# A question on the response of an RC circuit and pole of its transfer function

Below is an excerpt from an introductory level book about poles and zeros: 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);

ezplot(out,[0,0.2])
xlabel('t [sec]')
ylabel('Vc [V]')
grid on; 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?

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: - 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).

• 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? – user16307 Jul 9 '17 at 13:29
• You are quite correct about the physical response not being infinite. – Andy aka Jul 9 '17 at 15:13
• 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? – Verbal Kint Jul 10 '17 at 7:28
• 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.. – user16307 Jul 10 '17 at 9:53
• 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)".. – user16307 Jul 10 '17 at 9:55