# Getting an equation for an Output using S-Domain & Fourier

#### antseezee

Joined Sep 16, 2006
45
I'm going to keep this as simple as possible.

I have figured out the correct Fourier representation of a voltage or current source. It is stuck in a minor complex circuit (series-parallel or something along those lines).

I want to find the output voltage across a resistance or certain part of the circuit. Can I change my circuit into a s-domain representation, get the output equation in terms of s-values, then plug in n*j*w for s?

N = # of harmonic
j = complex
w = omega (2*pi*freq)

Is this applicable? Otherwise, am I forced to find my complex representation of the inductors/capacitors in terms of n*j*w by hand? I'm basically trying to simplify my circuit from the s-domain, and then substituting my Fourier expression & n*j*w expressions last.

I know our variable names may be off compared to what traditional E.E's use, however, any input is appreciated.

#### DrNick

Joined Dec 13, 2006
110
So what's the problem? You convert your circuit to the s-domain, solve for the voltage you want by nodal analysis, and perform an inverse transform. I am kindof confused by your notation. Here are standard transforms of components (assuming all initial conditions are zero).

Inductor of value L <=> s*L (ohms)
Capacitor of value C <=> 1/(s*C) (ohms)
Resistor of value R <=> R (ohms)
Step function (switch) u(t) <=> 1/s

After you transform the circuit components just treat them ans impedances and solve the problem with algebra. Once you have a solution perform a partial fraction expantion to inverse transform on your answer.

#### antseezee

Joined Sep 16, 2006
45
I'm trying to get my output equation in the form of:

Vo_n(t) = ....

Where n = # of harmonic. My final equation is suppose to have n's, any j's, omega's (where n*w gives the correct harmonic frequency), and t (for the time).

My question is, can I transform a circuit into the s-domain (calling the Fourier source a variable), get my output answer in terms of s, inverse transform it to the time domain, and THEN just place the n*w right next to the t variable? Or should I solve the circuit with phasor representations by hand?

I'm assuming the whole purpose of the S-domain is to make it easier to solve circuits without having to do phasor representations.

Here's an example we did in class:

We found the Fourier amount of a source. We found the total impedance of the circuit, and we narrowed down the impedance question to find Io_n(t). Our final equation looked like:

Io_n(t) = (Fourier voltage) * ((j*n*w)/(1+3*j*n*w))

You see how the impedance next to the Fourier voltage is in a frequency domain with the n variable next to it? Could I solve my for equation in the s-domain, inverse transform it, and just place the n*w next to the complex terms with j?

#### DrNick

Joined Dec 13, 2006
110
This is what you can do.
1) Find the transfer function of your circuit in the S-domain by transforming your circuit elements, and doing an analysis of that to get H(s).
2) Find the fourier series of your source.
3) Transform each term in the fourier series to the S domain. (so you'll get a bunch of transformed sins or cosines with different amplitudes).
4) multiply each harmonic you are interested in by the transfer function (i.e. convolution in time)
5) inverse transform each one.

This way you can find the harmonics you are interested in. Another thing you may be able to do is find your transfer function in s, transform to time, and convolve it with your series in summation form, to get an answer that is a function of n and time. Since you will probably be focusing on the lower order harmonics I would recommend doing the first method on the harmonics tht really matter.
And to answer your question about just placing n*w0 in front of all terms with an e^j***: that would not work.