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Old 22nd March 2014, 10:26 PM   #1
popilin is offline popilin  Argentina
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Default Design of transformers for valve amplifiers

Part 1- Fundamentals

WARNING!
If you don't like math, skip this post to the next with abbreviated equations and practical advices.

I would have liked to write a little more, but the forum server does not let me…
Just a few extra details to clarify (or obscure) some concepts.

Real Transformers -The Core Nightmare

In the attachment we have almost all the math to design transformers, but we need a core to transport energy from primary to secondary.
We need a ferromagnetic material, but in ferromagnetic materials, the relation between B and H exhibits both nonlinearity and hysteresis, B is not a single valued function of H.

B = μ H

Houston, we have a problem…

Magnetic anisotropy is a prerequisite for hysteresis in ferromagnetic materials, so μ is not a scalar, but a tensor.

We can still using previously derived equations in the attachment, on condition to renouncing the correct description of influence of the material medium, at others; they are an astonishingly accurate approximation of reality.

How transformers work

Following analysis, for simplicity and better comprehension, supposes no copper losses, our wire is an ideal conductor.

-No load
If we put an AC voltage, Up on the primary, it produces a magnetic field B with B given by eq (23) and it induces a voltage Us on the secondary, furthermore a current im flows on the primary, this current is related with magnetic field H with H given by eq (43) and is called magnetizing current.
However is not a sinusoidal current, even when Up and B both were sinusoidal.
Both fields B and H are interrelated by the magnetic hysteresis curve B=f(H), the area inside this loop represents core losses, and like magnetizing current, remains the same on transformer operation regardless of load.

-Full load
If on the secondary flows a current is , it needs a current ip on the primary to maintain the relation Np ip = Ns is, clearly responsible for this is the field H, however both fluxes, primary and secondary, are canceled and the field B remains the same, and fortunately, when Up is sinusoidal, currents are also sinusoidal, except magnetizing current which is usually negligible.
The magic to transfer energy from primary to secondary with the same field B is provided by field H, the energy transport depends on B.H

If we consider copper losses, the major difference is that B reach a maximum at no load when Up reaches its maximum, so core losses reach its maximum too.
The idea is counter intuitive, it even verges on not making sense, but is a fact, when exists primary resistance Rp it reduces primary voltage Up proportionally to primary current, so B is reduced with load.

Note1: Name the fields B and H (or E and D) as you want, is just semantic, but be in mind that they have different properties unless they were in vacuum.

Note2: AFAIK, all equations in the attachment are correct and double checked, maybe I screwed up something to derive eq (80), (81) for leakage inductance, its value differs in excess of about 15% from accepted value eq (82)
Also I apologize for drawings, made with my old DOS program to design PCBs.
If anyone finds any error, please let me notify
. Thanks.
Attached Files
File Type: zip Transformers.rar.zip (904.1 KB, 128 views)
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Old 22nd March 2014, 10:36 PM   #2
popilin is offline popilin  Argentina
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Part 2- Abbreviated equations

Core area
Actually, needed core area for a given power is pretty small, but for practical reasons there are plenty rules of thumb; for PT or PP OPT

S = 1000 √(P / fo Bmax) [cm²]
For SE OPT
S = 1500 √(P / fo Bmax) [cm²]
Where
P = Power [W]
fo = Lowest frequency at full power [Hz]
Bmax = Maximum B [Gauss]

Number of primary turns

Np = (Uac x 10) / [√2 π fo S Bac(max)]
With
Uac = √(P Zp) For OPT

S = ζA
Where
Uac = Primary RMS voltage [V]
Zp = Primary impedance []
A = Apparent core area [cm²]
ζ = Stacking factor, between 0.9 and 0.95 (thinner lamination the smaller value)

Number of secondary turns

Ns = (Np Us) / (η Up) For PT

Ns = Np √(η Zs / Zp) For OPT

Where
η = Efficiency factor 0.9 to 0.95
Zs = Secondary impedance []

AC magnetic field

Bac(max) = (Uac x 10⁸) / (√2 π fo S Np)

DC magnetic field

Bdc(max) = [4 π μ Np i(DC)] / (9 l)

Where
i(DC) = Primary DC current [A]
l = Magnetic circuit length [cm]

Primary Inductance

Lp = (4 π μ S Np²) / (9 l x 10)

Air Gap

lG = l (μ - μeff) / (μ μeff)

Where
l = Magnetic circuit length [cm]
μeff = Magnetic permeability for preceding equations work [cgs] dimensionless
μ = Magnetic permeability from lamination datasheet [cgs] dimensionless

Copper Losses

R = ρ N lm / (π r²)

Where
ρ = 0.017 (Ω mm2) / m , copper resistivity.
lm = average turn length around bobbin [m]
N = number of turns
r = wire radius [mm]


Ploss [%] = 100xRp/(Zp+Rp)

