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ENE 428Microwave Engineering

Lecture 9 Scattering parameters and their properties.

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

below,

n n n

n n n

V V V

I I I

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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

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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

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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

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• 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

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• 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

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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.

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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

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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.

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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.

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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)

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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)

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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

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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

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21 2111 12

21 22 11 11 2212

21 21

1

.

S

S ST T

T T S S SS

S S

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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

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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

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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

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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