Loop Gain Plots are the Key to Understanding Stability

Stability is determined by the loop gain, and when Aβ = –1 = |1| ∠–180° instability or oscillation occurs. If the magnitude of the gain exceeds one, it is usually reduced to one by circuit nonlinearities, so oscillation generally results for situations where the gain magnitude exceeds one.

Consider oscillator design, which depends on nonlinearities to decrease the gain magnitude; if the engineer designed for a gain magnitude of one at nominal circuit conditions, the gain magnitude would fall below one under worst case circuit conditions causing oscillation to cease. Thus, the prudent engineer designs for a gain magnitude of one under worst case conditions knowing that the gain magnitude is much more than one under optimistic conditions. The prudent engineer depends on circuit nonlinearities to reduce the gain magnitude to the appropriate value, but this same engineer pays a price of poorer distortion performance. Sometimes a design compromise is reached by putting a nonlinear component, such as a lamp, in the feedback loop to control the gain without introducing distortion.

Some high gain control systems always have a gain magnitude greater than one, but they avoid oscillation by manipulating the phase shift. The amplifier designer who pushes the amplifier for superior frequency performance has to be careful not to let the loop gain phase shift accumulate to 180°. Problems with overshoot and ringing pop up before the loop gain reaches 180° phase shift, thus the amplifier designer must keep a close eye on loop dynamics. Ringing and overshoot are handled in the next section, so preventing oscillation is emphasized in this section. Equation 5–14 has the form of many loop gain transfer functions or circuits, so it is analyzed in detail.

Figure 5–15. Magnitude and Phase Plot of Equation 5–14

The quantity, K, is the dc gain, and it plots as a straight line with an intercept of 20Log(K).

The Bode plot of Equation 5–14 is shown in Figure 5–15. The two break points, ω = ω 1 = 1/τ 1 and ω = ω 2 = 1/τ 2 , are plotted in the Bode plot. Each breakpoint adds –20 dB/decade slope to the plot, and 45° phase shift accumulates at each breakpoint. This transfer function is referred to as a two slope because of the two breakpoints. The slope of the curve when it crosses the 0 dB intercept indicates phase shift and the ability to oscillate. Notice that a one slope can only accumulate 90° phase shift, so when a transfer function passes through 0 dB with a one slope, it cannot oscillate. Furthermore, a two-slope system can accumulate 180° phase shift, therefore a transfer function with a two or greater slope is capable of oscillation.

A one slope crossing the 0 dB intercept is stable, whereas a two or greater slope crossing the 0 dB intercept may be stable or unstable depending upon the accumulated phase shift. Figure 5–15 defines two stability terms; the phase margin, φ M , and the gain margin, G M . Of these two terms the phase margin is much more important because phase shift is critical for stability. Phase margin is a measure of the difference in the actual phase shift and the theoretical 180° required for oscillation, and the phase margin measurement or calculation is made at the 0 dB crossover point. The gain margin is measured or calculated at the 180° phase crossover point. Phase margin is expressed mathematically in

Equation 5–15.

The phase margin in Figure 5–15 is very small, 20°, so it is hard to measure or predict from the Bode plot. A designer probably doesn’t want a 20° phase margin because the system overshoots and rings badly, but this case points out the need to calculate small phase margins carefully. The circuit is stable, and it does not oscillate because the phase margin is positive. Also, the circuit with the smallest phase margin has the highest frequency response and bandwidth.

Figure 5–16. Magnitude and Phase Plot of the Loop Gain Increased to (K+C)

Increasing the loop gain to (K+C) as shown in Figure 5–16 shifts the magnitude plot up. If the pole locations are kept constant, the phase margin reduces to zero as shown, and the circuit will oscillate. The circuit is not good for much in this condition because production tolerances and worst case conditions ensure that the circuit will oscillate when you want it to amplify, and vice versa.

Figure 5–17. Magnitude and Phase Plot of the Loop Gain With Pole Spacing Reduced

The circuit poles are spaced closer in Figure 5–17, and this results in a faster accumulation of phase shift. The phase margin is zero because the loop gain phase shift reaches 180° before the magnitude passes through 0 dB. This circuit oscillates, but it is not a very stable oscillator because the transition to 180° phase shift is very slow. Stable oscillators have a very sharp transition through 180°.

When the closed loop gain is increased the feedback factor, β, is decreased because V OUT /V IN = 1/β for the ideal case. This in turn decreases the loop gain, Aβ, thus the stability increases. In other words, increasing the closed loop gain makes the circuit more stable. Stability is not important except to oscillator designers because overshoot and ringing become intolerable to linear amplifiers long before oscillation occurs. The overshoot and ringing situation is investigated next.