Transmission-Line for Communication
Prof. Tzong-Lin Wu
EMC Laboratory
Department of Electrical Engineering
National Taiwan University
Prof. Tzong-Lin Wu / NTUEE
1
Introduction
Chapter 6 : transient state in time-domain, (digital circuits)
Chapter 7: steady state in frequency domain, (communication circuits)
In the steady state, the situation is equivalent
to the superposition of one (+) wave, which is the sum of all the transient (+)
waves, and one (-) wave, which is the sum of all the transient (-) waves.
Thus, the general solutions for the line voltage and line current in the sinusoidal
steady state are superposition of voltages and currents, respectively,
of sinusoidal (+) and (-) waves.
Prof. Tzong-Lin Wu / NTUEE
2
Introduction
7.1 Short-Circuited Line
7.2 Line Terminated by Arbitrary Load
7.3 Transmission-Line Matching
7.4 The Smith Chart: 1. Basic Procedures
7.5 The Smith Chart: 2. Applications
7.6 The Lossy Line
7.7 Pulses on Lossy Lines
Prof. Tzong-Lin Wu / NTUEE
3
7.1 Short-Circuited Line
General solution in the sinusoidal steady state
The general solutions for the line voltage and line current in the sinusoidal steady state
 
z
V ( z , t ) = A cos ω  t −
  v p
1
I ( z, t ) =
Z0
 
 
z
B
+
θ
+
cos
ω  t +
 
  v p
 

 
z
 A cos ω  t −
  v p

 
 + φ 
 
 
 
z
 + θ  − B cos ω  t +
  v p
 
  
 + φ  
  
The corresponding expressions for the phasor line voltage and phasor line current
+
−
V ( z ) = V e− jβ z + V e jβ z
I ( z) =
where
(
+
−
1
V e− jβ z − V e jβ z
Z0
)
β = ω /ν p
Prof. Tzong-Lin Wu / NTUEE
4
7.1 Short-Circuited Line
General solution in the sinusoidal steady state
It is convenient to use a distance variable d that increases as we go from the load toward the
generator as opposed to z, which increased from the generator to the load.
+
−
V (d ) = V e j β d + V e − j β d
I (d ) =
(
+
−
1
V e jβ d − V e− jβ d
Z0
)
We wish to determine the characteristics of the waves satisfying
the boundary condition at the short circuit.
What?
Prof. Tzong-Lin Wu / NTUEE
5
7.1 Short-Circuited Line
General solution in the sinusoidal steady state
B. C.
+
V (0) = 0
−
−
V (0) = V e j β (0) + V e − j β (0) = 0
V = −V
+
Particular solutions for the complex voltage and current
+
+
+
V (d ) = V e j β d − V e − j β d = 2 jV sin β d
)
(
+
+
+
1
V
I (d ) =
V e jβ d − V e− jβ d = 2
cos β d
Z0
Z0
The real voltage and current are then given by
V (d , t ) = Re V (d )e jωt 
(
+
= Re 2e jπ / 2 | V | e jθ sin β de jωt
)
+
= −2 | V | sin β d sin(ωt + θ )
λ
I (d , t ) = Re  I (d )e jωt 
 | V + | jθ

