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.
