Problem 2.27 At an operating frequency of 300 MHz, a lossless 50-Ω air-spaced
transmission line 2.5 m in length is terminated with an impedance ZL = (40 + j20) Ω.
Find the input impedance.
Solution: Given a lossless transmission line, Z0 = 50 Ω, f = 300 MHz, l = 2.5 m,
and ZL = (40 + j20) Ω. Since the line is air filled, up = c and therefore, from
Eq. (2.48),
ω
2π × 300 × 106
β=
=
= 2π rad/m.
up
3 × 108
Since the line is lossless, Eq. (2.79) is valid:
¶
·
¸
µ
(40 + j20) + j50 tan (2π rad/m × 2.5 m)
ZL + jZ0 tan β l
= 50
Zin = Z0
Z0 + jZL tan β l
50 + j(40 + j20) tan (2π rad/m × 2.5 m)
= 50 [(40 + j20) + j50 × 0] 50 + j(40 + j20) × 0
= (40 + j20) Ω.
Problem 2.32 A 6-m section of 150-Ω lossless line is driven by a source with
vg (t) = 5 cos(8π × 107t − 30◦ ) (V)
and Zg = 150 Ω. If the line, which has a relative permittivity εr = 2.25, is terminated
in a load ZL = (150 − j50) Ω, determine:
(a) λ on the line.
(b) The reflection coefficient at the load.
(c) The input impedance.
(d) The input voltage Vei .
(e) The time-domain input voltage vi (t).
(f) Quantities in (a) to (d) using CD Modules 2.4 or 2.5.
Solution:
vg (t) = 5 cos(8π × 107t − 30◦ ) V,
◦
Veg = 5e− j30 V.
150 Ω I~
i
Zg
~
Vg
+
Transmission line
+
+
~
Vi Zin
Z0 = 150 Ω
~
VL
~
IL
ZL (150-j50) Ω
Generator
-
z = -l
Zg
~
Vg
+
Load
l=6m
~
Ii
z=0
⇓
+
~
Vi
Zin
Figure P2.32: Circuit for Problem 2.32.
(a)
3 × 108
c
up = √ = √
= 2 × 108 (m/s),
εr
2.25
up 2π up 2π × 2 × 108
λ=
=
=
= 5 m,
f
ω
8π × 107
ω
8π × 107
β=
=
= 0.4π (rad/m),
up
2 × 108
β l = 0.4π × 6 = 2.4π
(rad).
Since this exceeds 2π (rad), we can subtract 2π , which leaves a remainder β l = 0.4π
(rad).
◦
ZL − Z0 150 − j50 − 150
− j50
(b) Γ =
=
=
= 0.16 e− j80.54 .
ZL + Z0 150 − j50 + 150 300 − j50
(c)
·
¸
ZL + jZ0 tan β l
Zin = Z0
Z0 + jZL tan β l
·
¸
(150 − j50) + j150 tan(0.4π )
= 150
= (115.70 + j27.42) Ω.
150 + j(150 − j50) tan(0.4π )
(d)
◦
Veg Zin
5e− j30 (115.7 + j27.42)
e
Vi =
=
Zg + Zin
150 + 115.7 + j27.42
¶
µ
− j30◦ 115.7 + j27.42
= 5e
265.7 + j27.42
◦
◦
= 5e− j30 × 0.44 e j7.44 = 2.2 e− j22.56
◦
(V).
(e)
◦
vi (t) = Re[Vei e jω t ] = Re[2.2 e− j22.56 e jω t ] = 2.2 cos(8π × 107t − 22.56◦ ) V.
Problem 2.42 A generator with Veg = 300 V and Zg = 50 Ω is connected to a load
ZL = 75 Ω through a 50-Ω lossless line of length l = 0.15λ .
(a) Compute Zin , the input impedance of the line at the generator end.
(b) Compute Iei and Vei .
(c) Compute the time-average power delivered to the line, Pin = 21 Re[Vei Iei∗ ].
(d) Compute VeL , IeL , and the time-average power delivered to the load,
PL = 21 Re[VeL IeL∗ ]. How does Pin compare to PL ? Explain.
(e) Compute the time-average power delivered by the generator, Pg , and the timeaverage power dissipated in Zg . Is conservation of power satisfied?
Solution:
50 Ω
~
Vg
Transmission line
+
75 Ω
Z0 = 50 Ω
Zin
-
l = 0.15 λ
Generator
z = -l
Zg
~
Vg
+
~
Ii
Load
z=0
⇓
+
~
Vi
Zin
Figure P2.42: Circuit for Problem 2.42.
(a)
βl =
2π
× 0.15λ = 54◦ ,
λ
Zin = Z0
·
¸
¸
·
75 + j50 tan 54◦
ZL + jZ0 tan β l
= (41.25 − j16.35) Ω.
