Problem Set 3 Solution
ECE 357
Winter 2015
1.
A 100 km telephone line has a series resistance of 4 /km, an inductance of 3 mH/km, a leakage
conductance of 1 S/km, and a shunt capacitance of 0.015 F/km, at an angular frequency  =
5000 rad/s. At the sending end there is a generator supplying 100 volts peak, at 5000 radians
per second, in series with a resistance of 300 . The load at the receiving end consists of a 200
 resistor. Find the voltage and current as functions of z, and calculate their values at the
midpoint of the line.
Solution:
2.
Consider a lossless transmission line.
a) Determine the line’s characteristic resistance so that it will have a minimum possible standingwave ratio for a load impedance of 40 + j30 .
b) Find this minimum standing wave ratio and the corresponding voltage reflection coefficient.
c) Find the location of the voltage minimum nearest to the load.
Solution:
c) At the voltage minimum: 𝜃Γ − 2𝛽𝑧 ′ = (2𝑛 − 1)𝜋 (where n is an integer)
𝜋
4𝜋
2
𝜆
At the nearest minimum, 𝑛 = 0: 𝜃Γ − 2𝛽𝑧 ′ = −
Solving for 𝑧 ′ gives 𝑧 ′ =
3𝜆
8
.
𝑧 ′ = −𝜋.
3.
The standing wave ratio on a lossless 200  transmission line terminated in an unknown load
impedance is 2.5, and the near voltage minimum is at a distance 0.4 from the load. Determine:
a) the voltage reflection coefficient  of the load
b) the unknown load impedance ZL
c) the equivalent length and terminating resistance of a line, such that the input impedance is
equal to ZL.
Solution:
4.
A 300  lossless air transmission line is connected to a complex load composed of a resistor in
series with an inductor, as shown in the figure below. At 5 MHz, determine: a) L, b) S, c) the
location of the voltage maximum nearest to the load, and d) the location of the current maximum
nearest to the load.
Solution:
|Γ| =
𝑆−1 1
=
𝑆+1 2
1
− 50| 1
|𝑍𝐿 − 𝑍0 |
𝑗𝜔𝐶
|Γ| =
=
=
1
|𝑍𝐿 + 𝑍0 |
2
|75 +
+ 50|
𝑗𝜔𝐶
|75 +
∴
1 + (25𝜔𝐶)2
1
=
1 + (125𝜔𝐶)2 4
𝜔 = 2𝜋(10 × 106 ) 𝑟𝑎𝑑/𝑠𝑒𝑐
𝐶 = 0.24 𝑛𝐹
6.
A 50  lossless line is terminated in a load impedance as shown in figure (a) below.
a) Find L and S
b) It has been proposed that by placing an appropriate selected resistor across the line at a
distance dmax from the load, as shown in figure (b), where dmax is the distance from the load to the
first voltage maximum, then it is possible to render Zin = Z0, thereby eliminating the reflection.
Show the proposed approach is valid and find the value of the shunt resistance.
Solution:
7.
The circuit below consists of a 100  lossless transmission line terminated in a load with ZL = 50 +
j100 . If the peak value of the load voltage was measured to be 12 V, determine:
a) the time-average power dissipated in the load
b) the time-average power incident on the line
c) the time-average power reflected by the load
Solution: