Understanding Equations From Merlin's Books And Deriving Values

[Gary_S]

Can anyone explain where the 6 is being derived from in the equation below from Merlin's power supplies book?

It's a calculation to work out minimum inductance and Merlin says it like this in the book: For a given resistance in series with the choke, the minimum value of  inductance required for proper regulation is:

                          Rs+Rl/6∏f

[dude]

I would think the 6 would be a multiplier (x) 

[Gary_S]

Yeah, obviously it's a multiplier (x ∏ x f) but where it's derived from is what i'm interested in. As in; why are we using that equation x6 as opposed to x2 or x4. The significance of it i'm talking about.

It must be something to do with the frequency of the sine wave or something as in 6 x ∏, as ∏r² represents the area of a circle which also corresponds to the full 360º of a sine wave.

[jazbo8]

The short answer: 6 = 2*3.  

The long answer:

The source of this equation is from O.H. Schade given in the footnote, which you can look up online. But the idea is basically finding the critical inductance for a full-wave rectifier circuit with series resistance, since the resistances from the rectifier and the inductor affect the amplitude and the conduction angle of the rectified waveform as shown in the graphs below:



The critical inductance equation for proper regulation is:
       
      Lo ~ (Rs+RL)/2*w*K (from an even older paper by Dellenbaugh & Quinby in 1932)

      K=1.5 (from the Schade paper), and w = 2*pi*F

      substituting w & K into the above equation then,

      Lo = (Rs+RL)/2*2*pi*F*K
          
          = (Rs+RL)/6*pi*F

Please note that the impedance of the inductor needs to be significantly larger than the impedance of the capacitor, i.e., 2*w*L >> 1/(2*w*C) for the above Lo estimation to be valid.

Jaz

[PRR]

Sub-notes:

In most systems, Rl is much larger than Rs (otherwise the Rl power is a not-large part of the total power; wasteful). So neglect Rs for pencil estimations. (You only need an estimate, since when you go to buy a choke you have to take the next-larger standard part.)

Rl is *not* the full-load condition.

Say an amp sucks 400V 200mA at full-roar. Rl=2K. But at idle it may only suck 400V 80mA: 5K. And at cold-start it may only suck with a 200K bleeder, or even no suck at all: Rl=infinity.

If it *always* ran 200mA Rl=2K, then 1.768H is critical, buy a 2H choke. But idling 80mA you need 4.42H; cold-start with 200K bleeder you need 177H, and if no load you need infinite inductance to maintain "choke input" mode.

If it isn't enough H to work "choke-input", it works "cap-input". Voltage tends to be 1.57 times higher than choke-input, and that affects your cap voltage rating (also choke insulation). If it is only a few seconds at start-up, a "450V" electrolytic may stand trying to charge to 628V for a while (it gets real leaky at 475V-525V and now we have an "Rl" and the choke has load; it takes time for a over-volted cap to blow-up and it may live until the finals warm-up and get the Rl down).

[jazbo8]

Sub-notes:
If it isn't enough H to work "choke-input", it works "cap-input". Voltage tends to be 1.57 times higher than choke-input, and that affects your cap voltage rating (also choke insulation). If it is only a few seconds at start-up, a "450V" electrolytic may stand trying to charge to 628V for a while (it gets real leaky at 475V-525V and now we have an "Rl" and the choke has load; it takes time for a over-volted cap to blow-up and it may live until the finals warm-up and get the Rl down).

Very good point. Since the chokes are more expensive than your typical non-audiophool capacitors, I found a good compromise is a CLC pi filter, with a small C first, which most tube rectifiers prefer, then a "normal" sized choke, followed by a large C.

[Gary_S]

Thanks guys! outstanding work. I liked that bit jazbo8 about the short answer is 2*3 and i knew there was a complicated bit coming after.

Thanks though because it's made me more aware of where he's deriving that figure from. I just hate seeing things written out and i don't know where it's coming from.

Now my next question! in the exact same area of the book (top of page 52) in this calculation for the choke voltage: 4√2Vrms/3∏ Now from what i've studied so far this equation is saying 4 x 1.414 x Vrms/3∏  the square root being removed by me because it's the 1.414 of 2. 1.414 x Vrms gives you the peak voltage, so it's 4 x that.

My thinking, and please set me straight if i'm off, is that the 4x is because it's double the mains frequency so thereby there's 4 peaks to the sine waves. As i say i may be way off base with that but that's the way i was thinking.

Also why 3∏?

Simple questions to some perhaps, but it's pretty difficult stuff until you totally get your head round it. But i always did like a challenge!

[jazbo8]

Now my next question! in the exact same area of the book (top of page 52) in this calculation for the choke voltage: 4√2Vrms/3∏

This one is easier, the Fourier series of a full-wave rectified sine wave is:



Ignoring the first term, which is the DC component, and the higher harmonics then,

     f(t) = 4/∏ * cos(2*2*∏*f*t) / (2^2-1)
           
           = 4/3∏

and the √2 term takes care of the rms to peak conversion. So, the AC component across the choke is:

     Vchoke = 4√2*Vrms / 3∏

[jojokeo]

Holy calculus, Batman. Don't let your friends drink and derive!