Sloss [%] = 100xRs/(Zs+Rs)
Or
InsertionLoss = 10 log {1 – [Rp/(Zp+Rp)] – [Rs/(Zs+Rs)]} dB

Core Losses
According to Steinmetz empirical equation
Hysteresis loss
Wh η f (Bmax)¹˙⁶ x 10⁻⁸
Eddy current loss
We ξ d² f² (Bmax)² x 10⁻¹¹

Where
[Wh] = [We] = W/cm³
f = Frequency [Hz]
d = Lamination thickness [mm]
η , ξ = Coefficients for the magnetic material

Leakage Inductance

Ls = [0.417 Np² lm (2 n c + a)] / (10 n² b)

Where
Np = number of primary turns
lm = average turn length around bobbin [mm]
n = number of dielectrics between windings
c = thickness of dielectric between windings [mm]
a = winding height [mm]
b = winding traverse [mm]


Capacitance
-Between primary and secondary windings

C (ε A) / (4πd)

Where
ε = Dielectric constant of insulator [cgs] dimensionless
A = Average winding area [cm²]
d = Insulator thickness [cm]
[C] = cm, 1cm 1.113 pF (I love cgs units)

-Between layers
Cl (ε A) / (4πd)

-Total winding capacitance Cw

(Cw)¹ ≈ (Cl)¹ ∑ {(4/3nı) [1 - (1/nı)]}¹ ı=1,...,NP

Where
n = Number of layers from +B to a layer midpoint (after/before insulation interphace)
NP = Total number of primary windings

-Total distributed capacitance

Cd ≈ C ∑ (nı/N)² ı=1,...,m

Where
n = Number of layers from +B to a layer midpoint (after/before insulation interphace)
N = Total number of layers in each primary winding
m = Number of insulation interphases between primaries and secondaries

-Shunt Capacitance
Cs = Cw + Cd

Insulation
Valve amps work with high voltages, we must very cautious with insulation, as my own rule of dumb I use

Vi = 3 (Upp + Udc)

Distortion
THD for silicon steel has been calculated by Dr.N.Partridge (see RDH4, p.215)

Vh/Vf = [(Shx10 l Ra)/(8 π² Np² S f)] [1 - (Ra/4 Zf)]

Where
Vh = Harmonic voltage across the primary
Vf = Fundamental voltage across the primary
Sh = Distortion coefficient of the magnetic material
l = Lenght of the magnetic path
Np = Number of primary turns
Ra = Resistance (or equivalent resistance) in series with the primary
S = Cross-sectional area of the core
f = Fundamental frequency in Hz
Zf = Primary impedance at fundamental frequency ≈ 2 π f L
L = Primary inductance at chosen field B
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Last edited by Mooly; 22nd April 2014 at 12:03 PM. Reason: Corrected text " ζ = Stacking factor, between 0.9 and 0.95 (thinner lamination the smaller value)"
 
Old 22nd March 2014, 11:28 PM   #3
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Transformers can be highly complex beasts.

I usually just buy off the shelf parts depending on the job I want it for.
I leave an experts job to the experts.
They also have the right kit for building the transformers quickly and accurately.

I have built a few for SMPS and learned a lot from that.
For parallel windings it is important to get them exactly the same or high currents can flow and cause extreme heating.
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Old 22nd March 2014, 11:45 PM   #4
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To improve accuracy I will collaborate with the following:

Stacking factor (depends on type of assembly as well as lamination thickness):
1:1 0.88
4:4 0.92
Butt joint 0.95
(Values for typical 0.35mm lamination, adjust accordingly)

Equivalent air gap of assembly types (cm):
1:1 0.012
4:4 0.025
8:8 0.05
16:16 0.1
Butt joint 0.12
(Values for typical 0.35mm lamination, adjust accordingly)

Source: Electronic designers handbook Mc-Graw Hill 1957
 
Old 22nd March 2014, 11:52 PM   #5
popilin is offline popilin  Argentina
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Quote:
Originally Posted by nigelwright7557 View Post
Transformers can be highly complex beasts.
I can vouch for that!

Quote:
Originally Posted by nigelwright7557 View Post
I leave an experts job to the experts.
No need to be an expert to build excellent transformers, just patience and perseverance.

Quote:
Originally Posted by nigelwright7557 View Post
For parallel windings it is important to get them exactly the same or high currents can flow and cause extreme heating.
When I need various secondaries paralleled, with an auxiliary core and an auxiliary winding, I measure carefully each secondary winding to mV tolerance, I do not want smoke.
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Old 22nd March 2014, 11:58 PM   #6
popilin is offline popilin  Argentina
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Quote:
Originally Posted by marcelop View Post
To improve accuracy I will collaborate with the following:

Stacking factor (depends on type of assembly as well as lamination thickness):
1:1 0.88
4:4 0.92
Butt joint 0.95
(Values for typical 0.35mm lamination, adjust accordingly)

Equivalent air gap of assembly types (cm):
1:1 0.012
4:4 0.025
8:8 0.05
16:16 0.1
Butt joint 0.12
(Values for typical 0.35mm lamination, adjust accordingly)

Source: Electronic designers handbook Mc-Graw Hill 1957
Gracias por tu colaboración !

