In the first installment of this series, I described various factors that affect the loop gain of a transimpedance amplifier (TIA) and demonstrated how to compensate a TIA to achieve a Butterworth (maximally flat) closed-loop response. In this installment, I will show you how to compensate a TIA for an arbitrary phase margin.

The closed-form equations to determine the closed-loop bandwidth and feedback capacitance (C_{F}) for TIA compensation can be quite involved. A simpler approach is to first compensate the TIA for a Butterworth response (Q = 0.707) using the theory presented in part 1 of this series and then, using Figure 1 below, determine the C_{F} to achieve the desired quality factor (Q). Figure 1 also gives the resulting closed-loop bandwidth of the amplifier for the desired Q factor.

**Figure 1: Scaling factor for bandwidth (f - 3dB) and C _{F} vs. Q factor**

Using the example for a Butterworth response from part 1 of this series, set C_{F} = 0.14pF, which results in f_{-3dB} = 10MHz. To achieve a Q = 0.6, from Figure 1 the resulting C_{F} = 0.14pF x 1.190 = 0.17pF and the resulting f_{-3dB} = 10MHz x 0.82 = 8.2MHz.

For more accurate results, you can download this Excel spreadsheet from the TI E2E™ Community. Figure 2 is a screenshot of two calculators in the spreadsheet. Calculators A and B determine the value of both f_{-3dB} and C_{F} for a Butterworth response and an arbitrary value of Q, respectively.

**Figure ****2****: Excel calculators to aid in TIA design**

The Q of a circuit is directly related to its phase margin (Φ_{M}), which determines the amplifier’s closed-loop frequency response and time-domain pulse response. An amplifier circuit with a low phase margin has a peaked frequency response and significant ringing in the pulse response. Conversely, a circuit with high phase margin has a flat frequency response and little or no overshoot in the pulse response.

Circuits with low phase margin extend the amplifier’s closed-loop bandwidth, while circuits with high phase margin reduce the closed-loop bandwidth. A Butterworth response, which has a Q = 0.707 and a maximally flat frequency response, has a phase margin of 65.5 degrees and about 4.3% overshoot in the pulse response. Figure 3 shows the Q and overshoot as a function of phase margin.

**Figure 3: Q and overshoot vs. phase margin**

To simplify the circuit design, the Excel spreadsheet also has Calculator C, with three separate calculators showing the relationships between Q, overshoot, phase margin and frequency-response peaking.

Calculator C1 gives the phase margin and overshoot for a given value of Q.

Calculator C2 calculates the Q and overshoot for a given value of phase margin. This calculator is useful for achieving a desired frequency-response shape for stability and for maximizing flatness in narrowband applications.

Calculator C3 is useful in time-domain applications when you want to target a certain value of overshoot in the pulse response.

**Figure 4: Calculator C relating Q, phase margin and overshoot**

Using the three different calculators, I calculated the frequency- and time-domain responses for the TIA discussed in part 1 of this series for different values of Q:

- Calculator A produced the C
_{F}and f_{-3dB}for a Butterworth response with the specified op amp transimpedance gain and input capacitance. - Calculator B gave the scaling factor for C
_{F}and the change in f_{-3dB}for different values of Q. - Calculator C predicted the peaking in the frequency response and the overshoot in the pulse response.

The simulated results in Figure 5 match the calculated values very closely.

**Figure 5: Simulated closed-loop frequency and transient responses as a function of Q**

You should now be very comfortable with compensating a TIA based on the specific application requirements. The calculators introduced in this post are intended for use with TINA-TI™ software to provide a starting point for TIA design. In the next installment of this series, I will describe the effects of the amplifier’s higher-order open-loop poles on the loop gain of a TIA and introduce the concept of decompensated amplifiers.

**Additional resources**

- Get online support in the TI E2E™ Community Amplifier forums.
- Read the first installment of this series,
- Download a free version of TINA-TI software.
- Learn about TI’s entire portfolio of amplifier ICs and find technical resources.