microwaves
Solutions of series 2
Problem 1
A transmission line has the following per unit length parameters :
L=0.2 µH/m
C=300pF/m
R=5Ω/m
G=0.01S/m
Compute the propagation constant and characteristic impedance of this line at 0.5 GHz.
Recompute these quantities in a lossless case.
Solution :
γ = α + jβ =
( R + jω L )( G + jωC ) = ( 5 +
j 628 )( 0.01 + j 0.94 ) =
24.3∠89.465° = 0.23 + j 24.3
Thus
α = 0.23 Np / m
β = 24.3 rad / m
Zo =
R + jω L
= 25.8 + 0.3 Ω
G + jω L
Lossless case : R=G=0
Z o = 25.8 Ω
α =0
β = 24.3
Problem 2 :
A lossless transmissionline is terminated with a 100 Ω load. If the SWR of the line is 1.5, find
the two possible values for the characteristic impedance of the line
Solution :
swr − 1 0.5
=
= 0.2
swr + 1 2.5
100 − Z o
Z − Zo
Γ = L
=
100 + Z o
Z L + Zo
moreover, we know that Zo is real, thus either
100 − Z o
= 0.2 ⇒ Z o = 66.7Ω
100 + Z o
or
100 − Z o
= −0.2 ⇒ Z o = 150Ω
100 + Z o
Γ =
Problem 3 :
A radio transmitter is connected to an antenna having an impedance 80+j40 Ω. with a 50 Ω
coaxial cable. If the transmitter can deliver 30W to a 50 Ω load, how much power is
delivered to the antenna ?
Solution :
Z − Z o 30 + j 40
Γ= L
=
= 0.367 ∠36°
Z L + Z o 130 + j 40
(
PLoad = Pin − Preflected = Pin 1 − Γ
2
) = 30 (1 − ( 0.367) ) = 25.9
2
Problem 4 :
Use the Smith chart to fond the following quantities for the transmission line circuit bellow :
a) the SWR of the line
b) the reflection coefficient at the load
c) the load admittance
d) the input impedance of the line
l=0.4λ
Zo=50
Zin
ZL=90+j60
Solution : see Smith Chart
a) SWR = 2.8
G= 0.45 ∠ 33°
c) yL=0.38-j0.25 => YL=7.6-j5 mS
d) zin=0.54+j0.63 => Zin=27+j32 Ω
b)