[Gary_S]

Thanks jazbo8 for helping me out with those. It's much appreciated i can tell you. 

I'm the kinda guy that likes to know, when someone gives a formula in a book, how exactly they came to get that formula and why i'm using it. It would be better if i could take it on blind faith and just use the formula without analysing it. However that's not me i suppose and i am curious of every little detail that comes up. Obsessive maybe even! It certainly makes a lot more work when you analyse everything you read but i'm always scared i miss something that's gonna be important.

Anyway that Schade footnote in the book; do you know where i can read about that? Is it available online or...?

Thanks mate.

[jazbo8]

I think I got the Schade paper from Scribd, if it is not there, let me know, and I will send you a copy.

[Gary_S]

I think I got the Schade paper from Scribd, if it is not there, let me know, and I will send you a copy.

Thanks jazbo8 i'll try there. 

[PRR]

> the square root being removed by me because it's the 1.414 of 2.

Remove _all_ the constants. In a universe where pi equals a number slightly over "3": 

0.424*(Vrms).

Why do you care about the choke voltage? (You care about choke voltage-to-ground, but not its end-to-end voltage so much.)

> the Schade paper

http://tubebooks.org/technical_books_online.htm
Electron Tubes (Volume II),  RCA, 1949, 71MB PDF file
first essay

[Gary_S]

> the Schade paper

http://tubebooks.org/technical_books_online.htm
Electron Tubes (Volume II),  RCA, 1949, 71MB PDF file
first essay

Thanks PRR for posting this, it's kind of you and i appreciate it. I'll have a good read through that. 

[Gary_S]

So i got that book downloaded and found the section by Schade. it's not on the page number Merlin quoted in the footnotes it's at the start of this online version in the rectifier section.

Anyway i started reading it last night. A lot of good stuff in here. But is it just me because i find the diagrams really hard to read! So many small symbols that aren't written clearly and it's quite difficult to decipher what they are.

Anyway it's all good stuff and i hope i can learn some more from it.

What do you guys recommend for this deciphering equations? Do i really need to go back and study the whole maths curriculum to get a handle on this stuff? At 50 years of age that's pretty daunting!

Basically i'm asking for advice about what you think is the best way to proceed in learning as regards equations for amp building.

Is it best to just take the equations in Merlin's book for example and just employ them without analysing them? or do you think it's essential to know where they come from and how they're derived?

To understand where they all come from is gonna take a lot of going back and studying up on all the maths symbols and what they mean. I'm certainly prepared to delve back into maths if that's what it takes. But am interested in what you guys think is needed.

Does every amp builder know all this stuff? 

[jazbo8]

For basic amplifier design, all one really need is algebra and occasionally trigonometry (when dealing with graphical solutions). But if you want to dig in deeper, for example on a physical level, say of tubes, transistors and transformers, then calculus is required. At the other extreme, there are many great builders and players that enjoy building their amps without knowing anything about electronics or math, all they need to do is to follow the assembly instructions or layout diagrams. So everybody works differently, it all depends on what you want to get out of it...

[Gary_S]

Thanks jazbo8. Yeah i've been studying that Schade paper. Pretty cool stuff though some of the diagrams are impossible to read they're so small, can't hardly make out any of the symbols. Do you have problems reading these as well? 

That other paper you mention, the Shellenbaugh one; can you get it on that link that PRR gave me? where i got Schade from? You might not know that.

Anyway on Schade i don't see where you get the ratio K from. I see the other one, the W for the 2∏fRl but i don't see the ratio one.

Also on these papers what does the N stand for in some of the equations? i haven't seen that one before.

Some of the terminology is a bit different to what's used now. They have E with a little sine above it i take it that's for AC and then there's E with a straight line, i think that's for DC but i've never seen anything written like that before so some of the symbols are a bit confusing!

[jazbo8]

Anyway on Schade i don't see where you get the ratio K from. I see the other one, the W for the 2∏fRl but i don't see the ratio one.
Also on these papers what does the N stand for in some of the equations? i haven't seen that one before.

Yes, the one in the RCA manual is hard to read, try the one on Scribd, it is easier to read. You can see the derivation for K in the inductance filter section. I am not sure where you are seeing the "N", is it from the same paper?

[Gary_S]

Yes, the one in the RCA manual is hard to read, try the one on Scribd, it is easier to read. You can see the derivation for K in the inductance filter section. I am not sure where you are seeing the "N", is it from the same paper?

Thanks jazbo8, yeah look on page 23 of the chapter and it has (n∏) i think it means number of cycles or something along those lines, i think i read that when i was skimming it but i'm not sure and just wanted some clarity as i've seen it used in a number of these types of things.

Your copy might be numbered differently to mine. I know in Merlin's notes he said the Schade thing was on pages 346 or something but in this copy i have they're right at the start in the first chapter.

