Telephone Wire with Load and Characteristic Impedance

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

Please see attachment.



2. Relevant equations

I(z)=I+(exp(-γz)) + I-(exp(γz))

V(z)=V+(exp(-γz)) + V-(exp(γz))





3. The attempt at a solution



I solved easily for the characteristic impedance and gamma (propagation constant), but am having a little difficulty putting all of the resistances together to set up the initial conditions. This is my logic so far:



Load Impedance is 300, total resistance due to the characteristic impedance is 4*100=400, gain impedance is 400. So is the total resistance in the circuit 1000? Or do I use my value of Z0 (characteristic impedance), which I found to be 502 - 37.4i. If so, then is the total resistance of the circuit 502+300+300? I also don't understand why the real part of the characteristic impedance doesn't equal the resistance per meter times the length.



Anyhow, then I figured if the peak voltage is 110, then the peak current is Vpeak/R-total=110/1000=0.11, or some other value, whatever ends up being the correct answer from the previous paragraph, but so I can ask my questions, let's say it's 0.11. The current will decay through the line due to the propagation constant. This is where I'm pretty sure I'm making an error.

I(z)=I+(exp(-γz)) + I-(exp(γz))= = If so,



These are my conditions:



V(0)= 110= V+ + V-



I(0)=V(0)/Z-total=110/1000=0.11= I+ + I-, so 0.11= V+ /Z0 - V+/Z0,



So V+ - V- = 0.11*Z0



Then I could just solve the system of equations and plug into the original. But surely I've made some poor assumptions along the way. Can you help? Thank you!