= 50
Z0 + jZL tan β l
50 + j75 tan 54◦
(b)
Veg
◦
300
=
= 3.24 e j10.16 (A),
Zg + Zin 50 + (41.25 − j16.35)
◦
◦
Vei = Iei Zin = 3.24 e j10.16 (41.25 − j16.35) = 143.6 e− j11.46 (V).
Iei =
(c)
◦
◦
1
1
Pin = Re[Vei Iei∗ ] = Re[143.6 e− j11.46 × 3.24 e− j10.16 ]
2
2
143.6 × 3.24
cos(21.62◦ ) = 216 (W).
=
2
(d)
ZL − Z0 75 − 50
=
= 0.2,
ZL + Z0 75 + 50
µ
¶
◦
1
143.6 e− j11.46
− j54◦
V0+ = Vei
=
◦
◦ = 150e
j54
−
j54
β
l
−
j
β
l
j
e
+ 0.2 e
e + Γe
◦
◦
+
−
j54
VeL = V (1 + Γ) = 150e
(1 + 0.2) = 180e− j54 (V),
Γ=
0
V0+
(V),
◦
◦
150e− j54
(1 − 0.2) = 2.4 e− j54 (A),
Z0
50
◦
◦
1
1
PL = Re[VeL IeL∗ ] = Re[180e− j54 × 2.4 e j54 ] = 216 (W).
2
2
IeL =
(1 − Γ) =
PL = Pin , which is as expected because the line is lossless; power input to the line
ends up in the load.
(e)
Power delivered by generator:
◦
1
1
Pg = Re[Veg Iei ] = Re[300 × 3.24 e j10.16 ] = 486 cos(10.16◦ ) = 478.4 (W).
2
2
Power dissipated in Zg :
1
1
1
1
PZg = Re[IeiVeZg ] = Re[Iei Iei∗ Zg ] = |Iei |2 Zg = (3.24)2 × 50 = 262.4 (W).
2
2
2
2
Note 1: Pg = PZg + Pin = 478.4 W.
Problem 2.45 The circuit shown in Fig. P2.45 consists of a 100-Ω lossless
transmission line terminated in a load with ZL = (50 + j100) Ω. If the peak value of
the load voltage was measured to be |VeL | = 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.
Rg
~
Vg
+
Z0 = 100 Ω
ZL = (50 + j100) Ω
−
Figure P2.45: Circuit for Problem 2.45.
Solution:
(a)
Γ=
◦
ZL − Z0 50 + j100 − 100 −50 + j100
=
=
= 0.62e j82.9 .
ZL + Z0 50 + j100 + 100
150 + j100
The time average power dissipated in the load is:
1
Pav = |IeL |2 RL
2
¯ ¯2
1 ¯¯ VeL ¯¯
= ¯ ¯ RL
2 ¯ ZL ¯
=
(b)
1
50
1 |VeL |2
RL = × 122 × 2
= 0.29 W.
2
2 |ZL |
2
50 + 1002
i
Pav = Pav
(1 − |Γ|2 )
Hence,
i
Pav
=
(c)
Pav
0.29
=
= 0.47 W.
2
1 − |Γ|
1 − 0.622
r
i
Pav
= −|Γ|2 Pav
= −(0.62)2 × 0.47 = −0.18 W.
Problem 2.66 A 200-Ω transmission line is to be matched to a computer terminal
with ZL = (50 − j25) Ω by inserting an appropriate reactance in parallel with the
line. If f = 800 MHz and εr = 4, determine the location nearest to the load at which
inserting:
(a) A capacitor can achieve the required matching, and the value of the capacitor.
(b) An inductor can achieve the required matching, and the value of the inductor.
Solution:
(a) After entering the specified values for ZL and Z0 into Module 2.6, we have zL
represented by the red dot in Fig. P2.66(a), and yL represented by the blue dot. By
moving the cursor a distance d = 0.093λ , the blue dot arrives at the intersection point
between the SWR circle and the S = 1 circle. At that point
y(d) = 1.026126 − j1.5402026.
To cancel the imaginary part, we need to add a reactive element whose admittance is
positive, such as a capacitor. That is:
ω C = (1.54206) ×Y0
1.54206 1.54206
=
=
= 7.71 × 10−3 ,
Z0
200
which leads to
C=
7.71 × 10−3
= 1.53 × 10−12 F.
2π × 8 × 108
Figure P2.66(a)
(b) Repeating the procedure for the second intersection point [Fig. P2.66(b)] leads
to
y(d) = 1.000001 + j1.520691,
at d2 = 0.447806λ .
To cancel the imaginary part, we add an inductor in parallel such that
1
1.520691
=
,
ωL
200
from which we obtain
L=
200
= 2.618 × 10−8 H.
