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: 𝜃
Γ
− 2𝛽𝑧 ′ = (2𝑛 − 1)𝜋
(where n is an integer)
At the nearest minimum, 𝑛 = 0
: 𝜃
Γ
− 2𝛽𝑧 ′ = 𝜋
2
−
4𝜋 𝜆 𝑧 ′ = −𝜋
.
Solving for 𝑧 ′ gives 𝑧 ′ =
3𝜆
8
.

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 Z
L c) the equivalent length and terminating resistance of a line, such that the input impedance is equal to Z
L
.
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
𝑆 + 1
=
1
2
|Γ| =
|𝑍
𝐿
|𝑍
𝐿
− 𝑍
0
|
+ 𝑍
0
|
=
|75 +
|75 +
1 𝑗𝜔𝐶 − 50|
1 𝑗𝜔𝐶 + 50|
=
1
2
1 + (25𝜔𝐶) 2
∴
1 + (125𝜔𝐶) 2
=
1
4 𝜔 = 2𝜋(10 × 10 6 ) 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 d max
from the load, as shown in figure (b), where d max
is the distance from the load to the first voltage maximum, then it is possible to render Z in
= Z
0
, 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 Z
L
= 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: