eliminates the ripple; only a small blip is visible in the orange spectrum on the right. The noise level is low, although appreciably worse than the iPhone.

HP TouchPad

There's no ripple visible in the HP charger spectrum on the right. The overall noise level is good.

Counterfeit iPhone

The output from this counterfeit charger is a wall of noise. In order to fit the waveform in the display, I had to double the scale on the left and increase it by a factor of 5 on the right, so the yellow curve is actually much worse than it appears. On the left, note the huge ripple with massive high-frequency noise on top. This output is not something you want to feed into your phone.

Monoprice

The output from this charger is very noisy, as you can see from the thickness of the yellow line. Note that the frequency spectrum (left) has very tall but narrow spikes at harmonics of the 28kHz switching frequency, showing a lot of high-frequency noise. On the positive side, there is hardly any ripple.

Counterfeit UK

This charger has very bad output. The large degree of ripple is visible in the waveform (yellow, left) and the very large spikes in the spectrum (orange, right). The thickness of the yellow waveform shows the large amount of high-frequency noise, which is also visible in the very high peaks in the spectrum (orange, left).

Counterfeit iPad

This counterfeit charger has so much noise in the output that I had to double the scale on the left to get it to fit. Note the very large spikes in the output (yellow). The spectrum (orange, left) is much higher everywhere, indicating noise at all frequencies. Surprisingly, it has only a moderate amount of ripple; the manufacturer seems to have done at least one thing right.

Belkin

The Belkin charger does well at eliminating ripple, but has a lot of noise otherwise. The spectrum (orange, left) shows large peaks. The yellow output is wide, showing a lot of noise, combined with many large voltage spikes of about 1/3 volt.

KMS

The KMS charger has fairly good output, with a small peak in the spectrum (orange, left) at the switching frequency. It has no detectable ripple. However, it has many large voltage spikes in the output, over half a volt, as can be seen on the right.

Motorola

The Motorola charger has a lot of spikes in the output (yellow) . The spectrum (orange, left) shows high frequency noise at the switching frequencies. There's a moderate amount of ripple (yellow, left and orange, right).

Summary

Power curve

	Voltage sag	Current sag	Regulation
Apple iPhone
Samsung oblong
Samsung cube
Apple iPad
HP TouchPad
Counterfeit iPhone
Monoprice
Counterfeit UK
Counterfeit iPad
Belkin
KMS
Motorola

The graphs in this section need a bit of explanation, which is provided in the diagram below. The voltage/current load curve shows the performance of the charger under different loads. Each point on the curve shows the current (X axis) and voltage (Y axis) produced by the charger under a particular load condition. Follow the yellow curve clockwise from the upper left to the lower left to see the effect of increasing load. The upper left point of the curve shows the voltage produced by the charger when there is no load on the charger. As the load increases, the charger is supposed to keep a constant voltage and increase the current (i.e. horizontal line), until it reaches the maximum power (upper right). If the load continues increasing, the charger switches to a constant current mode, dropping the voltage while continuing to provide the maximum current (i.e. vertical line).[14] At the lower right, the charger has reached its shutdown point due to excessive load, and rapidly drops to no output in the lower left corner to avoid damage.

[16]

Apple iPhone

The output from the Apple iPhone charger is surprisingly non-constant under load. The charger starts off with 5.2 volts with no load, dropping to 4.6 volts as the load increases, resulting in the downwards slope of the top yellow line. As the load increases, the current keeps increasing, resulting in the slope of the right yellow line. Note however that the yellow line is relatively thin, so the regulation is pretty good at each point.

Note that because this charger has a high current output, this chart has a different current (horizontal) scale than most of the charts to fit the whole trace in the image. Stretch it horizontally to compare with other graphs.

Samsung oblong

For this charger, the voltage is approximately flat, except for a bump under no load (upper left) which is probably a measurement artifact. The vertical yellow line shows the current stays nearly constant as the load increases. The charger shows good voltage and current stability under changing load. The yellow line is a bit wider than the iPhone charger, showing a bit less regulation for a fixed load.

Samsung cube

The voltage curve sags slightly under load. The right hand curve shows the current stays stable, but the line is moderately wide, showing a bit of weakness in regulation.

Apple iPad

Similar to the iPhone charger, the iPad charger shows a lot of voltage sag. The voltage is about 5.1 V unloaded, dropping to 4.4 volts and 2.3 A (10.1 W) at the corner. Unlike the iPhone charger, the iPad charger has pretty good current stability. The regulation is solid, as shown by the narrowness of the yellow trace. Note the scale change due to the high current output.

I'm puzzled by the steep voltage sag on both the iPhone and iPad charger. Since the designers of the Apple charger went to a great deal of effort to build a high quality charger, I conclude they must not consider voltage sag worth worrying about. Or, more interestingly, maybe they built this sag as a feature for some reason. In any case, the chargers lose points on this.

HP TouchPad

The charger has some voltage sag, but the current (vertical) is nice and constant. The yellow line is relatively thin, showing good regulation. Note the scale change due to the high current output.

Counterfeit iPhone

This counterfeit charger shows extremely poor regulation, as shown by the very wide yellow line. It's hard to fit a voltage-current curve to this picture. The amount of power supplied by this charger seems almost random.

Monoprice

The Monoprice charger shows reasonably straight voltage and current lines showing good constant voltage and current outputs. The vertical line shows some width and noise, suggesting the regulation isn't totally stable.

Counterfeit UK

For this charger, the upper line doesn't get very far, showing that this charger doesn't output much current. My suspicion is that it was only tested with 240 volts so it performs poorly with 120 volts, even though the label says it takes 100 to 240 volts. The width of the yellow line shows very poor regulation.

Counterfeit iPad

The output of this counterfeit charger is so poorly regulated that it's hard to tell exactly what's happening with the voltage and current. It looks like the voltage is roughly constant underneath all the noise.

Belkin

The Belkin charger shows voltage sag as the current increases. In addition, the output is fairly noisy.

KMS

The KMS charger shows a lot of voltage sag as the load increases. In addition, the output is all over the place, showing very poor regulation, more like what I'd expect from a counterfeit charger. Note the scale change due to the high current output.

Motorola

The Motorola charger shows a bit of voltage sag, but good current stability. The regulation is good but not perfect, as shown by the width of the yellow line. (The gaps in the vertical line are just measurement artifacts.) Note that the maximum current output of this charger is fairly low (as advertised).

Conclusions

P.S. How about some teardowns?

The above picture shows the internals of a counterfeit Apple iPhone cube charger. The two boards stack to form the compact cube shape. This charger blatantly tries to pass as a genuine Apple charger; unlike the "Designed by California" charger, this one exactly copies the "Designed by Apple in California" text from the real charger. Note the very simple circuitry[4] - there are no components on the other side of the board, no controller IC, and very little filtering. Also look at the terrible mounting of the transistor on the front right; clearly the build quality of this charger is poor. Finally, note the overall lack of insulation; this charger wouldn't meet UL safety standards and could easily short out. But on the plus side, this charger only cost a couple dollars.