e cos β de jωt 
= Re  2
 Z0



+
|V |
=2
cos β d cos(ωt + θ )
Z0
3λ/4
λ/2
Prof. Tzong-Lin Wu / NTUEE
λ/4
6
7.1 Short-Circuited Line
General solution in the sinusoidal steady state
The instantaneous power flow down the line is given by
P ( d , t ) = V (d , t ) I ( d , t )
+
4 | V |2
=−
sin β d cos β d sin(ωt + θ ) cos(ωt + θ )
Z0
+
| V |2
sin 2 β d sin 2(ωt + θ )
=−
Z0
Why do we call them the standing wave?Prof. Tzong-Lin Wu / NTUEE
7
7.1 Short-Circuited Line
Standing wave
Thus, the phenomenon on the short-circuited line is one in which the voltage, current, and power
flow oscillate sinusoidally with time with different amplitudes at different locations on the line,
unlike in the case of traveling waves in which a given point on the waveform progresses in
distance with time.
Since there is no feeling of wave motion down the line, these waves are known as standing waves.
In particular, they represent complete standing waves in view of the zero amplitudes of the voltage,
current, and power flow at certain locations on the line.
Complete standing waves are the result of (+) and (-) traveling waves of equal amplitudes.
Is there any power flow at any location d on the line?
Prof. Tzong-Lin Wu / NTUEE
8
7.1 Short-Circuited Line
Standing wave
Although there is instantaneous power flow at values of d between the voltage and
current nodes, there is no time-average power flow for any value of d.
Prof. Tzong-Lin Wu / NTUEE
9
7.1 Short-Circuited Line
Standing wave patterns
Take the amplitude
+
+
|V(d)|=2 | V || sin β d |= 2 | V || sin
+
2π
λ
d|
+
2 |V |
2 |V |
2π
|I(d)|=
| cos β d |=
| cos
d|
λ
Z0
Z0
They are the patterns of line voltage and line current one would obtain by connecting an ac
voltmeter between the conductors of the line and an ac ammeter in series with one of the
conductors of the line and observing their readings at various points along the line.
Standing-wave patterns should not be misinterpreted as the voltage and current remaining
constant with time at a given point. On the other hand, the voltage and current at every point on the
line vary sinusoidally with time.
Prof. Tzong-Lin Wu / NTUEE
10
7.1 Short-Circuited Line
Natural Oscillations
Since there is no power flow across a voltage node or a current node of the standing-wave
patterns, a constant amount of total energy is locked up in every section between two such
adjacent nodes with exchange of energy taking place between the electric and magnetic fields.
Electrical length of a line: its length in terms of wavelength, which is depends on the
frequency.
Let us now consider a line of length l, one end of which is open-circuited and the other end
short-circuited, and assume that some energy is stored in this line.
What are all the possible standing-wave patterns on this line?
Prof. Tzong-Lin Wu / NTUEE
11
7.1 Short-Circuited Line
Natural Oscillation
We note that the voltage across the short circuit must always be zero, and, hence, the current there
must have maximum amplitude. Similarly, the current at the open-circuited end must always be zero,
and, hence, the voltage there must have maximum amplitude. (boundary condition)
We also know that the standing-wave patterns are sinusoidal with the distance between successive
nodes corresponding to a half sine wave.
Thus, the least possible variation is a quarter cycle of a sine waveform.
L = λ/4
Prof. Tzong-Lin Wu / NTUEE
12
7.1 Short-Circuited Line
Natural Oscillation
Other possible patterns
(2 n − 1) λn
l=
4
n = 1, 2, 3,K
fn =
Prof. Tzong-Lin Wu / NTUEE
vp
λn
=
(2n − 1)v p
4l
n = 1, 2,3,....
13
7.1 Short-Circuited Line
Natural Oscillation
fn =
vp
λn
=
(2n − 1)v p
4l
n = 1, 2,3,....
These frequencies are known as the natural frequencies of oscillation.
The standing-wave patterns are said to correspond to the different natural modes of oscillation.
The lowest frequency (corresponding to the longest wavelength) is known as the fundamental
frequency of oscillation, and the corresponding mode is known as the fundamental mode. The
quantity n is called the mode number.
In the most general case of non-sinusoidal voltage and current distributions on the line, the
situation corresponds to the superposition of some or all of the infinite number of natural modes.
Prof. Tzong-Lin Wu / NTUEE
14
7.1 Short-Circuited Line
Input impedance
We define the ratio of phasor line voltage and the phasor line current as the line impedance,
Z(d), at that point seen looking toward the short circuit.
+
V (d )
2 jV sin β d
Z (d ) =
=
= jZ 0 tan β d
+
I (d ) 2(V Z 0 ) cos β d
2π f
Z in = jZ 0 tan β l = jZ 0 tan
l
vp
The input impedance of the short-circuited line is purely reactive.
As the frequency is varied from a low value upward, the input
reactance changes from inductive to capacitive and back to
inductive, and so on.
The input reactance is zero for values of frequency equal to multiples of v p / 2l . These are
the frequencies for which l is equal to multiples of λ/2 so that the line voltage is zero at the
input and hence the input sees a short circuit.
v / 4l
The input reactance is infinity for values of frequency equal to odd multiples of p
. These
are the frequencies for which l is equal to odd multiples of λ/4 so that the line current is zero
at the input and hence the input sees
open circuit.
Prof.anTzong-Lin
Wu / NTUEE
15
7.1 Short-Circuited Line
Input impedance
These properties of the input impedance of a short-circuited line (and, similarly, of an opencircuited line) have several applications.
• Determination of the location of a short circuit (or open circuit) in a line.
Since the difference between a pair of consecutive frequencies for which the input reactance
values are zero and infinity is v p / 4l
• Construction of resonant circuits at microwave frequencies.
The principle behind this lies in the fact that the input reactance of a short-circuited line of
a given length can be inductive or capacitive, depending on the frequency, and hence, two
short-circuited lines connected together form a resonant system.
How to derive characteristic equation for the resonant frequencies of such a system?
Prof. Tzong-Lin Wu / NTUEE
16
7.1 Short-Circuited Line
Input impedance
B.C. at the junction
V1 = V2
I1 + I 2 = 0
I1 I 2
+ =0
V1 V2
Y1 =
1
1
1
=
=
Z 1 jZ 01 tan β1l1 jZ 01 tan(2π f v p1 )l1
Y2 =
1
1
1
=
=
Z 2 jZ 02 tan β 2l2 jZ 02 tan(2π f v p 2 )l2
Y1 + Y2 = 0
characteristic equation for the resonance frequency
Z 01 tan
2π f
2π f
l1 + Z 02 tan
l2 = 0
v p1
vp2
Prof. Tzong-Lin Wu / NTUEE
17
7.2 LINE TERMINATED BY ARBITRARY LOAD
−
1 + jβ d
(V e − V e− j β d )
Z0
+
−
V (d ) = V e jβ d + V e − jβ d
I (d ) =
B. C. on load
V (0) = Z R I (0)
+
−
V +V =
−
ZR +
(V − V )
Z0
ΓR =
V
V
−
V =V
−
+
=
Z R − Z0
Z R + Z0
+
Z R − Z0
Z R + Z0
voltage reflection coefficient at the load
Prof. Tzong-Lin Wu / NTUEE
18
7.2 Line Terminated by Arbitrary Load
Generalized reflection coefficient
The solutions for V(d) and I(d) can then be written as
+
V (d ) = V e
jβ d
+ − jβ d
I (d ) =
+ Γ RV e
+
1 + jβ d
(V e − Γ R V e− j β d )
Z0
Generalized voltage reflection coefficient, the voltage reflection coefficient at any value of d,
as the ratio of the reflected wave voltage to the incident wave voltage at that value of d.
+
Γ( d ) =
Γ R V e− jβ d
+
V e jβ d
Γ(d ) = Γ R e − j 2 β d = Γ R
= Γ R e− j 2 β d
∠Γ(d ) = ∠Γ R + ∠e− j 2 β d = θ − 2β d
+
+
V (d ) = V e
jβ d
(1 + Γ R e
− j 2β d
)
I (d ) =
+
= V e j β d [1 + Γ(d )]
+
=
Prof. Tzong-Lin Wu / NTUEE
V jβ d
e (1 − Γ R e − j 2 β d )
Z0
V jβ d
e [1 − Γ(d )]
Z0
19
7.2 Line Terminated by Arbitrary Load
standing wave pattern
To study the standing-wave patterns
V (d ) = V
+
e j β d 1 + Γ(d )
I (d ) =
=
+
Z0
V
+
= V 1 + Γ Re− j 2 β d
V
e j β d 1 − Γ(d )
+
Z0
1 − Γ R e− j 2 β d
V+ is simply a constant, determined by the boundary condition at the source end.
Prof. Tzong-Lin Wu / NTUEE
20
7.2 Line Terminated by Arbitrary Load
standing wave pattern
From these constructions and assuming
−π ≤ θ ≤ π
, we note the following facts:
1.
Prof. Tzong-Lin Wu / NTUEE
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7.2 Line Terminated by Arbitrary Load
standing wave pattern
2.
Prof. Tzong-Lin Wu / NTUEE
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7.2 Line Terminated by Arbitrary Load
standing wave pattern
3.
Prof. Tzong-Lin Wu / NTUEE
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7.2 Line Terminated by Arbitrary Load
standing wave parameters (three parameters)
1. The standing-wave ratio, abbreviated as SWR.
+
V (1 + Γ R )
1+ ΓR
Vmax
SWR =
=
=
Vmin V + (1 − Γ R ) 1 − Γ R
also
+
Z 0 )(1 + Γ R ) 1 + Γ R
I max
=
=
I min ( V + Z )(1 − Γ R ) 1 − Γ R
0
(V
The SWR is a measure of standing waves
on the line.
It is an easily measurable parameter.
Prof. Tzong-Lin Wu / NTUEE
24
7.2 Line Terminated by Arbitrary Load
standing wave parameters (three parameters)
2.
d min =
θ +π λ
=
(θ + π )
2β
4π
Z R > Z0
θ =0
d min = λ / 4
Z R < Z0
θ = −π
d min = 0
Prof. Tzong-Lin Wu / NTUEE
Physical meaning ?
25
7.2 Line Terminated by Arbitrary Load
Determination of unknown load impedance
Conversely to the computation of standing-wave parameters for a given load impedance, an
unknown load impedance can be determined from standing-wave measurements on a line of
known characteristic impedance.
ΓR =
θ=
SWR − 1
SWR + 1
4π d min
λ
Z R = Z0
−π
1+ ΓR
1− ΓR
How to get the SWR?
Prof. Tzong-Lin Wu / NTUEE
26
7.2 Line Terminated by Arbitrary Load
Slotted-line measurements
A traditional method of performing standing-wave measurements in the laboratory is by using a
slotted line.
The slotted line is essentially a rigid coaxial line with air dielectric and having a length of about 1
meter (or at least a half wavelength long). The center conductor is supported by dielectric
inserts.
A narrow longitudinal slot is cut in the outer conductor,
Width of the slot is so small that it has negligible influence on
the current flow on the outer conductor
A probe of small length intercepts a portion of the electric
field between the inner and outer conductors, and a small
voltage proportional to the line voltage at the probe’s
location is developed between the probe and the outer
conductor.
Since the SWR is the ratio of to the quantity of interest is the ratio of the two readings
rather than the absolute values of the readings themselves. Therefore, absolute
calibration of the detector is not required.
Prof. Tzong-Lin Wu / NTUEE
27
7.2 Line Terminated by Arbitrary Load
Slotted-line measurements
Since it is not always possible to measure the distances of the standing wave pattern minima from
the location of the load, the following procedure is employed.
First, the line is terminated by a short circuit in the place of the load. One of the nulls in the
resulting standing-wave pattern is taken as the reference point, as shown in Fig. 7.12(a).This
establishes that the location of the load is an integral multiple of half-wavelengths from the
reference point.
Next, the short circuit is removed and the load is connected. The voltage minimum then shifts
away from the reference point, as shown in Fig. 7.12(b). By measuring this shift, either away from
the load or toward the load, the value of can be established.
Prof. Tzong-Lin Wu / NTUEE
28
7.2 Line Terminated by Arbitrary Load
Line impedance
The impedance at any value of d seen looking toward the load
+
V (d )
V e j β d [1 + Γ(d )]
Z (d ) =
= +
I (d ) (V Z 0 )e j β d [1 − Γ( d )]
= Z0
1 + Γ(d )
1 − Γ(d )
Prof. Tzong-Lin Wu / NTUEE
29
7.2 Line Terminated by Arbitrary Load
Line impedance
λ
5. The product of the line impedances at two values of d separated by 4 is
given by
λ 

