IIT Bombay Course Code : EE 611 Department: Electrical Engineering - - PowerPoint PPT Presentation

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IIT Bombay Course Code : EE 611 Department: Electrical Engineering - - PowerPoint PPT Presentation

May 18, 2023 205 likes •475 views

Page 1 IIT Bombay Course Code : EE 611 Department: Electrical Engineering Instructor Name: Jayanta Mukherjee Email: jayanta@ee.iitb.ac.in Lecture 4 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 2 Impedance of Loaded Transmission

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SLIDE 1

IIT Bombay

Course Code : EE 611 Department: Electrical Engineering Instructor Name: Jayanta Mukherjee Email: jayanta@ee.iitb.ac.in

EE 611 Lecture 4 Jayanta Mukherjee Page 1

Lecture 4

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SLIDE 2

Impedance of Loaded Transmission Lines

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 2

The impedance along a transmission line at position x is given by

x at located is V and V for plane reference The , : are I(x) current and V(x) voltage complex the where

  • =

− = + = =

+ − + + γx

  • x

γx

  • x
  • e

Z V e Z V x I e V e V V(x) x I x V x Z ) ( , ) ( ) ( ) (

γ γ Z0 γ=j β+α

Z(d) V+ V- l l d x

ZL

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SLIDE 3

Impedance Calculation

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 3

( ) ( )

) tan( ) tan( 2 1 ) ( ) ( ) ( ) ( ) ( l Z Z l Z Z Z ) I(x ) V(x l) Z( e I Z Z V e I Z Z e Z V e Z V l I I e V e V I Z l V V Z I V l I l V l Z

L L l L L l L L l l L l l L L L L L L

γ γ

γ γ γ γ γ γ

+ + = = = = − − = − + = − = = + = = = = = = =

+ − + + − − + − − +

3 and 2 Eqn in amplitudes V and V wave incident the ng Substituti 2 1 V

  • btain

we V and V amplitudes wave incident the for Solving (3) (2) have we solutions wave current and voltage the from Now Z impedance load the is l x position the at impedance The

  • L
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SLIDE 4

Lossless Case

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 4

For a loss free line we have γ=jβ and the impedance reduces to:

) tan( ) tan( ) ( d jZ Z d jZ Z Z d Z

L L

β β + + =

The impedance Z is then periodic function of frequency and position:

  • In terms of the electrical angle θ=βd the impedance Z repeats

every period π

  • In terms of position d it repeats every half wavelength λ/2 since

we have βd=(2π/λ)d

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SLIDE 5

Impedance of a Shorted Transmission Line

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 5

Z0 γ=j β

Z(d) V+ V- l l d x

ZL

) tan( ) tan( ) ( d jZ Z d jZ Z Z d Z

L L

β β + + =

  • For a short circuited line, ZL=0 and we have Z(d)=jZ0tan(βd)
  • For an open circuited line, and we have Z(d)=-jZ0cot(βd)
  • For a matched load, ZL=Z0, and we have Z(d)=Z0 for all values
  • f d

∞ =

L

Z

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SLIDE 6

Impedance of a Shorted Transmission Line

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 3 Jayanta Mukherjee IIT Bombay Page 6

  • The input impedance alternates between shorts (Z=0)

and opens ( )

  • The short is transformed into an open for d=λ/4

∞ = Z

Capacitance Inductance d Im[Z(d)] λ/2 λ λ/4 3/4λ

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SLIDE 7

Matched Line

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 7

  • When ZL=Z0 , the load is said to be matched

Z0 ZL Z0

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SLIDE 8

Impedance Matching vs Conjugate Impedance Matching

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 8

  • Impedance matching should be distinguished from conjugate

impedance matching ZG=ZL

*used for maximum power transfer

  • Both load matching and conjugate impedance matching can

happen when ZG = ZL

*= Z0 ZG ZL Z0 Z0 Z0

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SLIDE 9

Quarter Wave Transformer for a Resistive Load RL

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 9

  • For a line of one quarter of a wavelength (d=λ/2) we have

L G L 2 G in L 2 L L in

R R Z use we if R Z R Z : generator the to matched then is input line The R Z d) tan( jR Z d) tan( jZ R Z /4) Z(d Z is impedance line the /2) tan( since and = = = = + + = = = ∞ = = = β β λ π π λ λ π β 2 4 2 d

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SLIDE 10

Quarter Wave Transformer for a Resistive Load RL

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 10

RG RL β π/2 d= λ/4 d= Ζ0=(RGRL)

1/2

The impedance matching is only realized at the frequency where the transmission line length is quarter wavelength

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SLIDE 11

Reflection Coefficient

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 11

l) (x where (x) : wave incident the to wave reflected the

  • f

ratio the as x position a at t coefficien reflection the define We

L

= = = = = = =

− − + − − + − + − − + −

Γ Γ Γ Γ

γ γ γ γ γ γ d L d L L d l x x x

e e V V e V V e V V e V e V

2 2 ) ( 2 2

load V+(x) V-(x) d x=-d Γ(x) Γ L

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SLIDE 12

Reflection Coefficient Along a Line

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 12 α β α β

Γ Γ Γ Γ Γ

j d j L

e e

L L d 2j

  • L

define we if e (d) : as written be can t coefficien reflection the line less loss a For = = =

