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1RS
ENE 428Microwave
Engineering
Lecture 5 Discontinuities and the manipulation of transmission lines problems
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Review • Transmission lines or T-lines are used to guide propagation of
EM waves at high frequencies.
• Distances between devices are separated by much larger order of wavelength than those in the normal electrical
circuits causing time delay.
• General transmission line’s equation• Voltage and current on the transmission line
• characteristic of the wave propagating on the transmission
line
0 0
0 0
( )
( )
z z
z z
V z V e V e
I z I e I e
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Wave reflection at discontinuities• To satisfy boundary conditions between two
dissimilar lines
• If the line is lossy, Z0 will be complex.
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Reflection coefficient at the load (1)• The phasor voltage along the line can be shown
as
• The phasor voltage and current at the load is the sum of incident and reflected values evaluated at z = 0.
0
0
( )
( )
z j zi i
z j zr r
V z V e e
V z V e e
0 0
0 00 0
0
L i r
i rL i r
V V V
V VI I I
Z
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Reflection coefficient at the load (2)• Reflection coefficient
• A reflected wave will experience a reduction in amplitude and a phase shift
• Transmission coefficient
0 0
0 0
rjr LL
i L
V Z Ze
V Z Z
0 0
21 tjL L
Li L
V Ze
V Z Z
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Power transmission in terms of reflection coefficient
2
02 20 0,
00
20 0,
0
22 0 2
0
1 1 1Re Re cos2 2 2
( )( )1 1Re Re2 2
1cos
2
z zLavg i i i j
zL LLavg r r r j
zL
VV VP V I e e
ZZ e
V VP V I e
Z e
Ve
Z
2,
,
2,
,
1
Lavg rL
Lavg i
Lavg tL
Lavg i
P
P
P
P
W
W
W
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Total power transmission (matched condition)• The main objective in transmitting power to a
load is to configure line/load combination such that there is no reflection, that means
0
0
.L
LZ Z
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Voltage standing wave ratio
• Incident and reflected waves create “Standing wave”.
• Knowing standing waves or the voltage amplitude as a function of position helps determine load and input impedances
max
min
VVSWR
V
Voltage standing wave ratio
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Forms of voltage (1)
• If a load is matched then no reflected wave occurs, the voltage will be the same at every point.
• If the load is terminated in short or open circuit, the total voltage form becomes a standing wave.
• If the reflected voltage is neither 0 nor 100 percent of the incident voltage then the total voltage will compose of both traveling and standing waves.
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Forms of voltage (2)
• let a load be position at z = 0 and the input wave amplitude is V0,
0 0
0
0
( )
.
j z j zT L
jLL L
L
V z V e V e
Z Ze
Z Z
where
( )0( ) ( )j z j z
T LV z V e e
/ 2 / 2 / 20 ( )j j z j j z j
LV e e e e e
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Forms of voltage (3)
we can show that
/ 20 0( ) (1 ) 2 cos( ).
2j z j
T L LV z V e V e z
traveling wave standing wave
The maximum amplitude occurs when
The minimum amplitude occurs when standing waves become null,
0( ) (1 ).T LV z V
0( ) (1 ).T LV z V
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The locations where minimum and maximum voltage amplitudes occur (1)
• The minimum voltage amplitude occurs when two phase terms have a phase difference of odd multiples of .
• The maximum voltage amplitude occurs when two phase terms are the same or have a phase difference of even multiples of .
( ) (2 1) ; 0,1,2,...z z m m
min ( (2 1) )4
z m
( ) 2 ; 0,1,2,...z z m m
max ( 2 )4
z m
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The locations where minimum and maximum voltage amplitudes occur (2)
• If = 0, is real and positive
and
• Each zmin are separated by multiples of one-half wavelength, the same applies to zmax. The distance between zmin and zmax is a quarter wavelength.
• We can show that
min (2 1)4
z m
,max
,min
1.
1T L
T L
VVSWR
V
max .2m
z
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Ex1 Slotted line measurements yield a VSWR of 5, a 15 cm between successive voltage maximum, and the first maximum is at a distance of 7.5 cm in front of the load. Determine load impedance, assuming Z0 = 50 .
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Transmission lines of finite length (1)
• Consider the propagation on finite length lines which have load that are not impedance-matched.
• Determine net power flow.
