2 Port analysis of series Inductor (Simple)

1. Sep 8, 2015

Xyius

EDIT: Oops! I posted this in the wrong section! Meant to post in engineering section!!

I am not sure if I am doing this problem correctly or not. It is very simple.

1. The problem statement, all variables and given/known data

A 2 port network consists of just a series inductor with $Z=j 100 \Omega$ in a $50 \Omega$ system. Determine the corresponding S-parameters.

2. Relevant equations
S Parameter matrix equations
$$b_1 = S_{11}a_1 + S_{12}a_2$$
$$b_2 = S_{21}a_1 + S_{22}a_2$$
Where
$a_1=\frac{V_1+Z_0I_1}{2\sqrt{Z_0}}$,
$a_2=\frac{V_2+Z_0I_2}{2\sqrt{Z_0}}$,
$b_1=\frac{V_1-Z_0I_1}{2\sqrt{Z_0}}$,
$b_2=\frac{V_2-Z_0I_2}{2\sqrt{Z_0}}$.

3. The attempt at a solution

First I let $a_1=0$ by terminating the line with an impedance equal to the characteristic impedance so $V_1=-Z_0I_1$. Doing this allows me to use my first two equations above and solve for both $S_{22}$ and $S_{12}$.

$$S_{22}=\frac{b_2}{a_2}=\frac{V_2-Z_0I_2}{V_2+Z_0I_2}=\frac{Z_{out}-Z_0}{Z_{out}+Z_0}$$

When the circuit is terminated on the LHS with a resistor equal to the characteristic impedance, the output impedance is simply $Z+Z_0$. Plugging everything in, this gives me.
$$S_{22}=\frac{1}{2}(1+j)$$

This part wasn't a problem, getting $S_{12}$ is giving me confusion. The equation is, (we will use $I_1=I_2$)
$$S_{12}=\frac{b_1}{a_2}=\frac{V_1-Z_0I_1}{V_2+Z_0I_2}=\frac{\frac{V_1}{I_2}-Z_0}{\frac{V_2}{I_2}+Z_0}.$$

Because $a_1=0$ we know that $V_1=-Z_0I_1=-Z_0I_2$, also $V_2/I_2=Z_0$thus,
$$S_{12}=\frac{-2Z_0}{\frac{V_2}{I_2}+Z_0}=\frac{-2Z_0}{2Z_0}=-1.$$

This is what I am confused about. Can the transmission coefficient be negative? Would that just mean that the transmitted energy simply has a $\pi$ phase shift? Am I doing this right?

The other two parameters can be obtained by similar means.

2. Sep 13, 2015

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Sep 14, 2015

Xyius

Hey thanks! This was for a homework that I have since turned in. I don't believe I did it any differently than what is presented on this post. Although it is already turned in, I would still appreciate any comments as I want to learn this stuff. When I get the solution back I will post it for anyone else with a similar problem!

4. Sep 14, 2015

rude man

Since the inductor is imbedded in a Z0 = 50 ohm system you can
1. find the y parameters for the 2-port.
2. normalize to the 50 ohm environment (source and load = 50 ohms, possible xmsn line Z0 also = 50 ohms connecting the source to the input port and the output port to the load)
3. convert to s parameters using conversion tables.

This is probably not the approach used by seasoned microwave engineers but I like it because I understand y parameters better than s parameters.