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. 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

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 L_{L} = 0, then all of the flux is shared by the windings. A sad fact of life is that as f increases L_{L} increases.

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

L_{L} = u_{0}VN_{p}^{2}/CL^{2}, where V = volume between the windings, CL = coil length, N_{p} = number of turns.

Note that L_{L} is not dependent on the permeability of the iron or the induction levels. L_{L} decreases as the lamination size increases (increasing CL). L_{L} increases with the volume V between the windings, which is partly determined by the insulation thickness. A way to reduce V is to use bulk winding, as opposed to layered arrangements, which decreases the insulation between layers. L_{L} increases quickly as the number of turns increases.

So the opposing goals: increasing NP increases Lp and affords better bass (f_{L} decreases), but it also increases L_{L} which hurts treble (f_{U} decreases).

This can be summed up by the equation at the bottom right of the frequency curve above. This equation 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 (see above).

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}.