Below is an excerpt from an introductory level book about poles and zeros: enter image description here

The transfer function of the RC circuit system above for the output voltage is:

H(s) = s/(s+10)

The book claims that this system's response to e^-10t will be infinite because at s=-10 |H(s)| goes infinity.

So to check this out first I found the Laplace transform of an input e^-10t as:

X(s) = 1/(s+10)

So the output Y(s) must be

Y(s) = X(s) * H(s) = s / (s+10)^2

And for the time domain I take the inverse Laplace transform of Y(s) and plot it in MATLAB as follows:

syms Y s t
warning off;

H = s ./ (s+10); %Transfer function for the RC circuit system
X=1./(s+10); %Laplace transform for the input e^-10t 
Y=X.*H; %output

out = ilaplace(Y);

xlabel('t [sec]')
ylabel('Vc [V]')
grid on;

enter image description here

Above plots are the response of this system to e^-10t for 0,2 and 2 seconds interval. According to these above plots the response starts from 1 and then damps to zero.

What is infinite here? I underlined the claim of the book in yellow above. I don't see anything absurd or infinite in the response of this system for the e^-10t input. Where am I wrong here?


1 Answer 1


You are just producing plots that show the real world and not the underlying maths that lies behind the s-plane. Basically you are thinking in terms of time equivalence of bode plots when you should be thinking in terms of poles and zeros. Hopefully this picture of a 2nd order low pass filter will help: -

enter image description here

The three pictures along the top show the bode plots of three low pass filters where damping is getting less from left to right.

The bottom left picture shows a 3D view of the bode plot aAND the pole zero plot. The bottom right is just the traditional pole zero digram (as would be seen from above on the 3D plot).

  • \$\begingroup\$ What you pointed is what s-plane is about and how it reduces to bode plots on omega axis ect. But it is not what I'm asking. The book claims the actual response for the input e^-10t will be infinite. But it is not infinite according to plots in time domain. Or is it? What is infinite? Can we name it? \$\endgroup\$
    – user16307
    Jul 9, 2017 at 13:29
  • \$\begingroup\$ You are quite correct about the physical response not being infinite. \$\endgroup\$
    – Andy aka
    Jul 9, 2017 at 15:13
  • \$\begingroup\$ It is a strange statement indeed. Your Matlab analysis is correct and the voltage will dip to -135 mV at 200 ms which makes sense considering the differentiation function. I do not understand either the reference to infinity in the time domain with this \$RC\$ network. You can have a very high output voltage from a \$LC\$ network excited at \$\omega_0\$ if the insertion loss is small (poles are almost pure imaginary) and that is what I use to physically explain the manifestation of a pole. From what book or article did you extract this text? \$\endgroup\$ Jul 10, 2017 at 7:28
  • \$\begingroup\$ Here you can see the pdf of the book: scribd.com/doc/38628560/… I'm trying to understand what is the meaning of pole here for the circuit response in my question relation with the actual output. In other words "response of this system to e^-10t" should be unique and different comparing to any other inputs. What makes the response of input e^-10t different than any other responses in real world time domain? To that I still could not find an answer.. \$\endgroup\$
    – user16307
    Jul 10, 2017 at 9:53
  • \$\begingroup\$ Some kind of qualitative meaning: "When the input to this circuit is e^-10t then we obtain the most.... response in real world(observing the output with an ideal scope ect)".. \$\endgroup\$
    – user16307
    Jul 10, 2017 at 9:55

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