let's consider this important result of control theory for linear systems, called "Frequency Response Theorem" (reference):
Briefly, it says that under the hypotesis of stability and linearity, if the input signal is sinusoidal, the output signal will be the original sine with phase and amplitude variations respectively equal to phase and amplitude of the transfer function of that system.
Now, let's analyze a first order LTI system, whose transfer function may be written in this form:
\$H(s)=\frac{1}{s+b}\$
It is the transfer function for instance of a passive RC circuit whose output signal is taken from the capacitor:
Now, suppose the input signal to be a sine wave. Its Laplace transform will be the following one (table with Laplace transforms):
\$V_{in}(s)=\frac{a}{s^2+a^2}\$
The output signal in Laplace domain will be:
\$V_{out}(s)=\frac{a}{s^2+a^2}\cdot \frac{1}{s+b}\$
Now we may compute the inverse transform to find the time behaviour of the output signal:
\$V_{in}(s)=L^{-1} [ \frac{a}{s^2+a^2}\cdot \frac{1}{s+b} ]=\$
Suppose a = 5 and b = 10. We get the following result:
So, I have due questions:
1) You may see that there is a sine wave, but also an exponential term. It seems to be in contrast with the initial theorem. Which is the solution of this problem?
2) How do we see this exponential term in the simulation of the previous RC circuit? All simulations I have done with RC circuits determine behaviours like this:
I see it is a sine wave, so it is correct, according to the initial statement. But it is in contrast with the computation of the time domain behaviour.
