SDAA465 Application note

SDAA465 August 2026 LMH13000 , LMH32401

4.1 Stability Consideration of TIA

If the photodiode is assumed to be an ideal current source, the output voltage of TIA is given by Equation 3:

Equation 3.

V_{OUT} = \frac{- I_{PD} \times R_{F}}{(1 + j 2 {πfR}_{F} C_{F})}

Figure 4-1 Ideal Photodiode TIA Circuit

For a practical implementation, the photodiode cannot be treated merely as an ideal current source. Instead, the device is more accurately modeled as an ideal current source in parallel with an equivalent shunt resistance, R_D, and junction capacitance, C_D. The op amp input capacitance also cannot be considered insignificant and must be included as part of C_D.

Figure 4-2 Photodiode Electrical Model with TIA

Equation 4.

A_{CL} (f) = \frac{R_{F} + R_{D}}{R_{D}} \times \frac{1 + \frac{j 2 πf (R_{F} R_{D})}{(R_{F} + R_{D})} (C_{F} + C_{D})}{(1 + j 2 πf (R_{F} C_{F})}

Equation 5.

A_{CL} (f) = \frac{R_{F} + R_{D}}{R_{D}} \times \frac{(1 + j \frac{f}{f_{Z}})}{(1 + j \frac{f}{f_{P}})}

Equation 6.

Here, f_{P} = \frac{1}{{(2 πR}_{F} C_{F})}

Equation 7.

f_{Z} = \frac{1}{2 π (R_{F} {| | R}_{D}) (C_{F} + C_{D})}

Generally, R_D >> R_F, R_F||R_D = R_F and the DC gain also becomes unity.

Equation 8.

f_{Z} = \frac{1}{{2 πR}_{F} {(C}_{F} + C_{D})}

This implies f_Z is always lower than f_P.

For the system to be stable, the f_P must lie inside the open loop curve of op amp.

Figure 4-3 Open-Loop and Closed-Loop Response of TIA vs Frequency

Figure 4-3 shows three different scenarios for the intersection of the closed-loop response (inverse of the feedback factor) with the open loop gain curve. Instability occurs when the rate of closure of the two curves is 40dB. This happens when the f_P falls outside the open loop curve as in case of f_P1 and the circuit, in this case, oscillates. If f_P lies inside the open loop curve like in the case of f_P2, the transimpedance circuit is unconditionally stable, but the stability in this case is traded off for transimpedance bandwidth. The best practice is to place the f_P on the open-loop gain curve as shown for f_P3.

Since f_P is determined by the feedback network, judicious selection of C_F. is all that is necessary. This process can be greatly simplified by noting that the high frequency asymptote for the noise gain is determined by capacitance values alone.

Equation 9.

For (f > > f_{P}), ACL = \frac{C_{F} + C_{D}}{C_{F}}

For f_P to be on AOL curve,

Equation 10.

\frac{GBP}{f_{P}} = \frac{C_{F} + C_{D}}{C_{F}}

On solving for C_F:

Equation 11.

C_{F} = \frac{1}{{(4 πR}_{F} GBP)} \times (1 + \sqrt{1 + 8 {πR}_{F} C_{D} GBP})

Once the photodiode capacitance and op amp GBP are determined, the capacitance value of the feedback capacitor must be selected appropriate as per Equation 9 to Equation 11 to maintain stability.

Equation 12.

f_{P} = \sqrt{\frac{GBP}{2 {πR}_{F} C_{D}}}

This result indicates that, for a given op amp and photodiode, transimpedance bandwidth is inversely related to the square root of the feedback resistor. Thus, if bandwidth is a critical requirement, the best approach can be to opt for a moderate transimpedance gain stage followed by a broadband voltage gain stage.