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1RS
ENE 428Microwave
Engineering
Lecture 5 Discontinuities and the manipulation of transmission lines problems
2
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
3
Wave reflection at discontinuities• To satisfy boundary conditions between two
dissimilar lines
• If the line is lossy, Z0 will be complex.
4
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
5
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
6
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
7
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
8
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
9
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.
10
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
11
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
12
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
13
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
14
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 .
15
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
16
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
17
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
18
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
19
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
20
Ex2 From the circuit below, find
a) Power delivered to load
Vs Z0=300
300
30060 V 100 MHz
2 m
21
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
22
c) Where are the voltage maximum and minimum and what are they?
d) Express the load voltage in magnitude and phase?
23
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.
24
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
25
Smith chart derivation (1)
plane
26
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
27
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
28
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,
29
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
30
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)
31
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
32
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
33
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.
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