Stability as used in electronic circuit terminology is often defined as achieving a nonoscillatory state. This is a poor, inaccurate definition of the word. Stability is a relative term, and this situation makes people uneasy because relative judgments are exhaustive. It is easy to draw the line between a circuit that oscillates and one that does not oscillate, so we can understand why some people believe that oscillation is a natural boundary between stability and instability.

Feedback circuits exhibit poor phase response, overshoot, and ringing long before oscillation occurs, and these effects are considered undesirable by circuit designers. This chapter is not concerned with oscillators; thus, relative stability is defined in terms of performance. By definition, when designers decide what trade-offs are acceptable, they determine what the relative stability of the circuit is. A relative stability measurement is the damping ratio (ζ), and the damping ratio. The damping ratio is related to phase margin, hence phase margin is another measure of relative stability. The most stable circuits have the longest response times, lowest bandwidth, highest accuracy, and least overshoot. The least stable circuits have the fastest response times, highest bandwidth, lowest accuracy, and some overshoot.

Op amps left in their native state oscillate without some form of compensation. The first IC op amps were very hard to stabilize, but there were a lot of good analog designers around in the 1960s, so we used them. Internally compensated op amps were introduced in the late 1960s in an attempt to make op amps easy for everyone to use. Unfortunately, internally compensated op amps sacrifice a lot of bandwidth and still oscillate under some conditions, so an understanding of compensation is required to apply op amps.

Internal compensation provides a worst-case trade-off between stability and performance. Uncompensated op amps require more attention, but they can do more work. Both are covered here.

Compensation is a process of applying a judicious patch in the form of an RC network to make up for a less than perfect op amp or circuit. There are many different problems that can introduce instability, thus there are many different compensation schemes.

Internal compensation is accomplished in several ways, but the most common method is to connect a capacitor across the collector–base junction of a voltage gain transistor (see Fig. 8.1). The Miller effect multiplies the capacitor value by an amount approximately equal to the stage gain, thus the Miller effect uses small value capacitors for compensation.

Fig. 8.2 shows the gain/phase diagram for an older op amp (TL03X). When the gain crosses the 0 dB axis (gain equal to one) the phase shift is approximately 108 degrees, thus the op amp must be modeled as a second-order system because the phase shift is more than 90 degrees.

This yields a phase margin of ϕ = 180 degrees–108 degrees = 72 degrees, thus the circuit should be very stable. Referring to Fig. 8.3, the damping ratio is one and the expected overshoot is zero. Fig. 8.2 shows approximately 10% overshoot, which is unexpected, but inspecting Fig. 8.2 further reveals that the loading capacitance for the two plots is different. The pulse response is loaded with 100 pF rather than 25 pF shown for the gain/phase plot, and this extra loading capacitance accounts for the loss of phase margin.

Why does the loading capacitance make the op amp unstable? Look closely at the gain/phase response between 1 and 9 MHz, and observe that the gain curve changes slope drastically while the rate of phase change approaches 120 degrees/decade. The radical gain/phase slope change proves that several poles are located in this area. The loading capacitance works with the op amp output impedance to form another pole, and the new pole reacts with the internal op amp poles. As the loading capacitor value is increased, its pole migrates down in frequency, causing more phase shift at the 0 dB crossover frequency. The proof of this is given in the TL03X data sheet where plots of ringing and oscillation versus loading capacitance are shown.

Fig. 8.4 shows similar plots for the TL07X, which is the newer family of op amps. Notice that the phase shift is approximately 100 degrees when the gain crosses the 0 dB axis. This yields a phase margin of 80 degrees, which is close to unconditionally stable. The slope of the phase curve changes to 180 degrees/decade about one decade from the 0 dB crossover point. The radical slope change causes suspicion about the 90 degrees phase margin; furthermore the gain curve must be changing radically when the phase is changing radically. The gain/phase plot may not be totally false, but it sure is overly optimistic.

The TL07X pulse response plot shows approximately 20% overshoot. There is no loading capacitance indicated on the plot to account for a seemingly unconditionally stable op amp exhibiting this large an overshoot. Something is wrong here: the analysis is wrong, the plots are wrong, or the parameters are wrong. Fig. 8.5 shows the plots for the TL08X family of op amps, which are sisters to the TL07X family. The gain/phase curve and pulse response is virtually identical, but the pulse response lists a 100 pF loading capacitor. This little exercise illustrates three valuable points: first, if the data seem wrong it probably is wrong; second, even the factory people make mistakes; and third, the loading capacitor makes op amps ring, overshoot, or oscillate.

