# Math Help - Time-Varying Analysis...

1. ## Time-Varying Analysis...

Hi all,

I'm working on time-varying circuit analysis (i'm an electrical engineer), and i'm trying to come up with a frequency domain equation for a simple time-varying circuit. Basically i've come up with something similar to equation (1).

where Vout is the output, iw is the complex frequency, Iin is the input and iwX is another input frequency. However, I know that in this special case, i can combine R and α to form β(using circuit analysis techniques). Which leads to equation (2). The way to interpret equation (2) is that the output at frequency jw is equal to the input at frequencies iw-njwX multiplied by some coefficient. I used the Fourier transform to get these equations.

So my question is, "knowing that (1) can be rewritten as (2), is there a simple way to interpret (1)?" My problem is, if i want to find Vout(jw) in terms of Iin(jw-njwX) using equation (1), i have an extra term Vout(jw-njwX) which i don't know how to interpret...

This is kinda difficult to explain cos i don't know how familiar anybody else is with circuit theory, but i know there are some pretty good mathematicians here so...

any help is most welcome and thanks in advance,
Aaron

EDIT
ok assume i only want the n=1 term of Iin and n=0 term of Vout. Is it true that

Vout(jw) + Beta0.Vout(jw)/R = Beta1.Iin(jw - jwX) ???

therefore, Vout(jw)/Iin(jw - jwX) = Beta1.R/(Beta0 + R)

thanks,
Aaron

2. How do you come up to eq. 2 from eq1. It seems strange to me.

As well as implications of eq. 1. It seems to me that Vout is equal for any frequency j(w - nwx).

I'm familiar with circuit theory so you can write what the problem really is. If there is some mathematical problem i can give you some tips.

basically i have a current source (Iin) with its own parallel source resistance (R) feeding a shunt load resistor (RL).

______________Vout
| | |
| > >
Iin >R >RL
| > >
| | |
gnd gnd gnd

The load resistor, RL, is a periodically time-varying resistor of frequency wX, and Iin is a signal of a certain frequency. I wish to calculate the output voltage, Vout at frequency w when the input is w+wX.

This is what i did...

Step 1 is to expand the equation for RL using the fourier series expansion, and then take the fourier transform to find the frequency domain representation. You end up with the summation of Beta(n) multiplied by each harmonic of wX, nwX. So the first harmonic would be 2.pi.RL(1).Delta(w-wX) where Delta is the dirac delta function and RL(1) is the n=1 coefficient of the expanion of RL.

Step 2 is to write the time-domain expression for Vout versus Iin which is basically,

Vout(t) = (Iin(t) - Vout(t)/R).RL(t)

Step 3 is to take the fourier transform of this equation to find equation (1). Note that a multiplication in the time-domain is equivalent to a convolution in the frequency domain. In my equation (1), Alpha is the coefficients of RL.

However, if from step 1, we were to group R and RL(t) into an equavalent resistance RL'(t) (which has its own fourier series expansion), we would arrive at equation (2). i.e.

Vout(t) = Iin(t).RL'(t)

So how do equation (1) and equation (2) match? I guess I could try and igure it out the hard way...

Anyway, i'm not 100% sure i've done this correctly as its my first time dabbling in time-varying circuit theory. I belive others have started with the impulse response of the circuit rather than straightaway trying to write Vout(t) versus Iin(t).

Any help is greatly appreciated,
Aaron

BTW is there an easy way to paste equations?

4. Originally Posted by aaron_do
BTW is there an easy way to paste equations?
Use these LaTeX versions, all you have to do is click each equation, copy the text and put it into your post. It's pretty simple really:

Equation (1):

$V_{out}(j\omega) + \frac{1}{R}\sum_{n=-\infty}^{\infty}\alpha_n V_{out}(j\omega - nj\omega_x) = \sum_{n=-\infty}^{\infty}\beta_n I_{in}(j\omega - nj\omega_x)$

Equation (2):

$V_{out}(j\omega) = \sum_{n=-\infty}^{\infty}\beta_n I_{in}(j\omega - nj\omega_x)$

5. The point is we are dealing with two defferent frequencies. So how should we define Fourier series in such case? In fact the function:
$x(t) = \cos(\omega_0 t) + cos(\omega_1 t)$
is periodic only if ratio w0/w1 is rational.

I did some calculations in time domain but it leads to some recurrence equations which solution depends on R_L/R in a complicated way. If you want i can post it but I don't know if it finally leads to the solution.

Generally the problem is, when RL amplitude is to large there is a moment when the negative resistance RL and positive R gives 0 - we have short circuit and current is infinite.

Mathematicly current function might not have fourier transform in such case!

6. thanks for the help,

the method i'm trying to use is called Harmonic Balance, and i finally managed to get a book which describes the analysis well.

I'm still reading up on it, but apparently you need to use a two-dimensional fourier expansion which seems a little complicated. That said, if you neglect high order terms you can still come up with something meaningful...

Anyway I think i can get an answer if i read this book long enough...Its a Nonlinear microwave circuit analysis testbook...

thanks,
Aaron