1
+
Γ
(
d
±
)
 
λ    1 + Γ( d )  
4
[ Z (d )]  Z  d ±   =  Z 0
 Z0

4
1
(
d
)
−
Γ
 
 
  1 − Γ (d ± λ ) 

4 
1 + Γ(d )  1 + Γ(d )e m j 2 βλ / 4 
=Z 

m j 2 βλ / 4 
1
−
Γ
(
d
)
1
−
Γ
(
d
)
e




 1 + Γ(d )e m jπ 
2 1 + Γ(d )
= Z0 

m jπ 
1
−
Γ
(
d
)
1
−
Γ
(
d
)
e



2
0
1 + Γ ( d )   1 − Γ ( d ) 
= Z02 


1
−
Γ
(
d
)
1
+
Γ
(
d
)



Prof. Tzong-Lin Wu / NTUEE
 
λ 
[ Z (d )]  Z  d ±   = Z 02
4 
 
30
7.2 Line Terminated by Arbitrary Load
Input impedance
Z in = Z (l ) = Z 0
1 + Γ(l )
1 − Γ (l )
The input impedance is a useful parameter, because, for a given generator voltage and internal
impedance, the power flow down the line can be computed by considering the line voltage and
current at any value of d.
Ex: 7.3 (Please refer to textbook for detail).
Prof. Tzong-Lin Wu / NTUEE
31
7.2 Line Terminated by Arbitrary Load
Normalized impedance and Admittance
Define:
z (d ) =
Z (d ) 1 + Γ(d )
=
Z0
1 − Γ(d )
Γ(d ) =
z (d ) − 1
z (d ) + 1
Y (d ) =
1
1 1 − Γ(d )
=
Z (d ) Z 0 1 + Γ(d )
Admittance:
Y (d ) = Y0
y (d ) =
1 − Γ(d )
1 + Γ(d )
Y (d ) 1 − Γ(d )
=0
Y0
1 + Γ(d )
Prof. Tzong-Lin Wu / NTUEE
32
7.3 Transmission-Line Matching
When the load impedance is not equal to the characteristic impedance, the input impedance of
the line will vary with frequency
This sensitivity to frequency increases with the electrical length of the line. Why?
To show this, let the length of the line be l = nλ
If the frequency is changed by an amount ∆f , then the change in n is given by
 lf
l
∆n = ∆   = ∆ 
v
λ 
 p
 l
nλ
∆f
∆f = n
 = ∆f =
vp
f
 vp
It is desirable to eliminate standing waves on the line by connecting a matching device near the
load such that the line views an effective impedance equal to its own characteristic impedance
on the generator side of the matching device.
Does the matching device at the same time absorb any power?
It should be noted that matching, as referred to here, is not related to maximum power transfer
since the condition for maximum power transfer is that the line input impedance must be the
complex conjugate of the generator internal impedance.
Prof. Tzong-Lin Wu / NTUEE
33
7.3 Transmission-Line Matching
Quarter-Wave Transformer Matching
We try to decide dq and Zq.
Where is the possible locations of dq ?
Quarter wave transformer
Z1 Z 2 = Z q2
Z 2 = Z q2 / Z1 = Z q2 / Z 0
Prof. Tzong-Lin Wu / NTUEE
Real number
Do you know now where dq is?
34
7.3 Transmission-Line Matching
Quarter-Wave Transformer Matching
The line impedance is purely real at locations of voltage maxima and minima of the standing-wave
pattern.
Therefore, within the first half-wavelength from the load, there are two solutions for dq.
min
max
Prof. Tzong-Lin Wu / NTUEE
35
7.3 Transmission-Line Matching
Quarter-Wave Transformer Matching
If we choose a voltage minimum for the first solution,
d q(1) =
Z0 ⋅ Z0
1 − ΓR
1+ ΓR
λ
(θ + π )
4π
= Z q2
Z
(1)
q
= Z0
1− ΓR
1+ ΓR
For the second solution, the value of dq corresponds to the location of a voltage
maximum that occurs at ± λ / 4 from the location of the voltage minimum.
d
(2)
q
=d
(1)
q
±
λ
4
and
Z
(2)
q
= Z0
Prof. Tzong-Lin Wu / NTUEE
1+ ΓR
1− ΓR
36
7.3 Transmission-Line Matching
Single-stub Matching
In the single-stub matching technique, one stub is used and a match is achieved by varying the location
of the stub and the length of the stub.
We shall assume the characteristic impedance of the stub to be the same as that of the line.
normalized load impedance
normalized line admittances
just to the left of the stub
normalized line admittances
just to the right of the stub
normalized input susceptance
Prof. Tzong-Lin Wu / NTUEE
37
7.3 Transmission-Line Matching
Single-stub Matching
y1' = 1 − jb
Γ =
'
1
1 − y1'
1 + y1'
=
jb
2 − jb
' j2β d
s
1
ΓR = Γ e
=
ΓR =
b
4 + b2
b
4 + b2
=
jb j 2 β d s
e
2 − jb
e j ( ±π / 2 + tan
π
−1
b / 2 + 2 β d s + 2 nπ )
>
for b 0
<
b
θ = ± + tan
+ 2 β d s + 2nπ
2
2
−1
Prof. Tzong-Lin Wu / NTUEE
>
for b 0
<
38
7.3 Transmission-Line Matching
Single-stub Matching
b=±
2 ΓR
1− ΓR
λ
ds =
4π
2
π


−1 b
θ
π
−
−
n
m
tan
2


2
2


>
for b 0
<
• Two solutions are possible for b
• The integer value for n is chosen such that
Prof. Tzong-Lin Wu / NTUEE
0 ≤ ds < λ / 2
39
7.3 Transmission-Line Matching
Single-stub Matching
To find the solutions for the stub length
1
= j tan β ls
jb
so
tan β ls = −
Normalized input impedance of a short-circuited line
1
b
 2λπ  tan −1 ( − b1 ) + λ2
ls =  λ  −1 1 
 2π  tan ( − b )
Prof. Tzong-Lin Wu / NTUEE
for b > 0
for b < 0
40
7.3 Transmission-Line Matching
Double-stub Matching
In the single-stub matching technique, it is necessary to vary the distance between the stub and the
load, as well as the length of the stub, in order to achieve a match for different loads or for different
frequencies. This can be inconvenient for some arrangements of lines.
When two stubs are used, it is possible to fix their locations and achieve a match for a wide range of
loads by adjusting the lengths of the stubs.
Prof. Tzong-Lin Wu / NTUEE
41
7.3 Transmission-Line Matching
Double-stub Matching
y = y2 − jb2 = 1 − jb2
'
2
'
2
Γ1 = Γ e
j 2 β d12
jb2
=
e j 2 β d12
2 − jb2
Γ =
'
2
1 − y2'
1 + y2'
y1 =
=
jb2
2 − jb2
1 − Γ1
1 + Γ1
4 − j (4b2 cos 2 β d12 − 2b22 sin 2 β d12 )
=
4 − 4b2 sin 2 β d12 + 4b22 sin 2 β d12 )
y1' = y1 − jb1
=
1
1 − b2 sin 2 β d12 + b22 sin 2 β d12
 b22 sin 2 β d12 − 2b2 cos 2 β d12

