Before asking my questions I would like to write down briefly about my big picture understanding so far. I will go with an example RLC circuit below:
Above circuit’s transfer function H(s) can be written as H(s)=Vout(s)/Vin(s)
Since all is in series the same current I(s) passes through the components, this circuit can be treated as a voltage divider. So by using relevant Laplace transforms, the voltage across R which is Vout can be written in terms of Vin as:
Vout(s) = Vin(s)*R / (sL+R+1/sC)
So the transfer function H(s) can be simplified as:
H(s) = sR / (s^2*L+sR+1/C)
Now that we have the transfer function, we can find the response of this circuit for a given input in s domain and then transform it by taking inverse Laplace transform to plot it in time domain. One can also find the location of poles and zeros on the s plane.
Before that, first I will start with a step response of this circuit in LTspice to estimate the impulse response in time domain which is the transfer function. Impulse response of an LTI system in time domain is the transfer function. Since I cannot create a real Dirac function I apply a very sharp rising edge step input and see the response in LTspice. I also do the same thing by using MATLAB in s domain.
Since step input X(s)=(1/s) in s domain, the output can be written as:
Y(s) = X(s)*H(s)
Y(s) = R / (s^2*L+sR+1/C)
So taking the inverse Laplace transform of Y(s), L-1[Y(s)] yields a time domain response Vout(t).
And below plots shows the same result is obtained by using LTspice and MATLAB:
Ok so if we look at the above plots we can see that the circuit response to a sharp step input is a decaying sinusoidal with f=50Hz(ω = 2*pi*f= 314), in other words damped oscillation at a particular frequency.
Now the reason for the question I’m going to ask is about some observations or the relation between the transfer function and the time domain response.
First of all here is the 3D plot of |H(s)| of this circuit above the s plane:
At this point, from the transfer function H(s) = sR / (s^2*L+sR+1/C) one can already determine the two poles and the zero locations on the s plane.
First I look at the |H(s)| vs imaginary axis as below:
My first observation of the transfer function of the plot above is that the location of the poles are where ω = 314 on the imaginary axis. This value of the ω is exactly the same when we observed the step response or natural frequency of the circuit where it damped with an oscillation at ω = 314. So I think what pole’s imaginary part indicates is the impulse response’s frequency. Is that correct?
Second is that, if we look at the real axis which is σ axis of s=ωj+ σ plane as follows:
And as seen above, σ for the point of the pole is around -100. I think this negative sign indicates the decay rate of the sinusoid as e^-100. Is this correct?
And here is the top view of the s plane:
Now regarding these observations so far,(if you agree with them) my questions are as follows:
From the observations of the s plane, the locations of the poles turned out to be complex numbers which are carrying the knowledge of the time domain input response i.e the natural response. So as you see if it is true one can relate this particular pole points to something in time domain. But what about the other points on the s plane rather than a pole or a zero. Pick a random s point on the s plane and find out what |H(s)| for that point. What does that indicate in time domain?
Another observation is that this RLC circuit’s response is decaying. Does that mean this system is stable? If so, is there a passive network example where the system would be unstable? Or do we need an active component to observe such phenomena?
At the poles |H(s)| mathematically goes to infinity, not in MATLAB not in LTspice. Is that because applying a Dirac delta function is just a theoretical concept?