Today lecture on sensitivity. Definition of sensitivity: percentage change in magnitude of transfer function \(T\left ( s\right ) \) per one percent change of parameter \(\alpha \) in the transfer function. We normally make this parameter be \(\alpha \). This could be \(R\) (resistance) or \(C\) (capacitance) and so on. We call this \(S_{\alpha }^{T}\) which is read as the sensitivity of the \(T\left ( s\right ) \) with respect to changes in \(\alpha \).
Therefore \begin{align*} S_{\alpha }^{T} & =\frac{\frac{\Delta T}{T}}{\frac{\Delta \alpha }{\alpha }}\\ & =\frac{\Delta T}{\Delta \alpha }\frac{\alpha }{T}\\ & =\frac{dT}{d\alpha }\frac{\alpha }{T} \end{align*}
We then have to evaluate \(S_{\alpha }^{T}\) at the nominal value of the parameter \(\alpha =\alpha _{0}\). Therefore\[ \left . S_{\alpha }^{T}\right . _{\alpha =\alpha _{0}}=\left . \frac{dT}{d\alpha }\frac{\alpha }{T}\right . _{\alpha =\alpha _{0}}\]
\(\alpha _{0}\) is given numerical value. It is meant to be the value that the parameter \(\alpha \) fluctuate around and will be given in the problem to use.
For example, given this circuit
Let \(R\) be the parameter that will change and let amount of change be \(\Delta R\) and we want to find the sensitivity of change in the transfer function \(T\left ( s\right ) =\frac{V_{out}\left ( s\right ) }{V_{in}\left ( s\right ) }\) to changes in \(R\).
We know that \[ T\left ( s\right ) =\frac{RCs}{1+RCs}\] Say that \(R=R_{0}+\Delta R\) where \(R_{0}\) is the nominal value of resistance \(R\) and \(\Delta R\) is the amount of variation it has. Hence (1) becomes\[ T\left ( s\right ) =\frac{\left ( R_{0}+\Delta R\right ) Cs}{1+\left ( R_{0}+\Delta R\right ) Cs}\] To find \(S_{R}^{T}\) we let \(R=\alpha \) and apply the definition \(S_{\alpha }^{T}=\left . \frac{dT}{d\alpha }\frac{\alpha }{T}\right . _{\alpha =\alpha _{0}}\) Hence\begin{align*} S_{\alpha }^{T} & =\frac{dT}{d\alpha }\frac{\alpha }{T}\\ & =\frac{d}{d\alpha }\left ( \frac{\alpha Cs}{1+\alpha Cs}\right ) \frac{\alpha }{\frac{\alpha Cs}{1+\alpha Cs}} \end{align*}
Assume \(C=1\) and assume nominal value of \(R\) is \(1\) also. This means \(\alpha _{0}=1\). The above becomes\begin{align*} S_{\alpha }^{T} & =\frac{d}{d\alpha }\left ( \frac{\alpha s}{1+\alpha s}\right ) \frac{\alpha }{\frac{\alpha s}{1+\alpha s}}=\frac{s}{\left ( s\alpha +1\right ) ^{2}}\frac{1+\alpha s}{s}\\ & =\frac{1}{s\alpha +1} \end{align*}
Evaluate at \(\alpha =\alpha _{0}=1\) then\[ \left . S_{\alpha }^{T}\right . _{\alpha =\alpha _{0}}=\frac{1}{s+1}\] Next step is to replace \(s=j\omega \) and plot the magnitude in frequency domain\begin{align*} S_{\alpha }^{T} & =\frac{1}{j\omega +1}\\ \left \vert S_{\alpha }^{T}\right \vert & =\frac{1}{\sqrt{1+\omega ^{2}}} \end{align*}
A plot is
Examples below shows to calculate \(S_{\alpha }^{T}\) for difference parameters.
