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

IIT Bombay

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

EE 611 Lecture 8 Jayanta Mukherjee Page 1

Lecture 8

IIT Bombay

Topics Covered

• Impedance Matrix
• Properties of Loss Less and Reciprocal Devices
• Scattering Parameters

EE 611 Lecture 8 Jayanta Mukherjee Page 2

Impedance Matrix

EE 611 Lecture 8 Jayanta Mukherjee Page 3

ZI I I I Z Z Z Z Z Z Z Z Z Z Z Z Z V V V V

N NN N N N N N N N N N N

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

− − −

               

2 1 2 1 ) 1 ( 2 ) 1 ( 1 ) 1 ( 3 2 22 21 1 12 11 2 1

IIT Bombay

N Port Device IN VN + - I1 V1

• +

I2 V2

• +

ZN Z1 Z2

Admittance Matrix

EE 611 Lecture 8 Jayanta Mukherjee Page 4

1

• Z

Y Property = =                                 =                 =

− − −

YV V V V Y Y Y Y Y Y Y Y Y Y Y Y Y I I I I

N NN N N N N N N N N N N

               

2 1 2 1 ) 1 ( 2 ) 1 ( 1 ) 1 ( 3 2 22 21 1 12 11 2 1

IIT Bombay

N Port Device IN VN + - I1 V1

• +

I2 V2

• +

ZN Z1 Z2

Reciprocal and Non Reciprocal Materials

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• For non-reciprocal medium
• semiconductor + power supply
• plasma
• Ferrite + magnetic field

The dielectric constant ε and permeability μ are non- reciprocal matrix and we have Zij ≠ Zji

• For reciprocal medium we have Zij = Zji and [Z] t = [Z]

Reciprocal 2 Port network

EE 611 Lecture 8 Jayanta Mukherjee Page 6 IIT Bombay

• Circuit representation of a 2 port reciprocal network

     

22 12 12 11

Z Z Z Z

Z11-Z12 Z22-Z12 Z12

Lossless Networks

EE 611 Lecture 8 Jayanta Mukherjee Page 7 IIT Bombay

• If a network is lossless (i.e. no lossy lines, no lumped

resistors, no radiation loss etc.) then

• The total complex power Ptot dissipated by the N-port device

is given by

• For lossless devices Re {Ptot}=0
• Plugging in Ptot in the impedance matrix for Vn gives
• The real part of the above has to be zero regardless of the

values of In which excite the network

∑

=

= + ⋅ ⋅ ⋅ + + =

N n n n N N tot

I V I V I V I V P

1 * * * 2 2 * 1 1

2 1 2 1 2 1 2 1

∑ ∑

= =

      =

N n n N m m nm tot

I I Z P

1 * 1

2 1

• Consider that only one of the port currents is not zero. Say that

port is called n’ ; our previous equation is then so the diagonal entries of the matrix must be purely imaginary

• Now consider that only two currents are non zero. Call these

In’ and In”. We then get

Properties of a Lossless reciprocal impedance matrix

EE 611 Lecture 8 Jayanta Mukherjee Page 8 IIT Bombay

{ }

{ }

Re 2 1 Re 2 1

' ' 2 ' * ' ' ' '

= =

n n n n n n n

Z I I I Z

Properties of a Lossless reciprocal impedance matrix

EE 611 Lecture 8 Jayanta Mukherjee Page 9 IIT Bombay

{ } { } ( ) { }

Re{Z} have we devices lossless for Therefore real purely is s parenthesi in term the since imaginary purely be must Z the again that showing Re 2 1 Re 2 1 Re 2 1

nm * ' " * " ' " ' * ' " ' " * " ' " ' * ' " ' " * " ' " ' * " " " " * ' ' ' '

= = + ⇒ = + ⇒ = + + +

n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n

I I I I Z I I Z I I Z I I Z I I Z I I Z I I Z

Summary of Impedance matrix properties

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• If a network is reciprocal, the Z and Y matrices are symmetric
• If a network is lossless, the diagonal impedance or admittance

matrix elements are purely imaginary

• If a network is lossless and reciprocal, all impedance or

admittance matrix elements are purely imaginary

Scattering Parameters

EE 611 Lecture 8 Jayanta Mukherjee Page 11 IIT Bombay

• S-parameters can be measured with a network analyzer
• They have a natural relation with the flow of power
• S-parameters are readily represented by flow-graph
• The measurement of S-parameters relies on 50 ohm resistive