The above $2 charger is notable for its low-profile design; it's about as thin as you can make a charger and still fit the power prongs and the USB port. The transformer is very short to fit into this charger. Like the previous charger, it uses a very simple circuit,[4] has little filtering, and almost no safety insulation.

Finally, the above pictures show the internals of the Samsung cube charger, which has circuit boards packed with tiny components and is much more advanced than the counterfeits (although slightly less complex than the Apple charger). Despite being very similar to the Apple charger on the outside, the Samsung charger uses an entirely different design and circuitry internally. One interesting design feature is the filter capacitors fit through the cut-out holes in the secondary circuit board, allowing the large filter capacitors to fit in the charger.

Notes and references

[2] The charger selection may seem slightly eccentric; it is based on chargers I had previously acquired, chargers I could obtain at a reasonable price, chargers supplied by Gary F. and Anthony H. (thanks, guys!), and some counterfeit chargers for comparison.

[3] TI has an interesting new design for a 10 watt inch-cube charger. With this design a tablet charger could be as small as the iPhone charger.

[4] The cheap chargers all use a "ringing choke converter" circuit, which coincidentally is the same power supply topology used by the Apple II. These chargers use an extremely simple feedback mechanism in place of the control IC in higher-quality chargers. See a comic-book explanation or a technical explanation for details.

[5] Since the input AC has a frequency of 60 Hertz, you might wonder why the ripple in the output is 120 Hertz. The diode bridge converts the 60 Hz AC input to 120 Hz pulsed DC, as shown in the diagram below. The pulses are smoothed out with filter capacitors before being fed into the switching circuit, but if the filtering isn't sufficient the output may show some 120 Hz ripple.

[6] The chargers use specific voltages on the data pins to indicate the charger type to the device being charged. Because of this, an "incorrect" charger may be rejected by an iPhone with the message "Charging is not supported with this accessory".[7] Under the USB standard, a charger should short the two data pins together to indicate that it's a "dedicated" charger and not a real USB device. However, companies such as Apple, HP, and Sony have their own proprietary nonstandard techniques. The following table summarizes the voltages that appear on the D+ and D- lines for different chargers, and how the D+ and D- lines are configured internally.

Most of this data is based on Maxim USB Battery Charger Detectors, Adafruit's The mysteries of Apple device charging, TouchPad's USB Cable, XDA forum (Samsung), and TPS2511 USB Dedicated Charging Port Controller and Current Limiting Power Switch datasheet. The Apple 2A (i.e. iPad) information is a new result from my measurements. For details on USB charging protocols, see my references in my earlier posting.

Amusingly, semiconductor manufacturers have recently introduced chips that allow chargers to sequentially pretend to be different proprietary chargers until they trick the device into accepting the charger. It seems crazy that companies (such as Apple) design incompatible chargers, and then chip companies invent schemes to work around these incompatibilities in order to build universally compatible chargers. Two example chips are the TI TPS 2511 chip, and SMSC's USC1001 controller, which pretends to be nine different charger types.

[7] If you've wondered why some chargers cause the iPhone to give a "Charging not supported with this accessory" error, Silicon based annoyance reduction made easy describes how devices use proprietary protocols to limit the chargers they will work with.

[8] For the efficiency analysis I use 12 cents / kilowatt-hour as a typical residential energy price, which I got from US Energy Information Administration table 5.3.

[9] The official no-load charger star ratings are discussed at Meeting 30 mW standby in mobile phone chargers.

[10] There are many standards for energy consumption; see 5 W Cellular Phone CCCV (Constant Current Constant Voltage) AC-DC Adapter. For Energy Star ratings, a 5W charger must have under .5W no-load consumption, and 63% efficiency under load. A 10W charger must have under .75W no-load consumption, and 70% efficiency.

[11] Because switching power supplies use power in irregular waveforms, I used a complex setup to measure power consumption. I measured the AC input voltage and current with an oscilloscope. The oscilloscope's math functions multiplied the voltage and current at each instant to compute the instantaneous power, and then computed the average power over time. For safety and to avoid vaporizing the oscilloscope I used an isolation transformer. My measurements are fairly close to Apple's[15], which is reassuring.

You might wonder why I didn't just use a Kill A Watt power monitor, which performs the same instantaneous voltage * current process internally. Unfortunately it doesn't have the resolution for the small power consumptions I'm measuring: it reports 0.3W for the Apple iPhone charger, and 0.0W for many of the others. Ironically, after computing these detailed power measurements, I simply measured the input current with a multimeter, multiplied by 115 volts, and got almost exactly the same results for vampire power.

[12] The spike, noise, and ripple measurements come from the oscilloscope traces. The Spikes measurement is based on the maximum peak-to-peak voltage on the high frequency trace (the low frequency trace yields almost identical results). The Noise measurement is based on the RMS voltage on the high-frequency trace, and Ripple is based on the maximum dB measured in the low-frequency spectrum. These measurements appear on the right in the traces.

[13] In the power quality section, the high-frequency (left) images show 40 milliseconds of the waveform in yellow, and the frequency spectrum up to 234 kHz in orange. The low-frequency (right) images show 1 second of the output voltage in yellow and the frequency spectrum up to 600 Hz in orange. Because the frequency spectrum is measured in dBm, it is logarithmic; every division higher indicates 20 dB which is 10 times the voltage and 100 times the power.

[14] The chargers use a design called constant-voltage, constant-current (CVCC), since they provide a constant voltage (and increasing current) up to the maximum load and then a constant current (and decreasing voltage) if the load continues to increase. [15] The Apple 3GS Environmental Report gives some efficiency measurements for the Apple USB Power Adapter. It lists 0.23W no-load power and 75% efficiency. These values are reasonably close to my measurements of 0.195W no-load consumption and 73.6% efficiency.

[16] Measuring these curves was a bit tricky. I used a NTE2382 power MOSFET transistor as a variable load, manually varying the gate bias to generate the load curve. The transistor needed a large heat sink to dissipate 10 watts. A more complex dynamic load circuit is described here, but the simple circuit was sufficient for me.

The graphs were generated using the X-Y mode on the oscilloscope, with the load voltage as Y and the current as X. I used a .12&ohm; current sense resistor to measure the load current. This works out to 1/6 amp load current per division for the 20mV/div traces (most of them), and 5/12 amp load current per division for the 50mV/div traces (the high-current devices).

Note that increasing load corresponds to a decreasing resistance across the output: the upper left has infinite resistance (no load), the lower left has zero resistance (short circuit), and the resistance decreases in between. Since the power (in watts) is voltage * current, the maximum power is in the upper right corner, approximately 4W in this case. The load resistance can be computed by Ohm's law, e.g. middle of the upper curve: 5 V / .4 A = 12.5&ohm;, upper right corner 5 V / .8 A = 6.25 ohms. Middle of the right hand curve: 2.5 V / .8 A = 3&ohm;, overload point = .5 V / .8 A = .6&ohm;.

