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SLVAEX3 October 2020 TPS8802 , TPS8804

3.1.1 Photodiode Input Amplifier Model

Using a photodiode model consisting of a capacitor and current source, the photo input amplifier system can be modeled as a combination of linear circuit elements shown in Figure 3-1. To simplify the analysis, the PREF photo reference is approximated as a small-signal ground, and the two photo resistors and capacitors are combined in series. The simplified circuit is shown in Figure 3-2. The op-amp is modeled with frequency-dependent open-loop gain A(s). The op-amp open-loop gain is 1 at the gain-bandwidth frequency f_GBW.

Figure 3-1 Photo Input Amplifier Model

Figure 3-2 Simplified Photo Input Amplifier Model

Equation 1 shows the open loop transfer function from PDO to PDN. This equation provides insight on how to stabilize the feedback loop. The compensation capacitor C_PH must be sized to place the zero frequency below the closed loop unity gain frequency for sufficient phase margin. A phase margin of over 70° prevents overshoot and reduces noise on the output. This is achieved when the zero frequency is less than the unity gain frequency by at least a factor of 3. The zero frequency and unity gain frequency are calculated in Equation 2 and Equation 3. These two equations set a lower limit on the compensation capacitance, shown in Equation 4, using the assumption that C_PH is less than C_PD. The photo input amplifier has a minimum unity gain bandwidth f_GBW of 1 MHz. Set f_GBW to 1 MHz when calculating the minimum compensation capacitance.

Equation 1.

\frac{V_{P D N}}{V_{P D O}} = \frac{1 + s \times R_{P H} \times C_{P H}}{1 + s \times R_{P H} \times (C_{P H} + 2 \times C_{P D})}

Equation 2.

f_{Z} = \frac{1}{2 \times π \times R_{P H} \times C_{P H}}

Equation 3.

f_{U G} ≅ \frac{f_{G B W} \times C_{P H}}{2 \times C_{P D} + C_{P H}}

Equation 4.

C_{P H} > \sqrt{\frac{3 \times C_{P D}}{π \times R_{P H} \times f_{G B W}}}

The compensation capacitance sets the closed loop bandwidth of the amplifier. Solving the closed loop amplifier output PDO in terms of the input current I_PD results in Equation 5. The Equation 5 denominator approximately equals 1 at frequencies below the unity gain frequency calculated in Equation 3. In this condition, Equation 5 can be simplified to Equation 6. Equation 6 demonstrates that the photo amplifier output is approximately equal to the photocurrent scaled by the gain resistors and low-pass filtered by the compensation capacitors. Above the unity gain frequency, additional low-pass filtering occurs on the signal.

Equation 5.

V_{P D O (S I G)} = \frac{I_{P D} \times \frac{2 \times R_{P H}}{1 + s \times R_{P H} \times C_{P H}}}{1 + \frac{1 + s \times R_{P H} \times (C_{P H} + 2 \times C_{P D})}{(1 + s \times R_{P H} \times C_{P H}) \times A (s)}}

Equation 6.

V_{P D O (S I G)} ≅ I_{P D} \times \frac{2 \times R_{P H}}{1 + s \times R_{P H} \times C_{P H}}