AUDIO OUTPUT TRANSFORMERS

An audio transformer can be modeled as below. Image courtesy of Igor Popovich

r_{i} = anode resistance of output tube

C_{p} = capacitance of the primary winding

R_{p} = resistance of primary windings

L_{LP} = leakage inductance of the primary winding

R_{c} = core power losses, due to eddy currents and hysteresis

L_{p} = primary inductance

L_{LS}TR^{2} = leakage inductance of secondary reflected to the primary side. TR = turns ratio

R_{s}TR^{2} = secondary resistance reflected to the primary side

C_{s}/TR^{2} = parasitic capaticance of secondary reflected to the primary side

The parallel combination of Rc and Lp is the transformer itself. An ideal transformer doesn't present any impedance to the input voltage. The transformer is merely a device to transform the secondary resistances (by a factor of 1/T^{2}). Now, this is a very complicated model to analyze, so it will be analyzed for the Low frequency model, midfrequency model, and HF model.

MID-FREQUENCY MODEL

At mid-frequencies (100 - 5000Hz), all inductances and capacitances can be pretty much ignored, so you're left with just a simple voltage divider:

V_{o} = R_{L}TR^{2} / (r_{i} + R_{p} + R_{s}TR^{2} + R_{L}TR^{2}).

LOW FREQUENCY MODEL

At low frequencies (200Hz to 5kHz) the inductances and capacitances can be ignored except for L_{p}. So you're left with the following simplified model:

where R1 = r_{I} + R_{p} and R2 = (R_{s} + R_{L})TR^{2}

Now the frequency where the output voltage drops by 1/√2 __relative to the midband gain__ is the -3dB frequency f_{L}. This frequency is determined by the following:

f_{L} = [R_{1}R_{2}TR^{2} / (R_{1} + R_{2}TR^{2})]/2πL_{p}, or more simply

f_{L} = (R_{1} ∥ R_{2}TR^{2}) / 2πL_{p} ∥ means "parallel with"

Or, even more simply:

f_{L} = R_{PAR} / 2πL_{p}, where R_{PAR} = R_{1}∥R_{2}TR^{2}

Remembering that R_{2}TR^{2} = primary impedance of XFMR, also known as Z_{P}:

R_{PAR} = R_{1} ∥ Z_{P}. Now, r_{I} >> R_{p}, so R_{p} can be ignored (it's usually less than 100). Thus:

R_{PAR} = r_{I} ∥ Z_{P}

So, finally;

You want f_{L} as low as possible, so you want L_{P} above a certain minimum. Rearranging the above, gives:

L_{PMIN} = R_{PAR} / 2πf_{L}

L_{PMIN} is the minimum value of L needed to get the lower -3dB frequency f_{L} to the desired level to allow for adequate bass.

Looking at the above, a lower R_{PAR} means a lower (and easier to make) L_{PMIN}. This means that low impedance tubes (those with low r_{I}) are more desirable for the beginning choke maker.

HIGH FREQUENCY MODEL

For reasons I don't fully get, the shunt capacitances of the transformer model can be ignored, so you are left with the following model.

This is a simple low-pass LR filter and the upper -3dB frequency __relative to the midband gain__ is the frequency at which the reactance of the total __leakage__ inductance = resistance of the 3 resistors in series. In other words:

2πf_{U}L_{L} = R_{SER}, where L_{L} = L_{LP} + L_{LS}TR^{2} and R_{SER} = r_{I} + R_{1} + R_{2}TR^{2}. So...

f_{U} = R_{SER}/2πL_{L}

L_{LMAX} = R_{SER}/2πf_{U}

UNIVERSAL FREQUENCY AND PHASE RESPONSE CURVES

It turns out that if you look at values of R_{PAR} and R_{SER}, R_{SER} is usually about (and that's a very big about) 5X larger than R_{PAR}. So, looking at equations above, it's easy to see that:

f_{U} / f_{L} = 5L_{p} / L_{L}

All the above can be summed up in a graph below, found in Igor Popovich's book "Transformers for Tube amplifiers." Note that for both the high-frequency and the low-frequency halves, the gain has dropped to .707 (1/√2) when ωL = R

The equation at the very bottom right is derived as follows:

L_{P} = N_{P}^{2}μ_{R}μ_{0}A/ℓ_{MP} (see the Transformer Theory page)

L_{L} = μ_{0}N_{P}^{2}V_{W}/A, where V_{W} = volume of the windings.

So, f_{U} / f_{L} = 5L_{P} / L_{L} = 5μ_{R}A^{2} / (ℓ_{MP}V_{W})

This equation shows that increasing cross-sectional area A of the laminations seems to increase the bandwidth, but this is offset by the increase in ℓ_{MP} and V_{W}, so the only real way of increasing bandwidth is to increase permeability by using high quality iron.

Looking at the equations for f_{U} and f_{L}, the following is seen:

On the low frequency side, you want to minimize r_{I} and maximize L_{P}.

Unfortunately on the high frequency side you want to maximize r_{I}. This favors pentodes and tetrodes. L_{P} doesn't really matter, but you want to minimize L_{L}.

LEAKAGE INDUCTANCE

Leakage inductance L_{L} is due to magnetic flux between primary and secondary windings that is not coupled through the magnetic core. If LL = 0, then all of the flux is shared by the windings. A sad fact of life is that as f increases LL increases.

Without going through a derivation (which I don't know anyway), the following expression is given:

L_{L} = u_{0}V/CL^{2}N_{p}^{2}, where V =