Stability and frequency compensation

Every benefit of negative feedback depends on the feedback really being negative. At low frequencies it is. But each pole in the loop adds phase lag, up to 90° per pole, and at some frequency the total lag of the loop can reach 180°. There the returned signal adds to the input instead of subtracting from it. If the loop gain is still at least 1 at that frequency, the amplifier supplies its own input and oscillates. This lesson explains the condition with Bode plots of the loop gain, defines gain margin and phase margin, connects phase margin to the overshoot and ringing seen on an oscilloscope, and covers the standard ways of compensating a loop: dominant-pole, Miller, lead and lag compensation, and the special problem of capacitive loads.

The topic became urgent on Leo's bench. The first prototype of the APS-2 receiver used an op amp to drive a 10 nF filter capacitor at the converter input, and its output, which should have been a clean tone, showed a burst of ringing at every edge, at a frequency near 700 kHz. Nothing was wrong with the parts. The loop had too little phase margin.

When feedback becomes positive

The closed-loop gain A_f = A/(1 + T) becomes infinite if T = −1, that is, if the loop gain has a magnitude of 1 and a phase of −180° at the same frequency. In terms of poles, the closed-loop poles are the roots of 1 + T(s) = 0, and the amplifier is stable only if all of them lie in the left half of the s-plane. For loops with ordinary minimum-phase transfer functions, a practical test follows from the Bode plot of T: find the crossover frequency f_c where |T| = 1, and check the phase there. If the phase lag at f_c is less than 180°, the loop is stable.

The phase margin PM is how far the phase at crossover is from −180°; the gain margin GM is how far the loop gain is below 1 at the frequency f_180 where the phase reaches −180°, usually quoted in decibels. A loop with one pole can never reach −180° and is always stable. A loop with two poles approaches −180° only at infinite frequency, so it is stable but can have a very small phase margin. A loop with three or more poles can reach −180° at a finite frequency and oscillate if the loop gain is still above 1 there.

Phase margin and the step response

Phase margin is more than a pass-fail test; it predicts how the closed-loop amplifier will behave. A loop with a dominant pole and one other pole gives a closed-loop response like a second-order system, whose damping depends on the phase margin. Low phase margin means a peak in the closed-loop frequency response just below f_c and overshoot and ringing in the step response; very low phase margin means a long ring at a frequency near f_c, exactly what Leo saw. The code below computes, for a two-pole loop, the phase margin, the damping ratio, the step overshoot and the frequency-response peak as the second pole moves relative to the loop's unity-gain frequency.

import math, cmath
def loop(f, fu, p2):
    """Loop gain with a dominant pole far below and a second pole p2: T = fu/(jf) / (1 + jf/p2)."""
    return fu / (1j * f) / (1 + 1j * f / p2)
def crossover(fu, p2):
    lo, hi = fu * 1e-4, fu * 10
    for _ in range(200):
        m = math.sqrt(lo * hi)
        if abs(loop(m, fu, p2)) > 1: lo = m
        else: hi = m
    return m
fu = 1.0
print(" p2/fu   PM(deg)  zeta  overshoot  peaking")
for r in (0.25, 0.5, 0.7, 1, 2, 3, 4):
    fc = crossover(fu, r)
    pm = 180 + math.degrees(cmath.phase(loop(fc, fu, r)))
    zeta = 0.5 * math.sqrt(r)                 # closed loop: s^2/(wu*wp2) + s/wu + 1
    os_ = 100 * math.exp(-math.pi * zeta / math.sqrt(1 - zeta ** 2)) if zeta < 1 else 0.0
    pk = 20 * math.log10(1 / (2 * zeta * math.sqrt(1 - zeta ** 2))) if zeta < 0.707 else 0.0
    print(f"{r:5.2f}  {pm:7.1f}  {zeta:5.2f}  {os_:7.1f} %  {pk:5.2f} dB")

 p2/fu   PM(deg)  zeta  overshoot  peaking
 0.25     28.0   0.25     44.4 %   6.30 dB
 0.50     38.7   0.35     30.5 %   3.59 dB
 0.70     44.8   0.42     23.5 %   2.38 dB
 1.00     51.8   0.50     16.3 %   1.25 dB
 2.00     65.5   0.71      4.3 %   0.00 dB
 3.00     72.4   0.87      0.4 %   0.00 dB
 4.00     76.3   1.00      0.0 %   0.00 dB

