Adding a small inductor in series with the output of a power amplifier may sometimes be necessary to ensure stability of the overall feedback loop with a range of capacitive loads. The inductor plus capacitor load then form a series resonant LC circuit which can have a very small total series impedance at the resonant frequency. To reduce this effect a damping resistor is usually added in parallel with the inductor.
In the next diagram the open-loop output impedance is shown as an external 2 ohm resistor, this being a typical value for a lateral mosfet output stage at quiescent current 100mA as used in my MJR-6 design. This impedance is not a fixed value, it will fall in value as the output current increases because of the increase in mosfet gm at higher drain currents. The overall feedback is taken from a point before the inductor.
Vo is the open-circuit output voltage, and Vf is the output voltage with load connected. Suppose Vo is 10V, then Vf is reduced by the load impedance. For the next diagram Vf is shown as a function of frequency. For this example the load capacitor is 2uF and the inductor is 0.4uH. Vf is plotted for two different values of damping resistor Rp, these being 8 ohms and 0.33 ohms.
At very high frequencies the total load is close to the value of the damping resistor. An 8 ohm value is sometimes used in order to maintain the nominal 8 ohm speaker load impedance up to high frequencies with capacitive loads, while the usual RC Zobel network with 8 ohms in series with something around 0.1uF maintains the 8 ohm load impedance for inductive loads. Unfortunately an 8 ohm damping resistor is far too high for critical damping with a 2uF load, and it can be seen from the green trace that there is a notch in the output voltage around 180kHz. Reducing the damping resistor to about 0.33 ohms is just about sufficient to avoid a gain notch, as in the red trace, but this reduces the output voltage by a factor of 7 at high frequencies. Keeping the 0.33 ohm resistor but increasing the load above 2uF will again result in a notch in the gain, while reducing it below 2uF gives an over-damped result with no notch.
The phase shifts associated with the load effect are shown next, and it can be seen that both add a phase lag around 100kHz, the underdamped version being worse in this respect by almost 20 degrees.
So far we have only looked at the effect of the load on output stage gain. In a complete amplifier there is usually overall feedback and a -6dB / octave compensation. The next diagram combines the underdamped inductor effect with the compensation to show the loop gain as a function of frequency. Low frequency loop gain is taken as 60dB. Unity gain is shown by a blue line.
This result reveals another potential problem of using an underdamped inductor. The unity gain frequency is where instability can occur if the added phase lag round the feedback loop reaches 180 deg. In this diagram there are 3 unity gain frequencies shown as f1, f2 and f3. There is no problem at f2 because the output load adds a phase advance at this frequency, but at f1, around 180kHz, there is an 80 deg phase lag added by the load. The amplifier compensation will add more phase lag, and so there is some danger of instability at this frequency. At f3 there is the usual high frequency unity gain frequency, here just over 2MHz, and here again there may be several sources of phase shift which can be a problem. We now need to worry about stability at two different frequencies instead of the usual one. This particular problem can be reduced by using sufficiently high feedback loop gain at 180kHz so that the low load impedance at this frequency is not sufficient to reduce the loop gain to anywhere near unity. Near clipping the loop gain will fall, so there could then still be a problem.
Although it appeared from the results so far that using a critically damped inductor is the best solution for driving capacitive loads, there are other difficulties to consider. With the 0.33 ohm damping resistor the amplifier load is around this value at high frequencies, and the output stage gain is therefore reduced as shown in the second diagram on this page. Consider the MJR-6 circuit shown next in simplified form driving a 2uF load:
This circuit uses a conventional source-follower output stage with series inductor to reduce capacitive load effects, and without any added high frequency compensation. The output stage is driven from a high impedance, something like 200k in practice, and is a useful 'worst-case' example.
The output stage is represented by a unity gain stage with a 2 ohm output impedance, and the total combined mosfet gate-source capacitance, Cgs, is shown as 1500pF. Both output impedance and Cgs vary with output current level, and the values shown are about the worst case for the lateral mosfets used at Iq = 100mA. With an ideal current source driver stage the open-loop gain of the amplifier is proportional to the input impedance of the output stage, which is not just the impedance of Cgs because the full driver stage output voltage does not appear across this capacitor, only a reduced value determined by the potential divider formed by the 2 ohm output impedance and the load, and so we may expect the effective input capacitance to be fairly low at low frequencies, but increase with increased frequency because of the falling load impedance.
