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SLVAEX3 October 2020 TPS8802 , TPS8804

3.3.2 Two ADC Samples

Using one ADC sample for the base voltage and one ADC sample for the top voltage results in the SNR calculated in Equation 17. The top signal is calculated using the Equation 9 buffered photo signal pulse sampled at time t. The signal noise is calculated using the Equation 15 buffered AMUX output noise voltage divided by the square root of two. Sampling the signal at the peak maximizes the SNR. The peak of the signal is approximately t_LED, and by sampling the signal at time t_LED, Equation 17 simplifies to Equation 18. Equation 18 provides the key insight that increasing t_LED, τ₁ and τ₂ together by a constant factor improves the SNR by the square root of the factor.

Equation 17.

S N R_{2} = \frac{2 \times I_{P D} \times \sqrt{τ_{1} + τ_{2}}}{I_{N}} \times (V_{B U F (P U L S E)} (t) - V_{D C})

Equation 18.

S N R_{2} ≅ \frac{2 \times I_{P D} \times \sqrt{τ_{1} + τ_{2}}}{I_{N}} \times (1 + \frac{τ_{1}}{τ_{2} - τ_{1}} \times e^{\frac{- t_{L E D}}{τ_{1}}} + \frac{τ_{2}}{τ_{1} - τ_{2}} \times e^{\frac{- t_{L E D}}{τ_{2}}})

Improving the SNR by increasing t_LED, τ₁ and τ₂ comes at the cost of increased power consumption. Because low power consumption is essential for smoke alarms, it is desirable to optimize the SNR without increasing power consumption. The LED pulse energy consumption is calculated in Equation 19. Dividing Equation 18 by Equation 19 results in Equation 20, the SNR per unit energy (SNRE). The photodiode current is proportional to the LED current, therefore the SNRE is relatively unaffected by changes in LED current.

Equation 19.

E_{L E D} = t_{L E D} \times V_{L E D} \times I_{L E D}

Equation 20.

S N {R E}_{2} ≅ \frac{2 \times I_{P D} \times \sqrt{τ_{1} + τ_{2}}}{I_{N} \times t_{L E D} \times V_{L E D} \times I_{L E D}} \times (1 + \frac{τ_{1}}{τ_{2} - τ_{1}} \times e^{\frac{- t_{L E D}}{τ_{1}}} + \frac{τ_{2}}{τ_{1} - τ_{2}} \times e^{\frac{- t_{L E D}}{τ_{2}}})

Equation 20 is maximized in two steps. First, t_LED is held constant and the optimal τ₁ and τ₂ are numerically calculated using a nonlinear programming solver fminsearch in MATLAB. Setting τ₁ and τ₂ to 0.31 times the LED pulse length maximizes the SNR. Next, t_LED is varied with τ₁ and τ₂ set to 0.31 times t_LED. The SNRE increases as t_LED, τ₁ and τ₂ decrease, therefore the SNRE is maximized when t_LED is minimized. This result explains why shorter pulse lengths with higher LED currents can have a higher SNR than longer pulse lengths with lower LED currents. If τ₁ and τ₂ are equal and fixed while t_LED is varied, the optimal solution for t_LED is 1.8 τ₁.

Common methods set τ₁ and τ₂ below 0.2 t_LED to allow the signal to settle before taking the measurement or set τ₁ above t_LED to integrate the photodiode signal. Both of these methods do not have as high of an SNR compared to when τ₁ and τ₂ are set to 0.31 t_LED. Setting τ₁ and τ₂ to 0.2 t_LED compared to 0.31 t_LED has 14% higher amplitude and 25% higher noise. Setting τ₁ and τ₂ to t_LED compared to 0.31 t_LED has 58% lower amplitude and 44% lower noise.