Assume lossless line, at loadwe can write
0 0
0 0
( )
( ) .
j z j z
j z j z
V z V e V e
I z I e I e
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Input impedance (1)
Using and gives00 0 0
0
,L
VV V I
Z
0 0
0 0
( )( )
( )
j z j z
w j z j z
V e V eV zZ z
I z I e I e
00
0
VI
Z
0( ) .j z j z
Lw j z j z
L
e eZ z Z
e e
Using , we have0
0
LL
L
Z ZZ Z
00
0
cos sin.
cos sinL
wL
Z z jZ zZ Z
Z z jZ z
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Input impedance (2)
At z = -l, we can express Zin as
00
0
cos sin.
cos sinL
inL
Z l jZ lZ Z
Z l jZ l
I. Special case if then
II. Special case if then
; 0,1,2,.....2m
l m
.in L
l
Z Z
(2 1); 0,1,2,.....
4m
l m
20
( 1)2
( ) .4in
L
l m
ZZ l
Z
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Quarter wavelength lines
It is used for joining two TL lines with different characteristicimpedances
If
then we can match the junction Z01, Z02, and Z03 by choosing Quarter-wave matching
03 2 02 202
02 2 03 2
202
03
cos sincos sin
( 2) .
in
in
Z l jZ lZ Z
Z l jZ l
ZZ line
Z
01,inZ Z
02 01 03 .Z Z Z
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Complex loads
• Input complex impedance or loads may e modeled using simple resistor, inductor, and capacitor lump elements
For example, ZL = 100+j200 this is a 100 resistor in serieswith an inductor that has an inductance of j200 .
Let f = 1 GHz,
What if the lossless line is terminated in a purely reactive load?Let Z0 = R0 and ZL+jXL, then we have
that a unity magnitude, so the wave is completely reflected.
20032 .
jL nH
j
0
0
LL
L
jX RjX R
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Ex2 From the circuit below, find
a) Power delivered to load
Vs Z0=300
300
30060 V 100 MHz
2 m
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b) If another receiver of 300 is connected in parallel with the load, what is
b.1)
b.2) VSWR
b.3) Zin
b.4) input power
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c) Where are the voltage maximum and minimum and what are they?
d) Express the load voltage in magnitude and phase?
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Ex3 Let’s place another purely capacitive impedance of –j300 in parallel with two previous loads, find Zin and the power delivered to each receiver.
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Smith chart A graphical tool used along with Transmission lines and microwave circuit components
Circumventing the complex number arithmetic required in TL problems
Using in microwave design
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Smith chart derivation (1)
plane
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Smith chart derivation (2)
From
define
then
0
0
,LL
L
Z ZZ Z
0
LZzZ
1.1
LL
L
zz
Now we replace the load along with any arbitrary length of TL by Zin, we can then write
2
Re Im
1.1
,
j zL
ze
zj z r jx
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Smith chart derivation (3)
Re Im
Re Im
2 2Re Im
2 2Re Im
Im2 2
Re Im
11
11
1
(1 )
.(1 )
z
jr jx
j
r
jand jx
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Smith chart derivation (4)
2 2Re Im 2
2 2 2Re Im
1( )
1 ( 1)
1 1( 1) ( ) ( ) .
rr r
x x
We can rearrange them into circular equations,
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Normal resistance circle
2 2Re Im
1 1( )
2 4
Consider a normalized resistance r = 1, then we have
If r = 0, we have
so the circle represents all possible points for with || 1
2 2Re Im 1
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Normal reactance circle
2 2Re Im( 1) ( 1) 1
Consider a normalized resistance x = 1, then we have
The upper half represents positive reactance (inductance)
The lower half represents negative reactance (capacitance)
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Using the smith chart (1)
A plot of the normalized impedance The magnitude of is found by taking the distance from the center point of the chart, divided by the radius of the chart (|| = 1). The argument of is measured from the axis. Recall we see that Zin at Z = -l along the TL corresponds to
Moving away from the load corresponds to moving in a clockwise direction on the Smith chart.
2 ; 2jj zL Le e z
2 .z
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Using the smith chart (2)
Since is sinusoidal, it repeats for
every one turn (360) corresponds to
Note: Follow Wavelength Toward Generator (WTG)
Vmin and Vmax are locations where the load ZL is a pure resistance.
Vmax occurs when r > 1 (RL > Z0) at wtg = 0.25. Vmin occurs when r < 1 (RL < Z0) at wtg = 0.
je
2 2 ; 0,1,2,....
.2
z n n
nz
.2
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Using the smith chart (3)
The voltage standing wave ratio (VSWR) can be determined by reading the value of r at the = 0 crossing the constant-|L| circle.