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

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Page 1 IIT Bombay Course Code : EE 611 Department: Electrical Engineering Instructor Name: Jayanta Mukherjee Email: jayanta@ee.iitb.ac.in Lecture 9 EE 611 Lecture 9 Jayanta Mukherjee Page 2 IIT Bombay Topics Covered Properties of S

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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 9 Jayanta Mukherjee Page 1

Lecture 9

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SLIDE 2

IIT Bombay

Topics Covered

  • Properties of S parameters (Contd..)
  • Properties of 2 port networks
  • Signal flow graphs

EE 611 Lecture 9 Jayanta Mukherjee Page 2

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SLIDE 3

S Matrix for a Passive Reciprocal Lossless 2-port Junction

EE 611 Lecture 9 Jayanta Mukherjee Page 3

[ ] [ ] [ ]

      =         + + + + =       ⋅         = ⋅

  • =

     =

  • 1

1

2 12 2 22 12 * 22 11 * 12 22 * 12 12 * 11 2 12 2 11 22 21 12 11 * 22 * 12 * 12 * 11 22 21 12

S S S S S S S S S S S S S S S S S S S S U S S S S

H

S unitary be must S then lossless, is device the If device. the describe to lues complex va three leaves This symmetric. be must S since S S , reciprocal is device the If S S is device port 2 general a

  • f

matrix S The

21 12 11

IIT Bombay

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SLIDE 4

S Matrix for a Passive Reciprocal Lossless 2-port Junction

EE 611 Lecture 9 Jayanta Mukherjee Page 4

( ) ( ) ( )

( )

( )

frequency given a at system the ze characteri which θ and θ , S parameters

  • nly three

are there and that so

  • r

, gives this g Simplifyin S : get we equation third the From S S and S and S S define we ips, relationsh phase the establish To S and S : get we S : equations 2 first the From

2 1 11 2 1 12 12 12 12 22 11 11 12 11 11

1 2 2 2 1 1

12 2 1 2 12 12 1 12 2 1 12 12 2 1

2 12 22 11 12 * 22 11 12 11 2 11 22 2 12 2 22 2 12 2

+ − + = − = − =

  • =

+ = +

  • =

= =

  • −

= = = + = +

  • +

− − −

n e e e e e S e S e S e S S S S e e S e S S S S S

jθ θ θ j j

  • θ

θ j j j j j * j j j

π θ θ θ

θ θ θ θ θ θ θ θ θ

IIT Bombay

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SLIDE 5

Example of direct calculation of S Matrices

EE 611 Lecture 9 Jayanta Mukherjee Page 5 IIT Bombay jX Z1 Z2 Port 1 Port 2 jX Z1 Z2 Port 1 Port 2 a1 b1 a2 b2 Z2

  • To measure S11 and S21 we must terminate port 2 in a

matched load of impedance Z2

  • To find S11, terminate port 2 with Z2. Then we have

S11 = (b1/a1)

  • Consider the S matrix of the network shown below
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SLIDE 6

Example of direct calculation of S Matrices (2)

EE 611 Lecture 9 Jayanta Mukherjee Page 6 IIT Bombay

circuit port

  • 2

, reciprocal lossless, a for should it as S that can verify We S find to process same repeat the can We a b S so to identical is this port, same at the are a and b Since

11 22 1 1 11 1 1 22 2 1 2 1 1 2 1 2 1 1 1 1

. S Z jX Z Z jX Z Z jX Z Z jX Z V V /V V

in

  • =

+ + − + =

  • +

+ − + = = = =

  • +

− +

Γ

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SLIDE 7

Derivation of S21

EE 611 Lecture 9 Jayanta Mukherjee Page 7 IIT Bombay

( ) ( ) ( ) ( ) ( )

jX Z Z Z Z S Z Z a b S Z b S a S b a Z b a Z Z Z I I S

a

+ + = − = = ⇒ = − = =

  • −

− = − ⇒ − = − =

  • =

=

  • =

1 2 2 1 11 1 2 1 2 21 2 2 11 1 11 2 2 2 1 1 1 2 1 2 1

2 1 1 1 1

2

1 2 1 1 2 1 2 1 2 12 21

Z 1 : have we a fact that with the along matched, is port two when a b fact that the using Now I I