The output gives the rules of thumb that designers carry in their heads. A phase margin of about 28° gives over 40 percent overshoot and a 6 dB peak; about 45° gives roughly 23 percent overshoot and over 2 dB of peaking; about 52° gives 16 percent; about 65° gives only 4 percent and no peaking; and above 75° there is essentially no overshoot. Most op amp circuits are designed for 45° to 60°, power and audio amplifiers usually for 60° or more, so that tolerances, temperature and unexpected loads cannot erode the margin to zero. A margin of about 15°, as on Leo's prototype, gives ringing near the crossover frequency that dies away slowly.

Reading stability from the op amp's Bode plot

For an op amp circuit, β is set by the feedback resistors and is usually independent of frequency, so the loop gain is the op amp's open-loop gain shifted down by 1/β. On a log plot, the crossover is where the open-loop gain curve meets the horizontal line at the closed-loop gain 1/β, which is called the noise gain. The larger the closed-loop gain, the lower the crossover and the more phase margin there is, because the crossover moves away from the op amp's second pole. The worst case is the voltage follower, β = 1, which crosses over at the highest frequency. A unity-gain-stable op amp is compensated so that even the follower has adequate margin; a decompensated op amp, sold for higher bandwidth, is stable only at noise gains above a stated minimum, such as 5 or 10.

Capacitive loads

An op amp's open-loop output resistance R_o, tens of ohms, forms a pole with any load capacitance C_L at f_L = 1/(2πR_oC_L). That pole sits inside the feedback loop, because the feedback is taken from the capacitor's terminal, and it adds phase lag at crossover. A few hundred picofarads of cable can upset a fast op amp; 10 nF upsets almost any.

The isolation resistor is the simplest fix but makes the output voltage depend slightly on the load current, because R_iso is outside the loop. Where that matters, designers use in-loop compensation: DC feedback is taken from the load side of R_iso for accuracy, and a small capacitor from the op amp's output to its inverting input closes a high-frequency loop that does not include the load pole. Many op amps also specify a maximum load capacitance, and some are designed to drive any capacitance stably, at the cost of bandwidth.

Compensation techniques

When a loop lacks margin, the designer changes the loop gain's shape so that it crosses 1 before the extra poles add too much phase. The main techniques are:

Plain dominant-pole compensation is wasteful: to give a 60° margin it must place the unity-gain crossover below the first non-dominant pole, and if that pole comes from a slow transistor the whole amplifier becomes slow. Miller compensation achieves the same stability with a capacitor a hundred times smaller and a much higher crossover, because pole splitting pushes the non-dominant pole up as the dominant one moves down. That is why nearly every op amp and the power amplifier of lesson 12 use it.

Measuring stability

On the bench, the quickest check is the small-signal step response: drive the amplifier with a small square wave, small enough that nothing slews, and measure overshoot and ringing with an oscilloscope and a low-capacitance probe. The overshoot table from the code translates the result into a phase margin. A more direct method measures the loop gain itself by injecting a small signal through a resistor or transformer inside the loop and comparing the voltages on either side, a technique due to Middlebrook that network analyzers automate. Symptoms of outright oscillation include a fuzzy or thick trace, an op amp that runs warm with no load, supply current higher than the datasheet's quiescent value, and problems that change when a finger or a probe touches the circuit. Leo's prototype showed only ringing, not oscillation; with the isolation resistor in place, the step response shows less than 5 percent overshoot.