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SLVAEX3 October 2020 TPS8802 , TPS8804

3.2.2 Output Voltage Noise Model

The PDO voltage noise density is calculated using the op-amp and resistor noise models, shown in Equation 13. Independent noise sources are combined by taking the root-sum-of-squares. Equation 14 simplifies the output noise by dividing the Equation 13 noise voltage density by the signal voltage calculated in Equation 5. Equation 14 is used to identify the dominant source of noise in the amplifier.

Equation 13.

S_{P D O (N O I S E)} = \frac{\sqrt{(2 \times I_{N A M P}^{2} + {2 \times I}_{N R P H}^{2}) \times {(\frac{R_{P H}}{1 + s \times R_{P H} \times C_{P H}})}^{2} + V_{N A M P}^{2} \times {(\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}})}^{2}}}{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 14.

\frac{S_{P D O (N O I S E)}}{V_{P D O (S I G)}} = \frac{1}{I_{P D}} \times \sqrt{{(\frac{I_{N A M P}}{\sqrt{2}})}^{2} + {(\frac{I_{N R P H}}{\sqrt{2}})}^{2} + {(V_{N A M P} \frac{1 + s \times R_{P H} \times (C_{P H} + 2 \times C_{P D})}{2 \times R_{P H}})}^{2}}

The resistor noise and amplifier voltage noise is less than the amplifier current noise when R_PH is greater than approximately 200 kΩ, based on measurements of the voltage and current noise density. In most applications, R_PH is greater than 200 kΩ and the resistor noise and voltage noise are negligible.

The buffered photo RMS voltage noise is calculated in Equation 15 using the amplifier current noise as the only noise source. The amplifier current noise is assumed to be white. The buffered photo signal model in Equation 7 is used to calculate the buffered photo noise density in Equation 15. Integrating the noise density across the frequency spectrum calculates the RMS noise. The result demonstrates that the output noise increases with the amplifier gain and decreases with the root-sum of time constants.

Equation 15.

V_{B U F (N O I S E)} = \sqrt{\int_{0}^{\infty} 2 \times {(\frac{I_{N A M P} \times R_{P H} \times G_{P G A I N}}{(1 + j \times 2 \times π \times f \times τ_{1}) \times (1 + j \times 2 \times π \times f \times τ_{2})})}^{2} \times d f} = \frac{I_{N A M P} \times R_{P H} \times G_{P G A I N}}{\sqrt{2 \times (τ_{1} + τ_{2})}}