**Frequency Compensation Methods:**

**Phase-Lag and Phase-Lead Compensation –** Lag compensation and lead compensation are two Frequency Compensation Methods often employed to stabilize op-amp circuits. The phase-lag network in Fig. 15-7(a) introduces additional phase lag at some low frequency where the op-amp phase shift is still so small that additional phase lag has no effect. It can be shown that at frequencies where X_{C1} ≫ R_{2}, the voltage v_{2} lags v_{1} by as much as 90°. At higher frequencies where X_{C1} ≪ R_{2} no significant phase lag occurs, and the lag network merely introduces some attenuation. The effect of this attenuation is that the A_{v}/f graph is moved to the left, as illustrated in Fig. 15-7(b). Thus, the frequency (f_{x1}) at which A_{v}B = 1 [for a given closed-loop gain (A_{CL})] is moved to a lower frequency (f_{x2}), as shown. Because f_{x2} is less than f_{x1}, the phase shift at f_{x2} is less than that at f_{x1}, and the circuit is likely to be stable.

The network in Fig 15-7(c) introduces a phase lead. In this network, when X_{C1} ≫ R_{1}, the voltage v_{2} leads v_{1}. This phase lead cancels some of the unwanted phase lag in the operational amplifier θ/f graph, [see Fig. 15-7(d)], thus rendering the circuit more stable. Phase-lag and phase-lead networks are both used internally to Frequency Compensation Methods op-amp circuits. Both types of circuit can also be used externally.

**Manufacturer’s Recommended Compensation:**

Most currently-available operational amplifiers contain internal compensating components, and do not require additional external components. Some have internal compensating resistors, and need only a capacitor connected externally to complete a compensating network. For those that require Frequency Compensation Methods, IC manufacturers list recommended component values and connection methods on the op-amp data sheet. An example of this is illustrated in Fig. 15-8 for the LMI08.

When selecting standard value compensating capacitors the next larger values should be used. This is termed over-compensation and it results in better amplifier stability, but, it also produces a smaller circuit bandwidth.

**Miller Effect Compensation:**

Miller effect involves connecting a capacitor between the output and input terminals of an inverting amplifier. Miller-effect compensation of an op-amp circuit is very simple, and it is often the only external method available for stabilizing a circuit where the op-amp is internally compensated. A capacitor (C_{f}) is connected across the feedback resistor, as shown in Fig. 15-10(a) and (b). The capacitor value is calculated to have an impedance equal the feedback resistor value at the desired signal cutoff frequency (f_{2}).

This reduces the closed-loop by 3 dB at the selected frequency. So long as the op-amp is stable at this frequency, the circuit will not oscillate. The op-amp used should have an upper cutoff frequency much higher than f_{2}.