terminations: usually active devices do not oscillate for such terminations

• Devices are measured in the medium in which they are used

Normalized waves, voltages and currents

EE 611 Lecture 8 Jayanta Mukherjee Page 12 IIT Bombay

N Port Device IN VN + - I1 V1

• +

I2 V2

• +

ZN Z1 Z2

Normalized waves, voltages and currents

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( ) ( )

n n n

Z impedance stic characteri different handle to needed is ion Normalizat 1 : current Normalized : voltage Normalized b waves reflected and a incident Normalized

n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n

b a I Z I Z I b a Z Z V Z V I b a Z V V b a Z V V V Z V b Z V a − = = ⇒ = − = − = + = = ⇒ + = + = = =

− + − + − +

Normalized Z matrix

EE 611 Lecture 8 Jayanta Mukherjee Page 14 IIT Bombay

l symmetrica are Z znd both Z reciprocal is network port N a If : Property ly equivalent

• r

1 1 1 1 1 1

1 2 1 1 2 1 j i ij ij

Z Z Z Z Z Z Z Z Z Z Z Z =                                         =            

Definition of Scattering Parameters

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( ) ( ) ( )

j k for j k for ij

: can write ely we Alternativ j port except matched be must ports all S measure to Therefore

≠ = + − ≠ = − − −

+

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

2 1 2 1 1 2 1 1 1 3 2 22 21 1 12 11 2 1

k k

V i j j i ij a j i ij N NN N N N N N N N N N N

Z V Z V S a b S Sa a a a S S S S S S S S S S S S S b b b b                

2 port scattering parameters

EE 611 Lecture 8 Jayanta Mukherjee Page 16 IIT Bombay

matched input t with coefficien ion transmiss Reverse matched input t with coefficien reflection Output matched

• utput

t with coefficien ion transmiss Forward matched

• utput

t with coefficien reflection Input form matrix In . and between ip relationsh linear a have we signal)

• (small

device linear a For

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

1 1 2 2

= = = =

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

a a a a

a b S a b S a b S a b S a a S a a S S S S b b a S a S b a S a S b b's a's

S Matrix properties

EE 611 Lecture 8 Jayanta Mukherjee Page 17 IIT Bombay

• Reciprocal devices have symmetric impedance matrices, and

that lossless reciprocal devices have purely imaginary impedance matrices

• Let’s find similar identities for S matrix
• Our normalized impedance matrix equation is

[V] = [Z] [I]

• Now since [V] = [a] + [b] and since [I] = [a] - [b] we get

[U] ([a] + [b]) = [Z] ([a] - [b]) Where [U] is the identity matrix

S Matrix properties (2)

EE 611 Lecture 8 Jayanta Mukherjee Page 18 IIT Bombay

• We can also get another similar relationship using

[a] = (1/2) { [V] + [I] } = (1/2) { [Z] + [U] } [I] [b] = (1/2) { [V] – [I] } = (1/2) { [Z] – [U] } [I]

• Re-write as

{ [Z] + [U] } [b] = { [Z] – [U] } [a]

• Now since we define our S-matrix through [b] = [S] [a], we

find [S] = { [Z] + [U] }-1 { [Z] - [U] } (1) which is the relation between the normalized Z and S matrices

S Matrix properties (3)

EE 611 Lecture 8 Jayanta Mukherjee Page 19 IIT Bombay

• Solving the above equations for [b] in terms of [a], we find

[b] = { [Z] - [U] } { [Z] + [U] }-1 [a] So another formula for the S matrix is [S] = { [Z] – [U] } { [Z] + [U] }-1 (2)

• Taking the transpose of (1) we get

[S] t = { [Z] – [U] }t { [Z] + [U] }-1 t but [U] and [Z] are symmetric if the medium is reciprocal so [S]t = { [Z] – [U] } { [Z] + [U] }-1

S Matrix properties (4)

EE 611 Lecture 8 Jayanta Mukherjee Page 20 IIT Bombay

• Since this is the same as our equation (2) this means that the S

matrix is symmetric for a reciprocal medium [S] = [S]t

S Matrix for a lossless medium

EE 611 Lecture 8 Jayanta Mukherjee Page 21 IIT Bombay

• In a lossless device, the sum of all power entering and exiting

must be zero.

