lecture 8 terminated transmission lines - … · lecture 8 terminated transmission lines. ......

Lecture 8

Terminated Transmission Lines

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


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


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


• 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


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


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


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)


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


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


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


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


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,


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,


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


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


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


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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lecture 8 terminated transmission lines - … · lecture 8 terminated transmission lines. ......

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