3.1 Transmission lines outline
• Fundamentals: Lumped circuits, size vs
wavelength
• TL types
• TL equations and solution
• Reflection coefficient
• Input impedance
• VSWR
• Smith Chart and Matching
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Suggestive reading
Clayton R. Paul, “ Electromagnetics for
Engineers with applications”
D. M. Pozar, “Microwave Engineering” for
more in depth discussion
R. Ludwig, G Bogdanov, “RF Circuit
Design”
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Transmission lines
• Types of transmission lines
The two-conductor transmission line. (a) Electric field about a
transmission line caused by the voltage between the two conductors.
(b) Magnetic field about a transmission line caused by the current on
the conductors.
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Transmission lines
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Transmission lines: Introduction
Transmission Line (TL) theory
• Lumped circuit theory  assumes that all
elements in the circuit act as points in space
Elements don’t have any physical size, i.e. they are
ideal elements
R
L
Zin

C
1
Z in  R  j ( L 
)
C
R
L
Zin
C
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Transmission lines
• Microwaves are EM waves with frequency
range from 300MHz to 300GHz
Wavelength is 1m to 1mm
• At low frequencies, e.g audio cct at
3kHz, λ is 100km! Typical resistor
dimension is 1cm.
• At high freq., short λ e.g. in optical
fibres is about 1μm.
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Transmission lines
• Low freqs
Component
size << λ
Lumped
Microwaves
Optical
Component
Component
size λ
size >> λ
 distributed
 light ray
circ. theory
All of the above can be analysed by solving
Maxwell’s equations BUT analysing arbitrary
shaped ccts with ME is out of the question
We use distributed cct (or transmission line) theory
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3.2: Transmission line theory
• Analysis
Representing a transmission line as a distributed parameter circuit
consisting of cells of per-unit length inductance, l, and per-unitlength capacitance, c.
8
Transmission lines
• Transmission line theory
The per-unit-length equivalent circuit of a transmission line.
9
Transmission lines
• Writing Kirchhoff’s voltage low around the
outside loop gives:
I ( z , t )
V ( z  z , t )  V ( z , t )  l z
t
or for Δz 0,
V ( z , t )
I ( z , t )
 l
z
t
(1)
First transmission-line equation
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Transmission lines
• Writing Kirchhoff’s current low at the upper node
of the capacitor gives:
V ( z  z , t )
I ( z  z , t )  I ( z , t )  cz
t
or for Δz 0,
I ( z , t )
V ( z , t )
 c
z
t
(2)
Second transmission-line equation
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Transmission lines
• These lead to the Transmission line wave equations
(lossless):
Remember?
(3)
(4)
 2V ( z, t )
 2V ( z, t )
 lc
2
z
t 2
 2 Ex , y ( z , t )
 2 I ( z, t )
 2 I ( z, t )
 lc
2
z
t 2
 2 H x , y ( z, t )
z
2
z
2
  0 0
 2 Ex , y ( z , t )
 0 0
t 2
 2 H x , y ( z, t )
t 2
• Similar to the wave equation for PW. They explain
the wave propagation in the TL (time domain).
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Transmission lines
Frequency domain solution of TL equations
• To work in frequency domain, we restrict ourselves to
sinusoidal signals and work with phasors.
Remember from electric field free-space wave equation:
2 Ey
z 2
Solution:
E y ( z, t )  E0e
or
 2 o  o E y
 z
j t  
 c
 E0e 
j t z 
 E0e  jz e jt
Ey ( z)  E0e jz
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Transmission lines
Frequency domain solution of TL equations
• In frequency domain, for sinusoidal signals, we have:
 2V ( z )
2


lcV ( z )
2
z
(5)
2 I ( z)
2


lcI ( z )
2
z
(6)
Here we ignore the time dependence notation, and work with
phasors
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Transmission lines
Frequency domain solution of TL equations
• The general solution, for lossless Tx lines, is:
(7)
(8)
V ( z )  Vi e  jz  Vr e jz
I ( z )  I i e  jz  I r e jz
Wave incident on load, i.e.
travelling in +z direction
Phase constant:

   lc = rad/m

Wave reflected from load, i.e.
travelling in -z direction
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Transmission lines
Voltage and current waves on terminated transmission line
V(z) = Vi(z) + Vr(z), I(z) = Ii(z) + Ir(z)
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3.3: Transmission lines
Characteristic impedance
• What is the relationship between current and voltage waves?
• Recall the first Tx line equation in phasor form:
V ( z )
  jlI ( z )
z
• Substituting the incident part of the solution V(z) and I(z) into the
above, we get:
 j Vi e j z   jlI i e j z
Vi e j z l
l


 j z
Ii e

c
• The above has dimensions of impedance, and depends only on line
parameters, i.e. it is a characteristic of the line
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Transmission lines
Characteristic impedance
• We define the characteristic impedance of a lossless Tx
line, Z0 or ZC:
Vi ( z )
L
Z0 

Ii ( z)
C
• The reverse current and voltage waves are related by:
Vr ( z ) Vr e j z

