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TID - Part 2.

My first article about TID and related problems, Slew Rate and TID showed, as Fig.1, the overshoot at an amplifier input produced by a filtered step function input. A similar analysis using a step function was used in 1966 in an article by Daugherty and Greiner entitled 'Some Design Objectives for Audio Power Amplifiers' (March 1966, IEEE Transactions on Audio and Electroacoustics.) but they looked only at the percentage overshoot at the amplifier input and concluded that for a 20kHz input filter the amplifier needed to have a 20kHz open-loop -3dB frequency so that overshoot percentage is zero. Input stage distortion is however determined by its peak input signal amplitude, and if we look at ways to reduce this we come to a completely different conclusion.

Recently I decided to check my previous result using a Spice simulation. My original version was worked out around 1979 using a TI59 programable calculator. To my relief the Spice result for a step function was about the same. The circuit used for the Spice simulation is shown next. The feedback loop includes only two stages, but using three with the more common Miller compensated driver stage makes little difference to the final result.

R1 and C1 plus a unity gain buffer are the low-pass filter, with -3dB at 20kHz.
C2 is the high frequency compensation capacitor for the feedback loop, and the smallest value compatible with stability is normally chosen, so for a given amplifier there is not much choice about this. What we can choose is R3, a resistor in parallel with the compensation capacitor. R3 = 10k will make the amplifier open-loop -3dB frequency 20kHz.
What we are concerned about is the feedback loop input stage distortion, and this is determined by its input signal level, V4. To reduce input stage distortion we want to reduce the peak signal level it needs to handle. First see how V4 is affected by changing R3 from 10k (green) to 20k (red) when using a 1V input step:

With R3 increased to 20k the transient peak is reduced from 50mV to 45mV, and the steady state level from 50mV to 25mV. Distortion from the input stage will therefore be lower for both transient and steady state with the higher resistor. This is the same result I found from my earlier and more laborious computation.

If doubling R3 helps why not increase it further? Making the value 2M here is what we find:

The transient induced peak is slightly lower at 43mV and the steady state level becomes very small. From this we can conclude that adding any resistor in parallel with the compensation capacitor increases the signal amplitude the input stage must handle, and is therefore a bad idea.

Something I had not previously been aware of was the result of changing the step to go from -1V to +1V instead of 0V to +1V, equivalent to using a square wave input instead of a step. Now look at how V4 is affected by changing R3 from 10k (green) to 20k (red):

The peak to peak level is reduced, but the peak positive voltage now increases from 50mV to 64mV, so the peak input stage distortion will be higher if we increase R3. So what happens if we increase R3 even further? With the higher 2M value we get the next result:

This is even worse, the transient peak now reaches 85mV, an overshoot of 70% compared to the peak positive level with R3 = 10k.

Using a step we concluded that R3 should be as big as possible, but using a square wave we want R3 = 10k to give the same -3dB bandwidth as the input filter. How then can we choose R3 if different test signals have different optimum amplifier bandwidths?

It may be helpful to remember that the square wave has twice the peak slew rate of the step with the same input filter, so we would expect a greater input stage signal to achieve this, and looking at the red traces with R3 = 2M we do find an increase by a factor of 2 for the square wave compared to the step. The input signal slew rate appears to be the important factor rather than the signal bandwidth. My first TID article mentioned measurements of music signals from a CD source for which I found peak slew rates up to that of a full level 10kHz sinewave. To restrict our square wave to that slew rate we need a low pass filter with -3dB at 5kHz. To minimise V4 overshoots with these music signals we could therefore conclude that an open-loop bandwidth about 5kHz is a good choice.

More important is to ensure the input stage is designed to be adequately linear with whatever peak signal level it will encounter in practice. If we allow for a peak input stage signal level 70% higher than we would get with the optimum open-loop bandwidth then we can avoid the parallel resistors and the consequent increase in distortion for lower frequency signal components.

What ultimately matters is the signal at the amplifier output, but looking at this with a square wave input tells us little about linearity. For the present example the most obvious effect is a change in closed-loop gain resulting from changes in R3, with higher resistance giving higher output level. The outputs for the positive to negative transition are compared next with R3 = 10k (green) and R3 = 2M (red):

A Practical Example.

As it happens, my MJR6 and MJR7 mosfet amplifier designs have about 5kHz open-loop bandwidth. Using a simplified MJR7 simulation with input filter -3dB at 20kHz and a 1V square wave input this is the input stage voltage V4:

There is clearly an overshoot, but the peak positive level of V4 is just 0.65mV. Achieving good input stage linearity at this signal level is no problem. The peak slew rate is the same as a full level 40kHz sinewave, which should never happen in normal use, and with a music signal the peak level should be much smaller. Using a 5kHz filtered square wave to give a similar slew rate to typical music sources we find no overshoot and a peak level of about 140uV :

The MJR7 input stage can handle peak inputs up to about 10mV with good linearity, and the 140uV peak level is smaller than this by a factor of 70, giving a very good safety margin.

Simulations are inevitably simplifications, so it helps to check reality. Here is the MJR6 input stage signal with a square wave source about 740mV. Just the built-in low-pass filter is used, and -3dB was about 50kHz in this case, so the percentage overshoot is far higher than for the Spice version:

Even now the peak amplitude is only 420uV negative and 330uV positive, so still well under the 10mV input stage clipping level at which slew-rate limiting would occur. ( The positive and negative peaks differ because in the MJR6 the highly nonlinear mosfet capacitances are used for high frequency compensation. The MJR7 reduces this problem. )