[17] Most of these chargers aren't made by the companies that sell them, and there are some interesting facts about the manufacturers. The manufacturers of the chargers can be looked up from the UL certification number. The oblong Samsung is made in China by Korean RFTech, a manufacturer of mobile phone products. The Samsung cube is made in China by Korean power supply manufacturer Dong Yang E&P. The HP charger is made by Foxlink, who also makes the iPad charger for Apple. The counterfeit chargers are made by anonymous Chinese manufacturers, despite what they claim on the labels. The Monoprice is made by Golden Profit Electronics (formerly ShaYao Electric Factory Three - no word on what happened to factories One and Two). The Belkin charger is manufactured by the obscure company Mobiletec of Taiwan. The KMS charger doesn't give any clues as to the manufacturer, and I can't identify KMS as a company. The Motorola charger is built by Astec (now part of Emerson Network Power). Interestingly, Astec's big break was manufacturing power supplies for the Apple II, as I discuss in my article on the Apple II power supply.

Apple uses a dizzying variety of manufacturers for their chargers. The iPhone charger (A1265) is made by Flextronics, the UK charger (A1299) is made by Emerson Network Power (except the one I have is counterfeit), the iPad charger (A1357) is made by Foxlink Technologies, and the Magsafe (ADP-85) charger (not discussed in this article) is made by Delta Electronics. The A1385 iPhone charger often comes with the iPhone 5 and looks identical to the A1265 I measured, but is manufactured by Emerson Network Power instead of Flextronics. I am told that by using multiple manufacturers, Apple has more negotiating leverage, since they can easily switch manufacturers at any time if they're not happy with the price or quality.

Confusingly, Foxlink (Taiwan), Foxconn (Taiwan), and Flextronics (Singapore) are all manufacturers for Apple with similar names. Foxlink (the name for Cheng Uei Precision Industry) and Foxconn (the name for Hon Hai Precision Industry) are entirely independent companies aside from the fact that the chairmen of both companies are brothers and the companies do a lot of business with each other (statement, Foxlink annual report). Foxconn is the company with continuing controversy over employee treatment. Foxconn and Flextronics are the world's #1 and #2 largest electronics manufacturing companies according to the Circuits Assembly Top 50.

Getting started

Next, to start Spectral analysis, go to Math -> Spectral Setup.... Choose Magnitude, Channel 1, and Ok. This will display the spectrum of channel 1 as the Math1 curve.

You'll discover there are a dozen settings with complex interactions. There are three columns of controls: Time Controls (Record Length, Sample Rate, Duration, and Resolution), Gating Controls (Gate Position, Gate Duration, and Gate Length), and Frequency Controls (Center Frequency, Frequency Span, and Resolution Bandwidth).

The Time Controls select how much data is collected. Record Length is the number of data points collected, Sample Rate is the sampling frequency, Duration is the time interval collected, and Resolution is the time between samples.

The Gating Controls select which data is actually used for the FFT, since you can use less data than you are collecting. Gate Duration is the sub-region of the Duration that is processed, and Gate Length is the number of samples in the Gate Duration. The Gate Duration is the interval between vertical yellow lines in the display.

The Frequency Controls select what results are displayed. The displayed frequency spectrum is centered on Center Frequency with total width of Frequency Span. Clicking Full will make the Frequency Span as large as possible with the existing settings. The Resolution Bandwidth specifies how detailed the frequency spectrum is, so smaller is "better".

Controlling the parameters

Advice

Set the Sample Rate to at least twice the maximum frequency of interest. (Note that Sample Rate is basically in Hertz; s means samples per second, not seconds for Sample Rate. Resolution is the reciprocal of Sample Rate.)

Duration controls the Resolution Bandwidth; a longer Duration yields a smaller (better) Resolution Bandwidth. At least two cycles of the lowest frequency of interest must fit into the Duration. The Duration should include several cycles of the waveform of interest. Note that if the Duration is multiple seconds, it will inconveniently take a long time to collect data before showing you the waveform. And if the Record Length is long, the processing time will be long.

The Record Length will be determined from the Sample Rate and Duration, Generally around 20,000 is a good value. A much smaller value won't give you good frequency resolution, and a much larger value will slow down processing.

The Resolution control is usually the best Time control to adjust, since the others interact with each other in annoying ways.

For the Gating controls, I recommend leaving the Window Type at Gaussian and the Gate Position at 0 unless you know you want to do something else. you probably want the Gate Duration to be as large as the Duration, or else you're wasting data and getting a worse Resolution Bandwidth. If the Gate Duration is smaller than the Duration, it will be displayed as vertical yellow lines that show the part of the waveform that is getting processed.

For the Frequency Controls, you probably want Center Frequency to be half of Frequency Span; this means that you're seeing the full spectrum starting at 0. It's very easy for Center Frequency to get larger, which means you're looking at a random region in the middle of the middle of the spectrum, so you may need to continually set the Center Frequency back to the minimum value. It usually works best to modify freqency by increasing the Frequency Span and decreasing the Center Frequency, to keep the two in sync. If the frequency spectrum looks like sine waves rather than spikes, the Resolution Bandwidth is probably too large.

Be careful you don't clip the input waveform on the top of the screen, as this will mess up the spectrum. Set the vertical scale so it fits.

Example: Square wave frequency spectrum

The above picture shows a first attempt at generating the frequency spectrum (orange) of the square wave (yellow). The duration is too short - it only includes 4 cycles of the waveform, so the Resolution Bandwidth is large (500Hz). The result is the spikes are spread out into sine-like curves and the spectrum is not very distinct.

In the above picture, the duration has been increased to 40mS, so 40 cycles of the waveform are included. With a long duration, the Resolution Bandwidth is narrow (50Hz), so the harmonics are narrow spikes as desired. However, with a wide Frequency Span, the harmonics are crammed together and hard to see.

To see the spectrum in more detail (zoomed-in), decrease the Frequency Span, which gives more detail of the spectrum. However, note that the spikes are all almost the same size. since the Center Frequency is not half of the Frequency Span, we're looking at the middle of spectrum (3.125kHz to 18.75kHz), rather than starting at 0Hz. It is very easy to accidentally shift the spectrum range and get confused about what you're seeing, so be cautious.

Finally, with the Center Frequency set to half of the Frequency Span, the image above shows the spectrum starting at 0. The frequency scale (Math1) is 1.56kHz per grid line, so the spikes are at 1kHz, 3kHz, 5kHz, etc. as expected. (Inconveniently, I can't find a way to get the frequency scale to be round numbers.)