1.52 × 2π × 8 × 108
Figure P2.66(b)
Problem 2.31 A voltage generator with
vg (t) = 5 cos(2π × 109t) V
and internal impedance Zg = 50 Ω is connected to a 50-Ω lossless air-spaced
transmission line. The line length is 5 cm and the line is terminated in a load with
impedance ZL = (100 − j100) Ω. Determine:
(a) Γ at the load.
(b) Zin at the input to the transmission line.
(c) The input voltage Vei and input current I˜i .
(d) The quantities in (a)–(c) using CD Modules 2.4 or 2.5.
Solution:
(a) From Eq. (2.59),
Γ=
◦
ZL − Z0 (100 − j100) − 50
=
= 0.62e− j29.7 .
ZL + Z0 (100 − j100) + 50
(b) All formulae for Zin require knowledge of β = ω /up . Since the line is an
air line, up = c, and from the expression for vg (t) we conclude ω = 2π × 109 rad/s.
Therefore
2π × 109 rad/s 20π
β=
=
rad/m.
3 × 108 m/s
3
Then, using Eq. (2.79),
¶
µ
ZL + jZ0 tan β l
Zin = Z0
Z0 + jZL tan β l
"
¡
¢#
(100 − j100) + j50 tan 203π rad/m × 5 cm
¢
¡
= 50
50 + j(100 − j100) tan 203π rad/m × 5 cm
"
¢#
¡
(100 − j100) + j50 tan π3 rad
¢ = (12.5 − j12.7) Ω.
¡
= 50
50 + j(100 − j100) tan π3 rad
◦
(c) In phasor domain, Veg = 5 Ve j0 . From Eq. (2.80),
◦
Veg Zin
5 × (12.5 − j12.7)
=
= 1.40e− j34.0
Vei =
Zg + Zin 50 + (12.5 − j12.7)
and also from Eq. (2.80),
◦
◦
Vei
1.4e− j34.0
Iei =
=
= 78.4e j11.5
Zin (12.5 − j12.7)
(mA).
(V),
Problem 2.33 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 Fig. P2.33.
λ
0.2
0.3λ
75 Ω
(Antenna)
Zin1
Zin2
Zin
0.2
λ
75 Ω
(Antenna)
Figure P2.33: Circuit for Problem 2.33.
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 ZL′ , the effective load impedance of
the feedline.
(c) Calculate Zin of the feedline.
Solution:
(a)
Zin1
·
¸
ZL1 + jZ0 tan β l1
= Z0
Z0 + jZL1 tan β l1
½
¾
75 + j50 tan[(2π /λ )(0.2λ )]
= 50
= (35.20 − j8.62) Ω.
50 + j75 tan[(2π /λ )(0.2λ )]
(b)
ZL′ =
(c)
(35.20 − j8.62)2
Zin1 Zin2
=
= (17.60 − j4.31) Ω.
Zin1 + Zin2
2(35.20 − j8.62)
l = 0.3 λ
ZL'
Zin
Figure P2.33: (b) Equivalent circuit.
½
(17.60 − j4.31) + j50 tan[(2π /λ )(0.3λ )]
Zin = 50
50 + j(17.60 − j4.31) tan[(2π /λ )(0.3λ )]
¾
= (107.57 − j56.7) Ω.
Problem 2.39 A 75-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? Compare your result with that obtained through two successive
applications of CD Module 2.5.
Solution: The input impedance of the λ /4 section of line closest to the load is found
from Eq. (2.97):
Z2
502
= 33.33 Ω.
Zin = 0 =
ZL
75
The input impedance of the line section closest to the load can be considered as the
load impedance of the next section of the line. By reapplying Eq. (2.97), the next
section of λ /4 line is taken into account:
Zin =
Z02
1002
=
= 300 Ω.
ZL 33.33
Problem 2.43 If the two-antenna configuration shown in Fig. P2.43 is connected to
a generator with Veg = 250 V and Zg = 50 Ω, how much average power is delivered to
each antenna?
λ/2
50 Ω
λ/2
A
+
Zin
250 V
−
C
Lin
e2
ZL1 = 75 Ω
(Antenna 1)
Line 1
B
D
Generator
Lin
e3
λ/2
ZL2 = 75 Ω
(Antenna 2)
Figure P2.43: Antenna configuration for Problem 2.43.
Solution: Since line 2 is λ /2 in length, the input impedance is the same as
ZL1 = 75 Ω. The same is true for line 3. At junction C–D, we now have two 75-Ω
impedances in parallel, whose combination is 75/2 = 37.5 Ω. Line 1 is λ /2 long.
Hence at A–C, input impedance of line 1 is 37.5 Ω, and
Iei =
Veg
250
=
= 2.86 (A),
Zg + Zin 50 + 37.5
1
1
(2.86)2 × 37.5
∗
Pin = Re[IeiVei∗ ] = Re[Iei Iei∗ Zein
= 153.37 (W).
