# 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?

• 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