  • r

get we

  • pposite

but equal be must 2 and 1 ports entering current the ince

  • hms

by Z d terminate remains 2 port a b S S now calculate We

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SLIDE 8

Results Summary

EE 611 Lecture 9 Jayanta Mukherjee Page 8 IIT Bombay

( )

nations port termi about the careful be to have We 4 1 S that confirm can We jX Z Z Z Z 2 S symmetry By jX Z Z Z Z 2 S derived also We symmetry By jX Z

  • Z

S : derived we slides previous In the

2 2 2 1 2 1 2 11 2 12 2 21 21 2 1 2 1 12 2 1 2 1 21 1 2 2 1 22 1 2 1 2 11

  • +

+ = − = =

  • =

+ + =

  • +

+ =

  • +

+ + − =

  • +

+ + =

  • X

Z Z Z Z S S S jX Z Z jX Z Z S jX Z Z

jX Z1 Z2 Port 1 Port 2

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SLIDE 9

Input reflection coefficient of a 2 port network

EE 611 Lecture 9 Jayanta Mukherjee Page 9 IIT Bombay

2 22 1 21 2 22 1 21 2 2 12 1 11 2 12 1 11 1 2 2 L L 2 L 2 2 2 L

that us ls matrix tel S d generalize Our Z

  • Z

with b a at fixed is b and a between ip relationsh the , in Z d terminate is 2 port Since b Γ S a S a S a S b b Γ S a S a S a S b Z Z

L L L

+ = + = + = + =

  • +

= Γ Γ =

  • 2 Port Network

Z1 Z2 ZL a1 b1 a2 b2

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SLIDE 10

Input reflection coefficient

EE 611 Lecture 9 Jayanta Mukherjee Page 10 IIT Bombay

L L L

S S S S a b a Γ − Γ + = = Γ

       Γ Γ + =

  • Γ

=

  • 22

21 12 11 1 1 in L 22 21 L 12 11 1 1 1 1 1 22 21 2

1 a b : now is port two loaded the

  • f

t coefficien reflection input The S

  • 1

S S S a b :

  • f

in terms get

  • equation t

first

  • n the

work Now S

  • 1

S b get

  • equation t

second the solve can We

2 Port Network Z1 Z2 ZL a1 b1 a2 b2

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SLIDE 11

Attributes of Input Reflection Coefficient for Loaded 2-port Network

EE 611 Lecture 9 Jayanta Mukherjee Page 11 IIT Bombay

in

Γ

1 2 ZL

     

22 21 12 11

S S S S

  • ut

Γ

1 2 ZS

     

22 21 12 11

S S S S

S 11 12 S 21 22

  • ut

12 11 in L 11 in 22 21 L 12 11 in

S

  • 1

S S S t Coefficien Reflection Output device) l (unilatera for S load) (matched for S : Properties 1 S S S : t Coefficien Reflection Input Γ Γ + = Γ = = Γ

  • =

Γ = Γ

  • Γ

− Γ + = Γ S S

L

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SLIDE 12

Application: Matching using an attenuator

EE 611 Lecture 9 Jayanta Mukherjee Page 12 IIT Bombay

: by given is 5) (Chapter attenuator ideal an for matrix parameter S The power.

  • utput

reduced

  • f

cost at the match broadband a provide can power) the reducing (device attenuator An

  • hm).

50 (Z matching poor a with generator a Consider

S ≠

ZS

     

21 12

S S

+

  • Z1

Z2 Γs Γout ES

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SLIDE 13

Application: Matching using an attenuator

EE 611 Lecture 9 Jayanta Mukherjee Page 13 IIT Bombay

1

2 2 21 12 21 2 12 12

<< ≈ = ⇒ = + = + =       =

S S

  • ut

S att

S S S S S S Γ Γ Γ Γ Γ Γ Γ

21 S 11 12 S 21 22

  • ut

21 10

S for S

  • 1

S S S : is circuit attenuator generator by the provided match The (dB). S

  • 10log

L

  • f

n attenuatio an provides and matched is attenuator An

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SLIDE 14

Signal Flow Graphs

EE 611 Lecture 9 Jayanta Mukherjee Page 14 IIT Bombay

  • S-parameters were introduced in the previous lectures to

represent linear N-port networks

  • As we design microwave circuits we will need to analyze bigger

circuits realized with multiple building blocks Flow graph techniques will provide us:

  • an analysis technique applicable to S parameters
  • the means to visualize the power flow in a circuit
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SLIDE 15

Signal Flow Graphs

EE 611 Lecture 9 Jayanta Mukherjee Page 15 IIT Bombay

  • Each variable is designated as a node
  • The S parameters and reflection coefficients are represented

by branches

  • Branches enter dependent variable nodes and emanate from

independent variable nodes. The independent/dependent variable nodes are the incident/reflected waves respectively.

