ECE 451 Homework 4 Solutions
Problem 1
The voltage and current on a lossless transmission line are
() zz
Vz Ve Ve
γ
γ
+− −+
=+
()
() zz
o
Iz Y Ve Ve
γγ
+− −+
=−
Let port 1 be located at z=-l with voltage and current
1ll
VVe Ve
γ
γ
++ −−
=+
and
()
1ll
o
IYVe Ve
γγ
++ −−
=−
Let port 2 be located at z=0 with voltage and current
2
VVV
+−
=+
and
()
2o
I
yV V
+−
=−
where the current is reversed to take into account that positive current enters the DUT for
an admittance matrix. Solving for V+ and V- yields
22
2
o
VzI
V+−
=
and
22
2
o
VzI
V−+
=
Eliminating V+ and V- from the equations for port 1 and solving for the currents yields
22 22
122
ll
oo
VZI VZI
Ve e
γ
γ
+
−
−+
⎛⎞⎛⎞
=+
⎜⎟⎜⎟
⎝⎠⎝⎠
22 22
122
ll
oo
oo
VZI VZI
I
YeYe
γ
γ
+
−
−+
⎛⎞⎛⎞
=−
⎜⎟⎜⎟
⎝⎠⎝⎠
so that
12 2
cosh( ) sinh( )
o
VV l ZI l
γ
γ
=−