The frequency- and time-response plots for the TLV277X family of op amps is shown in Figs. 8.6 and 8.7. First, notice that the information is more sophisticated because the phase response is given in degrees of phase margin; second, both gain/phase plots are done with substantial loading capacitors (600 pF), so they have some practical value; and third, the phase margin is a function of power supply voltage.

At V_CC = 5 V, the phase margin at the 0 dB crossover point is 60 degrees, while it is 30 degrees at V_CC = 2.7 V. This translates into an expected overshoot of 18% at V_CC = 5 V, and 28% at V_CC = 2.7 V. Unfortunately the time-response plots are done with 100 pF loading capacitance, hence we cannot check our figures very well. The V_CC = 2.7 V overshoot is approximately 2%, and it is almost impossible to figure out what the overshoot would have been with a 600 pF loading capacitor. The small-signal pulse response is done with mV signals, and that is a more realistic measurement than using the full signal swing.

Internally compensated op amps are very desirable because they are easy to use, and they do not require external compensation components. Their drawback is that the bandwidth is limited by the internal compensation scheme. The op amp open-loop gain eventually (when it shows up in the loop gain) determines the error in an op amp circuit. In a noninverting buffer configuration, the TL277X is limited to 1% error at 50 kHz (V_CC = 2.7 V) because the op amp gain is 40 dB at that point. Circuit designers can play tricks such as bypassing the op amp with a capacitor to emphasize the high-frequency gain, but the error is still 1%. Keep Eq. (8.1) in mind because it defines the error. If the TLV277X were not internally compensated, it could be externally compensated for a lower error at 50 kHz because the gain would be much higher.

Other reasons for externally compensating op amps are noise reduction, flat amplitude response, and obtaining the highest bandwidth possible from an op amp. An op amp generates noise, and noise is generated by the system. The noise contains many frequency components, and when a high-pass filter is incorporated in the signal path, it reduces high-frequency noise. Compensation can be employed to roll off the op amp's high frequency, closed-loop response, thus causing the op amp to act as a noise filter. Internally compensated op amps are modeled with a second-order equation, and this means that the output voltage can overshoot in response to a step input. When this overshoot (or peaking) is undesirable, external compensation can increase the phase margin to 90 degrees where there is no peaking. An uncompensated op amp has the highest bandwidth possible. External compensation is required to stabilize uncompensated op amps, but the compensation can be tailored to the specific circuit, thus yielding the highest possible bandwidth consistent with the pulse response requirements.

The analysis starts by looking into the capacitor and taking the Thevenin equivalent circuit.

Then the output equation is written.

Rearranging terms yields Eq. (8.5).

When the assumption is made that (Z_F + Z_G) >> Z_O, Eq. (8.5) reduces to Eq. (8.6).

Eq. (8.7) models the op amp as a second-order system. Hence, substituting the second-order model for a in Eq. (8.6) yields Eq. (8.8), which is the stability equation for the dominant pole compensation circuit.

Several conclusions can be drawn from Eq. (8.8) depending on the location of the poles. If the Bode plot of Eq. (8.7), the op amp transfer function, looks like that shown in Fig. 8.10, it only has 25 degrees phase margin, and there is approximately 48% overshoot. When the pole introduced by Z_O and C_L moves toward the zero frequency axis, it comes close to the τ₂ pole, and it adds phase shift to the system. Increased phase shift increases peaking and decreases stability. In the real world, many loads, especially cables, are capacitive, and an op amp like the one pictured in Fig. 8.10 would ring while driving a capacitive load. The load capacitance causes peaking and instability in internally compensated op amps when the op amps do not have enough phase margin to allow for the phase shift introduced by the load.

Prior to compensation, the Bode plot of an uncompensated op amp looks like that shown in Fig. 8.11. Notice that the breakpoints are located close together thus accumulating about 180 degrees of phase shift before the 0 dB crossover point; the op amp is not usable and probably unstable. Dominant-pole compensation is often used to stabilize these op amps. If a dominant pole, in this case ω_D, is properly placed, it rolls off the gain so that τ₁ introduces 45 degrees phase at the 0 dB crossover point. After the dominant pole is introduced the op amp is stable with 45 degrees phase margin, but the op amp gain is drastically reduced for frequencies higher than ω_D. This procedure works well for internally compensated op amps, but is seldom used for externally compensated op amps because inexpensive discrete capacitors are readily available.

