Circuit ODE with multiple short voltage impulses

  • #1
ishan_ae
1
0
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 ?
 
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  • #2
:welcome:
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:

1715796011657.png

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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