ELCT564
Spring 2012
Chapter 2: Transmission Line
Theory
4/8/2015
ELCT564
1
The Lumped-Element Circuit Model of T-Line
Transmission line theory bridges the gap between field analysis and basic circuit theory
Voltage and current definitions of an incremental length of transmission line
R: Series resistance per unit length (Ω/m)
L: Series inductance per unit length (H/m)
G: Shunt conductance per unit length (S/m)
C: Shunt capacitance per unit length (F/m)
Lumped-element equivalent circuit of an incremental length of transmission line
4/8/2015
ELCT564
2
The Lumped-Element Circuit Model of T-Line
Kirchhoff’s voltage law
Kirchhoff’s current law
Telegrapher equations
4/8/2015
ELCT564
3
Wave Propagation on a Transmission Line
4/8/2015
ELCT564
4
Wave Propagation on a Lossless Line
4/8/2015
ELCT564
5
Field Analysis of Transmission Lines
Field lines on an arbitrary TEM transmission line
Time-average stored magnetic energy
Time-average stored electric energy
Power loss per unit length in
lossy dielectric
Power loss per unit length due to conductor
4/8/2015
ELCT564
6
Terminated Lossless Transmission Line
A transmission line terminated in a load impedance ZL
A superposition of an
incident and a reflected
wave: standing waves
Return loss
Standing Wave Ratio
Input impedance
4/8/2015
ELCT564
7
Short Terminated Lossless Transmission Line
Г=-1
Impedance
Voltage
4/8/2015
Current
ELCT564
8
Open Terminated Lossless Transmission Line
Г=1
Impedance
Voltage
4/8/2015
Current
ELCT564
9
Two Transmission Lines
Insertion Loss
Decibels and Nepers
Ratio of power
levels
dBm
4/8/2015
ELCT564
10
The Smith Chart
4/8/2015
ELCT564
11
The Smith Chart: Resistance Circle
If Zo is 50 Ohm, indicate the
position of 10, 25, 50 and
250 Ohm in the plot
If Zo is 100 Ohm, indicate
the position of 10, 25, 50
and 250 Ohm in the plot
4/8/2015
ELCT564
12
The Smith Chart: Reactance Curves
If Zo is 50 Ohm, indicate the
position of j50, j10, -j25 in
the plot
4/8/2015
ELCT564
13
The Smith Chart
If Zo is 50 Ohm, indicate the
position of 25+j50, 50+j100,
10-j25 in the plot
4/8/2015
ELCT564
14
The Smith Chart: SWR Circles
4/8/2015
ELCT564
15
The Smith Chart: Example 1
Suppose we have a transmission line with a characteristic impedance of 50Ω
and an electrical length of 0.3λ. The line is terminated with an impedance
having a resistive component of 25Ω and an inductive reactance of 25Ω.
What is the input impedance to the line?
Basic Steps using Smith Chart:
• Normalize and plot a line input/load impedance and construct a constant
SWR circle
• Apply the line length to the wavelengths scales
• Read normalized load/input impedance, and convert to impedance in ohms
4/8/2015
ELCT564
16
The Smith Chart: Example 2
Suppose we have a measured input impedance to a 50Ω of 70-j25 Ω. The line
is 2.35λ long, and is terminated in an antenna. What is the antenna feed
impedance?
4/8/2015
ELCT564
17
The Slotted Line
The following two step procedure has been carried out with a 50 Ω coaxial
slotted line to determine an unknown load impedance:
•
•
A short circuit is placed at the load plane, resulting in a standing wave on the line with infinite
SWR, and sharply defined voltage minima recorded at z=0.2 cm, 2.2cm, 4.2cm
The short circuit is removed, and replaced with the unknown load. The SWR is measured as 1.5,
and voltage minima are recorded at z=0.72cm, 2.72cm, 4.72cm.
Find the load impedance.
4/8/2015
ELCT564
18
The Quarter-Wave Transformer
Consider a load resistance RL=100Ω to be matched to a 50Ω line with a quarter-wave
transformer. Find the characteristic impedance of the matching line section and plot the
magnitude of the reflection coefficient versus normalized frequency, f/fo, where fo is the
frequency at which the line is λ/4 long.
4/8/2015
ELCT564
19
Transform of a complex
load impedance into a
real impedance?
4/8/2015
ELCT564
20
The Multiple-Reflection Viewpoint
Zo
4/8/2015
Z1
ELCT564
21
The Quarter-Wave Transformer:
Bandwidth Performance
l=λ/4 at frequency f0
Bandwidth
4/8/2015
ELCT564
22
The Quarter-Wave Transformer:
Bandwidth Performance
Design a single-section quarter-wave matching
transformer to match a 10Ω load to a 50Ω ;ome. At
f0=3GJz/ Determine the percent bandwidth for which
the SWR≤1.5.
Zo
4/8/2015
Z1
Z2
ELCT564
23
Generator and Load Mismatches
4/8/2015
ELCT564
24
Generator and Load Mismatches
Load matched to line
Generator matched to loaded line
Conjugate matching
4/8/2015
ELCT564
25
Lossy Transmission Line
The low-loss line
4/8/2015
ELCT564
26
The Distorionless Line
When the phase term is not a linear function of frequency, the various frequency components
of a wideband signal will travel with different phase velocities and arrive the receiver end of the
transmission line at slight different times. This will lead to dispersion.
Distortionless line
4/8/2015
ELCT564
27
The Terminated Lossy Line
4/8/2015
ELCT564
28
Additional Examples
Use the Smith Chart to find the shortest lengths of a short-circuited 75Ω line to give the
following input impedance:
1. Zin = 0
2. Zin = infinity
3. Zin = j75 Ω
4. Zin = -j50 Ω
1.
2.
3.
4.
0 or 0.5 λ
0.25 λ
0.125 λ
0.406 λ
4/8/2015
ELCT564
29