Two port network model and analysis

[polarmystery]

Homework Statement

Greetings. I am having difficulty with a couple of parts to a homework problem. Mainly coming up with the transfer functions to plot V_o. See the picture for the problem statement:
[PLAIN]http://img827.imageshack.us/img827/5481/problempa.jpg

Homework Equations

$$l \leqq v_{ph}*T_{rise} / 10$$

$$Z_o = Z_c = 75\Omega$$

I've also included the model parameters for the two-port network:
[PLAIN]http://img8.imageshack.us/img8/1712/modelsz.jpg

The Attempt at a Solution

I found T_rise to be 6.66ns. I am able to calculate L and C for the line using the characteristic impedance equation from Z so L = 0.5*10^-6 H/m and C = 88.88*10^-12 F/m. I'm having difficulty with the transfer functions. I can find it for the 7 model:

$$\frac{1/LC}{s^2+s\frac{R}{L}+\frac{1}{LC}} = H(s)$$

But not for the other two. Also. In order to plot Vo correctly, we have to incorporate a delay operator with the ramp function as such:

$$\frac{1-e^{sT_{rise}/0.8}}{T_{rise}/0.8} = D(s)$$

So our final plot would be the step response (or ramp, as my colleagues and I think the professor wants) of D(s)*H(s).

Any and all help would be greatly appreciated! Other examples for this type of problem is very difficult to find on the internet.

 

[rude man]

I don't know about your various models, but it should be helpful to know the following: if an ideal transmission line is source-terminated in its characteristic impedance and the other end is open-circuited, there is no distortion of Vi at Vo. There is only a time-delay.

You can show this easily by using ABCD parameters for the resistor and for the line, then concatenating.

 

[polarmystery]

I don't know about your various models, but it should be helpful to know the following: if an ideal transmission line is source-terminated in its characteristic impedance and the other end is open-circuited, there is no distortion of Vi at Vo. There is only a time-delay.

You can show this easily by using ABCD parameters for the resistor and for the line, then concatenating.

I think that's what the problem is. We have to show that measured delay in the line with the given parameters.

 

[rude man]

I am addressing only your parts 4 and 5. The 2nd and 3rd are standard lumped-parameter exercises. Not sure about the first. I would have said there is no finite rise time that will repoduce exactly at the output with a lumped-parameter circuit.

The delay time is simply physical line length divided by phase velocity = 0.1/1.5e8 s.
.

 

[DeeNos]

Hello,

It seems like you are on the right track with your calculations for L and C. As for the transfer functions, you can use the given model parameters to find the transfer functions for the other two models. For the first model, you can use the equation for the input impedance (Zin) and the output impedance (Zout) to find the transfer function:

H(s) = \frac{Zin}{Zout} = \frac{R1 + sL1}{R2 + sL2}

Similarly, for the second model, you can use the equation for the voltage gain (Av) and the current gain (Ai) to find the transfer function:

H(s) = \frac{Av}{Ai} = \frac{R2 + sL2}{R1 + sL1}

Once you have these transfer functions, you can use them to plot Vo as you have mentioned. Just remember to use the delay operator (D(s)) as well.

I hope this helps. Good luck with your homework!