With the Gate Duration decreased, note the vertical yellow lines showing the gate region. Only the signal between the lines is being processed. This increases the Resolution Bandwidth (i.e. makes it worse), so the harmonics are now considerably wider than the spikes seen previously.

The theory behind dB and dBm

For the dBm scale, the offset voltage is set to 223.6mV. Although this value may seem random, there is an explanation. The definition of dBm is 10*log(power), where power is the RMS power in milliwatts. dBm is defined with a fixed impedance of 50&ohm;. (Other fields use different impedances for dBm, for example audio work uses 600&ohm;.) Working through some math yields V_ref = sqrt(50&ohm;*1mW) = 223.6mV. (See Wikipedia for more explanation of dBm.)

The oscilloscope's dB and dBm settings

The scale values for Math1 are used to display the dBm curve. The vertical scale shows the number of dBm per division, and the horizontal scale shows the frequency per division. The M1 indicator on the left shows the position of 0 dBM. Note that dBm values can be positive or negative.

In the Math controls, the controls are Pos and Scale - Pos is the number of divisions that M1 (0 dBm) is above the center line, so Level = (4 - pos) * Scale.

On the oscilloscope, there are few differences between the dB and dBm settings. dBm defaults to centering the spectrum on the screen (which generally puts M1 high up), and using 223.6mV as the offset. dB defaults to centering M1 on the screen (which generally puts the spectrum lower). If you adjust the Level, Offset, and Scale to be equal, the dBm and dB settings give the same results, so they aren't fundamentally different. The default Level and Scale are usually ugly numbers; you can adjust them to round values.

The square wave spectrum gives an example of dBm in use. Looking at the theoretical values for the square wave, the first harmonic of 0.45V is 20*log(450mV/223.6mV) = 6.1dBm and the second odd harmonic of 0.15V is 20*log(150mV/223.6mV) is -3.5dBm. Looking at the measured harmonics, the first is about 0.4 divisions above the M1 mark, and the second is about 0.2 divisions below the mark. Since the scale is 15.1dBM / division, this yields measured values of 6dBm and -3dBm, close to the theoretical values.

Understanding the linear scale

The linear scale displays the RMS voltage of each harmonic. The M1 arrow at the left indicates the 0 position (which can be adjusted). The Math1 Scale shows how many mV per vertical division, and how many kiloHertz per horizontal division. Position shows the number of divisions M1 is above the center line.

Changing the scale to Linear shows how the harmonics in the example drop off rapidly as 1/N: 1, 1/3, 1/5, 1/7, etc. (Note that higher-order harmonics are much harder to see with the linear scale than with the dB scale.) Aligning cursors with the first two harmonics indicate the first harmonic is 1kHz at 445.4mV, and the third harmonic is 3kHz at 144.6mV. These values are close to the theoretical values of 450mV and 150mV. The values can also be determined from the display using the scale. In this example, the Math1 scale is 117mV per vertical division and 2.5kHz per horizontal division. The first harmonic is about 3.8 divisions above the 0 level, which works out to 445mV.

Reading voltage and frequency values can be inconvenient because the scale usually ends up with inconveniently non-round values per division. Using a math cursor, as above, make this easier. The cursors are accessed under Cursors > Cursor Setup, and then moved to the desired positions.

Some relations between the controls

• 1 / Sample Rate = Resolution.
  Resolution is just the time between samples.
• Record Length = Sample Rate * Duration.
  Record Length is the total number of samples collected.
• Gate Length = Gate Duration * Sample Rate.
  Gate Length is the total number of samples processed.
• Gate Duration ≤ Duration.
  The Gate Duration must be smaller than the total Duration.
• Gate Length ≤ Record Length.
  The Gate Length must be smaller than the total Length.
• Center Frequency ≥ Frequency Span / 2.
  The minimum frequency in the range must be positive.
• Resolution Bandwidth = 2 / Gate Duration.
  The slowest waveform must fit into the duration at least twice.
• Freq Span ≤ Sample Rate / 2.
  The Nyquist limit requires at least two samples at the highest frequency.
• Spectral Level = (4 - Math Pos) * Scale
  Level is measured from the top of the screen, 4 divisions above the center line.

For the official Tektronix documentation, see Defining a Speactral Math Waveform in the TDS5000B Series Oscilloscopes Online Help, page 291. Also see TDS5000B Series Quick Start User Manual, page 75.

This tutorial is mostly for my own reference, so I can remember how to use the oscilloscope in the future, but hopefully it will be of benefit to some other readers.

Creating a part is surprisingly tricky - Eagle is one of those software packages with a GUI that looks more intuitive than it is. To create a component, there are three abstractions to deal with: Symbol, Package, and Device. The Symbol is the symbol as it appears in the schematic. It must be tied to a Package, which describes the shape of how the component is physically mounted on a circuit board, in particular the pads for the pins. Finally, the Device holds the complete description of a device, including the symbol and potentially multiple packages. Thus, even if you just want a schematic symbol, you must also deal with the package and device. The following image shows important parts of the Device screen.

Example: Zener diode

First, find a part in an existing library you want to modify, e.g. use Edit > Add, then Search for something similar. Note the name of the library and the device, e.g. "diode" and "IN4004".

Next, create your custom library by going to the Control Panel and selecting File > New > Library, or open your existing library.

In Control Panel expand the library that has the component you want to copy and find the package you want. Right click and select "Copy to Library".

Next, open the new symbol in your library: Go to Library > Symbol and select the Symbol. Or from the Device screen right click on the crosshair in the middle of the symbol and select Edit Symbol.

Finally, you can edit the symbol using the standard Eagle editor functions. To make a Zener diode, I needed to change the grid resolution to 0.025, select the angular wire bend, and then use Wire to add the "fins" to make the symbol look like a Zener diode.

Go to File > Save As..., and save the new library.

To use this new library in your schematic, select Library > Use, and add your new library.

Improving the Symbol

To rename your symbol, go to Library > Rename, and enter a new name for the symbol.

To add a description, click on the Description link and enter a description using HTML. Typically it has <b>the title</b> <p>, and then a description.

To move the >NAME or >VALUE, use move and click on the crosshair at the lower left. You might want to temporarily reduce the grid size to get more control over the position.

Improving the Device

To rename the device, use Library > Rename to rename the Device. You can also change the description as above, which is useful for searching.

To change the symbol's prefix (e.g. D1, D2, D3 for diodes), click on Prefix and enter the desired prefix.

You probably just want to have one package, so delete others by right clicking on them on the Device screen and selecting Delete.

To rename the package, go to Library > Package, and select the package. Then use Library > Rename to rename the Package. You can also change the description.

Adding new pins

Add the pins using Draw > Pin. You may need to rotate the pin. Right click the pin and select Properties to change the length or other property. You can enter a pin name or set Visible: Off if you don't want the pin name to show up.

To add pads, go to the Device, right click the Package, and select Edit Package. Add a Pad (anywhere) using the green circle icon.