Given the signal graph
\begin{align*} T\left ( s\right ) & =\frac{\frac{s}{1+s}}{1-\alpha \left ( \frac{s}{s+1}\right ) }\\ & =\frac{s}{1+s\left ( 1-\alpha \right ) } \end{align*}
hence \begin{align*} S_{\alpha }^{T} & =\frac{\alpha }{T}\frac{dT}{d\alpha }=\frac{\alpha }{\frac{s}{1+\left ( s-\alpha \right ) }}\left ( \frac{-\left ( -s\right ) s}{\left ( \left ( 1-\alpha \right ) s+1\right ) ^{2}}\right ) \\ & =\frac{\alpha s}{\left ( \left ( 1-\alpha \right ) s+1\right ) } \end{align*}
Let \(s=j\omega \) and let \(\alpha =\alpha _{0}=3\) then above becomes\begin{align*} S_{\alpha }^{T} & =\frac{3j\omega }{1-2j\omega }\\ \left \vert S_{\alpha }^{T}\right \vert & =\frac{3\omega }{\sqrt{1+4\omega ^{2}}} \end{align*}
The plot is
The above curve give the percentage of change in \(T\) when \(\alpha \) changes by one percentage. We just need to determine the magnitude plot, we normally do not worry above phase when doing sensitivity analysis. The above plots says that \(T\left ( s\right ) \) is not sensitive to changes in \(\alpha \) when the frequency is near \(DC\), and as \(\omega \) increases, the sensitivity increases. For \(1\%\) change in \(\alpha \), at high \(\omega \), the magnitude of \(T\left ( s\right ) \) changes by \(1.5\%\)
Given the circuit
Let us see how \(T\left ( s\right ) =\frac{V_{out}\left ( s\right ) }{V_{in}\left ( s\right ) }\) changes when \(L\) changes. So we make \(L\) as our \(\alpha \) here. The nominal \(\alpha _{0}=1\) and we also take \(C=1\).\[ T\left ( s\right ) =\frac{Ls}{Ls+\frac{1}{Cs}}=\frac{\alpha s}{\alpha s+\frac{1}{s}}=\frac{\alpha s^{2}}{\alpha s^{2}+1}\] Hence\[ S_{\alpha }^{T}=\frac{\alpha }{T}\frac{dT}{d\alpha }=\frac{1}{\alpha s^{2}+1}\] Let \(s=j\omega \) \(\alpha =\alpha _{0}=1\) and the above becomes\[ S_{\alpha }^{T}=\frac{1}{1-\omega ^{2}}\] Hence\[ \left \vert S_{\alpha }^{T}\right \vert =\frac{1}{\left \vert 1-\omega ^{2}\right \vert }\] Notice that at \(\omega =1\) there is resonance. Here is the plot
The above says that when \(\omega \) near \(1\), then the transfer function is very sensitive to changes in \(L\). For \(1\%\) change in \(L\), the magnitude of the transfer function become very large at that frequency. This can cause problems, so we need to avoid getting close to \(\omega =1\) and must stay above it for safe operations.
Given this circuit
Now we will find the \(S_{\alpha }^{T}\) for each parameter in the circuit. These are \(R,L,C\), each time we fix all the parameters, except the one in interest, and call that one \(\alpha \) and repeat the steps we did in the earlier examples.\[ T\left ( s\right ) =\frac{V_{out}\left ( s\right ) }{V_{in}\left ( s\right ) }=\frac{R}{R+\frac{1}{Cs}+Ls}=\frac{RCs}{RCs+LCs^{2}+1}\] Let \(\alpha =R\), and let \(C=1,L=1\) and let \(\alpha _{0}=1\) as well. Hence\[ T\left ( s\right ) =\frac{\alpha s}{s^{2}+\alpha s+1}\] Hence\begin{align*} S_{\alpha }^{T} & =\frac{\alpha }{T}\frac{dT}{d\alpha }\\ & =\frac{1+s^{2}}{s^{2}+\alpha s+1} \end{align*}
We now switch to \(\omega \) domain\begin{align*} S_{\alpha }^{T} & =\left . \frac{1-\omega ^{2}}{1-\omega ^{2}+\alpha j\omega }\right . _{\alpha =1}\\ & =\frac{1-\omega ^{2}}{1-\omega ^{2}+j\omega } \end{align*}
Hence\[ \left \vert S_{\alpha }^{T}\right \vert =\frac{\left \vert 1-\omega ^{2}\right \vert }{\sqrt{\left ( 1-\omega ^{2}\right ) ^{2}+\omega ^{2}}}\] Be careful to use \(\left \vert 1-\omega ^{2}\right \vert \) above and not just \(1-\omega ^{2}\) since these are norms. Plotting the above gives
We see from above that \(T\left ( s\right ) \) is least sensitive to changes in \(R\) when \(\omega =1\) and that the maximum change is \(1\%\)
We now repeat the above, but for \(C\). Hence now \(\alpha =C\), and \(L=1,R=1\). Therefore\begin{align*} T\left ( s\right ) & =\frac{RCs}{RCs+LCs^{2}+1}\\ & =\frac{\alpha s}{\alpha s+s^{2}+1} \end{align*}
Hence\begin{align*} S_{\alpha }^{T} & =\frac{\alpha }{T}\frac{dT}{d\alpha }\\ & =\frac{1}{\alpha s^{2}+\alpha s+1} \end{align*}
We now switch to \(\omega \) domain and set \(\alpha =1\) which gives\[ S_{\alpha }^{T}=\frac{1}{1-\omega ^{2}+j\omega }\] Hence\[ \left \vert S_{\alpha }^{T}\right \vert =\frac{1}{\sqrt{\left ( 1-\omega ^{2}\right ) ^{2}+\omega ^{2}}}\] Plotting the above gives
The maximum occurs near \(\omega =0.7\). Finally, we now look at sensitivity against changes in \(L\). Hence now \(\alpha =L\), and \(C=1,R=1\). Therefore\begin{align*} T\left ( s\right ) & =\frac{RCs}{RCs+LCs^{2}+1}\\ & =\frac{s}{\alpha s+s^{2}+1} \end{align*}
Hence\begin{align*} S_{\alpha }^{T} & =\frac{\alpha }{T}\frac{dT}{d\alpha }\\ & =\frac{-\alpha s^{2}}{\alpha s^{2}+\alpha s+1} \end{align*}
We now switch to \(\omega \) domain and set \(\alpha =1\) which gives\[ S_{\alpha }^{T}=\frac{-\left ( -\omega ^{2}\right ) }{1-\omega ^{2}+j\omega }\] Hence\[ \left \vert S_{\alpha }^{T}\right \vert =\frac{\omega ^{2}}{\sqrt{\left ( 1-\omega ^{2}\right ) ^{2}+\omega ^{2}}}\] Plotting the above gives
The maximum occurs near \(\omega =1.14\).
