2 Port analysis of series Inductor (Simple)

[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. Homework Statement
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.

Homework 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
$a_1=\frac{V_1+Z_0I_1}{2\sqrt{Z_0}}$,
$a_2=\frac{V_2+Z_0I_2}{2\sqrt{Z_0}}$,
$b_1=\frac{V_1-Z_0I_1}{2\sqrt{Z_0}}$,
$b_2=\frac{V_2-Z_0I_2}{2\sqrt{Z_0}}$.

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.

 

[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.