EMG2016 Electromagnetic Theory
Tutorial 1
1. Calculate the line parameters R’, L’, G’, and C’ for a coaxial line with an inner
conductor diameter of 0.5 cm and an outer conductor of 1 cm, filled with an
insulating material where µ=µo, εr=2.25, and σ=10-3 S/m. The conductors are made
of copper with µc=µo and σc=5.8x107 S/m. The operating frequency is 1 GHz.
Ans: R’=0.788Ω/m; L’=139nH/m; G’=9.1mS/m; C’=181pF/m
2. A lossless transmission line with Zo=60 Ω is being operated at 60 MHz. The
velocity on the line is 3x108 m/s. If the line is short-circuited at z=0; find Zin at z=:
a)-1m; b)-2m; c)-2.5m; d)-1.25m.
Ans: a) j184.6 Ω ; b) -j43.6 Ω; c)0 ; d)j∞
3. Using a slotted line, the voltage on a lossless transmission line was found to have a
maximum magnitude of 1.5 V and a minimum magnitude of 0.8 V. Find the magnitude
of the load’s reflection coefficient.
Ans: 0.3
4. A 50- Ω lossless transmission line is terminated in a load with
impedance Z L = ( 30 – j50 )Ω . The wavelength is 8 cm. Find the followings:
(a) The reflection coefficient at the load,
(b) The standing-wave ratio on the line,
(c) The position of the voltage maximum nearest the load,
(d) The position of the current maximum nearest the load.
Ans: a) 0.57 -79.8º ; b) 3.65 ; c) 3.11cm; d) 1.11cm
5. Two half-wave dipole antennas, each with an impedance of 75 Ω, are connected in
parallel through a pair of transmission lines, and the combination is connected to a
feed transmission line, as shown in Figure below. All lines are 50 Ω and lossless.
(a) Calculate Zin1, the input impedance of the antenna-terminated line, at
the parallel juncture,
(b) Combine Zin1 and Zin2 in parallel to obtain Z’L, the effective load
impedance of the feedline,
(c) Calculate Zin of the feedline.
Ans: a) 35.2-j8.62 Ω ; b) 17.6-j4.31 Ω ; c) 107.57-j56.7 Ω
,EMG2016 Electromagnetic Theory
6. A 60- Ω resistive load is preceded by a λ/4 section of a 50- Ω lossless line, which
itself is preceded by another λ/4 section of a 100- Ω line. What is the input impedance?
Ans: 240 Ω
7. A load, ZL=(25+j75) Ω is located at z=0 on a lossless two-wire line for which
Zo=50 Ω and v=c.
a) If f=300 MHz, find the shortest distance d(z=-d) at which the input admittance
has a real part equal to 1/Zo and a negative imaginary part.
b) What value of capacitance C should be connected across the line at that point to
provide unity standing wave ratio on the remaining portion of the line?
Ans: a) 39.6cm ; b) 24pF
8. A lossless 50- Ω transmission line is terminated in a load with ZL = (50+j25) Ω.
Use the Smith chart to find the followings:
a. The reflection coefficient,
b. The standing wave ratio,
c. The input impedance at 0.35λ from the load,
d. The input admittance at 0.35λ from the load,
e. The shortest line length for which the input impedance is purely resistive,
f. The position of the first voltage maximum from the load.
Ans: a) 0.24 76º; b) 1.65; c) (30 – 1.25j) ; d) (33 + 1j) mS; e) 0.105 ; f)0.105
9. A lossless 50- Ω transmission line is terminated in a short circuit. Use the Smith
chart to find the followings:
a. The input impedance at a distance 2.3λ from the load,
b. The distance from the load at which the input admittance is Yin = -j0.04 S.
Ans: a) -155j ; b) 0.074
10. A 50- Ω lossless line is to be matched to an antenna with ZL=(100+j50) Ω using a
shorted stub. Use the Smith chart to determine the stub length and distance
between the antenna and stub.
Ans: d=0.199 , l=0.125 and d=0.375, l = 0.375
, d = 0 5 cm .
↑c = To Rs =
/I
D =
I cm -c =
5 . 8 x107
8 25 x 10 - 3
↑ Po
7
f IGHz =
-
4TX 10
.
= = =
Er =
2 . 25
-
3
- =
10
= OX106)
-I
Zin = -E jtan(BH)
2
+
up - 3 x100
B
1+
jzLTan(BL) => B =2
Zin (1) =
jEoTan(BU) Zin [ 2 5)
.
= D
=
j184. 6 d
Zin( -
2)
=
-
j 43 .
6
, In the form of S
Vmax = 1 .
5
s =
max = 1 .
85 5
Umin =
0 8
+
.
s =
= 1 .
85
1+ 1P1 = 1 875
.
-
1 . 875/P/
6 .
875/M1 =
0 . 875
IP1 = 0 .
3
Zo
zc
=
50 d
130
(a) P =
E =
4 = 0 .
1 -j0 :
e
%
j50) 57 L-79 8
=
0
-
=
E
. .
* =
8 cm
2 =
= 0 .
6 -
j
(b) s =
00
= A =
10
.
(dmax + + .
0
=
3 11cm
.
