Transimpedance amplifiers (TIAs) translate the current output of sensors like photodiode-to-voltage signals, since several circuits and instruments can only accept voltage input. An operational amplifier with a feedback resistor from output to the inverting input is the most straightforward implementation of these TIAs. However, even this simple TIA circuit requires careful tradeoffs among noise gain, offset voltage, bandwidth, and stability.

Clearly stability in a TIA is essential for good, reliable performance. Thus, it’s important to understand the empirical calculations for assessing stability, and carefully select the feedback phase-compensation capacitor.

**Wild Oscillations: Why Do They Happen? **

One basic TIA circuit is commonly found in dual-supply systems *(Fig. 1)*. It can be modified slightly to fit in single-supply applications *(Fig. 2)*. The resistive-divider formed by R1 and R2 ensures that the op amp’s output node is higher than the Output Voltage Low specification during a no-light condition, when only a small dark current flows through the photodiode. By ensuring that the op amp’s output stage operates in the linear region, this offset improves both photo-detection in low-light conditions and response speed.

However, care must be taken to keep a small bias voltage on the In+ pin. Otherwise, reverse-leakage current in the photodiode can degrade linearity and increase offset drift over temperature.

In some applications, the photodiode is placed directly across the input terminals of the op amp *(Fig. 3)*. This circuit avoids the reverse bias across the photodiode, although it requires a buffered reference. The reference must be fast enough to sink the photodiode current as required by the application. This, in turn, implies that amplifier A1 must be as fast as amplifier A2.

Like any op-amp circuit with feedback, each of the aforementioned circuits can be separated into an amplifier with open-loop gain (A_{VOL}) and a feedback network comprised of the resistance and the photodiode.

Figure 4 shows the equivalent circuit of the photodiode in Figure 1, Figure 2, and Figure 3.^{1} For most photodiodes, R_{SERIES} = 0 and R_{SHUNT} = Infinity is a fair approximation. Consequently, the simplified model reduces to the short-circuit current source in parallel with the junction capacitance. This simplified photodiode model will be used for subsequent stability analysis.

To understand why the circuits in Figures 1 to 3 might oscillate, it’s useful to plot the frequency of the open-loop gain and the feedback factor. For example, the open-loop gain response of the op amp is constant from dc until the dominant-pole corner frequency; it decreases at 20 db per decade thereafter until it reaches the second-pole corner *(Fig. 5)*. Mathematically, the single-pole response can be represented as:

(Eq. 1)

where A_{VOL} = dc open-loop gain; A_{VOL}(jw) = open-loop gain corresponding to frequency, w; and w_{PD} = dominant-pole frequency in radians.

Using the simplified equivalent circuit for the photodiode, the feedback network is simply a one-pole RC filter comprised of the feedback resistance (R_{F}) and the total input capacitance (C_{i}), which is the junction capacitance of the photodiode in parallel with the op amp’s input capacitance. The feedback factor is given as:

(Eq. 2)

Therefore, the reciprocal of the feedback factor is:

(Eq. 3)

Figure 5 also plots the response curve for 1/β(jw). At low frequencies, the curve remains flat at unity gain, as expected from the unity-gain resistive feedback. It then rises at 20 db/dec starting from the pole-corner frequency (f_{F}).

From the Barkhausen stability criterion, oscillation can result if the closed-loop TIA circuit doesn’t have sufficient phase margin for Aβ ≥ 1. Hence, the intersection of the A_{VOL}(jw) response curve with the 1/β (jw) curve denotes a critical intercept fundamental for stability analysis. The phase margin at this intersection frequency can be determined by observing the rate of closure between the two response curves, A_{VOL}(jw) and 1/β(jw). If the rate of closure of the two response curves is 40 dB *(Fig. 5, again)*, the circuit will be unstable.

There is another intuitive way to understand this scenario. At lower frequencies, the phase shift in the feedback signal is 180° due to the inverting nature of the negative feedback. As the frequency increases well into the −20-dB/dec slope region of A_{VOL}, the dominant pole of the op amp can add up to 90° of phase shift.

Similarly, the pole introduced by the feedback network can add another 90° of phase shift, thus producing a phase shift of about 360° at Aβ = 1. If the phase shift is 360°, self-sustaining oscillations will result. If the phase shift is close to 360°, heavy ringing is observed. In either case, some form of phase compensation scheme will be required to stabilize the circuit.

**No Evil Is Without Its Compensation **

It’s common knowledge that adding a bypass capacitor in parallel with the feedback resistance provides the requisite compensation to guarantee sufficient phase margin *(Fig. 6)*. A key factor is to calculate the value of the feedback capacitor required to provide optimal compensation. To account for the added phase-compensation capacitor, substitute Z_{F} in Equation 2 with R_{F} || C_{F}. The feedback factor now becomes:

(Eq. 4)

Comparing Equation 2 and Equation 4 shows that adding C_{F} introduces a zero in the feedback factor, besides modifying its pole. The zero compensates for the phase shift introduced by the feedback network *(Fig. 7)*. If the phase shift is overcompensated by choosing a large feedback capacitor, then the rate of closure can be reduced to 20 dB/dec (90° phase margin). However, overcompensation also reduces the TIA’s usable bandwidth.

While a reduced bandwidth may not be an issue with low-frequency photodiode applications, high-frequency or low-duty-cycle pulsed photodiode circuits definitely need to maximize the available bandwidth. For such applications, the goal is to find the minimum value of the feedback compensation capacitor (C_{F}) needed to eliminate oscillation and minimize ringing. However, it’s always a good idea to overcompensate the TIA circuit slightly. Overcompensation is recommended to provide sufficient guardband to account for up to ±40% variation in an op amp’s bandwidth over process corners and the feedback capacitor’s tolerance.