  • A node is equal to the sum of the branches entering it

a2 b2 a1 b1

     

22 21 12 11

S S S S

Z1 Z2 a1 b2 b1 a2 S21 S22 S11 S12

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SLIDE 16

Signal Flow Graph Examples

EE 611 Lecture 9 Jayanta Mukherjee Page 16 IIT Bombay

  • A length of transmission line of length l satisfies b2=a1e-jβl,

b1=a2e-jβl so the corresponding signal flow graph is

  • Suppose we have a load with reflection coefficient ΓL connected

to port 2. Then we know a2 = ΓLb2 and we can include this as a “branch” in a signal flow graph.

l a1 b1 a2 b2 Z0 a1 b1 a2 b2 e-jβl e-jβl

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SLIDE 17

Loaded Port Network driven by a Generator

EE 611 Lecture 9 Jayanta Mukherjee Page 17 IIT Bombay

  • A basic circuit we need to be analyzed is given below.

ΓL ΓS ZS Z0 Z0 ZL

     

22 21 12 11

S S S S

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SLIDE 18

Flow Graph for a Generator

EE 611 Lecture 9 Jayanta Mukherjee Page 18 IIT Bombay

  • A basic circuit we need to be analyzed is given below.

( )

S S S S b S S S S S S S S S g

b b Z Z Z Z b E Z Z Z a Z Z b Z E Z Z a b a Z Z Z E b a Z Z I Z Z E Z V

S S

Γ + = + − + + =       − + =       + − − = + − = =

Γ 1 1 1 1 1 1 1 1 1 1 S 1 S g

1 1 Z by divide and Z I

  • E

V : from Starting          

ZS Z0 Z0 ZL

     

22 21 12 11

S S S S

+

  • +
  • Vg

a1 b1 a2 b2 ES bS a1 b2 a2 b1 b2 S21 S22 S12 S11 ΓL ΓS 1 1

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SLIDE 19

Flow Graph for a Generator

EE 611 Lecture 9 Jayanta Mukherjee Page 19 IIT Bombay

  • A basic circuit we need to be analyzed is given below.

( )

S S S S b S S S S S S S S S g

b b Z Z Z Z b E Z Z Z a Z Z b Z E Z Z a b a Z Z Z E b a Z Z I Z Z E Z V

S S

Γ + = + − + + =       − + =       + − − = + − = =

Γ 1 1 1 1 1 1 1 1 1 1 S 1 S g

1 1 Z by divide and Z I

  • E

V : from Starting          

ZS Z0 Z0 ZL

     

22 21 12 11

S S S S

+

  • +
  • Vg

a1 b1 a2 b2 ES bS a1 b2 a2 b1 b2 S21 S22 S12 S11 ΓL ΓS 1 1

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SLIDE 20

Analysis Method for Flow Graphs

EE 611 Lecture 9 Jayanta Mukherjee Page 20 IIT Bombay

Simplification of signal flow graphs can be performed using two methods:

  • “reduction” techniques – good for simple systems, not for

complex systems

  • “Mason’s gain rule” - Mathematically involved but convenient

for large systems

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SLIDE 21

Reduction Method

EE 611 Lecture 9 Jayanta Mukherjee Page 21 IIT Bombay

T1 T1 T1T2 T1 T2 T1+T2 T3 T1 T2 T4 T1T3 T1T2 T4 T1 T4 T2 T3 T4 T2/(1-T3) T1/(1-T3)

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SLIDE 22

Reduction Method (2)

EE 611 Lecture 9 Jayanta Mukherjee Page 22 IIT Bombay T4 T1 T2 T3 T2 T3 T4 T1 T1 T1 T4 T1 T2 T3 T2 T3 T4 T1 T1 T1

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SLIDE 23

Example using Reduction Method

EE 611 Lecture 9 Jayanta Mukherjee Page 23 IIT Bombay

ZL Z0 Z0 I1

     

22 21 12 11

S S S S

Γin S12 S22 S21 S11 a1 a1 b1 b1 a2 ΓL b2 1 1 a1 b1 a2 b2 S12 S22 S21 S11 a1 b1 a2 ΓL b2 1 1 ΓL a2 node split a1 b1 S12 S21 S11 a1 a1 b1 b1 a2 ΓL b2 1 1 1-ΓLS22 a1 b1 Γin