+ j
−
b
1
2
2
2
2
sin
2
2
sin
)
−
b
d
+
b
d
β
β
2
12
2
12


Prof. Tzong-Lin Wu / NTUEE
42
7.3 Transmission-Line Matching
Double-stub Matching
1
= g'
1 − b2 sin 2 β d12 + b22 sin 2 β d12 )
sin 2 β d12 ± sin 2 2 β d12 − 4(1 − 1/ g ') sin 2 β d12
b2 =
2sin 2 β d12
cos β d12 ± 1/ g '− sin 2 β d12
b2 =
sin β d12
Prof. Tzong-Lin Wu / NTUEE
43
7.3 Transmission-Line Matching
Double-stub Matching
From the equation for the imaginary parts of
given by
y1′ , the corresponding values of b1 are then
b22 sin 2 β d12 − 2b2 cos 2β d12
b1 =
−b'
2 − 2b2 sin 2 β d12 + 2b22 sin 2 β d12
Finally, the lengths of the two stubs are computed from b1 and b2
Prof. Tzong-Lin Wu / NTUEE
44
7.3 Transmission-Line Matching
Bandwidth
For any transmission-line matched system, the match is disturbed as the frequency is varied
from that at which the various electrical lengths and distances are equal to the computed
values for achieving the match.
For example, in the QWT matched system, the electrical length of the QWT departs from onequarter wavelength as the frequency is varied from that at which the match is achieved, and the
system is no longer matched even if the load does not vary with frequency.
tolerable value of SWR
Prof. Tzong-Lin Wu / NTUEE
45
7.3 Transmission-Line Matching
SWR versus frequency computation
To discuss a procedure by means of which the SWR versus frequency curve can be computed for
all three types of matching techniques discussed, let us consider a transmission-line system
having n discontinuities
We shall consider a specification of zero for the length of the stub to mean no stub is present
instead of a stub of zero length.
Prof. Tzong-Lin Wu / NTUEE
46
7.3 Transmission-Line Matching
SWR versus frequency computation
For a given f/f0, the procedure consists of starting at the load and computing in succession the line
admittance to the left of the stub at each discontinuity, from a knowledge of the line admittance at
the output of the line section to the right of that stub, until the line admittance to the left of the last
discontinuity is found and used to compute the required SWR.
To calculate the frequency response, the following formulation is commonly used.
1 − Γi 1 − Γ 0 e −2 β l
yi =
=
1 + Γ i 1 + Γ 0 e −2 β l
1 − [(1 − y0 ) /(1 + y0 )]e −2 β l
=
1 + [(1 − y0 ) /(1 + y0 )]e−2 β l
yi =
j sin β l + y0 cos β l
cos β l + j y0 sin β l
Prof. Tzong-Lin Wu / NTUEE
47
7.4 The Smith Chart: Basic Procedure
We begin with the relationship for the reflection coefficient in terms of the normalized line
impedance as given by
Γ(d ) =
r + jx − 1 ( r − 1) + jx
Γ( d ) =
=
r + jx + 1 (r + 1) + jx
z (d ) − 1
z (d ) + 1
1
2
 ( r − 1) + x 
Γ(d ) = 
≤1
2
2
 (r + 1) + x 
2
2
r + jx − 1 r 2 − 1 + x 2
2x
Γ=
=
+
j
r + jx + 1 (r + 1) 2 + x 2
(r + 1) 2 + x 2
r 2 − 1 + x2
Re(Γ) =
(r + 1) 2 + x 2
Im(Γ) =
Prof. Tzong-Lin Wu / NTUEE
2x
( r + 1) 2 + x 2
48
7.4 The Smith Chart: Basic Procedure
Z is purely real
Z is purely imaginary
Prof. Tzong-Lin Wu / NTUEE
49
7.4 The Smith Chart: Basic Procedure
Center:
Z is complex and its real part is constant
Z is complex and its imaginary part is constant
Center:
(r /(r + 1), 0)
(1,1/ x)
radius
Prof. Tzong-Lin Wu / NTUEE
50
7.4 The Smith Chart: Basic Procedure
http://www.sss-mag.com/smith.html
Smith chart Collection
Prof. Tzong-Lin Wu / NTUEE
51
7.4 The Smith Chart: Basic Procedure
example 7.4
1.
2. SWR = Vmax / Vmin
= (1 + |Γ|) / (1 - |Γ|)
= V / (Z0*I)
at the position of voltage maximum B
= Rmax / Z0
= Normalized input impedance at voltage maximum
= 4 in this case
Prof. Tzong-Lin Wu / NTUEE
52
7.4 The Smith Chart: Basic Procedure
example 7.4
Prof. Tzong-Lin Wu / NTUEE
53
7.4 The Smith Chart: Basic Procedure
example 7.4
Prof. Tzong-Lin Wu / NTUEE
54
7.4 The Smith Chart: Basic Procedure
example 7.4
λ
 Z (d )   Z (d + )  = Z 02

