1 ene 428 microwave engineering lecture 9 scattering parameters and their properties

1

ENE 428Microwave Engineering

Lecture 9 Scattering parameters and their properties.

2

Impedance and Admittance Matrices• Consider an arbitrary N-port network

below,

n n n

n n n

V V V

I I I

3

The impedance [Z] matrix relates voltages and currents.

So we can write [V] =[Z][I]V1 = Z11I1 + Z12I2V2 = Z21I1 + Z22I2, etc.

1 111 12 1

2 221

1

N

N NNN N

V IZ Z ZV IZ

Z ZV I

4

The admittance [Y] matrix relates currents and voltages.

So we can write [I] =[Y][V]I1 = Y11V1 + Y12V2

I2 = Y21V1 + Y22V2, etc.

1 111 12 1

2 221

1

N

N NNN N

I VY Y YI VY

Y YI V

5

and

• Zij can be found by driving port j with the current Ij, open-circuiting all other ports and measuring the open-circuit Voltage at port i.

• Yij can be found by driving port j with the voltage Vj, short-circuiting all other ports and measuring the short-circuit current at port i.

Zij or Yij can be found by o/c or s/c at all other ports

0i

ij I for k jkj

VZ

I 0i

ij V for k jkj

IY

V

6

• Many practical networks are reciprocal (not containing any nonreciprocal media such as ferrites or plasmas, or active devices)

• Impedance and admittance matrices are symmetric, that is

and

Reciprocal Network

ij jiZ Z

.ij jiY Y

7

• If the network is lossless, then the net real power delivered to the network must be zero. Thus, Re{Pav} = 0.

• Then for a reciprocal lossless N-port junction we can show that the elements of the [Z] and [Y] matrices must be pure imaginary

where m, n = port index.

Lossless Network

Re{ } 0mnZ

8

Single- and Two-port Networks• The analysis can be done easily through simple input-

output relations. • Input and output port parameters can be determined

without the need to know inner structure of the system. • At low frequencies, the z, y, h, or ABCD parameters are

basic network input-output parameter relations.• At high frequencies (in microwave range), scattering

parameters (S parameters) are defined in terms of traveling waves and completely characterize the behavior of two-port networks.

9

Basic definitions

• Assume the port-indexed current flows into the respective port and the associated voltage is recorded as indicated.

Two-portnetwork

Port 1 Port 2

V1

+

-

V2

+

-

I1 I2

10

Ex of h and ABCD parameters for two-port network• H parameters

• ABCD parameters

1 11 12 1

2 21 22 2

V H H I

I H H V

1 2

1 2

V VA B

I IC D

These two-port representations (Z, Y, H, and ABCD) are very useful at low frequencies because the parameters arereadily measured using short- and open- circuit tests at the terminals of the two-port network.

11

Two-port connected in series

1 1 1 11 11 12 12 1

2 22 2 21 21 22 22

a b a b a b

a b a b a b

v v v z z z z i

v iv v z z z z

12

Two-port connected in shunt

1 1 1 11 11 12 12 1

2 22 2 21 21 22 22

a b a b a b

a b a b a b

i i i y y y y v

i vi i y y y y

13

Two-port connected in cascade fashion

1 1 2 2

1 1 2 2

a a ba a a a b b

a a ba a a a b b

v v v vA B A B A B

i i i iC D C D C D

14

Disadvantages of using these parameters at RF or microwave frequency• Difficult to directly measure V and I• Difficult to achieve open circuit due to stray

capacitance• Active circuits become unstable when terminated

in short- and open- circuits.

15

Scattering Matrix (1)

• The scattering matrix relates the voltage waves incident on the ports to those reflected from the ports

• Scattering parameters can be calculated using network analysis techniques or measured directly with a network analyzer.

1 111 12 1

2 221

1

N

N NNN N

V VS S S

V VS

S SV V

16

Scattering Matrix (2)

• A specific element of the [S] matrix can be determined as

• Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads.

• Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads.

0.i

ij V for k jkj

VS

V

17

Reciprocal networks and lossless networks• [S] matrix for a reciprocal network is symmetric,

[S]=[S]t.

• [S] matrix for a lossless network is unitary that means

1[ ] {[ ] } .tS S

18

Ex1 A two-port network has this following scattering matrix

Determine if the network is reciprocal, and lossless

0.15 0 0.85 45[ ]

0.85 45 0.2 0S

19

Introduction of generalized scattering parameters (S parameters)1.Measure power and phase2.Use matched loads3.Devices are usually stable with matched loads.

S- parameters are power wave descriptors that permits us to define input-output relations of a network in terms of incident and reflected power waves

20

Introduction of the normalized notation (1)

0

0

00

00

( )( )

( ) ( )

( )( ) ( )

( )( ) ( ).