Thanks for your reply !

And for the air gap, the idea is to adjust magnetic permeability to desired primary inductance in order to not exceed Bdc(max) in SE OPTs.
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Old 23rd March 2014, 01:57 AM   #7
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I applaud the above dissertation, but for practical purposes designers would rather use tables and design charts taking care of much of the spadework, not to mention committed cad programmes available these days. I still use the Crowhurst charts published in RDH-4 (as mentioned) I know of at least one cad program derived from that and including properties of steels, which is even more convenient.

Quote:
Originally Posted by popilin View Post
No need to be an expert to build excellent transformers, just patience and perseverance.
Respectfully, I would disagree with that, Poplin. There are practical factors like winding curvature, end-of-layer, core assembly etc. which cannot just be rushed into. Initial experience is necessary. Thus, yes; one can build an excellent transformer - but don't expect it to come out right after the first try. Unless you are singularly fortunate that it did ... in which case you would not have learnt much.

Quote:
When I need various secondaries paralleled, with an auxiliary core and an auxiliary winding, I measure carefully each secondary winding to mV tolerance, I do not want smoke.
I do not fully understand that; do you in the middle of winding move the lot to a different core and first measure each winding?? In my experience it is quite sufficient to begin and end similar windings on the same imaginary cross-line. That is easily accomplished to within 1 mm; most commercial transformers are made that way.
 
Old 23rd March 2014, 02:29 AM   #8
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Quote:
Originally Posted by Johan Potgieter View Post
I applaud the above dissertation, but for practical purposes designers would rather use tables and design charts taking care of much of the spadework, not to mention committed cad programmes available these days. I still use the Crowhurst charts published in RDH-4 (as mentioned) I know of at least one cad program derived from that and including properties of steels, which is even more convenient.
Yes, agreed, but everyone has their own method, there are lots over there, I like mine.

Not a dissertation, but my humble collaboration to share at least the equations, the correct ones, I hate to read 4.44, instead of √2 π

Quote:
Originally Posted by Johan Potgieter View Post
Respectfully, I would disagree with that, Poplin. There are practical factors like winding curvature, end-of-layer, core assembly etc. which cannot just be rushed into. Initial experience is necessary. Thus, yes; one can build an excellent transformer - but don't expect it to come out right after the first try. Unless you are singularly fortunate that it did ... in which case you would not have learnt much.
Maybe I'm a rara avis, my first set of transformers worked wonderfully from the first time, maybe I had luck, but with my winding method there is only one chance, a silver bullet.

Quote:
Originally Posted by Johan Potgieter View Post
I do not fully understand that; do you in the middle of winding move the lot to a different core and first measure each winding?? In my experience it is quite sufficient to begin and end similar windings on the same imaginary cross-line. That is easily accomplished to within 1 mm; most commercial transformers are made that way.
That I meant is about counting turns, with a cheap calculator, a reed switch and a magnet, everytime that finish a winding I must check if all was OK.
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Old 23rd March 2014, 01:51 PM   #9
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Quote:
Originally Posted by fullrangeSR View Post
thank you but no. if i wanted a headache inducing formulaic arcana, there's always RDH4 - why paraphrase it? if i wanted a free wealth of simple and useful how-to explanation there's the unbeatable mr. turner from down under. and if i wanted an automated freebie there's the yvesm from dissident audio... your level of contribution on this subject can't really match any of the above three... are you trying to get back at your nemesis from another opt thread here?
Turner is a good source for practical transformer design and it is good in that respect. This thread (as I see it) was to put together every formula someone could need to build a transformer, which I personally (as other people) find useful. If you think it´s useless don´t comment and let the thread slide by.

With respect to the thread if someone could add the RDH4 leakage inductance and capacitance charts by crowhurst it would add a lot.
 
Old 23rd March 2014, 02:33 PM   #10
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Quote:
Originally Posted by marcelop View Post
Turner is a good source for practical transformer design and it is good in that respect. This thread (as I see it) was to put together every formula someone could need to build a transformer, which I personally (as other people) find useful. If you think it´s useless don´t comment and let the thread slide by.

With respect to the thread if someone could add the RDH4 leakage inductance and capacitance charts by crowhurst it would add a lot.
Bien Marcelo, un poco de buena onda al thread!

All who make transformers had visited sometime Patrick Turner's page, very useful and detailed.

That's precisely the idea, unfortunately words are not my thing and I must write equations, with bigger characters because of super indexes, my eyes are not as before...

BTW, RDH4 here on Pete Millett's page

Technical books online
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Last edited by popilin; 23rd March 2014 at 02:53 PM.
 

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