 
4 
λ
 z (d )   z (d + )  = 1

 
4 
λ
y (d ) = z (d + )
4
Prof. Tzong-Lin Wu / NTUEE
55
7.4 The Smith Chart: Basic Procedure
example 7.4
1.
4.
0.325 – 0.185 = 0.14λ
2.
r=1
3.
Prof. Tzong-Lin Wu / NTUEE
56
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.5 Single-stub matching solution
To achieve a match,
• The stub must be located at a point on the line at which the real part of the normalized line
admittance is equal to unity;
• The imaginary part of the line admittance at that point is then canceled by appropriately
choosing the length of the stub.
Prof. Tzong-Lin Wu / NTUEE
57
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.5 Single-stub matching solution
Prof. Tzong-Lin Wu / NTUEE
58
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.5 Single-stub matching solution
y = 1 + j 1.16
C
B
The location of the stub from the load is
Prof. Tzong-Lin Wu / NTUEE
59
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.5 Single-stub matching solution
E
D
Is there second solution?
Prof. Tzong-Lin Wu / NTUEE
60
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.5 Single-stub matching solution
Second solution
Prof. Tzong-Lin Wu / NTUEE
61
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.6 double-stub matching solution
Auxiliary circle
Auxiliary circle
y1
has to be designed on the auxiliary circle.
Prof. Tzong-Lin Wu / NTUEE
62
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.6 double-stub matching solution
A–B–C–D
So,
Prof. Tzong-Lin Wu / NTUEE
63
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.6 double-stub matching solution
The second solution:
Prof. Tzong-Lin Wu / NTUEE
64
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.6 double-stub matching solution
Before proceeding further, we recall from Section 7.3 that in the double stub
matching technique, it is not possible to achieve a match for loads that result in the
real part of y1′ being greater than 1/ sin 2 β d
12
In this case, d12 is 3/8λ, no solution for the case of
r > (1/ sin 2 β d12 ) = 2
this shaded region
Why?
The shaded region is therefore called the forbidden
region of y1′ for possible match.
How to solve?
Prof. Tzong-Lin Wu / NTUEE
65
7.5 THE SMITH CHART: 2. APPLICATIONS
ex: 7.6 double-stub matching solution
Prof. Tzong-Lin Wu / NTUEE
66
7.5 THE SMITH CHART: 2. APPLICATIONS
Transformation of across a discontinuity
y1 =
Y1 Y2 Yd
=
+
Y01 Y01 Y01
Y Y
Y
= 02 ( 2 + d )
Y01 Y02 Y02
1 − Γ1
1 − Γ2
= a(
+ yd )
1 + Γ1
1 + Γ2
Γ1 =
(1 + a − a yd )Γ 2 + (1 − a + a yd )
(1 − a + a yd )Γ 2 + (1 + a + a yd )
= a ( y2 + yd )
Prof. Tzong-Lin Wu / NTUEE
67
7.5 THE SMITH CHART: 2. APPLICATIONS
Transformation of across a discontinuity
Bilinear transformation between two complex planes, a property of which is that circles in one
plane are transformed into circles in the second plane.
Γ1 =
(1 + a − a yd )Γ 2 + (1 − a + a yd )
(1 − a + a yd )Γ 2 + (1 + a + a yd )
Since a circle is defined completely by three points.
It is therefore sufficient if we use any three points on the locus of y2 and find the corresponding
three points for y1.
Prof. Tzong-Lin Wu / NTUEE
68
7.5 THE SMITH CHART: 2. APPLICATIONS
Transformation of across a discontinuity
Ex 7.7
We wish to
• find the locus of the normalized admittance y1 to the left of the discontinuity as the susceptance
slides along the line, and then
• determine the location, nearest to the load, of the susceptance for which the SWR to the left of it is
minimized.
Prof. Tzong-Lin Wu / NTUEE
69
7.5 THE SMITH CHART: 2. APPLICATIONS
Transformation of across a discontinuity
A, B, C points on y2 transform to
C, D, E, respectively, on y1 based on
y1 = y2 +j 0.8
Where is the minimum SWR point on Locus
of y1?
Prof. Tzong-Lin Wu / NTUEE
70
7.5 THE SMITH CHART: 2. APPLICATIONS
Transformation of across a discontinuity
Point G because of minimum reflection
coefficient.
Back to y2 by the bilinear transform
y2 = y1 - j 0.8
The slide distance from the load is from point A
to H with
Prof. Tzong-Lin Wu / NTUEE
71
7.6 The Lossy Line
Distributed Equivalent Circuits
The power dissipation in the conductors is taken into account by a resistance in series with the
inductor, whereas the power dissipation in the dielectric is taken into account by a conductance in
parallel with the capacitor.
∂I ( z , t )
∂t
∂V ( z , t )
I ( z + ∆z , t ) − I ( z , t ) = −G ∆zV ( z , t ) − C ∆z
∂t
V ( z + ∆z , t ) − V ( z , t ) = − R∆zI ( z , t ) − L∆z
Prof. Tzong-Lin Wu / NTUEE
72
7.6 The Lossy Line
Transmission Line Equations and Solutions
∂V ( z, t )
∂I ( z , t )
= − RI ( z , t ) − L
∂z
∂t
∂I ( z , t )
∂V ( z , t )
= −GV ( z , t ) − C
∂z
∂t
The corresponding equations in terms of phasor voltage and current
∂V
= −( R + jω L) I
∂z
∂I
= −(G + jωC )V
∂z
∂ 2V
∂I
=
−
(
R
+
j
ω
L
)
∂z 2
∂z
= ( R + jω L)(G + jωC )V
∂ 2V
= γ 2V
2
∂z
γ = α + j β = ( R + jω L)(G + jωC )
Prof. Tzong-Lin Wu / NTUEE
73
7.6 The Lossy Line
Transmission Line Equations and Solutions
V ( z ) = Ae −γ z + Beγ z
I (z) = −
=−
1
∂V
R + jω L ∂z
I ( z) =
1
 −γ V + e −γ z + γ V − eγ z 

R + jω L 
where
1  + −γ z
V e + V − eγ z 

Z0 
Z0 =
R + jω L
G + jωC
characteristic impedance of the line, which is now complex.
Prof. Tzong-Lin Wu / NTUEE
74
7.6 The Lossy Line
Low Loss Line
For the special case of the low-loss line:
γ=
jω L(1 +
≈ jω LC 1 +
R
jω L
) jωC (1 +
G
)
jωC
ωC >> G
ω L >> R
jω L(1 +
Z0 =
R
G
+
jω L jωC
R
)
jω L
G
jωC (1 +
)
jωC
 1 R
G 
≈ jω LC 1 + (
+
)
 2 jω L jωC 
≈
L
R
G
(1 +
) (1 −
)
C
jω L
jωC
1
C
L
≈ (R
+G
) + jω LC
L
C
2
≈
L
R
G
−
(1 +
)
C
jω L jωC
1
C
L
α ≈ (R
+G
)
2
L
C
≈
L
C
β ≈ ω LC
ω
1
vp = ≈
β
LC
 1 R
G 
1 + 2 ( jω L − jωC ) 


L
C
Thus, for the low-loss line, the expressions for β and Z0 are essentially
the same as those for a lossless line.
≈
Prof. Tzong-Lin Wu / NTUEE
75
7.6 The Lossy Line
Experimental Determination of βand Z0
The experimental technique is based on the measurements of the input impedance of the line
for two values of load impedance.
where
V (d ) = V + eγ d 1 + Γ( d ) 
V ( d ) = V + eγ d + V − e − γ d
1  + γd
I (d ) =
V e + V − e −γ d 

