Cable Effect On Amplifier Stability
Some time ago I read the Bateman article about the effect of cables on amplifier stability. I didn't see any reason to worry about that sort of thing, my reasoning being that at any given frequency any passive load such as a speaker plus cable would have just a simple impedance with resistance plus some reactance, either inductive or capacitive, and provided the amplifier is designed for stability into any reasonable capacitive load there should be no problem. The total load impedance can have no more than 90deg phase angle (plus or minus), for anything beyond that there would be a component of negative resistance, and that is impossible for a passive load. (There would be problems with conservation of energy at least). Here I will demonstrate how my MJR7 circuit is affected by cables, and why there is no serious problem. Not all amplifiers are happy with even 'normal' cable and speaker combinations, and some of the more extreme cables may be worse.
There are some potentially significant line effects, primarily at very high frequencies where the line length reaches a quarter wavelength. Information on quarter-wave effects can be found at Quarter-wave impedance transformer.
For 3 metres of cable the quarter wave frequency is under 25MHz, but probably not low enough to be a problem for any audio amplifier. Much longer cables could have effects at a few MHz, causing problems for some amplifiers with feedback loop unity gain frequency in that area, but the problem is easily avoided by adding a R+C Zobel at the speaker end.The starting point is to look at the load impedance seen by the amplifier with a 3m 100R cable feeding a simple full range drive unit. As usual I am assuming a lossless line with light speed propagation, which is not entirely realistic, but just putting in plausible figures for these things adds nothing useful to the results, the conclusions are still about the same. More complex speakers with crossover networks may have lower and more resistive impedance at high frequencies and so be less of a problem. The example shown is the same one used on my 'Cable Impedance' page, and we can detect the signal across a 0.01 ohm resistor to observe the magnitude and phase of the current. A high current corresponds to a low impedance, so it is the admittance we are looking at rather than impedance, and the sign of the phase angle is also reversed.
What we find is that up to about 300kHz the line has little effect, the plots look about the same with the line reduced to just 1cm in length. At 350kHz however something dramatic appears to happen, the admittance remains very low (i.e. high impedance) but the phase angle jumps from +90 to -90 (remember the plot reverses the sign). Up to that point the 2mH inductance of the speaker is dominant, but the line has an equivalent capacitance about 100pF and this causes a parallel resonance at 350kHz and the impedance is then capacitive at higher frequency. As stated earlier the phase angle never goes beyond 90deg in either direction.
This demonstrates why the capacitance of the cable can be important for amplifier stability if the speaker is primarily inductive at high frequencies. If we used a cable with characteristic impedance 10R instead of 100R to try to more closely match the nominal speaker impedance, then the transition to capacitive impedance will occur at a lower frequency, and the capacitance will be higher.Before concluding that all is well if we have designed for stability into small capacitances, look what happens with a much longer 100R line, here 30m:
The transition to capacitive impedance has now moved close to 100kHz, the line has 10 times the length and as we may expect its capacitance is therefore 10 times higher. (The resonance frequency where the transition takes place is inversely proportional to the square root of the capacitance.) But look what happens at 2.5MHz and 7.5MHz, the phase jumps from capacitive to inductive, but more seriously the admittance is off the scale (actually a long way off the scale) so there are severe dips in the impedance to a fraction of an ohm. That's easy to fix, all we need is to add a 100R resistor across the speaker end of the line to match its impedance and the dips vanish. (Usually we would use 100R in series with a 100n or 10n capacitor to reduce power dissipation in the resistor at audio frequencies). Maybe we don't know the exact impedance of the line, but even a very inaccurate resistor value helps a lot, for example using 1k instead of 100R the impedance dips are still there but only drop to around 8R so probably not a big problem for the amplifier.
An explanation for the impedance dips can be found at the link given earlier at Quarter-wave impedance transformer, where we learn that at a quarter wavelength a high impedance output load causes a low input impedance. The equivalent speaker circuit has a series inductor which has a relatively high impedance at 2.5MHz. Similar effects occur at any odd number of quarter waves, the one at 7.5MHz appears on the plot, but there will also be a series at higher frequencies.
But what is the effect of all this on amplifier stability? There are seemingly endless amplifier designs in existance, and some will be largely immune to cable effects while others may turn to smoke. Just adding that 100R termination resistor will help in some cases. Here I will just show how the cable affects my MJR7 feedback loop stability.
Luckily we don't need to do a full simulation of the amplifier, all we need to include are the L//R and C+R networks at the output and also the open-loop output impedance, about 2R at low signal levels. That reduces at higher output so 2R is the worst case, unless the quiescent current has been set too low. What we need to look at is the phase shift and attenuation added at the point where the feedback is taken from, as in the next diagram, together with the result with no line effect included.
Now add the line effect, in this case the 30m line to include those nasty quarter-wave effects:
There are still some small glitches, but nothing we would expect to cause stability problems. As previously just adding a 100R termination to the line totally removes even these small effects. So, my conclusion is that the MJR7 has good immunity to cable effects and even with 30m cables there is little to worry about. Adding the 100R + 100n across the speaker terminals when using a long cable may not be essential but still looks like a good idea, it may give useful protection under fault conditions, for example if the output stage quiescent current is too low.
Footnote 1:
One thing I should add is the effect of those things I ignored, such as resistive losses and propagation velocity. The velocity is easy enough to include, e.g. the quarter-wave effects I show for a 30m cable with lightspeed propagation will be similar for a 24m cable with 0.8c velocity. Resistive losses are in practice unimportant if we are looking at the effect of 3m of reasonably heavy gauge copper wire, something like a 0.01 ohm series resistance and some small conductance through the insulation are not going to seriously affect the result at audio frequencies.
Using the full equation for characteristic impedance including R and G in addition to L and C leads to what appear to be extreme effects at low frequencies, I saw one example showing Zo = 5k at 20Hz, but remember the definition of 'characteristic impedance', it is the impedance of an infinite line, or more realistically a finite line terminated by a resistance equal to Zo. If we connect a 5k resistor to the far end of a 3m cable we will not be surprised to measure an input impedance of 5k at low frequencies. If we have really good measuring equipment we may expect to find a very small effect from R and G, maybe the resistance will measure 5.00001k, but although a 0.01 ohm series resistance will increase the figure, the parallel resistance corresponding to G will reduce it. What is special for a load equal to Zo is that these effects will cancel and we will find exactly 5k. With a smaller than 5k resistor there would be a small increase, and for a larger than 5k resistor there will be a small decrease. So Zo does have some meaning even at very low frequencies, it is then the load at which the small effects of R and G cancel when measuring the input impedance.
What I also ignored is that L, C, R, G, Zo and propagation velocity all can vary with frequency in a real cable.Footnote 2:
I stressed that the phase angle of the line input impedance 'at a given frequency' never exceeds 90deg. We could argue that there are time delays involved, whatever we apply to the input has to travel along the line and back to the input before affecting the impedance we see there, and with a long line and high frequency a time delay can certainly involve far more equivalent phase shift than 90deg. This overlooks what we mean by a single frequency, which is a purely mathematical concept, with a sine wave extending in time from minus infinity to plus infinity. We therefore are only concerned with a steady state. Our usual practical test methods involve applying a sinewave generator to the line for a finite time interval, maybe a few minutes, but this is not a single frequency, it is a toneburst with a wide range of frequencies involved. Analysing the effect on a toneburst would be more difficult, but I see no reason to expect it to tell us anything useful about amplifier stability. After a few cycles it will settle down to something close to the steady state condition of an ideal theoretical sinewave.