The potential divider just mentioned is not however a resistive divider because the load has a reactive component, and so it does not just reduce the voltage across Cgs, it also changes its phase, so the effect is something more complex than just an increase in capacitance at higher frequencies. With a fixed capacitance the resulting open-loop gain reduction would be the required -6dB/octave, but with this increasing capacitance the rate is greater. The phase shift of a pure capacitance, even if it had different values at different frequencies, would be 90 degrees, but the input impedance of the output stage is not just a pure capacitance, and adds a phase shift which can approach 180 degrees with consequent risk of instability. For the phase shift to actually reach 180 deg. the input impedance of the output stage would need to be a negative resistance rather than a capacitance.
To see the extent of the problem the phase shift was plotted using AIM-Spice, and the next diagram shows the phase shift at input, V2, and output, V4.
The phase shift at the input of the output stage is the red trace, and this reaches a maximum in excess of 160 deg between 6kHz and 10kHz. This is not what we would expect if our mosfet output stage has a capacitive input impedance, the phase shift could then only reach 90 deg. There is evidently a negative resistance component also. (This is not a consequence of using an output inductor, it is still present with this shorted. The capacitive load is the problem.)
The negative resistance is not itself a serious problem because it has a series capacitance, but if the output stage were to be driven from an inductive source so that the capacitance is cancelled at some frequency then the result would be similar to a Colpitts oscillator, and oscillation is then possible. If wiring inductance is kept low this should not be a problem, but the effect of the phase shift on loop stability may still be important.
The phase shift at the output at the point where the overall negative feedback is taken is shown by the green trace, which goes perilously close to 180 deg. Even if we can avoid any further phase lags in the input stage or the feedback network we are already close to the point where the feedback loop may not be unconditionally stable. Allowing the phase shift to exceed 180 deg well below the unity gain frequency may cause problems near clipping where the loop gain falls, but the phase shift can also change near clipping and predicting the overall result is not always easy.
In my mosfet designs the driver stage normally clips before the output stage, resulting in a lower impedance drive to the output stage and reduction in phase shift in the region where it may otherwise be excessive with a capacitive load, and so stability is maintained. The more widely used 'Miller' method of high frequency compensation can be less well behaved, for example feedforward through the compensation capacitor can increase phase shift near clipping, making matters worse.
We can now try a few variations to see if the situation can be improved. First the damping resistor. Increasing this above the 0.33 critical damping value extends the frequency range at which phase lag approaches 180 deg, which is bad, but pulls back the phase shift at higher frequencies. The smaller resistor will both reduce the unity gain frequency because of the lower impedance and also allow greater phase lag at that frequency, which is also bad, so a slightly higher resistance may be better. In the next picture a 1 ohm damping resistor gives the red trace for the output voltage phase shift, and the reduction in phase lag at higher frequencies may be judged an advantage.
Another change we could make is adding a damping resistor in parallel with the load. In practice no speaker load is likely to be anywhere near a pure capacitance, and the common 2uF in parallel with 8 ohms test load may be more realistic. Keeping the 1 ohm damping resistor the next two traces are for 100 ohms (green) and 8 ohms (red) in parallel with the 2uF load.
The 100 ohms makes a small improvement, but the 8 ohms is rather more helpful. With a real speaker instead of a theoretical pure capacitor there may be no serious problem.
Other component changes have effects which are not entirely helpful, for example increasing the inductor value to 2uH does little to reduce the maximum phase lag, but does reduce the frequency range over which it is a problem.
Adding the usual Zobel RC using 100nF in series with 8 ohms has almost zero effect with a capacitor load, but this is intended only to help with inductive loads.
We have only looked at the effect with a 2uF load. With a lower capacitance the maximum phase shift reduces, but the effect moves further up the frequency range, and the optimum inductor damping resistance increases also, so for the whole range of capacitance the original 7R5 damping resistor may not be such a bad choice.
My MJR-6 and MJR-7 circuits were found to have no obvious stability problems when built and tested, but even so some further adjustment of component values could at least give a theoretical improvement, and may be added at a later date. With the existing circuits I suggest avoiding high signal source impedance, particularly when driving highly capacitive loads. If driven from a volume control which is itself driven from a low impedance then a 10k log control or lower value can be used.