microwave network analysis lecture 1: the … network analysis lecture 1: ... (transmission) matrix...

Microwave Network Analysis Lecture 1: The Scattering Parameters

ELC 305a – Fall 2011

Department of Electronics and Communications Engineering

Faculty of Engineering – Cairo University

Review on Network Parameters

Impedance and Admittance Matrices

ABCD (Transmission) Matrix

Example: Transmission Line Section

The Scattering Matrix

Introduction

The Scattering Parameters

Examples

Relation to Other Network Parameters

Network Properties

Shift in Reference Planes

Circuit Analysis

© Islam A. Eshrah, 2008

2

Outline

1

2

Review on Network Parameters The Scattering Matrix






Introduction


Examples


Network Properties


Circuit Analysis


3

Outline

1

2




4

Definitions

n n n

n n n

V V V

I I I Terminal (Reference) Plane

Linear Network

V Z I I Y V

What is the difference between these matrices and the conventional ones?




5

How to Compute the Matrix Elements

0k

iij

j I k j

VZ

I Terminal (Reference) Plane

Linear Network

1Y Z

The impedance and admittance matrices are the inverses of each other.

0k

iij

j V k j

IY

V




6

Network Properties

ii jj

ii jj

Z Z

Y Y

How to prove the losslessness condition?

ij ji

ij ji

Z Z

Y Y

Re 0

Re 0

ij

ij

Z

Y

Symmetric

Lossless Reciprocal

Linear Network


Transmission (ABCD) Matrix


7

Two-Port Transmission Parameters

• How to compute the ABCD parameters? • What is the advantage of using the ABCD matrix? • How is the ABCD matrix related to the Z (or Y) matrix? • What are the conditions for reciprocity, symmetry, and losslessness?

1 2

1 2

V VA B

I IC D




8

Relation to Impedance Matrix

11

12

21

22

1

AZ

C

AD BCZ

C

ZC

DZ

C

11

21

21

21

22

21

1

ZA

Z

ZB

Z

CZ

ZD

Z

Similar relations can be derived for the admittance parameters.




9

Example

0,Z β

Determine the impedance, admittance, and transmission parameters for the lossless transmission line section shown.

0

0

cos sin

sin cos

β jZ βA B

jY β βC D







Introduction


Examples


Network Properties


Circuit Analysis


10

Outline

1

2


Introduction


11

What are the Scattering Parameters?

A set of reflection and transmission coefficients defined at the reference planes of the network ports. They relate the incident and outgoing “waves” at the network ports under arbitrary loading/excitation conditions.


Introduction


12

The Wave Amplitudes and Power

j

j j j

j

VP I Z

Z

2

2

0

0

1 1

2 2

j

j j j

j

Va I Z

Z0

0

j jP a21

2

j

j j j

j

Vb I Z

Z0

0

j jP b21

2

j j j j j j

j j j j j

j

V V V Z a b

I I I a bZ

0

0

1

0

0

0

0

1

2

1

2

j j j j

j

j j j j

j

a V I ZZ

b V I ZZ

Incident/Reflected Power on/from Port j

Incident Wave Amplitude

Reflected Wave Amplitude

How to compute the total power absorbed by the network?




13

The Incident and Reflected Wave Amplitudes

sV 1

sR 1

snV

snR

i i

i

ij j in n

b S a

S a

S a S a

1 1

2 2

i si i sia b aΓ

Incident Wave on Port i

Outgoing Wave from Port i

Reflection Coefficient of Source i

Wave Generated by Source i

iij

j

bT

a i

i

i

b

aΓ

Transmission Coefficient (from Port j to Port i)

Reflection Coefficient (at Port i)

What is the difference between Sij and Tij, Sii and Γi?




14

Evaluation of the Scattering Parameters

b S a

0k

iij

j a k j

bS

a

To evaluate the elements of any column of the scattering matrix, a source should be connected to the corresponding port and all the other ports should be terminated in matched loads (why?).

0

00

0 0

000

k

k

jiij

j ia k j

j i i iij

j j jia k j

ZVS

V Z

Z V I ZS

V I ZZ


Examples


15

Scattering Matrix for a TL Section

0,Z β

0

0

jβ

jβ

eS

e

0Z 0Z


Examples


16

Scattering Matrix for a T- and Π-Network

Z3

Z1 Z2

Z3

Z1

Z2


Examples


17

Different Port Characteristic Impedances

0,Z β1Z 2Z




18

Transformation from S to Z Matrix

1

2

1

2

n n n

n n n

a v i

b v i

0

0

n n n

n n n

v V Z

i I Z

12

12

0

0

v Z V

i Z I

1 12 2

0 0

1

V Z I

v Z Z Z i z i

a b z a b

b z U z U a

1

S z U z U

1

z U S U S

01 02 00, , , NZ diag Z Z Z


Network Properties


19

Reciprocity, Losslessness, and Matching

1 *T

S S

0iiS

Reciprocal

Lossless

Internal Matching

T

S S

How to prove the losslessness condition?


Shift in Reference Plane


20

Transformation Matrix

S PSP

1 2, , , ,Njθ jθ jθ

n n nP diag e e e θ β


Circuit Analysis


21

Analysis of a Two-Port Network Using the Z Parameters

Two-Port Network

I1 I2

1 2 V1 V2 Vg

Zg

ZL

Z

Zin=?

1 11 1 12 2

2 21 1 22 2

V Z I Z I

V Z I Z I 2 2 LV I Z

1 12 2111

1 22

in

L

V Z ZZ Z

I Z Z


Circuit Analysis


22

Analysis of a Two-Port Network Using the S Parameters

Two-Port Network

1 2 Vg

Zg

ZL

S

Γin=?

a1

b1

a2

b2

1 11 1 12 2

2 21 1 22 2

b S a S a

b S a S a2 2ΓLa b

1 12 2111

1 22

ΓΓ

1 ΓL

in

L

b S SS

a S

The scattering parameters and their use in the analysis of microwave circuits.

The relation between the S matrix and the other network parameters.

Determining the network properties (reciprocity, symmetry, losslessness, and matching) from the S matrix.

The effect of shifting the reference plane on the S matrix.

Circuit analysis using the scattering parameters.


23

Conclusion

microwave network analysis lecture 1: the … network analysis lecture 1: ... (transmission) matrix...

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