A Circuit ODE with multiple short voltage impulses

[ishan_ae]

I am trying to understand some introductory material on applications of Green's functions and the book which I am following uses the example of an electric circuit subjected to multiple voltage impulses.

This is the Google drive viewing link to the few pages I am referring to: . I had to post this link because description of the problem would have been too long.

I am able to understand the derivation when there is a single voltage impulse for a duration $\Delta \tau$. This is represented by Eq.2.7.9 and I have derived it. However, I am not able to derive Eq. 2.7.10 which represents the case of multiple voltage impulses.

This is what I have tried to do myself for the interval of $\tau_1 < t < \tau_2$. At $t=\tau_1$, $i(\tau_1) = \frac{V_0}{L} e^{ -\frac{R}{L}(t-\tau_0)}$ and, at $t=\tau_2=\tau_1+\Delta \tau$, $i(\tau_1+\Delta \tau) = I_0 e^{-\frac{R}{L}}$. These values are substituted in the equation $L[i(\tau_1+\Delta \tau) - i(\tau_1)] = V_1$ from Eq.2.7.6, to solve for $I_0$. With the value of $I_0$ obtained, it can be substituted in Eq 2.7.3 to get $i(t)$. However, this does not lead to what is given in Eq. 2.7.10. Can someone please point out where I could be going wrong ?

 

[BvU]

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The response of the system is simply a summation of the responses to the pulses from the past: $V_0$ at $\tau_0$, so $t-\tau_0$ time ago, $V_1$ at $\tau_1$, so $t-\tau_1$ time ago, etc. etc.

Figure 2.7.3 is not very helpful in bringing this across. I expected a figure somewhat like this:

Red line is response to first pulse, green to second, purple to third (took $V_0 = V_1 = V_2$). Dashed line is the actual system response: the sum. That's all there is to it in eq 2.7.10 !

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