Next you need to connect the pins and pads. Go to Library > Device and select your device. You will notice in the package pane an exclamation point in a circle. Underneath, click Connect, which will bring up the Connect panel. Select a Pin and a Pad and click Connect, until all pins are connected, and click Ok. Badk at the Device screen, you should now see a checkmark next to the Package. If you don't connect the pad to the pin, you will get "Error: Device ... has unconnected pin (G$1/P$1)!" when you try to add it to the schematic.

Creating an IC (or other component) from scratch

First, create a package with the right number of pins. Go to Library > Package, enter your package name (e.g. DIP8) after New, and click Ok. Using the green pad, drop 8 (for example) pads onto the package - the positions don't matter if you're not generating a PCB. Use Properties on each one (right click) to give the pads names 1, 2, etc.

Go to Library > Symbol, enter your symbol name after New (e.g. 555), and click Ok. (For this example, I'll create a 555 timer from scratch even though the library has one.) Put down all the pins for your IC, leaving plenty of room horizontally for the labels, rotating the pins as necessary. Under properties, give each pin the desired name and set length to short. Use four wires to create the outline. Put Text >NAME on the name layer (95) and Text >VALUE on the value layer (96). Add a description if you want.

Go to Library > Device, enter your device name after New (e.g. 555) and click Ok. Click on Add, select your new symbol, and add your symbol to the device. Under the package pane, click New, select your package, and click Ok. Click Connect, and carefully connect all the pins to the right pads, so the pin numbers show up okay. (Tip: do the pins in numerical order.) Click Ok. Add a Prefix and Description if you want.

Save your library with File > Save, and it should be ready to use.

Conclusion

Here's the transcript:

SCHMIDT: Now, Senator, you're here at Google and I like to think of the presidency as a job interview. Now, it's hard to get a job as president. And--I mean, you're going to do a great job. It's also hard to get a job at Google. We have questions and we ask our candidates questions. And this one is from Larry Schwimmer. What--you guys think I'm kidding, it's right here.[1] What is the most efficient way to sort a million 32-bit integers?

OBAMA: Well...

SCHMIDT: Maybe--I'm sorry...

OBAMA: No, no, no, no. I think--I think the bubble sort would be the wrong way to go.

SCHMIDT: [facepalm] Come on. Who told him this? Okay. I didn't see computer science in your background.

OBAMA: We've got our spies in there.[2]

If you're wondering about the complete answer to the sorting question, see a detailed explanation.[3]

All in all, it was a very unexpected answer from Obama with perfect delivery.

Notes

[1] Nervous laughter greeted the mention of a Larry Schwimmer question, because he once asked Jimmy Carter about his UFO encounter.

[2] Eric Schmidt had asked the sorting question to John McCain on a different visit, with the expected result. See the YouTube clip of McCain's visit.

[3] The pedantic may note that Obama's question is slightly different from the Stack Overflow question since it involves 32-bit integers vs. 8 digit integers, and the memory constraint was omitted.

One puzzle about this charger is it's unclear who makes it and what model it is. The case says it's the KMS AC-09 but the circuit board says "TC09-new-V4.2". Amazon lists the brand as "Cosmos®", but I couldn't find any sign that KMS or Cosmos are actual companies. After some web searches, I think the charger is built by Guangzhou Panyu Qiaonan Saidi Electronic Factory (more) as the TC09 charger for $5.30 wholesale, or maybe HK Yingjia International, a consumer electronics manufacturer in Shenzhen (more). In any case, I'll call this the "KMS charger" since I need to call it something.

In my previous lab analysis of 12 chargers, I compared a dozen different chargers in 9 different categories, rating them from 1 to 5 'bolts' and the KMS charger came in about average in terms of performance. The results for the KMS charger are summarized below. For details on these measurements, see my previous article A dozen USB chargers in the lab).

Overall rating
Vampire (idle) power usage
Efficiency under load
Achieves power rating
Spikes in output
High-frequency noise in output
Ripple in output
Voltage sag
Current sag
Regulation quality

The good and the bad

The charger provides more filtering than basic chargers, from the large input choke to the multiple output inductors. It includes X and Y capacitors for filtering.

The charger looks mostly safe, although it doesn't have UL certification and I suspect it would fail certification. The 6mm clearance between the primary and secondary looks solid. However, the transformer windings are only separated by 3mm, rather than 6mm, as I show below. (This is still much superior to the $2 chargers that have almost no separation.)

One interesting feature of the power supply is the power plug can be interchanged for use in different countries. (Some other chargers such as the HP TouchPad and Apple iPad are similar.)

The charger has some quality issues. The power quality measurements I did in my previous article show the KMS charger has fairly poor quality output, with a lot of noise in the output.

The IC datasheet recommends 200 mm² of foil on the IC output pins to provide cooling. I measured about 18 mm² (less than 10% of recommended), which suggests the charger may overheat under full load.

The above photo shows that the build quality of the charger is not extremely high. The inductor at the front right is very crooked, and the optocoupler at the left is somewhat crooked. While this doesn't affect the performance, it shows the assembly was rapid rather than careful. More concerning, some of the solder joints appear to be almost bridged, which could cause catastrophic failure of the charger. I also found a government report of a KMS charger catching fire, apparently due to a loose wire in the power plug.

One unique feature of the charger is the blue LEDs which cause it to emit an eerie blue glow when in use. A lot of users dislike this though (according to reviews), because the light is distracting at night.

The circuit

For readers interested in circuits, I have prepared the above approximate schematic (click for a larger view). The circuit is pretty straightforward compared to other chargers (look at my iPhone charger schematic for comparison). Starting at the upper left, the input AC is converted to DC by the diode bridge, and then filtered by a simple inductor-capacitor filter. This high-voltage DC is connected to the flyback transformer primary. The THX203H control IC switches the other side of the flyback transformer to ground through the current-sense resistors R12A and R12B and inductor L3. (Most chargers use a separate switching transistor, but in this charger, the transistor is inside the control IC.) The snubber circuit R2, C3, and D6 absorbs some of the high-frequency switching spikes (although looking at the output below, this circuit isn't entirely successful). The auxiliary transformer winding and D7 and C4 provide the DC power to the control IC. The optocoupler provides feedback to the IC, indicating the output voltage level.

On the secondary side, the high-speed Schottky diodes (D5) convert the transformer output to DC. This is then filtered through an inductor-capacitor filter that smooths it out. The output voltage feedback is generated by the TL431A regulator and fed into the optocoupler.[1]

Finally, the actual USB output circuitry has more components than you'd expect. For each pair of ports, four resistors set the D+ and D- voltages to indicate to devices that the charger is (pretending to be) an Apple 2A charger. Each port has a small bypass capacitor to smooth out power transients. Finally there are two blue LEDs with current-limiting resistors to provide the blue glow.