 Z0
j z
Ir ( z) Ir e
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Transmission lines
Impedance at z
• We can also define a position-dependent impedance Z(z)
as:
total voltage at point z V ( z ) Vi ( z )  Vr ( z )
Z ( z) 


total current at point z I ( z ) I i ( z )  I r ( z )
• After substituting from (9), (10) we get:
Vi ( z )  Vr ( z )
Z ( z)  Z0
Vi ( z )  Vr ( z )
(11)
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Transmission lines
Impedance at z
• Clearly Z(z) is not the same as Z0. For a given Tx line, Z0 is
constant and independent of z.
• However, for the special case where there is no reflected
wave, Z(z) does equal Z0.
• This happens when the Tx line is terminated at a load
impedance ZL=Z0 (perfect matching)
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Transmission lines
Terminated Tx lines - Reflection coefficient
• Consider an infinitely long Tx line supplied by a source
• Clearly, there is no reflected wave
• However, since the incident wave “sees” an impedance of Z0 at every
point in the line, we can replace the above with ZL = Z0
• Conclusion: For a perfectly matched Tx line (ZL = Z0) there is no
reflected wave
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Transmission lines
Terminated Tx lines - Reflection coefficient
• We should expect a reflected wave for a mismatched line
• We define the voltage reflection coefficient as:
Vr ( z )
( z ) 
Vi ( z )
(12)
• The value of Γ(z) at the load (z=l) is usually of interest.
Denote Γ(l)=ΓL.
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Transmission lines
Terminated Tx lines - Reflection coefficient
• At the load we have:
V (l )
Z L  Z (l ) 
I (l )
(13)
• From (11), (12) and (13), we have:
Vi (l )  Vr (l )
Z L  Z0
Vi (l )  Vr (l )
1 L
 Z0
1 L
(14)
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Transmission lines
Terminated Tx lines - Reflection coefficient
• Re-arrange to give the reflection coefficient at the load:
Z L  Z0
L 
Z L  Z0
(15)
• We can identify three special cases:
– Perfect match: ZL=Z0  ΓL= 0
– Short circuit:
ZL=0  ΓL= 1
– Open circuit:
ZL=  ΓL= 1
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3.4: Transmission lines
Input impedance
• We can now calculate the input impedance at z = 0, from
equations (11) and (12) .
Vi (0)  Vr (0)
1  (0)
Z in  Z (0)  Z 0
 Z0
Vi (0)  Vr (0)
1  (0)
(16)
• An expression in terms of ZL is preferable
ZL
Zin z=0
d=l
z=l
d=0
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Transmission lines
Input impedance
• From previous equations it can be shown that:
 Z L  jZ 0 tan  l 
Zin  Z 0 

Z

jZ
tan

l
L
 0

(17)
• When ZL= Z0 , then Zin= Z0 as expected
ZL
Zin z=0
d=l
z=l
d=0
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Transmission lines
Input impedance
• Note that the impedance at z looking towards the load is:
 Z L  jZ 0 tan  d 
Z ( z)  Z0 

Z

jZ
tan

d
L
 0

(18)
• Equation (18) is the impedance transforming equation, and
it indicates how the load impedance can be transformed
using lengths of Tx line (important in matching networks).
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Transmission lines
Input impedance
• Remember that
tan( x  n )  tan x , n  1, 2,...
• Hence, equation (18) shows that Z(z) is periodic in z with a
period of λ/2
2
 l  n 
l  n


ln
2
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Transmission lines
Input impedance
• (a) At distances from the
load that are multiples of
a half wavelength,
Zin=ZL.
• (b) The input impedance
replicates for distances
that are multiples of a
half wavelength.
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Transmission lines
Input impedance
• We can identify some special cases:
– Short-circuited line (ZL=0): Zin  jZ 0 tan  l
– Open-circuited line (ZL=∞): Zin   jZ 0 cot  l
By changing the length of the line we can synthesise the
type of reactance we want
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Transmission lines
Input impedance
• We can construct
capacitors and
inductors using shortcircuited lines.
• The input impedance
to a short-circuited
line whose length is
given in wavelengths
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3.5: Transmission lines
Quarter-wave transformer
– Quarter-wave (λ/4) line:
Gives βl=π/2, tan βl∞
2
Z0
Z in 
ZL
i.e. if the load is an o/c, Zin is s/c and vice-versa.
Also, if the load is inductive, Zin is capacitive and viceversa
Quarter-wavelength transformer is useful for match
resistive loads to a Tx line.
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Transmission lines
Quarter-wave transformer
• Effect of a quarter
wavelength line.
• (a) The input to a quarter
wavelength line that has a
short-circuit load appears
to be an open circuit.
• (b) The input to a quarter
wavelength line that has
an open-circuit load
appears to be a short
circuit.
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Transmission lines
Quarter-wave transformer
• Matching antenna feed lines
with the quarter-wave
transformer. (a) Illustration
of the effect of the feed line.
(b) The quarter-wave
transformer
• We want:
' 2
C
Z
ZC 
ZL
Z C'  Z C Z L
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Transmission lines
Voltage standing waves
• We know that:
V ( z )  Vi (l )e jd  Vr (l )e  jd
ZL
Zin z=0
d=l
z=l
d=0
• Standing waves are formed by adding up incident and
reflected waves
V ( z ) max  Vi (l )  Vr (l )
V ( z ) min  Vi (l )  Vr (l )
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Transmission lines
Voltage standing waves
• We define the voltage standing wave ratio (VSWR) as:
•
VSWR

•
•
•
•
•
V ( z ) max
V ( z ) min

Vi (l )  Vr (l )
Vi (l )  Vr (l )
or
1  L
VSWR 
1 L
VSWR is a quantitative measure of the degree of mismatch
If ZL=0 or ∞, then VSWR= ?
If ZL=Z0 , then VSWR=
?
In general 1  VSWR  
Industry considers a line to be matched if VSWR<2
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