By Robert Pease

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**Sample text**

Find the frequency where the magnitude of a(s)f(s) drops to ϩ1. This is the crossover frequency c. 3. The difference between the angle at the crossover frequency and Ϫ180° is the phase margin m. 4. ) is deﬁned as the change in open-loop gain required to make the system unstable. Systems with greater gain margins can withstand greater changes in system parameters before becoming unstable in closed loop. 5. The phase margin is deﬁned as the negative change in open-loop phase shift required to make a closed-loop system unstable.

Using MATLAB, the bandwidth and phase margin are calculated, as shown in Figure 1-18. The results show a phase margin of 52° and a crossover frequency of 786,150 radians/second (125 kHz). 8 μs (Figure 1-19). 35/fc, where fc is the crossover frequency in Hz. w ww. 828 deg. 8615e ϩ 005 rad/sec) 100 50 0 Ϫ50 Ϫ100 Ϫ150 101 102 103 104 105 106 Frequency (rad/sec) Figure 1-18: Phase margin and bandwidth calculation for gain of ؉1 op-amp circuit. 2 ϫ10Ϫ5 Figure 1-19: Step response for gain of ؉1 ampliﬁer.

T . 1 R (a) (b) Figure 1-20: Gain of ϩ10 op-amp circuit. (a) Circuit. (b) Block diagram. 323 deg. (at 99509 rad/sec) 50 0 Ϫ50 Ϫ50 Ϫ100 Ϫ150 101 102 103 104 105 Frequency (rad/sec) 106 Figure 1-21: Phase margin and bandwidth calculation for gain of ϩ10 op-amp circuit. 8 kHz). 35/15,800 ϭ 22 μs, as shown in Figure 1-22. Therefore the system is well damped and there is no overshoot in the step response (Figure 1-23). Example 1-3: Integral control of reactive load Consider the op-amp driving a reactive load as shown in Figure 1-24.