Assuming that Z_O << Z_F, the closed-loop transfer function is easy to calculate because C_L is enclosed in the feedback loop. The ideal closed-loop transfer equation is the same as Eq. (6.11) for the noninverting op amp and is repeated below as Eq. (8.9).

When a ⇒ ∞ Eq. (8.9) reduces to Eq. (8.10)

As long as the op amp has enough compliance and current to drive the capacitive load, and Z_O is small, the circuit functions as though the capacitor was not there. When the capacitor becomes large enough, its pole interacts with the op amp pole causing instability. When the capacitor is huge, it completely kills the op amp's bandwidth, thus lowering the noise while retaining a large low-frequency gain.

The original loop gain curve for a closed-loop gain of one is shown in Fig. 8.12, and it is or comes very close to being unstable. If the closed-loop noninverting gain is changed to 9, then K changes from K/2 to K/10. The loop gain intercept on the Bode plot (Fig. 8.12) moves down 14 dB, and the circuit is stabilized.

Gain compensation works for inverting or noninverting op amp circuits because the loop gain equation contains the closed-loop gain parameters in both cases. When the closed-loop gain is increased, the accuracy and the bandwidth decrease. As long as the application can stand the higher gain, gain compensation is the best type of compensation to use. Uncompensated versions of normally internally compensated op amps are offered for sale as stable op amps with minimum gain restrictions. As long as gain in the circuit you design exceeds the gain specified on the data sheet, this is economical and a safe mode of operation.

The loop equation for the lead-compensation circuit is given by Eq. (8.12).

The compensation capacitor introduces a pole and zero into the loop equation. The zero always occurs before the pole because R_F > R_F∥R_G. When the zero is properly placed, it cancels out the τ₂ pole along with its associated phase shift. The original transfer function is shown in Fig. 8.14 drawn in solid lines. When the R_FC zero is placed at ω = 1/τ₂, it cancels out the τ₂ pole causing the Bode plot to continue on a slope of −20 dB/decade. When the frequency gets to ω = 1/(R_F∥R_G)C, this pole changes the slope to −40 dB/decade. Properly placed, the capacitor aids stability, but what does it do to the closed-loop transfer function? The equation for the inverting op amp closed-loop gain is repeated below:

When a approaches infinity, Eq. (8.13) reduces to Eq. (8.14).

Substituting R_F∥C for Z_F and R_G for Z_G in Eq. (8.14) yields Eq. (8.15), which is the ideal closed-loop gain equation for the lead-compensation circuit.

The forward gain for the inverting amplifier is given by Eq. (8.16). Compare Eq. (8.13) with Eq. (6.5) to determine A.

The op amp gain (a), the forward gain (A), and the ideal closed-loop gain are plotted in Fig. 8.15. 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 R_F/(R_G + R_F), and it contains a high-frequency pole. The ideal closed-loop gain follows the ideal curve until the 1/R_FC breakpoint (same location as 1/τ₂ breakpoint), and then it slopes down at −20 dB/decade. Lead compensation sacrifices the bandwidth between the 1/R_FC breakpoint and the forward gain curve. The location of the 1/R_FC pole determines the bandwidth sacrifice, and it can be much greater than shown here. The pole caused by R_F, R_G, 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.

The forward gain for the noninverting op amp is a; compare Eqs. (6.11) to (6.5). The ideal closed-loop gain is given by Eq. (8.17).

The plot of the noninverting op amp with lead compensation is shown in Fig. 8.16. 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 R_F > R_F∥R_G. 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.

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.

Op amps having high input and feedback resistors are subject to instability caused by stray capacitance on the inverting input. Referring to Eq. (8.18), when the 1/(R_F∥R_GC_G) pole moves close to τ₂ the stage is set for instability. Reasonable component values for a complementary metal–oxide–semiconductor (CMOS) op amp are R_F = 1 MΩ, R_G = 1 MΩ, and C_G = 10 pF. The resulting pole occurs at 318 kHz, and this frequency is lower than the breakpoint of τ₂ for many op amps. There is 90 degrees of phase shift resulting from τ₁, the 1/(R_F∥R_GC) pole adds 45 degrees phase shift at 318 kHz, and τ₂ starts to add another 45 degrees phase shift at about 600 kHz. This circuit is unstable because of the stray input capacitance. The circuit is compensated by adding a feedback capacitor as shown in Fig. 8.18.