Z0 
Γ(d ) =
V + γd
I (d ) =
e 1 − Γ(d ) 
Z0
V − (d )
+
V (d )
=
V − e−γ d
V + eγ d
= Γ R e−2γ d = Γ R e−2α d e− j 2 β d
The line impedance is given by
Z (d ) =
V (d )
1 + Γ(d )
= Z0
I (d )
1 − Γ(d )
1 + Γ R e −2γ d
= Z0
1 − Γ R e −2γ d
Prof. Tzong-Lin Wu / NTUEE
76
7.6 The Lossy Line
Experimental Determination of βand Z0
The input impedance of a line of length l terminated by a load impedance ZR is
1 + Γ R e −2 γ l
Z in = Z (l ) = Z 0
1 − Γ R e −2γ l
1 + ( Z R − Z 0 ) /( Z R + Z 0 )  e−2γ l
= Z0
1 − ( Z R − Z 0 ) /( Z R + Z 0 )  e−2γ l
( Z R + Z 0 ) + ( Z R − Z 0 ) e −2 γ l
= Z0
( Z R + Z 0 ) − ( Z R − Z 0 ) e −2 γ l
= Z0
Z R cosh γ l + Z 0 sinh γ l
Z R sinh γ l + Z 0 cosh γ l
Z in = Z 0
Z R + Z 0 tanh γ l
Z R tanh γ l + Z 0
Prof. Tzong-Lin Wu / NTUEE
77
7.6 The Lossy Line
Experimental Determination of βand Z0
s
s
in
Z = Z 0 tanh γ l
Ex:
0
Z in = (30 + j 40)Ω
0
Z in = Z 0 coth γ l
s
in
Z0 = Z Z
Z 0 = (30 − j 40)(30 + j 40) = 50Ω
0
in
30 − j 40
50∠ − 53.13°
tanh γ l =
=
30 + j 40
50∠53.13°
s
tanh γ l =
Z in
Z
α l = 0.3466
α = 0.3466 / l
β l = nπ − π / 4
Z in = (30 − j 40)Ω
= 1∠ − 53.13° = 0.6 − j 0.8
0
in
γ l = tanh −1 (0.6 − j 0.8)
n = 0, 1, 2…
1 1+ x
tanh −1 x = ln
2 1− x
1 1.6 − j 0.8 1 1.789∠ − 26.565
= ln
γ l = ln
2 0.4 + j 0.8 2
0.894∠63.435
1
1
= ln 2∠90° = ln  2e j (2 nπ −π / 2) 
2
2
1
= [ ln 2 + j (2nπ − π / 2) ] = 0.3466 + j ( nπ − π / 4)
2
However, if the approximate value of β is
known, then the correct value of n, and
hence of β can be determined.
Prof. Tzong-Lin Wu / NTUEE
78
7.6 The Lossy Line
Experimental Determination of βand Z0
s
Z in = Z 02
Z R + Z in
s
Z R Z in + Z 02
s
Z0 =
Z R Z in Z in
s
in
Z R + Z − Z in
solve
s
in
Z = Z 0 tanh γ l
Prof. Tzong-Lin Wu / NTUEE
79
7.6 The Lossy Line
Power Flow
Power flow down the line.
1
P = Re V (d ) I * (d ) 

2 

1  + γ d
(V + )* γ *d 
= Re V e 1 + Γ(d ) 
e 1 − Γ* ( d )  


2 
Z 0*

 +2

V
2
1 

= Re  * e2α d 1 − Γ(d ) + Γ( d ) − Γ* ( d )  

 
2  Z0


2
2
1
= Re V + Y0*e 2α d 1 − Γ(d ) + j 2 Im Γ( d ) 


2
{
P =
}
{
}
2
1 + 2 2α d
V e
G0 1 − Γ(d )  + 2 B0 Im Γ(d )


2
where
Y0 =
1
= G0 + jB0
Z0
Prof. Tzong-Lin Wu / NTUEE
80
7.6 The Lossy Line
Power Flow (example)
Please compute the time-average power delivered to the input of the line, the time-average
power delivered to the load, and the time-average power dissipated in the line.
ΓR =
Z R − Z 0 150 − 50
= 0.5
150
+
50
ZR + Z0
Γ(l ) = Γ R e −2γ l = Γ R e −2α l e − j 2 β l
= 0.5e −0.204 e − j 40.8π
= 0.4077∠ − 144°
Prof. Tzong-Lin Wu / NTUEE
81
7.6 The Lossy Line
Power Flow (example)
The input impedance of the line is given by
Z in Z (l ) = Z 0
1 + Γ(l )
1 − Γ(l )
1 + 0.4077∠ − 144°
1 + (−0.33 − j 0.24)
= 50
=
50
1 − 0.4077∠ − 144°
1 − (−0.33 − j 0.24)
0.7117∠ − 19.708°
= 50
= 26.33∠ − 29.937°
°
1.3515∠10.299
= (22.817 − j13.140)Ω
The current drawn from the voltage generator can be obtained as
Vg
100∠0°
I (l ) =
=
Z g + Z in (30 − j 40) + (22.817 − j13.140)
100∠0°
100∠0°
=
=
52.817 − 53.140 74.923∠ − 45.175°
= 1.3347∠45.175°
Prof. Tzong-Lin Wu / NTUEE
82
7.6 The Lossy Line
Power Flow (example)
The voltage at the input end of the line is given by
V (l ) = Z in I (l ) = 26.33∠ − 29.937° × 1.3347∠45.175°
= 35.143∠15.238°
The time-average power flow at the input end of the line is given by
P(l ) =
1 
Re V (l ) I * (l ) 

2 
1
Re 35.143∠15.238° ×1.3347∠ − 45.175° 
2
1
= × 35.143 ×1.3347 × cos 29.937°
2
= 20.32W
=
Prof. Tzong-Lin Wu / NTUEE
83
7.6 The Lossy Line
Power Flow (example)
2 P(l ) e −2α l
+
V =
2
G0 1 − Γ(l ) 


2 × 20.32 × e −0.204
=
0.02(1 − 0.4077 2 )
= 44.58V
The time-average power delivered to the load is then given by
P(0) =
(
1 +2
V G0 1 − Γ R
2
2
)
1
= × 44.582 × 0.02(1 − 0.25)
2
= 14.91W
Finally, the time-average power dissipated in the line is
Prof. Tzong-Lin Wu / NTUEE
Pd = P (l ) − P(0)
= 20.32 − 14.91
= 5.41W
84
7.7
PULSES ON LOSSY LINES
Since αand νp are in general functions of frequency, the different frequency components of an
arbitrarily time-varying signal undergo different amounts of attenuation and different amounts of
phase shift. Hence, as the signal propagates down the line, it gets distorted.
For the general case, the analysis can be performed by employing Fourier techniques.
There are, however, two special cases of importance that permit solution without the use of
Fourier techniques: distortionless transmission and diffusion.
Prof. Tzong-Lin Wu / NTUEE
85
7.7
PULSES ON LOSSY LINES
Distortionless Line
As the signal propagates down the line, its shape versus time remains the same, but it diminishes
in magnitude. The situation arises for the condition:
R G
=
L C
γ = ( R + jω L)(G + jωC )
= R(1 + j
ωL
R
)G (1 + j
ωC
G
)
ωL 