The controller IC poses a bit of a mystery. It's labeled as the THX 203H controller, which turns out to be manufactured by NanJing TongHuaXin Electronic Co, Ltd., a Chinese switching power supply chip company (details). The datasheet for this part is very hard to understand, as it is machine-translated from Chinese, for example:

The startup circuit inside IC is designed as a particular current inhalation way, so it can start up with the magnification function of the power switch tube itself.

The controller chip is a basic 8-pin current-mode PWM controller chip. It includes a built-in NPN power transistor, which reduces the charger part count. The chip can produce 12 watts output power.

Circuit board

The above picture shows the KMS charger circuit board on the left and a circuit board from the HP TouchPad charger on the right. Compact phone chargers such as the iPhone or TouchPad chargers go to amazing effort to pack the components as tightly as possible. The KMS charger on the other hand has a much more spacious design with a lot of wasted space. Since any charger with 4 USB ports is going to be fairly large, they probably figured it's not worth the effort to make the rest of the circuitry compact. The difference in density between the two circuit boards is striking, though.

A key safety feature of the KMS charger is visible in the middle of the circuit board - note the angular cut-out slot, and the empty vertical region with no circuitry. This isolates the high-voltage circuits on the right from the low-voltage output circuits on the left. The KMS charger has a safe 6mm gap and the cut-out provides additional creepage distance. Counterfeit chargers usually skip this critical safety feature, with only a millimeter or two keeping the high voltage from reaching the output and shocking the user.

You might wonder how the charger works if the high voltage and low voltage circuits are separated by a gap. The key is that any components that cross this gap must be specially designed to avoid electrical hazards. The key component is the flyback transformer, which transfers the power through magnetic fields, avoiding any direct electrical connection between the two sides. The feedback signal passes from the secondary to the primary through an optocoupler, which transmits the feedback through a light signal, again avoiding an electrical connection. Finally, a Y safety capacitor connects the primary and secondary grounds to reduce electrical noise. The design of a Y capacitor ensures it won't pass dangerous electrical currents, and won't short out even under fault conditions.

Transformer teardown

The black circle on top of the transformer seen earlier is simply a foam disk, which helps reduce transformer noise by padding the transformer against the case. If a charger makes a high-pitched noise, it's usually coming from the transformer. Power supplies are usually designed with switching frequencies higher than people can hear, but in some circumstances it's still audible, especially if you are young and haven't lost high frequency hearing.

Under the first layers of insulating tape is a copper 'belly band' which surrounds the transformer to provide noise shielding from eddy currents in the transformer.[2] This copper shielding is omitted from super-cheap transformers, showing that this charger goes beyond the minimum.

The windings are all separated by insulating tape. Under the belly band and insulating tape is the auxiliary winding, which provides power to the control IC. You might wonder why the IC needs a separate power supply instead of using the USB power output, but this wouldn't be safe because the USB output would no longer be isolated from the input. This winding is 9 turns of wire; since the IC requires low current, the wire is fairly thin.

Above you can see half of the primary winding, which is fed by the input power. This winding has 40 turns of wire.

An interesting safety feature is the 3 mm "margin tape"[3] to the lower right of the winding, which ensures that the primary winding stays 3 mm away from the edge. I was interested to see this, since other transformers I've disassembled use triple-insulated wire instead of boundary tape. To ensure safe electrical isolation between the primary and secondary windings, either the secondary wires need to be triple-insulated, or there needs to be at least 6mm of distance between the windings. Super-small chargers don't have 3mm of extra room, so they use the more expensive triple-insulated wire. But since the KMS is larger, it uses the 3mm margin tape. I'm not an expert on safety requirements, but it looks like this transformer doesn't quite meet the requirements. Normally, the margin tape is put on both sides, so there's a total of 6mm creepage distance between the windings.[4][5] But since the tape is only on one side, the windings only have half of the required distance.

The secondary winding provides the low-voltage high-current output with 8 turns of wire. In order to support 2 amps, this winding has thick wire with four strands in parallel. I haven't seen parallel strands like this before, probably because the KMS charger supplies higher power. Note the 3mm margin tape keeping the winding away from the edge.

Finally, the second half of the primary winding forms the innermost layer of the transformer; this is also 40 turns of wire. The primary winding is split into two layers that surround the secondary winding for better electrical properties. Note that the primary winding is 80 turns, while the secondary output winding is 8 turns. To oversimplify a bit, this means the output will be 10 times the current of the input at 1/10 the voltage, which is how the high voltage low current input results in the low voltage high current output. The above picture gives a good view of the 3mm margin tape at the right that keeps the wire away from the edge of the core.

Measuring the charger in use

The charger is a switching power supply using a flyback transformer. How this works is the high voltage DC is switched on and off tens of thousands of times a second by the control IC. These pulses of DC are sent into the flyback transformer. A flyback transformer is different from normal transformers in that the output diode blocks power from flowing out of the transformer while power is flowing in. Instead, as the current increases, power is stored in the transformer as a magnetic field. When the input current switches off, the stored power then flows out of the transformer, providing the desired output.

By looking at the output voltage and frequency spectrum, we can determine a fair bit about how the device operates. I measured a constant 60 kHz switching frequency above 1 amp output load, but a dropping frequency for lower loads. The datasheet gives some clues to this behavior. The power supply normally operates using PWM (pulse width modulation). The switching frequency is constant, but the amount of time the power transistor is on varies. The longer it is on, the more power into the transformer and the more output power. This matches the observed behavior from 1 amp to 3.5 amps. The datasheet also describes how the switching frequency drops under low power, which matches what I observed below 1 amp.

The above oscilloscope trace illustrates the behavior when producing 2 amps. The frequency spectrum shows narrow peaks (orange) at the 60 kHz switching frequency and harmonics. The yellow output voltage shows a bunch of large spikes due to the power switching on and off - this indicates that the charger isn't filtering the output very well, letting these spikes get into the connected device.

The diagram below zooms in to show the output in more detail. Each spike is when the switching transistor turns on at 60 kHz. The output power drops as the current through the flyback transformer increases (since the transformer secondary is blocked by the diode at this time). The output then climbs when the transistor switches off and the power is transferred to the secondary.

As the charger load increases above 3 amps, the quality of the output significantly decreases, and large 120 Hz ripple appears in the output (yellow). This is probably because the input capacitors can't store enough power to provide a constant output at this high load. Since the charger is only rated to provide 2.1 amps of output, I don't consider this a design flaw, but it's interesting to see this behavior in the output. The key result here is not to overload the charger, because the power quality gets much worse.

The charger is designed to reduce the switching frequency under low load for efficiency. I found this feature kicks in at loads under 1 amp, with the switching frequency smoothly dropping from 60 kHz to 29 kHz at 250 mA load and even lower under no load. The graph below shows the frequency spectrum at 250 mA load. Note that the spikes are wider than the previous case since the frequency becomes more unstable when it is reduced.