The loop gain with C_F added is given by Eq. (8.19).

If R_GC_G = R_FC_F Eq. (8.19) reduces to Eq. (8.20).

The compensated attenuator Bode plot is shown in Fig. 8.19. Adding the correct 1/R_FC_F breakpoint cancels out the 1/R_GC_G breakpoint; the loop gain is independent of the capacitors. Now is the time to take advantage of the stray capacitance. C_F can be formed by running a wide copper strip from the output of the op amp over the ground plane under R_F; do not connect the other end of this copper strip. The circuit is tuned by removing some copper (a razor works well) until all peaking is eliminated. Then measure the copper and have an identical trace put on the printed-circuit board.

The inverting and noninverting closed-loop gain equations are a function of frequency. Eq. (8.21) is the closed-loop gain equation for the inverting op amp. When R_FC_F = R_GC_G, Eq. (8.21) reduces to Eq. (8.22), which is independent of the breakpoint. This also happens to the noninverting op amp circuit. This is one of the few occasions when the compensation does not affect the closed-loop gain frequency response.

Referring to Fig. 8.21, a pole is introduced at ω = 1/RC, and this pole reduces the gain 3 dB at the breakpoint. When the zero occurs prior to the first op amp pole, it cancels out the phase shift caused by the ω = 1/RC pole. The phase shift is completely canceled before the second op amp pole occurs, and the circuit reacts as if the pole was never introduced. Nevertheless, Aβ is reduced by 3 dB or more, so the loop gain crosses the 0 dB axis at a lower frequency. The beauty of lead-lag compensation is that the closed-loop ideal gain is not affected as shown below. The Thevenin equivalent of the input circuit is calculated in Eq. (8.24), the circuit gain in terms of Thevenin equivalents is calculated in Eq. (8.25), and the ideal closed-loop gain is calculated in Eq. (8.26).

Eq. (8.26) is intuitively obvious because the RC network is placed across a virtual ground. As long as the loop gain, Aβ, is large, the feedback will null out the closed-loop effect of RC, and the circuit will function as if it were not there. The closed-loop log plot of the lead-lag-compensated op amp is given in Fig. 8.22. Notice that the pole and zero resulting from the compensation occur and are gone before the first amplifier poles come on the scene. This prevents interaction, but it is not required for stability.

Dominant-pole compensation is often used in IC design because it is easy to implement. It rolls off the closed-loop gain early; thus, it is seldom used as an external form of compensation unless filtering is required. Load capacitance, depending on its pole location, usually causes the op amp to ring. Large load capacitance can stabilize the op amp because it acts as dominant-pole compensation.

The simplest form of compensation is gain compensation. High closed-loop gains are reflected in lower loop gains, and in turn, lower loop gains increase stability. If an op amp circuit can be stabilized by increasing the closed-loop gain, do it.

Stray capacitance across the feedback resistor tends to stabilize the op amp because it is a form of lead compensation. This compensation scheme is useful for limiting the circuit bandwidth, but it decreases the closed-loop gain.

Stray capacitance on the inverting input works with the parallel combination of the feedback and gain setting resistors to form a pole in the Bode plot, and this pole decreases the circuit's stability. This effect is normally observed in high-impedance circuits built with CMOS op amps. Adding a feedback capacitor forms a compensated attenuator scheme that cancels out the input pole. The cancellation occurs when the input and feedback RC time constants are equal. Under the conditions of equal time constants, the op amp functions as though the stray input capacitance was not there. An excellent method of implementing a compensated attenuator is to build a stray feedback capacitor using the ground plane and a trace of the output node.

Lead-lag compensation stabilizes the op amp, and it yields the best closed-loop frequency performance. Contrary to some published opinions, no compensation scheme will increase the bandwidth beyond that of the op amp. Lead-lag compensation just gives the best bandwidth for the compensation.

The circuit bandwidth can often be increased by connecting an external capacitor in parallel with the op amp. Some op amps have hooks that enable a parallel capacitor to be connected in parallel with a portion of the input stages. This increases bandwidth because it shunts high frequencies past the low-bandwidth gm stages (see Appendix B), but this method of compensation depends on the op amp type and manufacturer.

The compensation techniques given here are adequate for the majority of applications. When the new and challenging application presents itself, use the procedure outlined here to invent your own compensation techniques.