An op amp's input capacitance can often be the bugaboo that prevents the frequency stability of a circuit. The reason is that input capacitance along with the resistive feedback network creates a pole in an inverting amplifier that introduces phase lag in the open-loop response. In noninverting op amps, the input capacitance can produce a zero in the closed-loop response, which causes high-frequency peaking, which can degrade common-mode rejection. Knowledge about phase-compensation techniques can help you counteract these deleterious effects of an op amp's input capacitance.

Fig 1 illustrates an inverting amplifier with some typical values for the input capacitance. C_idrepresents the differential capacitance between the amplifier's two inputs. The two C_icm components represent the common-mode input capacitance for each op-amp input to ground. In the inverting configuration, two of the three input capacitances shunt the feedback network. A capacitor in the feedback path C_f compensates for the effects of the input capacitance. To examine the effects of the input capacitance when C_f is 0, consider the uncompensated feedback voltage divider:

where [smlbeta]₀=R₁/(R₁+R₂) and C_i=C_id+C_icm. The op-amp input capacitance introduces a feedback pole at f=1/2[pi](R₁|| R₂)C_i.

The significance of the feedback pole produced by C_i on frequency stability depends on the values of the feedback resistors. Large resistor values and op amps with wide open-loop bandwidths cause stability concerns. The addition of C_f in the feedback path introduces a zero to compensate for the feedback pole. Fig 2 is a bode diagram that demonstrates the stabilizing effect of C_f. The interception of the op amp's open-loop gain curve and the 1/[smlbeta] response determine the circuit's frequency stability. The 1/[smlbeta] response is:

where [smlbeta]₀=R₁/(R₁+R₂).

The bode diagram shows a rise in the frequency response of 1/[smlbeta] due to the zero at f_z=1/2[pi](R₁||R₂)(C_i+C_f). Without the pole in the 1/[smlbeta] response at f_p=1/2[pi]R₂C_f, the open-loop gain A_OL and the 1/[smlbeta] curve would intersect with a 40-dB/decade slope difference, which would produce 180° phase shift at the intercept point and loop oscillations. Placing f_p at the intercept point introduces 45° of phase correction and extends the closed-loop bandwidth to f_i. The phase shift at the intercept point reduces from 180 to 135&3176;, producing a 45° phase margin. Placing the f_p closer to the f_z would further increase the phase margin. However, because C_f shunts the feedback resistor, the C_f increase required for this phase margin would restrict the inverting circuit's bandwidth.

To quantify f_i and C_f using geometric analysis, note the triangle in the bode diagram formed by the rise in the 1/[smlbeta] response, the fall in the A_OL response, and the projected dash extension of the 1/[smlbeta]0 curve. The intercept point, f_i, lies midway between the f_z and B₀f_c endpoints of the base of the triangle. The logarithm of the midpoint is given as Log f_i=(Log f_z+Log b₀f_c)/2. Solving for f_i yields:

Setting f_p=f_i=1/2pR₂C_f yields the value of the compensation capacitor, C_f. Another convenient way to determine C_f is to assume a fictitious capacitor that creates the amplifier crossover frequency, f_c. The fictitious capacitor is given by Cc=1/2pR2fc. Solving for the compensation capacitor in terms of the fictitious capacitor yields:

where

For a crossover frequency of f_c=16 MHz and a total in- put capacitance of C_i=14 pF, the compensation capacitor is C_f=3 pF.

The test to determine whether C_f for the noninverting amplifier is needed comprises the following steps. First, compute f_z=1/2[pi]R₁||R₂C_i, where C_i=C_id+C_icm. Compare the result with [smlbeta]₀f_c, where [smlbeta]₀=/R₁+R₂ and f_c is the crossover frequency of the op amp. If f_z is greater than [smlbeta]₀f_c, then no compensation is required and there is no peaking in the closed-loop response. When f_z is less than [smlbeta]₀f_c, however, you calculate the compensation capacitor by:

where

For the component values shown in Fig 1, C_i=15 pF and R₁||R₂=10 k[smlomega], which makes f_z=1.06 MHz. Also [smlbeta]₀=0.5 and f_c=16 MHz, so the calculated value for [smlbeta]₀f_c=8 MHz. Because f_z<[smlbeta]₀f_c, the circuit requires phase compensation. In this case, the wide bandwidth of the OPA627 op amp and the large value of 20 k[smlomega] for feedback resistors contribute to the need for compensation. The closed-loop bandwidth for the circuit is BW=1.4f_i. For the values shown, f_i=2.7 MHz, so the BW=3.8 MHz.

In the noninverting op-amp configuration (Fig 3), one of the input capacitors (C_icm) shunts the feedback resistor (R₁). Although the feedback factor [smlbeta] is the same as the inverting amplifier, at high frequencies C_icm and R₁ introduce a zero in the closed-loop response, which causes peaking. This effect does not occur in the inverting configuration because the inverting input is at a virtual ground. The phase-compensation capacitor C_f can also compensate for the peaking in the noninverting configuration. An appropriate choice of C_f creates a pole that cancels the zero produced by C_icm. Cancellation occurs when C_fŚR₂=C_icmŚR₁. Solving for C_f:

To compensate for peaking in the closed-loop response, use:

Because the calculated capacitance for peaking compensation is a larger value than the calculated phase-compensation capacitor, you can use the larger value to extend the closed-loop bandwidth.

How well the pole cancels the zero depends on the variation in the input capacitance due to production variances and voltage sensitivity. C_icm is dependent on voltage; therefore, the signal level at e₁ can introduce high-frequency distortion, which affects the frequency compensation. However, the input-capacitance variation is generally small, so high-frequency distortion is generally slight whether or not you employ a compensation capacitor.