− ) 2 (

Toward the load Toward the generator α −2β d Γ(d) for d>0 Re[Γ] Im[Γ] | ΓL ΓL | e j α = Γ plane

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SLIDE 13

Reflection Coefficient Along a Line

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 14

  • As we move along the line, Γ(d) moves along a circle of radius

|ΓL|

  • The reflection coefficient Γ(d) rotates clock wise as d increases

and we move towards the generator

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SLIDE 14

Power Dissipated at the Load

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 15

{ } { } ( )

power) reflected minus (incident Z V P : load the by dissipated Power

L

) 1 ( Re Re 2 1 Re

2 2 2 2 2 * * L amplitude rms

P P P Z V Z V Z V Z V Z V V V VI VI Γ − = − = − = − =                   − + = = =

+ − + − + − + − + − +

γ= jβ Z0

L

ZL Γ

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SLIDE 15

Relation between impedance and Reflection Coefficient

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 16

load V+(x) V-(x) d x=-d Γ(x) ΓL ZL Z(x) γ= j β Z0

) reflection (no have we and Z Z load matched a for : Note : load the at ly particular and Z Z(x) Z

  • Z(x)

(x) : Inverting

L L L

= = + − = + = − + = − + = =

− − + − − +

Γ Γ Γ Γ Γ

γ γ γ γ

) ( 1 ) ( 1 ) ( ) ( ) ( Z Z Z Z x x Z e Z V e Z V e V e V x I x V x Z

L L x x x x

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SLIDE 16

The Smith Chart

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 17

Bilateral Transform connecting the impedance Z and the Reflection coefficient Γ . The smith chart maps the x-plane on the Γ plane

jx r z z Z Z Z Z Z Z Z Z + = = + − = + − = + − = Z Z z with 1 1 1 1 Γ

0.5 1 1 0.5

  • 0.5
  • 1

j 0.5 1 1

  • 1

0.5

  • 0.5

1

  • 1
  • j

Open Short x r

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SLIDE 17

Extended Smith Chart

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 18

For negative resistance r<0 we have |Γ|>1

jx r z z Z Z Z Z Z Z Z Z + = = + − = + − = + − = Z Z z with 1 1 1 1 Γ

0.5 1 1 0.5

  • 0.5
  • 1

j 0.5 1 1

  • 1

0.5

  • 0.5

1

  • 1
  • j

Open Short x r

  • 0.5
  • 0.5

For r=-1 (Re{Z}=-50 ohms) we have Γ=infinity

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SLIDE 18

Active and Passive Load

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 19

Im[Γ

L]

Re[Γ

L]

| Γ

L |

Active Devices >1 | Γ

L |

Passive Devices <1

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SLIDE 19

Inductance and Capacitance on Smith Chart

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 20

Locus of the reflection coefficient for an inductor and a capacitor In a Z Smith chart

ω=0 Open Short infinity ω= Short Open ω= ω=0 infinity

Γ plane Z smith chart

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SLIDE 20

Y Smith Chart

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 21

Open Short Short Open

1 1 1 1 1 1 1 1 1 1 1 1 + − − = + − − = + − = + − = + − = Y Y Y Y y y y y z z Z Z Z Z

L L L L L

: Y

  • f

terms in expressing by

  • btained

be can chart smith

  • Y

The Γ Γ

Γ plane

  • Γ plane

Normal Rotated

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SLIDE 21

Y Smith Chart

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 22

  • The Y Smith Chart is obtained by inverting the Z smith chart
  • In the rotated Y-Smith Chart the short and open are exchanged
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SLIDE 22

Impedance and Admittance Smith Charts

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 23 Z Smith Chart ZY Smith Chart Y Smith Chart Open Short Open Open Short Short Γ plane Γ plane −Γ plane Short Open Y Smith Chart Γ plane

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SLIDE 23

Voltage Standing Wave Ratio (VSWR)

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 24

The voltage wave inside a transmission line can be written using

( )

( ) ( )

Γ Γ

β β β β

  • 1

V V Γ 1 V V between varies voltage The Γe 1 e V e V e V V(z)

min max z j2 j z j z j

  • +

+ − + − + + −

= + = + = + = =

z

V V

The voltage standing wave ratio (VSWR) is defined as:

Γ Γ − + = = 1 1

min max

V V VSWR

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SLIDE 24

Voltage Standing Wave Ratio (VSWR)

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 25

  • A perfect matching (|Γ|=0) corresponds to VSWR of 1
  • VSWR should be less than 2
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SLIDE 25

Calculation of VSWR with Smith Chart

EE 611 Lecture 1 EE 611 Jayanta Mukherjee Lecture 1 EE 611 Lecture 4 Jayanta Mukherjee IIT Bombay Page 26

Γ α Read VSWR