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22nd March 2014, 10:26 PM  #1 
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Join Date: Oct 2010
Location: Traslasierra

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.
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I do not suffer from insanity, I enjoy every minute of it.  Edgar Allan Poe He has the most who is most content with the least.  Diogenes of Sinope 
22nd March 2014, 10:36 PM  #2 
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Join Date: Oct 2010
Location: Traslasierra

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] S = 1500 √(P / fo Bmax) [cm²] P = Power [W] fo = Lowest frequency at full power [Hz] Bmax = Maximum B [Gauss] Number of primary turns Np = (Uac x 10⁸) / [√2 π fo S Bac(max)] Uac = √(P Zp) For OPT S = ζA Uac = Primary RMS voltage [V] Zp = Primary impedance [Ω] 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 η = Efficiency factor ∼ 0.9 to 0.95 Zs = Secondary impedance [Ω] AC magnetic field Bac(max) = (Uac x 10⁸) / (√2 π fo S Np) Bdc(max) = [4 π μ Np i(DC)] / (9 l) i(DC) = Primary DC current [A] l = Magnetic circuit length [cm] Primary Inductance Lp = (4 π μ S Np²) / (9 l x 10⁸) Air Gap lG = l (μ  μeff) / (μ μeff) 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) 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) 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 We ≈ ξ d² f² (Bmax)² x 10⁻¹¹ [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)] / (10⁹ n² b) 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) 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) (Cw)⁻¹ ≈ (Cl)⁻¹ ∑ {(4/3nı) [1  (1/nı)]}⁻¹ ı=1,...,NP 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 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 (Up₋p + Udc) Distortion THD for silicon steel has been calculated by Dr.N.Partridge (see RDH4, p.215) Vh/Vf = [(Shx10⁹ 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 = Crosssectional 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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I do not suffer from insanity, I enjoy every minute of it.  Edgar Allan Poe He has the most who is most content with the least.  Diogenes of Sinope 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)" 
22nd March 2014, 11:28 PM  #3 
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Join Date: Apr 2008
Location: Carlisle, England

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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22nd March 2014, 11:45 PM  #4 
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Join Date: Jul 2012

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 McGraw Hill 1957 
22nd March 2014, 11:52 PM  #5 
diyAudio Member
Join Date: Oct 2010
Location: Traslasierra

I can vouch for that!
No need to be an expert to build excellent transformers, just patience and perseverance. 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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I do not suffer from insanity, I enjoy every minute of it.  Edgar Allan Poe He has the most who is most content with the least.  Diogenes of Sinope 
22nd March 2014, 11:58 PM  #6  
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Join Date: Oct 2010
Location: Traslasierra

Quote:
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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I do not suffer from insanity, I enjoy every minute of it.  Edgar Allan Poe He has the most who is most content with the least.  Diogenes of Sinope 

23rd March 2014, 01:57 AM  #7  
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Join Date: May 2005
Location: Pretoria, South Africa

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 RDH4 (as mentioned) I know of at least one cad program derived from that and including properties of steels, which is even more convenient.
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23rd March 2014, 02:29 AM  #8  
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Join Date: Oct 2010
Location: Traslasierra

Quote:
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:
Quote:
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I do not suffer from insanity, I enjoy every minute of it.  Edgar Allan Poe He has the most who is most content with the least.  Diogenes of Sinope 

23rd March 2014, 01:51 PM  #9  
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Join Date: Jul 2012

Quote:
With respect to the thread if someone could add the RDH4 leakage inductance and capacitance charts by crowhurst it would add a lot. 

23rd March 2014, 02:33 PM  #10  
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Join Date: Oct 2010
Location: Traslasierra

Quote:
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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I do not suffer from insanity, I enjoy every minute of it.  Edgar Allan Poe He has the most who is most content with the least.  Diogenes of Sinope Last edited by popilin; 23rd March 2014 at 02:53 PM. 

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