Course Notes for EE1266 Application of Fields and Waves
Chapter 1: Lecture 3
1
NOTES for Transmission Lines III
This lecture covers Chapter 6.3
1. Short- and open-circuits T-line, applications
2. Quarter-wave transformer
3. Power flow on a lossless T-line
1). Short-circuit T-line
• ZL=0, Γ=-1.
• Voltage and current
( )
[
]
(
)
( )
[
]
(
)
z
Z
V
e
e
Z
V
z
I
z
jV
e
e
V
z
V
z
j
z
j
SC
z
j
z
j
SC
β
β
β
β
β
β
cos
2
sin
2
0
0
0
0
0
0
+
−
−
+
+
−
−
+
=
+
=
−
=
−
=
• Input impedance:
(
)
(
)
(
)
l
jZ
l
I
l
V
Z
SC
SC
SC
in
β
tan
0
=
−
−
=
• A few more observations, depending on the length of
short-circuited T-lines, the SC T-line can be positive
(inductive) or negative (capacitive).
Course Notes for EE1266 Application of Fields and Waves
Chapter 1: Lecture 3
2
• If tanβl≥0, the SC T-line is inductive, the equivalent
inductance is:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
=
−
0
1
0
tan
1
tan
Z
L
l
l
Z
L
eq
eq
ω
β
ω
β
• If tanβl≤0, the SC T-line is capacitive, the equivalent
inductance is:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
−
=
−
0
1
0
1
tan
1
tan
1
Z
C
l
l
Z
C
eq
eq
ω
β
π
β
ω
Example: Choose a shorted 50-Ω lossless T-line such that its
input impedance at 2.25 GHz is equivalent to the a capacitor
with capacitance Ceq=4 pF, assuming the wave velocity on the
line is 0.75c.
2). Open-circuit T-line
• ZL=∞, Γ=11.
• Voltage and current
( )
[
]
(
)
( )
[
]
(
)
z
Z
jV
e
e
Z
V
z
I
z
V
e
e
V
z
V
z
j
z
j
OC
z
j
z
j
OC
β
β
β
β
β
β
sin
2
cos
2
0
0
0
0
0
0
+
−
−
+
+
−
−
+
−
=
−
=
=
+
=
• Input impedance:
(
)
(
)
(
)
l
jZ
l
I
l
V
Z
SC
SC
OC
in
β
cot
0
−
=
−
−
=
Course Notes for EE1266 Application of Fields and Waves
Chapter 1: Lecture 3
3
Example: Find Z0 and β of a 57-cm lossless T-line whose SC
input impedance is j40.42 Ω and its OC input impedance is –
j121.24 Ω. We also know that the line is between 3 and 3.25
λ.
3). λ/2 line, λ/4 line