lecture 8 terminated transmission lines - … · lecture 8 terminated transmission lines. ......
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Terminated TL• terminations of TLs cause reflections analogous to the reflections of
plane waves from material interfaces at normal incidence
2LECTURE 08: TERMINATED TRANSMISSION LINES
( 0) 0 00 0
0 0( 0) 0 0
0 0
0.5( )
0.5( )
z LL L
z L L L
V V V VV V Z I
V VI I V V Z IZ Z
• the incident and reflected voltage at the load (z = 0) can be expressed in terms of the total voltage and current at the load
inVGV
GZ
LV LZ
inZ Lz L
0( , )Z
inI
LI0
00
( ) , ( )z zVV z V e I z eZ
00
0( ) , z zVV z V e I e
Z
0z
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Reflection Coefficient in a Terminated TL• reflection coefficient Γ and SWR are defined in the same way as with
plane-wave reflection at normal incidence
• Γ is the ratio of reflected and incident voltage at the load
( 0) 0 0 0
0 0( 0) 0
( / )( / )
z L L L L
L L L Lz
V V V Z I V I ZV Z I V I ZV V
011LZ Z
3LECTURE 08: TERMINATED TRANSMISSION LINES
0
0
L
L
Z ZZ Z
( 0)
( 0)
zLL
L z
VVZI I
• return loss – shows how much of the incident power is “lost” to transmission (we want large RL, RL = 10 dB, ≈ 90% of power is delivered to the load)
010 10
0RL 20log 20log | |, dBV
V
LZ
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Reflection Coefficient at Generator and Input Impedance
( ) 20( )
( ) 0
Lz L L
g z L Lz L
V V e eV V e
• one can define the reflection coefficient at the generator’s terminals as well
• the relation between Γg and Zin is the same as for Γ and ZL
011
gin
gZ Z
4LECTURE 08: TERMINATED TRANSMISSION LINES
LZ
LZinZ
0z z L
0Zg
L
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at generator at load
at generator
Standing Wave Ratio in a Terminated TL• the relation between the SWR and Γ is derived in the same manner
as for plane waves1 | | 11 | |
SWR
max
min
| ( ) || ( ) |V zSWRV z
5LECTURE 08: TERMINATED TRANSMISSION LINES
• locations of the voltage minima (current maxima) are found in the same way as for plane waves [see Lecture 05]
l
min,1lmin,2lmin,3lmin,4lz
load
min, (2 1) , 0,1,4 4g g
nl n n
transmission line
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Slotted Line
allows for sampling of the E field along a terminated TL
allows to determine the load impedance by measuring• E-field’s envelope minima and maxima• the position of the first minimum with respect to load terminals
6LECTURE 08: TERMINATED TRANSMISSION LINES
waveguide slotted line[Pozar, Microwave Engineering, 3rd ed.]
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Principles of Slotted Line Measurement Procedure
max
min
| ( ) | 1 | || ( ) | 1 | |V zSWRV z
1 | |1
SWRSWR
locations of voltage minima respective to load terminals are given by
min,2 (2 1) , 0,1,nl n n st
minfor 0 (1 minimum) and ,2gl
min2 l
load impedance is calculated as
0 01 1 | |1 1 | |
j
L jeZ Z Ze
7LECTURE 08: TERMINATED TRANSMISSION LINES
min, (2 1) , 0,1,4 4g g
nl n n
4 2g
calculate reflection coefficient magnitude from SWR
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Visual Aid for Slotted Line Example
8LECTURE 08: TERMINATED TRANSMISSION LINES
z
z
l
l
effective line termination
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toward generatortoward load
toward generatortoward load
Power Delivered to Load over a TL
• complex voltage and current at the load
( 0) 0
0( 0)
0
(1 )
(1 )
L z
L z
V V V
VI IZ
• complex power at the load0.5 , WL L LP V I
• time-average (active) power delivered to the load
( ) Re{ } Re{0.5 }L av L L LP P V I
9LECTURE 08: TERMINATED TRANSMISSION LINESElecEng4FJ4
Power Delivered to Load over a TL – 2
2 2
20 0
imaginary0 0number
| | | |1 1 1 (1 | | )2 2L L L L
V VP V I PZ Z
20 2
0( ) (1 | || )|
2L avV
ZP
20
0
22 0
0
| |1| |1( )2
| |2L av
VZ Z
P V
incident power reflected power
(( ( )) )r avL iav avP PP
power delivered to load
(assume Z0 is real, i.e., TL is low-loss)
10LECTURE 08: TERMINATED TRANSMISSION LINES
2( ) (1 | |( ))i aL av vP P
• power balance
iP
210 10
( )10log 10log 1 | |( )
L av
i av
PTLP
• transmission loss
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Power at the Input of the TL
complex voltage and current at input terminals of TL
( ) 0
0( )
0
( )
( )
L Lin z L
L Lin z L
V V V e eVI I e eZ
2
0
0
| |12 2
L j L L j L L j L L j Lin in in
VP V I e e e e e e e eZ
22 2 2 2 20
0 imaginary
| |1 ( | | )2
L j L j L Lin
VP e e e eZ
complex power at input terminals
power delivered to input of TL (assuming Z0 is real)
2 22( ) Re ( ) | |in av in iL
avLe eP P P
11LECTURE 08: TERMINATED TRANSMISSION LINES
iP
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at generator
Power Loss in a Low-loss TL (Z0 assumed real)
2 2 2( ) Re ( ) | |L Lin av in i avP P P e e
12LECTURE 08: TERMINATED TRANSMISSION LINES
loss( ) ( ) 0in av L avP P P
2( ) ( ) (1 | | )L av i avP P
the power dissipated in TL is the difference between the input power and the power delivered to the load
loss ( ) ( )in av L avP P P
2 2 2loss ( ) | | (1 ) 1L L
i avP P e e
loss-free case
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delivered to load
delivered by generator
Input Impedance Zin of Terminated TL (review)
• Zin describes the equivalent load that the loaded TL presents at the generator terminals
• it allows for simple equivalent-circuit models
• it is a function of the load ZL, the TL length L, Z0, and γ
13LECTURE 08: TERMINATED TRANSMISSION LINES
GZ
inZinV
inI
GV
z L
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Input Impedance of Terminated TL (review) – 2
at the location of the generator, z = −L, the input impedance is
0( ) 0
0
tanh( )tanh( )
Lin z L
L
Z Z LZ Z ZZ Z L
for a lossless TL0 and 0 0,R G j
00
0
tan( )tan( )
Lin
L
Z jZ LZ ZZ jZ L
14LECTURE 08: TERMINATED TRANSMISSION LINES
or2
0 211
j L
in j LeZ Ze
Zin is a periodic function of βL – it repeats itself every half-wavelength, i.e., making a line half-wavelength shorter or longer does not change its input impedance
cosh sinhcosh sinh
x
xe x xe x x
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Input Impedance of Short-circuited Loss-free TL
00 tan( )L
in ZZ jZ L
• Zin is purely reactive • if L = λ/4, 3λ/4, …, input reactance is infinity (like an open circuit)• if L = λ/2, λ, …, input reactance is 0 (like a short circuit)• there is periodicity with a period of λ/2• for L < λ/4, reactance is inductive,
15LECTURE 08: TERMINATED TRANSMISSION LINES
Im 0inZ
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Input Impedance of Open-circuited Loss-free TL
00 0
0
tan( )lim cot( )tan( )L L
Lin Z Z L
Z jZ LZ Z jZ LZ jZ L
• Zin is purely reactive • if L = λ/4, 3λ/4, …, input reactance is 0 (like a short circuit)• if L = λ/2, λ, …, input reactance is infinite (like an open circuit)• there is periodicity with a period of λ/2• for L < λ/4, reactance is capacitive,
16LECTURE 08: TERMINATED TRANSMISSION LINES
Im 0inZ
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Input Impedance of a TL of length L = nλ/2
00/2
0
tan( )tan( )
Lin LL
L
Z jZ nZ Z ZZ jZ n
• every λ/2, a TL reproduces the load impedance ZL regardless of its own characteristic impedance Z0
• this was already observed in the particular cases of short-circuited (ZL = 0) and open-circuited (ZL → ∞) loss-free TLs
17LECTURE 08: TERMINATED TRANSMISSION LINES
tan( ) 0n
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Input Impedance of a TL of length L = λ/42
0 00/4
0
tan( / 2)tan( / 2)
Lin L
L L
Z jZ ZZ ZZ jZ Z
this lines are used as quarter-wave impedance transformers – the characteristic impedance of the line is chosen so that
0 in LZ Z Z
one can obtain any desired input impedance for a given load and achieve impedance match at a given frequency
the same result holds for a line of length
, 1,2,4 2
L n n
18LECTURE 08: TERMINATED TRANSMISSION LINES
known load
desired input impedance
tan( / 2)
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Input Impedance of a Matched/Infinite TL
we say that a TL is terminated in a matched load if(no reflection from load!)
0LZ Z
0inZ Z
the result does not depend on the length L of the line
Zin of a matched TL is identical to Zin of an infinitely long TL and both are equal to Z0
• there is no reflected wave in a matched or an infinite TL (Γ = 0)
00
0
0
( )( ) .( )
z
z
V eV zZ z Z constI z V e
Z
• the impedance is the same regardless of the position along the TL
19LECTURE 08: TERMINATED TRANSMISSION LINESElecEng4FJ4
Summary a TL is said to be matched if ZL = Z0; then its input impedance is
simply Z0 regardless of its length
20LECTURE 08: TERMINATED TRANSMISSION LINES
if ZL ≠ Z0, the input impedance depends on the line’s length L, on its propagation constant γ, on Z0 and on ZL
0( ) 0
0
tanh( )tanh( )
Lin z L
L
Z Z LZ Z ZZ Z L
the ratio reflected-to-incident power is equal to |Γ|2 (reflection loss is this ratio in dB with a minus sign)
the ratio delivered-to-incident power is equal to (1 − |Γ|2) (transmission loss is this ratio in dB with a minus sign)
TL of length L = nλ/2 has always input impedance equal to the load
TL of length L = λ/4 can serve as a simple narrow-band impedance matching network provided 0 in LZ Z Z
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