Given this circuit
Let the nominal values be \(C=1,L_{1}=1\) and \(L_{2}=1\). We will find \(S_{\alpha }^{T}\) for each of these parameters now one at a time as in the above example.\[ T\left ( s\right ) =\frac{L_{2}s}{\left ( L_{1}+L_{2}\right ) s+\frac{1}{Cs}}=\frac{L_{2}Cs^{2}}{\left ( L_{1}+L_{2}\right ) Cs^{2}+1}\] When \(\alpha =L_{1}\) then (after putting \(C=1,L_{2}=1\)) the above becomes\[ T\left ( s\right ) =\frac{s^{2}}{\left ( \alpha +1\right ) s^{2}+1}\] Hence\begin{align*} S_{\alpha }^{T} & =\frac{\alpha }{T}\frac{dT}{d\alpha }\\ & =\frac{-\alpha s^{2}}{\left ( \alpha +1\right ) s^{2}+1} \end{align*}
We now switch to \(\omega \) domain and set \(\alpha =1\) which gives\[ S_{\alpha }^{T}=\frac{\omega ^{2}}{1-2\omega ^{2}}\] Hence\[ \left \vert S_{\alpha }^{T}\right \vert =\frac{\omega ^{2}}{\left \vert 1-2\omega ^{2}\right \vert }\] The plot is
We see that \(\left \vert S_{\alpha }^{T}\right \vert \) blows up at \(\omega =\frac{1}{\sqrt{2}}\) and at \(\omega =1,\left \vert S_{\alpha }^{T}\right \vert =1\). We now consider \(\alpha =L_{2}\) then (after putting \(C=1,L_{1}=1\)) the transfer function becomes\[ T\left ( s\right ) =\frac{\alpha s^{2}}{\left ( 1+\alpha \right ) s^{2}+1}\] Hence\begin{align*} S_{\alpha }^{T} & =\frac{\alpha }{T}\frac{dT}{d\alpha }\\ & =\frac{s^{2}+1}{\left ( \alpha +1\right ) s^{2}+1} \end{align*}
We now switch to \(\omega \) domain and set \(\alpha =1\) which gives\[ S_{\alpha }^{T}=\frac{1-\omega ^{2}}{1-2\omega ^{2}}\] Hence\[ \left \vert S_{\alpha }^{T}\right \vert =\frac{\left \vert 1-\omega ^{2}\right \vert }{\left \vert 1-2\omega ^{2}\right \vert }\] The plot is
We see that now at low frequency \(T\left ( s\right ) \) is sensitive to \(L_{2}\) while it is not sensitive to changes in \(L_{1}\). Finally, looking at \(\alpha =C\) then (after putting \(L_{2}=1,L_{1}=1\)) the transfer function becomes\[ T\left ( s\right ) =\frac{\alpha s^{2}}{2\alpha s^{2}+1}\] Hence\begin{align*} S_{\alpha }^{T} & =\frac{\alpha }{T}\frac{dT}{d\alpha }\\ & =\frac{1}{2\alpha s^{2}+1} \end{align*}
We now switch to \(\omega \) domain and set \(\alpha =1\) which gives\[ S_{\alpha }^{T}=\frac{1}{1-2\omega ^{2}}\] Hence\[ \left \vert S_{\alpha }^{T}\right \vert =\frac{1}{\left \vert 1-2\omega ^{2}\right \vert }\] The plot is
We see that \(\left \vert S_{\alpha }^{T}\right \vert \) blows up at \(\omega =\frac{1}{\sqrt{2}}\).
In conclusion, we can use these plots to determine how each component affect the transfer function. We do not want the transfer function to be sensitive to changes in components. If we know the range of operating frequencies, we can now know which components can cause most problems and may be spend more money to buy better quality component for that specific one.