76 Lead Compensation
Figure 7-13. Lead-Compensation Circuit
transfer function? The equation for the inverting op amp closed-loop gain is repeated below.
-aZF
- 1/RfC 1/RfIIRgC
Figure 7-14. Lead-Compensation Bode Plot
When a approaches infinity, Equation 7-13 reduces to Equation 7-14.
V IN ZIN
Substituting RF||C for ZF and RG for ZG in Equation 7-14 yields Equation 7-15, which is the ideal closed-loop gain equation for the lead compensation circuit.
The forward gain for the inverting amplifier is given by Equation 7-16. Compare Equation 7-13 with Equation 6-5 to determine A.
The op amp gain (a), the forward gain (A), and the ideal closed-loop gain are plotted in Figure 7-15. The op amp gain is plotted for reference only. The forward gain for the inverting op amp is not the op amp gain. Notice that the forward gain is reduced by the factor Rf/(Rg +Rf), and it contains a high frequency pole. The ideal closed-loop gain follows the ideal curve until the 1/RFC breakpoint (same location as 1/t2 breakpoint), and then it slopes down at -20 dB/decade. Lead compensation sacrifices the bandwidth between the 1/RfC breakpoint and the forward gain curve. The location of the 1/RFC pole determines the bandwidth sacrifice, and it can be much greater than shown here. The pole caused by Rf, Rg, and C does not appear until the op amp's gain has crossed the 0-dB axis, thus it does not affect the ideal closed-loop transfer function.
- Figure 7-15. Inverting Op Amp With Lead Compensation
The forward gain for the noninverting op amp is a; compare Equation 6-11 to Equation 6-5. The ideal closed-loop gain is given by Equation 7-17.
The plot of the noninverting op amp with lead compensation is shown in Figure 7-16. There is only one plot for both the op amp gain (a) and the forward gain (A), because they are identical in the noninverting circuit configuration. The ideal starts out as a flat line, but it slopes down because its closed-loop gain contains a pole and a zero. The pole always occurs closer to the low frequency axis because RF > RF||RG. The zero flattens the ideal closed-loop gain curve, but it never does any good because it cannot fall on the pole. The pole causes a loss in the closed-loop bandwidth by the amount separating the closed-loop and forward gain curves.
- Figure 7-16. Noninverting Op Amp With Lead Compensation
Although the forward gain is different in the inverting and noninverting circuits, the closed-loop transfer functions take very similar shapes. This becomes truer as the closed-loop gain increases because the noninverting forward gain approaches the op amp gain. This relationship cannot be relied on in every situation, and each circuit must be checked to determine the closed-loop effects of the compensation scheme.
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