[jazbo8]

I am still not quite sure if we are looking at the same thing, but the reference for n that I see is shown in the picture below, is this what you are referring to? If so, the n in this case, defines the two conditions for the waveforms - half-wave and full-wave.

[PRR]

> it's not on the page number Merlin quoted

The book I pointed you to is a "greatest Hits" compilation. Not the first publication of that essay but a reprint. AFAIK it is the same stuff. And the last word on the subject: nobody else has covered it better. TI's power supply manual from the 1970s reprinted Schade's graphs.

[Gary_S]

Thanks jazbo8, no that's not it. I'll see if i can copy it and post it up here. If not i'll post the full equation the way it's written. But that's good info you gave me just there about the waveforms. All info is good for me!

On this copy i'm reading the bit you posted is on page 31. The page i'm referring to is back by 8 pages and is on page 23. It says on the screen, the very first words on the page are "The initial value of it (0) of the second transient is increased by the value i (∏) flowing in the common circuit inductance L"

Then another paragraph and then the equations i'm talking about with the n's in them.

Thanks PRR, just read your comment before i posted this.

Thanks man.

[jazbo8]

On this copy i'm reading the bit you posted is on page 31. The page i'm referring to is back by 8 pages and is on page 23. It says on the screen, the very first words on the page are "The initial value of it (0) of the second transient is increased by the value i (∏) flowing in the common circuit inductance L"

Then another paragraph and then the equations i'm talking about with the n's in them.

I see now says the blind-man  , so "n∏" is referring to time, see the x-axis scale on Fig. 20 (others as well where the time scale are used), it is in increments of ∏.

[Gary_S]

Thanks man i appreciate it. The page numbers are just different 'cos i'm using that different copy!

When they're talking pi in these publications well obviously everyone remembers pi as 3.14 whatever, can't remember the other numbers! But what does that represent in terms of the sine wave? Is it the full AC sine wave positive and negative? or does it represent the positive alternation or vice versa the negative alternation?

[jazbo8]

When you see ∏, 2∏, n∏... used, they are usually referring to angular frequency, they key thing to remember is one revolution is equal to 2π radians, so ∏ is half-a-cycle, or half-wave and 2∏ is a full-cyle or full-wave, then the cycle starts all over again like this

Also I would suggest that you re-read Merlin's book again, before getting too deep into Schade's paper, there is just too much information to digest if you have not gotten the basic concepts down first (that's why it was in the footnote in the first place).

[Gary_S]

When you see ∏, 2∏, n∏... used, they are usually referring to angular frequency, they key thing to remember is one revolution is equal to 2π radians, so ∏ is half-a-cycle, or half-wave and 2∏ is a full-cyle or full-wave, then the cycle starts all over again like this

Also I would suggest that you re-read Merlin's book again, before getting too deep into Schade's paper, there is just too much information to digest if you have not gotten the basic concepts down first (that's why it was in the footnote in the first place).

Thanks jazbo8. Yeah i'm reading Merlin's book all the time. I study it every day but if a term is not explained and i can't reason it out i usually have to study up or revise it from somewhere else or ask someone about it.

The thing is i think much of the complicated maths in Schade might not be needed in most cases because Merlin has those charts up where you can just calculate the DC voltage based on a % of the AC or RMS, taking into consideration the 2∏fCRl and Rs/Rl figures. I see what Merlin meant when he said in the book there's some extensive maths goin' on in calculating the DC from a rectifier!

Cheers jazbo8 you've really helped me out.

[jjasilli]

Yes, charts are the solution unless you really want the theory.  This is never ending across all discipline's.  E.g., if you want to mount something to the chassis, you have to drill a hole.  What size hole? That's an engineering question for which there are complex equations.  But we have standard size screws & bolts, with charts for tight and loose hole sizes.  If the hole is not to be tapped you could just eyeball the drill size.  But if you want to know how to derive the value of the hole's diameter, the math is there.  On this Forum the derivation of an inductor's value is probably more interesting, but you get the point. 

The depth of skill and knowledge of the members of this Forum never ceases to be amazing.   

[Gary_S]

The depth of skill and knowledge of the members of this Forum never ceases to be amazing.   

Yes agreed, the best of anywhere i've seen. Without a doubt. And willing to educate us who are newer to the game, which i think is important too

You have to listen to people who are more knowledgeable about the subject than yourself and ask questions and learn from what they say and their experience.

[jazbo8]

thanks for the kind words, now I just wish that I could figure out how to actually build amplifiers without butchering the chassis and cabinet 

Anyway, the Schade regulation charts after more than 70 years are still being used today for modern power supply design because there's none better. So sometimes, the graphical method is the best and quickest for certain designs, speaking of... if you want to see a lot more handy charts for tube amp design, the Radiotron Designer's Handbook, aka RDH4 is hard to beat. I refer to it all the time, there is a wealth of information to be learned and digested.

[jjasilli]

thanks for the kind words, now I just wish that I could figure out how to actually build amplifiers without butchering the chassis and cabinet

I feel your pain! which is worsened by seeing photo's of the fine soldering, chassis and cabinet work posted here.