(d) Amin = dmax- * = 1 . 11 cm
·
Tutorial 1
1. Calculate the line parameters R’, L’, G’, and C’ for a coaxial line with an inner
conductor diameter of 0.5 cm and an outer conductor of 1 cm, filled with an
insulating material where µ=µo, εr=2.25, and σ=10-3 S/m. The conductors are made
of copper with µc=µo and σc=5.8x107 S/m. The operating frequency is 1 GHz.
Ans: R’=0.788Ω/m; L’=139nH/m; G’=9.1mS/m; C’=181pF/m
2. A lossless transmission line with Zo=60 Ω is being operated at 60 MHz. The
velocity on the line is 3x108 m/s. If the line is short-circuited at z=0; find Zin at z=:
a)-1m; b)-2m; c)-2.5m; d)-1.25m.
Ans: a) j184.6 Ω ; b) -j43.6 Ω; c)0 ; d)j∞
3. Using a slotted line, the voltage on a lossless transmission line was found to have a
maximum magnitude of 1.5 V and a minimum magnitude of 0.8 V. Find the magnitude
of the load’s reflection coefficient.
Ans: 0.3
4. A 50- Ω lossless transmission line is terminated in a load with
impedance Z L = ( 30 – j50 )Ω . The wavelength is 8 cm. Find the followings:
(a) The reflection coefficient at the load,
(b) The standing-wave ratio on the line,
(c) The position of the voltage maximum nearest the load,
(d) The position of the current maximum nearest the load.
Ans: a) 0.57 -79.8º ; b) 3.65 ; c) 3.11cm; d) 1.11cm
5. Two half-wave dipole antennas, each with an impedance of 75 Ω, are connected in
parallel through a pair of transmission lines, and the combination is connected to a
feed transmission line, as shown in Figure below. All lines are 50 Ω and lossless.
(a) Calculate Zin1, the input impedance of the antenna-terminated line, at
the parallel juncture,
(b) Combine Zin1 and Zin2 in parallel to obtain Z’L, the effective load
impedance of the feedline,
(c) Calculate Zin of the feedline.
Ans: a) 35.2-j8.62 Ω ; b) 17.6-j4.31 Ω ; c) 107.57-j56.7 Ω
,EMG2016 Electromagnetic Theory
6. A 60- Ω resistive load is preceded by a λ/4 section of a 50- Ω lossless line, which
itself is preceded by another λ/4 section of a 100- Ω line. What is the input impedance?
Ans: 240 Ω
7. A load, ZL=(25+j75) Ω is located at z=0 on a lossless two-wire line for which
Zo=50 Ω and v=c.
a) If f=300 MHz, find the shortest distance d(z=-d) at which the input admittance
has a real part equal to 1/Zo and a negative imaginary part.
b) What value of capacitance C should be connected across the line at that point to
provide unity standing wave ratio on the remaining portion of the line?
Ans: a) 39.6cm ; b) 24pF
8. A lossless 50- Ω transmission line is terminated in a load with ZL = (50+j25) Ω.
Use the Smith chart to find the followings:
a. The reflection coefficient,
b. The standing wave ratio,
c. The input impedance at 0.35λ from the load,
d. The input admittance at 0.35λ from the load,
e. The shortest line length for which the input impedance is purely resistive,
f. The position of the first voltage maximum from the load.
Ans: a) 0.24 76º; b) 1.65; c) (30 – 1.25j) ; d) (33 + 1j) mS; e) 0.105 ; f)0.105
9. A lossless 50- Ω transmission line is terminated in a short circuit. Use the Smith
chart to find the followings:
a. The input impedance at a distance 2.3λ from the load,
b. The distance from the load at which the input admittance is Yin = -j0.04 S.
Ans: a) -155j ; b) 0.074
10. A 50- Ω lossless line is to be matched to an antenna with ZL=(100+j50) Ω using a
shorted stub. Use the Smith chart to determine the stub length and distance
between the antenna and stub.
Ans: d=0.199 , l=0.125 and d=0.375, l = 0.375
, d = 0 5 cm .
↑c = To Rs =
/I
D =
I cm -c =
5 . 8 x107
8 25 x 10 - 3
↑ Po
7
f IGHz =
-
4TX 10
.
= = =
Er =
2 . 25
-
3
- =
10
= OX106)
-I
Zin = -E jtan(BH)
2
+
up - 3 x100
B
1+
jzLTan(BL) => B =2
Zin (1) =
jEoTan(BU) Zin [ 2 5)
.
= D
=
j184. 6 d
Zin( -
2)
=
-
j 43 .
6
, In the form of S
Vmax = 1 .
5
s =
max = 1 .
85 5
Umin =
0 8
+
.
s =
= 1 .
85
1+ 1P1 = 1 875
.
-
1 . 875/P/
6 .
875/M1 =
0 . 875
IP1 = 0 .
3
Zo
zc
=
50 d
130
(a) P =
E =
4 = 0 .
1 -j0 :
e
%
j50) 57 L-79 8
=
0
-
=
E
. .
* =
8 cm
2 =
= 0 .
6 -
j
(b) s =
00
= A =
10
.
(dmax + + .
0
=
3 11cm
.
(d) Amin = dmax- * = 1 . 11 cm
·