V xv x

Z

i x Z I x

V xa x Z I x

Z

V xb x Z I x

Z

we can write Let’s define

( ) ( ) ( )

( ) ( ) ( )

v x a x b x

i x a x b x

and( ) ( ) ( ).b x x a x

21

Introduction of the normalized notation (2)

We can also show a(x) and b(x) in terms of V(x) and I(x) as

00

1 1( ) [ ( ) ( )] [ ( ) ( )]

2 2 a x v x i x V x Z I x

Z

and

00

1 1( ) [ ( ) ( )] [ ( ) ( )]

2 2 b x v x i x V x Z I x

Z

22

Normalized wave generalization• For a two-port network, we can generalize the

relationship between b(x) and a(x) in terms of scattering parameters. Let port 1 has the length of l1 and port 2 has the length of l2, we can show that

1 1 11 1 1 12 2 2

2 2 21 1 1 22 2 2

( ) ( ) ( )

( ) ( ) ( )

b l S a l S a l

b l S a l S a l

or in a matrix form,

1 1 11 12 1 1

2 2 21 22 2 2

( ) ( )

( ) ( )

b l S S a l

b l S S a l

Observe that a1(l1), a2(l2), b1(l1), and b2(l2) are the values of in-cident and reflected waves at the specific locations denoted as port 1 and port 2.

23

The measurement of S parameters (1)

• The S parameters are seen to represent reflection and transmission coefficients, the S parameters measured at the specific locations shown as port 1 and port 2 are defined in the following page.

Two-portnetwork

Input port

Output port

Z01

Port 1x1=l1

a1(x)

b1(x)

a1(l1)

b1(l1)

Port 2x2=l2

Z02

a2(x)

b2(x)

a2(l2)

b2(l2)

24

The measurement of S parameters (2)

2 2

2 2

1 1

2 1

1 111 ( ) 0

1 1

2 221 ( ) 0

1 1

2 222 ( ) 0

2 2

1 112 ( ) 0

2 2

( )|

( )

( )|

( )

( )|

( )

( )|

( )

a l

a l

a l

a l

b lS

a l

b lS

a l

b lS

a l

b lS

a l

(input reflection coefficient with output properly terminated)

(forward transmission coefficient with output properly terminated)

(output reflection coefficient with input properly terminated)

(reverse transmission coefficient with input properly terminated)

25

The advantages of using S parameters• They are measured using a matched termination.

• Using matched resistive terminations to measure the S parameters of a transistor results in no oscillation.

Two-portnetwork

Port 1x1=l1

a1(l1)

b1(l1)

Port 2x2=l2

a2(l2)=0

b2(l2)E1

+

-

Z2=Z02

ZOUT2 2

1 111 ( ) 0

1 1

( )( ) a l

b lS

a l

Z1=Z01

Z01 Z02

26

The chain scattering parameters or scattering transfer parameters (T parameters) (1)• The T parameters are useful in the analysis of cascade

connections of two-port networks.

• The relationship between S and T parameters can be developed. Namely,

1 1 11 12 2 2

1 1 21 22 2 2

( ) ( )

( ) ( )

a l T T b l

b l T T a l

22

21 2111 12

21 22 11 11 2212

21 21

1

.

S

S ST T

T T S S SS

S S

27

The chain scattering parameters or scattering transfer parameters (T parameters) (2)

21 21 1222

11 1111 12

21 22 12

11 11

.1

T T TT

T TS S

S S T

T T

and

We can also write

21 11 12 11 12

1 221 22 21 22

.

x x y yyx

x x y yx y

ba T T T T

b aT T T T

28

Review (2)• Normalized notation of the incident a(x) and reflected waves b(x) are defined as

• The relationship between the incident and reflected waves and the scattering matrix of the two-port network,

( )( ) ( )

( )( ) ( )

00

00

V xa x Z I x

Z

V xb x Z I x

Z

( ) ( )

( ) ( )1 1 11 12 1 1

2 2 21 22 2 2

b l S S a l

b l S S a l

29

Shifting reference planes

• S parameters are measured using traveling waves, the positions where the measurements are made are needed to be specified. The positions are called reference planes.

Two-portnetwork

Port 1x1=l1

a1(0)

b1(0)

a1(l1)

b1(l1)

Port 2x2=l2

a2(0)

b2(0)

a2(l2)

b2(l2)

Port 1'x1=0

Port 2'x2=0

q1bl1 q2bl2

Reference planes

30

Scattering matrix of the shifting planes• At the reference planes at port 1 and port 2, we write the

scattering matrix as

and at port 1’ and port 2’ as

• We can show that

1 1 1 111 12

21 222 2 2 2

( ) ( )

( ) ( )

b l a lS S

S Sb l a l

' '1 111 12

' '2 221 22

(0) (0)

(0) (0)

b aS S

b aS S

1 1 2

1 2 2

2 ( )1 111 12

( ) 22 221 22

(0) (0).

(0) (0)

j j

j j

b aS e S e

b aS e S e

q q q

q q q