The output waveform below at 250 mA is similar to the previous (2A) case, except at a lower frequency. Note that the output still has large spikes when the transistor switches on. The output voltage drops while the switching transistor is on and then rises while the transistor is off (due to the flyback design), so you can see below that the transistor is off most of the time at low power.

Power consumption

Measuring the power consumption of a charger is tricky because the charger doesn't use power like a normal resistive load, but uses a nonlinear part of the input current. This results in a power factor lower than unity. (You might expect that the poor power factor is because the charger switches on and off thousands of times a second, but actually it's the fault of the diode bridge.) I measured the power consumption of the charger under load by measuring the instantaneous line voltage and current, computing the instantaneous power, and then computing the real power from this.[6] In the following diagrams, the input line voltage is shown in yellow, and the input current is in cyan. The instantaneous power is graphed in orange at the bottom - simply the product of the voltage and current.[7]

The oscilloscope output below shows the power usage of the charger under no load. The line input voltage (yellow) is a nice sine wave, but the current (cyan) is very irregular. There is a bump corresponding to the voltage peaks as the input diodes conduct and re-charge the filter capacitors. The remaining current oscillations are unusual - I haven't seen them in other chargers, and I expect they are due to the large input choke. From the orange line you can see that the power usage has small spikes at 120 Hz. Taking the power factor into account and computing real power shows the charger uses 180 mW when idle which is fairly high, but actually lower than the Apple iPhone charger.

With load applied to the charger, the power usage shoots up as shown below. I compute the power usage as 6.4 watts, while the charger is supplying 4.4 watts to the output, for an efficiency of 69%. The shape of the current curve (cyan) and power curve (orange) shows that the charger is taking line power about half the time (the big curved peaks), and not for the other half (the flat oscillations in between). This illustrates the bad power factor that switching power supplies have. (PC power supplies often use power factor correction (PFC) circuits to improve the power factor.)The yellow input voltage curve is somewhat distorted, probably due to the lame isolation transformer I used.

You might wonder what happens if you short-circuit the output of the charger. It is designed to shut down before damage occurs, rather than self-destruct. After the internal voltage drops, the charger will start up again, and repeat this cycle until the problem goes away. This is called "hiccup mode", since the charger generates hiccups of power. The oscilloscope trace below shows the power consumption of the KMS charger when shorted. Note the pulses as it start up and shuts down every 250 milliseconds.

Components

The diodes are manufactured by MIC (Master Instrument Corporation, Shanghai). Most chargers use a diode bridge to convert the AC to DC, but this charger uses four independent diodes, which are 1N4007 700V diodes. The secondary rectification uses two Schottky diodes (SR360 3 amp 60V) from MIC. The circuit board uses the unusual mounting of two diodes on top of each other soldered into the same holes. The charger also uses FR107 700V fast recovery diodes.

Like most power supplies, the charger uses a TL431A for the voltage feedback.[1] This TL431A is produced by Wing Shing Computer Components The optocoupler is an ORPC 817B optocoupler from Shenzen Orient Technology Co., Ltd. (I don't want to speculate on the cultural significance of their raising the flag over Iwo Jima company logo.)

Conclusion

Notes and references

[1] To summarize the feedback circuit: R17 and R18 form a resistor divider on the output voltage. If the output voltage is above 5.125 volts, the TL431 control input will be above 2.5 and the TL431 conducts. This energizes the optocoupler, providing current pulling the FB pin lower. Low FB increases the duty cycle, increasing the maximum transformer current, and increasing the output voltage. If the output voltage is considerably too high, or overtemperature is sensed, the switching frequency is decreased, reducing the power transferred to the output. (This is over-simplified; the frequency response of the feedback control loop is controlled via R13, R16, C8, and C9.) An alternative is to sense voltage from the primary side, so the feedback circuit can be eliminated. This reduces the total charger cost by about 20 cents according to a report.

[2] The use of a copper "belly band" in flyback transformers is discussed in Flyback Transformer Design for the UCC28600 (page 2). It provides an electromagnetic radiation shield. The article mentions that the belly band may cause difficulties with creepage requirements and that seems to be the case with the KMS, since there is only 3mm creepage between the primary-grounded belly band and the secondary wiring.

[3] A lot of interesting information about flyback transformer design and construction is in Cookbook for do-it-yourself transformer design

[4] A discussion of how to achieve 5-6mm creepage distance by using 2.5 or 3mm margin tape is in Flyback Transformer Design for the IRIS40xx Series. Note that the margin tape must be on both sides of the winding to achieve this distance, while the KMS transformer only uses the tape on one side.

[5] Safety Considerations in Power Supply Design provides a detailed explanation of safety requirements for power supplies. It explains creepage and clearance

[6] See Understanding power factor and input current harmonics in switched mode power supplies for details on power factor, power supplies have poor power factors, and why poor power factors are a bad thing. Briefly, the power factor is due to the non-linear current through the diodes at peaks, not due to a phase shift. Real power can be measured with an oscilloscope as the average value of the instantaneous power, see Power - Real And Apparent: A Tutorial On Basic Line Power Measurements or Measuring power using the DL750.

[7] For the input power measurements it is very important to use an isolation transformer to avoid destroying your oscilloscope or shocking yourself. For my measurements, a resistor voltage divider reduced the input line voltage - the actual voltage is 11.06 times the displayed probe 1 voltage (C1, yellow). The current was measured through a 5.2 ohm shunt resistor, so the current is 1/5.2 times the displayed probe 2 voltage (C2, cyan). Combining these, the power in watts is 2.13 times the measured C1*C2 value (M1, orange).

While traveling over Thanksgiving I was thinking about how the 6502 microprocessor works and wanted to analyze some Boolean logic circuits. A trivial programming task but the only computer I had was my phone.

I searched for programming languages available on Android. Python for Android looked way too complex. The Clojure REPL intrigued me but I didn't want to learn Clojure right now. Other languages seemed limited or buggy. Then I was struck by the obvious choice for a powerful and fully-supported language with graphics capabilities: JavaScript. I could run JavaScript programs in the browser if I had a way to enter them.

I downloaded DroidEdit Pro which gave me a fullscreen editor for files on my phone. Typing HTML on the phone was painful until I downloaded the Hacker's Keyboard, which makes it much easier to type special characters. The picture below shows these tools in use.

My development cycle is:

• Edit the code in DroidEdit and save it to a local .html file.
• Select 'Preview in Browser' from DroidEdit and test the program.
• Upload the file to my web server using DroidEdit's SFTP support when ready.

For debugging, the trick is to use the default browser, not Chrome. Enter about:debug in the URL bar to open the JavaScript console, which is vital for debugging.

Obviously this environment isn't as powerful as a full-size keyboard and monitor and powerful editor, but it lets me program no matter where I am. I haven't got the hang of cut-and-paste in the editor, but shift-arrow seems to work better than tapping.