]=
2
2
2
This is divided equally between the two antennas. Hence, each antenna receives
153.37
= 76.68 (W).
2
Problem 2.74 A 25-Ω antenna is connected to a 75-Ω lossless transmission
line. Reflections back toward the generator can be eliminated by placing a shunt
impedance Z at a distance l from the load (Fig. P2.74). Determine the values of Z
and l.
l=?
B
Z0 = 75 Ω
A
ZL = 25 Ω
Z=?
Figure P2.74: Circuit for Problem 2.74.
Solution:
0.250 λ
1.0
1.2
1.6
7
0.1
1.8
0.6
3
2.0
0.5
2
0.2
40
0.3
3.0
0.6
1
9
0.8
0.2
30
SWR Circle
0.2
4.0
0.28
1.0
5.0
0.2
20
8
0.
0.25
0.26
0.24
0.27
0.23
0.25
0.24
0.26
0.23
COEFFICIENT IN
0.27
REFLECTION
DEGR
LE OF
EES
ANG
0.6
10
0.1
0.4
20
0.2
50
20
10
5.0
4.0
3.0
2.0
1.4
1.2
1.0
0.9
0.8
0.7
0.6
0.5
0.4
B
50
0.3
50
A
1.8
4
0.
0.3
1.0
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
20
0.4
0.1
10
0.6
8
-20
0.
1.0
0.47
5.0
1.0
4.0
0.8
9
0.6
3.0
2.0
1.8
0.2
1.6
-60
4
1.4
-70
0.15
0.35
1.2
6
0.3
0.14
-80
0.36
0.9
0.1
1.0
7
3
0.7
0.1
0.3
0.8
8
2
0.6
0
0.1
0.3
0.1
0.4
1
-110
0.0
9
0.4
2
0.0
CAP
-1
8
A
2
0
C
ITI
VE
0.4
RE
3
AC
0.0
TA
7
NC
-1
EC
30
O
M
PO
N
EN
T
(-j
0.5
31
0.
-5
-90
0.12
0.13
0.38
0.37
0.11
-100
0.39
0.4
06
0.4
9
0.
1
0.
0.3
0
-4
44
0.2
4
0.
0.2
0.2
1
-30
0.3
0.28
0.22
0.2
0.
0.22
0.0 —> WAVELEN
0.49
GTHS
TOW
ARD
0.48
— 0.0
0.49
GEN
D LOAD <
ERA
OWAR
0.48
± 180
HS T
T
TO
G
170
0
R—
-17
EN
0.47
VEL
>
A
W
0.0
6
160
4
<—
0.4
-160
0.4
4
.0
6
0
IND
o)
Y
U
0.0
/
C
5
15
TIV
0
(-jB
5
0
0.4
ER
-15
CE
N
0.4
EA
A
5
CT
5
PT
0.0
AN
CE
0.1
CE
US
S
CO
VE
14
0
M
TI
4
0
C
PO
-1
DU
N
EN
IN
R
T
O
(+
),
jX
o
Z
/Z
0.2
X/
8
0.3
50
0.4
1.6
06
0.
44
0.1
0.3
60
0.2
31
0.
4
0.
0.500 λ
6
0.3
19
R
,O
o)
0.1
70
0.
3
0.4
0
13
0.35
80
)
/Yo
0
0
12
(+jB
CE
AN
PT
CE
S
SU
VE
TI
CI
PA
CA
.42
7
0.15
0.36
90
1.4
0.0
0.0
110
0.14
0.37
0.38
0.7
8
0.8
0.4
9
0.0
1
0.4
0.39
100
0.9
0.1
0.13
0.12
0.11
0.750 λ
The normalized load impedance is:
zL =
25
= 0.33
75
(point A on Smith chart)
The Smith chart shows A and the SWR circle. The goal is to have an equivalent
impedance of 75 Ω to the left of B. That equivalent impedance is the parallel
combination of Zin at B (to the right of the shunt impedance Z) and the shunt
element Z. Since we need for this to be purely real, it’s best to choose l such that
Zin is purely real, thereby choosing Z to be simply a resistor. Adding two resistors in
parallel generates a sum smaller in magnitude than either one of them. So we need
for Zin to be larger than Z0 , not smaller. On the Smith chart, that point is B, at a
distance l = λ /4 from the load. At that point:
zin = 3,
which corresponds to
yin = 0.33.
Hence, we need y, the normalized admittance corresponding to the shunt
impedance Z, to have a value that satisfies:
yin + y = 1
y = 1 − yin = 1 − 0.33 = 0.66
1
1
z= =
= 1.5
y 0.66
Z = 75 × 1.5 = 112.5 Ω.
In summary,
λ
,
4
Z = 112.5 Ω.
l=