The 1-rst order ($R_s=0$) detuned unit step response is found as:
\begin{equation} \mu_{t}=\frac{R_{b}}{R_{a} + R_{b}} - \frac{\left(- C_{a} R_{a}^{2} R_{b} + C_{b} R_{a} R_{b}^{2}\right) e^{- \frac{t \left(R_{a} + R_{b}\right)}{R_{a} R_{b} \left(C_{a} + C_{b}\right)}}}{R_{a} R_{b} \left(C_{a} + C_{b}\right) \left(R_{a} + R_{b}\right)} \end{equation}The 1-rst order ($R_s \neq 0,\, C_b=C_a \frac{R_a}{R_b}$) tuned unit step response is found as:
\begin{equation} \mu_{t}=\frac{R_{b}}{R_{a} + R_{b} + R_{s}} - \frac{R_{b} e^{- \frac{t \left(R_{a} + R_{b} + R_{s}\right)}{C_{a} R_{a} R_{s}}}}{R_{a} + R_{b} + R_{s}} \end{equation}The settling error to the final value of a wrongly tuned probe can be written as a function of time:
\begin{equation} \epsilon_{t}=\frac{\left(C_{a} R_{a} - C_{b} R_{b}\right) e^{- \frac{t \left(R_{a} + R_{b}\right)}{R_{a} R_{b} \left(C_{a} + C_{b}\right)}}}{\left(C_{a} + C_{b}\right) \left(R_{a} + R_{b}\right)} \end{equation}After equating this value with a 1 LSB error, we find the settling time as function of the relative positivedetuning $\delta$ of $C_b$:
\begin{equation} \tau_{s}=\frac{9 C_{a} R_{b} \left(9 \delta + 10\right) \log{\left(\frac{9 \cdot 2^{n} \delta}{9 \delta + 10} \right)}}{10} \end{equation}
The settling error to the final value of a wrongly tuned probe can be written as a function of time:
\begin{equation} \epsilon_{t}=\frac{\left(C_{a} R_{a} - C_{b} R_{b}\right) e^{- \frac{t \left(R_{a} + R_{b}\right)}{R_{a} R_{b} \left(C_{a} + C_{b}\right)}}}{\left(C_{a} + C_{b}\right) \left(R_{a} + R_{b}\right)} \end{equation}After equating this value with a 1 LSB error, we find the settling time as function of the relative negativedetuning $\delta$ of $C_b$:
\begin{equation} \tau_{s}=\frac{9 C_{a} R_{b} \left(9 \delta + 10\right) \log{\left(- \frac{9 \cdot 2^{n} \delta}{9 \delta + 10} \right)}}{10} \end{equation}
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Last project update: 2022-02-16 10:55:49