Here's my program in action. It wont get any style points - I rapidly lost my enthusiasm for whitespace with the tiny keyboard - but it got the job done.

I also used this development environment to show my nephew how to make web pages with HTML. He thought it was very cool that he could type HTML into the phone, hit Control-S to save, and immediately load the web page on his iPad. He's now busily learning HTML and building his own web pages.

I hope these tips help you program while on the road. Leave a comment if you have tips of your own.

The 6502 is an 8-bit microprocessor that was very popular in the 1970s and 1980s, powering popular home computers such as the Apple II, Commodore PET, and Atari 400/800. The 6502 instruction set includes 8-bit addition and subtraction operations. Various status flags (carry, zero, negative, overflow) are set based on the result of the operation. Most of the flags (carry, zero, negative) are straightforward, but the meaning of the overflow (V) flag is harder to understand. If the result of a signed add or subtract won't fit into 8 bits, the overflow flag is set. (The overflag is affected in a couple other cases - the BIT operation, and the SO pin on the chip. These are discussed in detail in the excellent article The overflow flag explained, so I won't discuss them here.)

Addition on the 6502

The carry flag is used as the carry-in for the operation, and the resulting carry-out value is stored in the carry flag. The carry flag can be used to chain together multiple ADC operations to perform multi-byte addition.

Ones-complement and twos-complement

The twos complement of a number is the ones complement of the number plus 1. That is, the twos complement of N is 256-N. Thw twos complement is very useful because adding M and the twos complement of N is the same as subtracting N from M. For example, to compute 80 - 112, simply take the twos complement of 112 (binary 10010000) and add it to 80 (binary 01010000), yielding (binary 11100000). This result is the twos complement of 32, indicating -32.

Note that 80+144 and 80-112 had exactly the same bit-level operations - only the interpretation of the bits was different. This is why twos complement numbers are so useful - the same addition circuit works with them.

To see why twos complement numbers work this way, consider M + (-N) or M - N

Thus, adding the twos complement is the same as subtracting. (With the exception of the carry bit, which is affected by the extra 256. This will be discussed later)

Twos-complement signed numbers

The nice thing about signed numbers is that regular binary arithmetic yields the expected results (in most cases). That is, the processor adds or subtracts the numbers as if they are unsigned binary numbers, and the right answer occurs just by interpreting them as signed.

Another example shows that the carry is ignored with signed addition. In this case, 80 and -48 are added, yielding 32. Since 80 + (256-48) = 256 + (80-48), the "extra" 256 ends up in the carry bit.

Unfortunately, problems can happen. For instance, 80 + 80 = 160 with unsigned arithmetic, but with signed arithmetic the result is unexpectedly -96. The problem is that 160 will fit into a byte as an unsigned number, but it is too big to store in a byte as a signed number. Since the top bit is set, it is interpreted as a negative number. To indicate this problem, the 6502 sets the overflow flag.

The table that explains everything about overflow

This section explores all the possible ways that overflow can occur. The following examples consider the addition of two signed numbers M and N. It is only necessary to consider the top bits of the numbers and the carry from bit 6, as shown in the diagram below, since the lower bits don't affect overflow (except by causing a carry from bit 6).

There are 8 possibilities for these bits, as expressed in the table below. For each set of input bits, the table shows the carry out (C₇), the top bit of the sum (S₇), which is the sign bit, and the overflow bit V. This covers the 4 possibilities for sign of the arguments (positive + positive, positive + negative, negative + positive, negative + negative), with and without carry from bit 6. The table shows an example sum for each line, first expressed in hexadecimal, and then interpreted as unsigned addition and signed addition.

A few interesting things can be noted from this table. Signed overflow (V=1) happens in two of the eight cases - when the result of adding two positive numbers overflows and ends up negative, and when the result of adding two negative numbers overflows and ends up positive. These rows are highlighted. Signed overflow will never happen when adding a positive number and a negative number, since the result will have a smaller magnitude. Unsigned carry (red in the unsigned column) happens in four of the eight cases, and is independent of signed overflow.

Formulas for the overflow flag

A common definition of overflow is V = C6 xor C7. That is, overflow happens if the carry into bit 7 is different from the carry out.

A second formula simply expresses the two lines that cause overflow: if the sign bits (M₇ and N₇) are 0 and the carry in is 1, or the sign bits are 1 and the carry in is 0:
V = (!M7&!N7&C6) | (M7&N7&!C6)

The above formula can be manipulated with De Morgan's laws to yield the formula that is actually implemented in the 6502 hardware:
V = not (((m7 nor n7) and c6) nor ((M7 nand N7) nor c6))

Overflow can be computed simply in C++ from the inputs and the result. Overflow occurs if (M^result)&(N^result)&0x80 is nonzero. That is, if the sign of both inputs is different from the sign of the result. (Anding with 0x80 extracts just the sign bit from the result.) Another C++ formula is !((M^N) & 0x80) && ((M^result) & 0x80). This means there is overflow if the inputs do not have different signs and the input sign is different from the output sign (link).

Subtraction on the 6502

Defining the borrow bit in this way makes the hardware implementation simple. SBC simply takes the ones complement of the second value and then performs an ADC. To see how this works, consider M minus N minus borrow B.

The following table shows the overflow cases for subtraction. It is similar to the previous table, with the addition of the B column that indicates if a borrow resulted. Unsigned operation resulting in borrow are shown in red, as are signed operations that result in an overflow.

Comparing the above table with the overflow table for addition shows the tables are structurally similar if you take the ones-complement of N into account. As with addition, two of the rows result in overflow. However, some things are reversed compared with addition. Overflow can only occur when subtracting a positive number from a negative number or vice versa. Subtracting positive from positive or negative from negative is guaranteed not to overflow.

The formulas for overflow during addition given earlier all work for subtraction, as long as the second argument (N) is ones-complemented. Since internall subtraction is just addition of the ones-complement, N can simply be replaced by 255-N in the formulas.

Overflow myths

The most common myth is that just as the carry bit indicates a carry (or overflow) from bit 7, the overflow bit indicates a carry (or overflow) from bit 6 (example, example, example). As can be seen from the table above, sometimes a carry from bit 6 causes an overflow and sometimes it doesn't.

Another myth is that for multi-byte signed numbers, you use the overflow flag instead of the carry flag to carry from one byte to another (example). In fact, carry is still used to add/subtract multi-byte signed numbers, the same as with unsigned numbers.

It is sometimes claimed that the overflow bit is set if a result is too large to be represented in a byte (example, example). This omits the critical word signed - a signed result can be too large to fit in a byte, even if the unsigned result fits, and vice versa. Examples are in the table above.

Another confusing explanation is that the overflow flag is set when the sign bit is affected (example). The table shows that sometimes there is overflow when the sign bit is affected by bit 6 carry, and sometimes there is overflow when the sign bit is not affected.

Conclusions