= RG 1 + j

R 

= RG + jω L
G
R
= RG + jω L
C
L
= RG + jω LC
Prof. Tzong-Lin Wu / NTUEE
86
7.7
PULSES ON LOSSY LINES
Distortionless Line
α = RG
Z0 =
β = ω LC
ω
1
vp = =
β
LC
=
R + jω L
R(1 + jω L / R)
=
G + jωC
G (1 + jωC / G )
R
L
=
G
C
α and νp are both independent of frequency,
provided, of course, that R, L, C, G are independent
of frequency.
Prof. Tzong-Lin Wu / NTUEE
independent of frequency
87
7.7
PULSES ON LOSSY LINES
Diffusion
This case pertains to the historically important noninductive, leakage-free cable first investigated
by Lord Kelvin in 1855, but is also relevant to modern lines with large skin-effect losses and to
many other physical phenomena, such as heat flow.
The noninductive, leakage-free cable is characterized by
L=G=0 so that the distributed equivalent circuit consists
simply of series resistors and shunt capacitors,
∂V ( z , t )
= − RI ( z , t )
∂z
∂I ( z , t )
∂V ( z , t )
= −C
∂z
∂t
∂ 2V
∂V
=
RC
∂z 2
∂t
diffusion equation
∂ 2τ 1 ∂τ
=
2
∂z
D ∂t
Prof. Tzong-Lin Wu / NTUEE
88
7.7
PULSES ON LOSSY LINES
Diffusion
∂ 2τ 1 ∂τ
=
2
∂z
D ∂t
Solution:
τ = A∫
( 1/ 4 Dt ) z
0
e −Y dY + B
2
where A and B are arbitrary constants to be evaluated from boundary conditions, and Y is a dummy
variable.
∂ 2V
∂V
=
RC
∂z 2
∂t
Solution:
V ( z , t ) = A∫
( RC / 4 t ) z
0
e −Y dY + B
2
What are A and B?
Prof. Tzong-Lin Wu / NTUEE
89
7.7
PULSES ON LOSSY LINES
Diffusion
V (0, 0+ ) = V0
V ( ∞, t ) = 0
0
A∫ e −Y dY + B = V0
2
t >0
for
B = V0
0
∞
A∫ e −Y dY + V0 = 0
0
2
A
π
2
+ V0 = 0
Prof. Tzong-Lin Wu / NTUEE
A=−
2V0
π
90
7.7
PULSES ON LOSSY LINES
Diffusion
Thus the particular solution
V ( z, t ) = −
2V0
∫
π
2

= V0 1 −
π

( RC / 4 t ) z
0
∫
( RC / 4 t ) z
0
(
= V erfc ( RC / 4t z )
V ( z , t ) = V0 1 − erf

e−Y dY + V0
2
2

e −Y dY 

error function (erf)
)
RC / 4t z 

0
http://mathworld.wolfram.com/Erf.html
erfc is the complementary error function.
the corresponding solution for I (z, t)
I ( z, t ) = −
= V0
1 ∂V ( z , t )
R ∂t
C − ( RC / 4t ) z 2
e
π Rt
Prof. Tzong-Lin Wu / NTUEE
d
2 − z2
erf ( z ) =
e
dz
π
91
How to solve the diffusion equation?
Laplace transform
∂ 2V
∂V
=
RC
∂z 2
∂t
V% ( z , s ) = Ae −
∂2 %
V ( z, s ) = RCsV% ( z , s )
2
∂z
sRC z
+ Be
Step function excitation at z = 0
B. C.
V = 0 at z = infinity
sRC z
V
V% (0, s ) = 0
s
V% (∞, s ) = 0
V0
s
B=0
A=
It is known the Laplace transform of erfc is (proof is complicated)
erfc(
a
2 t
)
e− a
s
s
V ( z , t ) = V0 erfc
Prof. Tzong-Lin Wu / NTUEE
(
RC / 4t z
)
92
7.7
PULSES ON LOSSY LINES
Diffusion
Error function
Complementary Error function
Erf(z) is the "error function" encountered in integrating the normal distribution (which is a
normalized form of the Gaussian function).
erf ( z ) =
2
π ∫
z
0
e − t dt
2
erf ( z ) + erfc( z ) = 1
Prof. Tzong-Lin Wu / NTUEE
93
7.7
PULSES ON LOSSY LINES
Diffusion
Since the argument involves both z and t in the manner RC / 4t z , this sketch represents the
shape of the line voltage variation with z for any fixed value of t.
V ( z, t )
= erfc
V0
(
RC / 4t z
Thus, we note that, immediately after closure of the switch, there is voltage everywhere on
the line. This corresponds to the phenomenon of diffusion—as distinguished from
propagation, which is characterized by a well-defined velocity.
Prof. Tzong-Lin Wu / NTUEE
94
)
7.7
PULSES ON LOSSY LINES
Diffusion
As t increases, a given point on the sketch corresponds to larger and larger values of z, indicating
that as time progresses, the line voltage at all values of z increases.
For example, since the values of z are doubled as t is quadrupled, the voltage at a given distance
from the source and at a particular time after closure of the switch is the same as the voltage at
half that distance and at one-fourth of that time.
Prof. Tzong-Lin Wu / NTUEE
95
7.7
PULSES ON LOSSY LINES
Diffusion
We can also obtain the time-variations of the line voltage for fixed values of z, by noting that for a
fixed z, the numbers on the abscissa can be converted to values of time.
Prof. Tzong-Lin Wu / NTUEE
96
7.7
PULSES ON LOSSY LINES
Diffusion
Note:
As the value of z is increased, the attainment of
the maximum of the pulse is delayed and the
value of the maximum is reduced.
Prof. Tzong-Lin Wu / NTUEE
97