Fig 4 is a bode diagram for a compensated noninverting amplifier. The figure shows the op amp's open- (A_OL) and closed-loop (A_CL) responses, along with the 1/[smlbeta] response curve. The dashed curve superimposes the 1/[smlbeta] response for the previously considered inverting amplifier. Using a larger compensation capacitor C_f than the noninverting case eliminates any overshoot in the closed-loop response.

The interception of the 1/[smlbeta] curve and the A_OL curves determines the closed-loop bandwidth B_Hf_c. The closed-loop bandwidth f_i* for the noninverting amplifier is wider than the inverting amplifier f_i. Also note that the noninverting compensation capacitor levels off the 1/[smlbeta] curve at a lower frequency than the intercept point with the A_OL curve. The noninverting closed-loop bandwidth is given as:

where

For the values shown in the figures, the bandwidth for the noninverting amplifier is 13 MHz vs 3.8 MHz for the inverting amplifier.

A differential op-amp configuration poses conflicting phase-compensation demands because of the combined effect of the noninverting and inverting configurations. Fortunately, a judicious choice of circuit resistance can compensate for the conflict. Fig 5 depicts a difference amplifier with the op amp's input capacitances brought out for circuit analysis. At first glance, it appears that the value of the feedback compensation capacitor C_f conflicts with frequency stability and gain-peaking compensation. In addition, the value of C_icm in parallel with R₄ provides an extra pole to the e₂ path that doesn't occur in the e₁ path.

Analysis of the circuit's closed-loop response, however, shows that the compensation for gain peaking resolves the phase-compensation difference for both paths to the op amp. With no feedback compensation capacitor (C_f=0), the closed-loop response is given by:

To balance the dc gain for each input path, set the circuit resistor ratios so that R₄/R₃=R₂/R₁. Maintaining these resistor ratios reduces the closed-loop gain to:

When s=0, the dc gain for both input paths are the same. However, the ac closed-loop response for the e₂ path has a pole at f_p2=1/2[pi](R₃||R₄)C_icm and a zero at f_z2=1/2pR₁||R₂C_icm. Although both singularities depend on C_icm, the capacitances are at the opposite op-amp inputs. Because the capacitors are closely matched, you can make the pole and zero cancel by allowing R₃=R₁ and R₄=R₂. Under these conditions, the closed-loop gain is:

The previous closed-loop analysis of the difference amplifier ignores the effect of the op amp's open-loop response (A_OL), which must be phase-compensated. To phase-compensate the difference amplifier, you must bypass R₂ with C_f as described for the inverting and noninverting amplifiers. Although both the e₁ and e₂ gain paths experience the same pole at 1/2[pi]R₂C_fs, the feedback capacitor alters the f_z2 zero for the e₂ path without equivalently affecting a zero in the e₁ path. The uncompensated f_z2 zero is caused by the resistive feedback network of R₁||R₂ reacting with C_icm capacitor at the negative op-amp input. By bypassing R₂ with a compensation capacitor C_f, you modify the f_z2 zero to
be 1/2[pi](R₁||R₂)(C_icm+C_f).

The modified f_z no longer cancels the pole at f_p2=1/2[pi] (R₁||R₂)C_icm. To realign the pole and zero cancellation, you must bypass R₄ with an equivalent compensation capacitor equal in value to C_f. The second compensation capacitor generates a pole at f_p2=1/2[pi](R₁||R₂)(C_icm+C_f), which effectively cancels the zero at f_z2. The resulting closed-loop gain is given as:

The pole-zero cancellation described for the difference amplifier suggests an alternative compensation method for the noninverting case. Grounding the e₁ input for the difference amplifiers converts the circuit into a noninverting configuration. However, the input impedance is R₃+R₄, which is not as large as the input impedance for the connection shown in Fig 3. However, because there is only gain to the e₂ path in the noninverting amplifier, you need only the resistance ratio R₃||R₄=R₁||R₂ to maintain the pole and zero cancellation. To accommodate large source impedances (R_s), make R₄=[infinity] and R₃=R₁||R₂-Rs to achieve a large input impedance.

The previous analysis of the noninverting amplifier in Fig 3 ignored the effect of the source impedance's reacting with
the C_icm capacitance, causing an uncompensated pole in the closed-loop response. The addition of R₃ and the addition of C_f from the op-amp noninverting input to ground provides compensation for this pole, as described in the difference amplifier configuration. The pole-zero matching also removes a potential distortion effect. The voltage levels at the op-amp inputs cause C_icm to vary slightly; however, the matching of the pole and zero causes the distortion products to cancel.

Equal signal voltages on the two C_icm capacitors maintain the pole-zero cancellation, which removes any high-frequency distortion to which the circuit in Fig 3 is susceptible. In addition, by making the impedances for the dc currents for each op-amp input the same, you minimize the effect of input offset currents on the amplifier's overall output offset voltage.

1. Graeme, J, "Circuit Options Boost Photodiode Bandwidth," EDN, May 21, 1992, pg 155.
2. Graeme, J, "Feedback Plots Offer Insight into Operational Amplifiers," EDN, January 19, 1989, pg 131.
3. Graeme, J, "Bode Plots Enhance Feedback Analysis of Operational Amplifiers," EDN, February 2, 1989, pg 163.
4. Graeme, J, "Feedback Models Reduce Op-Amp Circuits to Voltage Dividers," EDN, June 20, 1991, pg 139.