( )

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ][ ]

{ } [ ][ ] { }

[ ] [ ][ ] [ ] [ ] [ ] [ ]

* * * * * * * 1 , , 1 2 1 1 2 , , , , 2 2

as notation in vector written re be can which get we ports the all for up these Adding 2 1 a S S a a U a a S a S a U a b b a a P P b a P P P P b a P

t t t t t t t N n tot

• ut

n

• ut

N n n N n N n n n in tot in n

• ut

n in n n n

= ⇒ = ⇒ = = = = = = − = − =

∑ ∑ ∑ ∑

= = = =

S Matrix for a lossless medium

EE 611 Lecture 8 Jayanta Mukherjee Page 22 IIT Bombay

• On the previous slide we obtained

[a]t [U] [a]* = [a]t [S]t [S]* [a]*

• Simplifying we get [a]t { [U] – [S]t [S]* } [a]* = 0
• Since this is true regardless of what values the elements of [a]

take , we have [S]t [S]* = [U] which is equivalent to saying that S is “unitary” : [S]H = [S]-1

• This is most conveniently expressed in terms of elements of S

as

• therwise

zero j, i if

• ne

is where

j i,

= =

∑

=

δ δ

j i N k j k i k S

S

, 1 * , ,

Reverse Time Symmetry

EE 611 Lecture 8 Jayanta Mukherjee Page 23 IIT Bombay

• Refers to the concept that if time could be reversed then the

incident and reflected waves would main the exact same relation at any two ports

• Since however the direction of propagation is reversed we need

to take the conjugate and so the relation between [a] and [b] becomes [a]* = [S] [b]*

bk=VL,k

• exp(βkd)

ak=VL,k

+ exp(-βkd)

al=VL,l

+ exp(-βld)

[ ] [ ] ( ) [ ]

t t b S a − = = ' ) (

bl=VL,l

• exp(βld)

[ ]

S

d

Reciprocity of Lossless Devices

EE 611 Lecture 8 Jayanta Mukherjee Page 24 IIT Bombay

• We have demonstrated that [S] is unitary for lossless devices
• Using time reversal symmetry: if [b] = [S] [a] then [a]* = [S] [b]*
• It results that:

[a]* = [S] [b]* = [S] [S]* [a]* → [S]* = [S] -1

• Now since the device is lossless we also have:

[S]* = [S] -1 = [S]H → [S] = [S] t

• It results that a lossless device whose response is invariant

under time reversal symmetry has a reciprocal [S] matrix

Shifting reference planes for lossless lines

EE 611 Lecture 8 Jayanta Mukherjee Page 25 IIT Bombay

a1 b1 a’1 b’1 x1 a’2 b’2 a2 b2

     

22 21 12 11

S S S S

New Old Z1 Z2 x2 Old New

Shifting reference planes for lossless lines

EE 611 Lecture 8 Jayanta Mukherjee Page 26 IIT Bombay

) correction (phase with with are and by planes reference shift the ly respective after we port at wave reflected and port at ave incident w new The . port at wave reflected and port at ave incident w an Consider

) ( ' ' ' ' ' ' ' ' '

' l k l i l k l i l l l k k k

j kl a j l j k a l k kl l l l j l x j l l k k k j k x j k k k l k l k l

e S e a e b a b S x β θ e a e a a x β θ e b e b b l l k b l a k b l a

θ θ θ θ θ β θ β + − = − = − −

= = = = = = = = =

≠ ≠

Shifting reference planes for lossless lines

EE 611 Lecture 8 Jayanta Mukherjee Page 27 IIT Bombay

( ) ( )

[ ] [ ]

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

− − − − − − + − + − = =

≠ ≠ N N n m n n m m m k m k

j j j j j j n n n j nm x x j a m n a m n nm

e e e s e e e d β θ e S e a b a b S

θ θ θ θ θ θ θ θ β β

      S' as ly convenient more written be can matrices between ip relationsh The with become now quantities primed the

• f

in terms parameters scattering

• ur

that means This

2 1 2 1 '

' ' '

3 Port example

EE 611 Lecture 8 Jayanta Mukherjee Page 28 IIT Bombay

S

New Old Z1 Z2 1 2 3 Z3 θ1 θ2 θ3

( )

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

− − − − − − + −

3 2 1 3 2 1

33 32 31 23 22 21 13 12 11 ' 33 ' 32 ' 31 ' 23 ' 22 ' 21 ' 13 ' 12 ' 11 ' θ θ θ θ θ θ θ θ j j j j j j j kl kl

e e e S S S S S S S S S e e e S S S S S S S S S e S S

l k