L L in

S S S S a b Γ Γ Γ

22 12 21 11 1 1

1− + = =

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SLIDE 24

Relative Advantages of Various Small-Signal Parameters

EE 611 Lecture 9 Jayanta Mukherjee Page 24 IIT Bombay

The linear response of a N-port circuit can be represented by various small-signal parameters:

  • S parameters (used for experimental and final data)
  • Z parameters (facilitate computation for circuits in series)
  • Y parameters (facilitate computation for circuits in shunt)
  • ABCD parameters (facilitate computation for circuits in cascade)
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SLIDE 25
  • Z parameters are useful for N-port networks connected in

series Z1+2=Z1+Z2

  • Y parameters are useful for N Port networks in parallel

Y1//2=Y1+Y2

Z and Y parameters

EE 611 Lecture 9 Jayanta Mukherjee Page 25 IIT Bombay

+

  • Port 1

v1 +

  • v2

Port 2 i1 i2 2 Port Network + +

  • Port 2

Port 1 v1 v2 i1 i2 2 Port Network #1 2 Port Network #2 +

  • Port 1

v1 +

  • v2

Port 2 i1 i2 2 Port Network + +

  • Port 2

Port 1 v1 v2 i1 i2 2 Port Network #1 2 Port Network #2

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SLIDE 26

ABCD Parameters

EE 611 Lecture 9 Jayanta Mukherjee Page 26 IIT Bombay +

  • Port 1

v1

  • v2

Port 2 i1 i2 2 Port Network + +

  • Port 2

Port 1 v1 v2 i1 i2 2 Port Network #1 2 Port Network #2

        =                 =        

  • ×

=

     =             =      

  • ×

2 2 2 2 1 1 2 2 2 2 1 1

i v M i v D C B A i v i v M i v D C B A i v definition parameter ABCD Normalized M M M cascade in networks port

  • 2

for two Useful measured directly be cannot parameters ABCD definition parameter ABCD

2 1 2 1

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SLIDE 27

Sample ABCD Matrices

EE 611 Lecture 9 Jayanta Mukherjee Page 27 IIT Bombay

     

    

  • 1

1 Y 1 becomes matrix ABCD the two, and

  • ne

ports between Y admittance shunt element lumped a For 1 becomes matrix ABCD the two, and

  • ne

ports between Z impedance series element lumped a For Z

Z i1 i2 v1 v2 +

  • +
  • i1

i2 v1 v2 +

  • +
  • Y
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SLIDE 28

Sample ABCD Matrices (2)

EE 611 Lecture 9 Jayanta Mukherjee Page 28 IIT Bombay

tan j S define we if is matrix the d, length electrical

  • f

line ion transmiss

  • f

piece a For θ θ θ θ β θ =       − =       =

  • 1

1 1 1 cos sin sin cos

2

S Y S Z S jY jZ θ

Z0 l

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SLIDE 29

Sample ABCD Matrices (2)

EE 611 Lecture 9 Jayanta Mukherjee Page 29 IIT Bombay

( )

1 C B

  • D

A requires S S networks reciprocal for that Note and and table) conversion a provides 211 page (pozar follows as parameters Z and S to parameters ABCD relate can We : by related are parameters ABCD normalized and normalized

  • n

12 21

= =

  • +

+ + + + = + + + = + + + = + + + + =

     =        

  • D

C B A D C

  • B

A S D C B A S D C B A C B

  • D

A S D C B A D

  • C
  • B

A S D CZ Z B A D C B A N

22 21 12 11

2 2 /

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SLIDE 30

Sample ABCD Matrices (2)

EE 611 Lecture 9 Jayanta Mukherjee Page 30 IIT Bombay

impedance. in changes any cause not will line h wavelengt half since matrix identity simply the is This : be then would lines /4 two

  • f

cascade a for matrix ABCD The inverted are voltage and current normalized : inverter a is This M : is line ve quarter wa single a for matrix ABCD the degrees, 90 length electrical has line ve quarter wa the Since lines n ransmissio velength t quarter wa two

  • f

cascade a Consider

     − =       − − =            

     =         = ⇒       =         =

  • 1

1 1 1 90 cos 90 sin 90 sin 90 cos Y Z Y Z jY jZ jY jZ j D C B A M jY jZ jY jZ λ

   