A good design compromise is to target 45° of phase margin at the intercept of the A_{VOL}(jw) and 1/ β (jw) curves. This margin requires the optimum value of C_{F} to be calculated so that the added zero in the feedback factor, β(jw), is located at the frequency corresponding to Aβ = 1 *(Fig. 7)***.** One equation for the intercept frequency is:

(Eq. 5)

Equation 5 has two unknowns, the intercept frequency (f_{i}) and C_{F}. To solve for C_{F}, we need to find another simultaneous equation. One way to obtain the second equation is to equate the A_{VOL}(jw_{i}) and 1/β(jw_{i}) curves. The resulting equation is complicated and doesn’t lend itself to an easy solution.

The graphical approach for solving C_{F} is a more convenient alternative.^{2} Observing Figure 7, both curves have a slope of 20 dB/dec. Therefore, the approximate triangle formed by both curves with the horizontal axis is isosceles. Hence, f_{i} is the average of the other two vertices. The frequency is plotted in the logarithmic scale, therefore:

(Eq. 6)

Here:

(Eq. 7)

where f_{GBWP} = unity gain bandwidth of the op amp. To account for the variation in unity-gain bandwidth over process corners, select f_{GBWP} to be 60% of the value specified on the op amp’s data sheet.

For decompensated op amps, use f_{GBWP} to equal 60% of the frequency where the projection of the −20-dB A_{VOL}(jw_{i}) slope intersects the 0-dB x-axis line. With some algebraic manipulation, Equation 6 can be rewritten as:

(Eq. 8)

Equation 8 shows that f_{i} is equal to the geometric mean of f_{GBWP}, and the f_{F} of β(jw).

When substituting for f_{F} from Equation 7:

(Eq. 9)

When equating Equations 5 and 9 and squaring, it becomes:

(Eq. 10)

The above quadratic equation can be easily solved to calculate the following value of C_{F}:

(Eq. 11)

The calculated value of feedback capacitor C_{F} is valid for both large- and small-area photodiodes.

**Alright…Give Us the Scope Now **

TIAs are finding homes in numerous applications, such as 3D goggles, compact-disc players, pulse oximeters, IR remote controls, ambient light sensors, night-vision equipment, and laser range finding. To illustrate their usage, consider a rain-sensor application example. Rain sensors are presently used in high-end automobiles to automatically adjust the wiper speed depending on the presence and intensity of rain.

Usually optical rain sensors operate on the principle of total internal reflection. The sensor is generally located behind the driver’s rear-view mirror. An infrared light laser source beams the light pulses at an angle to the windshield. If the glass is not wet, then most of the light comes back to the photodiode detector. If the glass is wet, then some of the light becomes refracted and the sensor tuning on the wiper detects less light. Wiper speed is based on how fast the moisture builds up between the sweeps.

In order to detect the change in moisture for wiper adjustment while rejecting low-frequency, ambient-light IR content, the rain sensor must operate at a pulse frequency over 100 Hz. For instance, consider the problem of designing a rain-sensor TIA with the following specifications:

- Photodiode IR-current-pulse peak amplitude = 50 nA up to 10 μA, depending on reflected light content
- On-time duration = 50 μs
- Duty cycle = 5%
- R
_{F}= 100 kΩ - Usage of BPW46 photodiode

The op amp of choice for TIA circuits typically features low noise, a CMOS input, and a high-bandwidth-to-supply-current ratio. This rain-sensor design example employs the MAX9636 op amp. The MAX9636 also suits other battery-powered, portable equipment, since its design offers a good tradeoff between lower quiescent current and noise performance. For higher-bandwidth applications, op amps like the MAX4475 and MAX4230 might be a better fit.

To calculate the estimated value of feedback capacitance, substitute the following parameters in Equation 11:

- C
_{i}= junction capacitance of photodiode (70 pF) + 2 pF input capacitance of the MAX9636 = 72 pF - f
_{GBWP}= 0.9 MHz

Gain bandwidth, which isn’t a trimmed parameter, can vary ±40% over process corner for any op amp. Consequently, even though the data sheet specifies the typical unity-gain bandwidth to be 1.5 MHz, the unity-gain bandwidth in this case is considered to be 60% of this typical value to account for process variations.

Here, R_{F} = 100 kΩ.Therefore, the calculated value of C_{F} = 15.6 pF. The capacitor’s next highest standard value is 18 pF.

Not surprisingly, oscillation is observed at the output of a TIA that doesn’t include a compensation feedback capacitor and incorporates the circuits shown in Figure 1, Figure 2, and Figure 3 *(Fig. 8)*. If C_{F} = 10 pF is used, then ringing stops, although an overshoot is still visible *(Fig. 9)*.

Next, the feedback capacitor value is increased to the recommended calculated value of 18 pF. In this instance, no ringing or oscillation is observed for the C_{F} = 18 pF case, thus validating the theoretical analysis presented earlier *(Fig. 10)*. The corresponding small signal-step response with 50-nA amplitude of photodetector current also is provided *(Fig. 11)*.

**References:**

1. Jiang, H., and Yu, P. K. L., “Equivalent Circuit Analysis Of Harmonic Distortions In Photodiode,” *IEEE Photonics Technology Letters*, vol. 10, no. 11, Nov. 1998, pp. 1608-1610.

2. Graeme, Jerald, “Photodiode Amplifiers: Op amp Solutions,” The McGraw-Hill Companies, Inc., ISBN 0-07-024247-X, pp. 47-50.