Homework # 2
Ch. 2 # 11 - 14, 17, 18
Pozar
Dr. Ray Kwok
2.11
A 100 Ω tranmission line has an effective dielectric constant
of 1.65. Find the shortest open-circuited length of this line
that appears at its input as a capacitor of 5 pF at 2.5 GHz.
Repeat for an inductance of 5 nH.
5 pF at 2.5 GHz
j
ZOC = − jZo cot β l = −
ωC
1
100 cot βl =
2π(2.5 ⋅109 )(5 ⋅10 −12 )
cot β l = 0.1273
rad
βl = 82.74o = 1.444
β=
2 π 2π
2πf
=
εr =
λ λo
c
5 nH at 2.5 GHz
ZOC = − jZo cot βl = jωL
− 100 cot βl = 2π(2.5 ⋅109 )(5 ⋅10 −9 )
cot βl = −0.785
βl = −0.905 + π = 2.2366
εr
2π(2.5 ⋅109 )
β=
1.65 = 67.25
8
(3 ⋅10 )
1.444 1.444
mm
l=
=
= 21.4
β
67.25
rad
2π 2πf
β=
=
ε r = 67.25
λ
c
2.2366 2.2366
l=
=
= 33.2
β
67.25
mm
2.12
A radio transmitter is connected to an antenna having an
impedance 80 + j40 Ω with a 50 Ω coaxial cable. If the 50 Ω
transmitter can deliver 30 W when connected to a 50 Ω load,
how much power is delivered to the antenna?
Reflection at the load
ΓL =
Z L − Zo 80 + j40 − 50
=
= 0.367∠36o
Z L + Zo 80 + j40 + 50
2
% power delivered
1 − ΓL = 1 − (0.367 ) = 86.5%
total power delivered
(30W )(86.5%) = 25.9W
2
2.13
A 75 Ω coaxial transmission line has a length of 2 cm and is
terminated with a load impedance of 37.5 + j75 Ω. If the
dielectric constant of the line is 2.56 and the frequency is 3
GHz, find the input impedance to the line, the reflection
coefficient a the load, the reflection coefficient at the input,
and the SWR on the line.
λ= v/f = c/(nf) = (3 x 108)/[(√2.56)(3 x 109)]= 6.25 cm
2π
2
l = 2π
= 2.01 = 115o
λ
6.25
ZL + j tan β l
0.5 + j1 + j tan(115o )
Zin =
=
= 0.253 − j0.275
o
1 + jZL tan β l 1 + j(0.5 + j1) tan(115 )
βl =
Zin = Zo Zin = 19.0 − j20.6Ω
Z L − Zo 37.5 + j75 − 75
ΓL =
=
= 0.62∠83o
Z L + Zo 37.5 + j75 + 75
Γin =
Zin − Zo 19 − j20.6 − 75
=
= 0.62∠ − 147 o
Zin + Zo 19 − j20.6 + 75
VSWR =
1 + ρ 1 + 0.62
=
= 4.26
1 − ρ 1 − 0.62
2.14
Calculate SWR, ρ, and return loss values to complete the
entries in the following table.
1+ ρ
VSWR =
1− ρ
VSWR − 1
ρ=
VSWR + 1
RL = −20 log ρ
ρ = 10
− RL / 20
VSWR
ρ
RL (dB)
1.00
0.00
∞
1.01
0.005
46
1.02
0.01
40
1.05
0.024
32
1.065
0.032
30
1.10
0.048
26.4
1.20
0.09
20.8
1.22
0.10
20
1.50
0.20
14.0
1.925
0.32
10
2.00
0.33
9.5
2.50
0.43
7.4
2.17
Consider the transmission line circuit shown below. Compute
the incident power, the reflected power, and the power
transmitted into the infinite 75 Ω line. Show that power
conservation is satisfied.
Zg=50Ω
Prefl
Pinc
50Ω
10V
50Ω, λ/2
Vg=10V
Ptrans
75Ω
75Ω
Pinc
Power delivered by source
Psource =
1
1
 10 
Vg I g = (10 )
 = 0.40W
2
2
 50 + 75 
2
Power dissipated at source impedance
Ploss −g
1 2
1  10 
= Ig R g = 
 (50 ) = 0.16 W
2
2  50 + 75 
2
Power delivered to the 75Ω line
Ptrans
1 2
1  10 
= I RL = 
 (75) = 0.24 W
2
2  50 + 75 
2
Power incident to the 50Ω line before reflection
1 2
1  10 
Pinc = I o+ Zo = 
 (50 ) = 0.25W
2
2  50 + 50 
2
Power reflected at the load
Power conservation
Power conservation (without multiple reflection)
 75 − 50 
Prefl = ΓL Pinc = 
 (0.25) = 0.01W
 75 + 50 
Ptrans = Psource − Ploss −g = 0.40 − 0.16 = 0.24W
2
Ptrans = Pinc − Prefl = 0.25 − 0.01 = 0.24 W
2.18
A generator is connected to a transmission line as shown
below. Find the voltage as a function of x along the
transmission line. Plot the magnitude of this voltage V(x).
Zg=100Ω
V100Ω, 1.5λ
Vg=10V
V+e-jβx
80-j40Ω
x min
V+
x→
(
)
x max
V( x ) = Vo+ 1 + ρe j( θ+ 2βx )
Zo
100
= 10
=5
Zo + Zg
100 + 100
V( x ) = 51 + 0.24e
j( θ + 2 β x )
6.5
6
5.5
|V(x)| [volts]
Z − Zo 80 − j40 − 100
ΓL = L
=
= 0.24∠256o
Z L + Zo 80 − j40 + 100
Vo+ e − jβx = Vg
λ
(n = 0)
4π
π  λ

= − 256
= −0.356λ

180
4
π


x max = −[θ m 2nπ]
V ( x ) = Vo+ e − jβx + Vo− e jβx
V ( x ) = Vo+ e − jβx 1 + ΓL e 2 jβx
λ
(n = 0)
4π
π

 λ
= − 256
− π
= −0.106λ
180
4
π


x min = −[θ m (2n + 1)π]
5
4.5
4
3.5
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
x [lam da]
-0.3
-0.2
-0.1
0