Unit-II

S - PARAMETERS

Scattering matrix parameters:

Definition: the scattering matrix of an m-port junction is a square matrix of a set of elements which

relate incident and reflected waves at the port of the junction. The diagonal elements of the s-matrix

represents reflection coefficients and off diagonal elements represent transmission coefficients.

Characteristics of s-matrix:

It describes any passive microwave component.

It exists for linear passive and time invariant networks.

It gives complete information on reflection and transmission coefficients.

Explanation:

A scattering matrix represents the relationship between the parameters an’s(incident wave

amplitude) bn’s(reflected wave amplitude).

an=vn+/√ o ; bn=vn-/√ o

vn+ and vn- represent incident and outgoing waves along the line connected to the nth port.

Zon- characteristic impedence of the line.

Relationship between bn’s and an’s:

for the network may be written as,

b1 = s11a1 + s12a2.

b2 = s21a1 + s22a2.

Using matrix notation,

[b] = [s] [a].

For n-port network,

b1 s11 s12 … s1N a1

[ b2 ] = [ s21 s22 … s2N ] [ a2 ]

: : :

bN sN1 sN2 … sNN aN

The matrix s is known as scattering matrix,

S11= b1/a1|a2=0 => reflection coefficient of port 1 when port 2 is terminated with match load (a2=0).

S22= b2/a2 |a1=0 => reflection coefficient of port 2 when port 1 is terminated with match load (a1=0).

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S12= b1/a2 |a1=0=> transmission coefficients from port 2 to port 1.

S21= b2/a1|a2=0 => transmission coefficients from port 1 to port 2.

(transmission coefficient->attenuation)

Losses in microwave circuits:

Insertion loss(dB)

Reflection loss(dB)

Return loss(dB)

In a two port network power fed at port 1 is Pi, power reflected at the same port is Pr, and

the output power at port 2 is Po.

Insertion loss(dB) = 10 log pi/po

Input power at the nth port Pin = ½|an|2.

Reflected power at the nth port Pr = ½|bn|2.

=10 log ½|an|2/½|bn|2.

=10 log |a1|2/|b1|

2.

= 20 log 1/|s21|.

Transmission loss(dB) = 10 log Pi-Pr/Po.

Reflection loss(dB) = 10 log Pi/Pi-pr.

Return loss(dB) = 10 log pi/pr.

In terms of s-parameter:

Insertion loss(dB) = 20 log 1/|s21|.

Transmission loss(dB) = 10 log 1-|s11|2 /|s12|2.

Reflection loss(dB) = 10 log 1/1-|s11|2.

Return loss(dB) = 20 log 1/|s11|.

PROPERTIES OF S-PARAMETER

1.)Zero diagonal element for perfect matched network :

For an ideal N-port network with matched terminations Sii=0, since there is no reflection

from any.

2.)SYMMETREY OF [S] FOR A RECIPROCAL NETWORK:

A reciprocal device has the same transmission characteristics in either direction of a pair of

ports and is characterised by a symmetric scattering matrix

Sij=Sji (i=j)

Which results [St]=[S]. This property is known as symmetry property of S-matrix.

Proof:

2

The impedance Z of a network is given by

[V] = [Z][I] ----------------------(1)

The average power flowing in to the port n may be evaluated using the following relation

an = Vn+ / √ ------------------(2) where Vn+ = incident

bn = Vn- / √ -------------------(3) Vn- = outgoing wave

Using above relation Vn and In can be written as

Vn = (Vn+) + ( Vn-) = √ (an + bn) ------------------(4)

In = {(Vn+) + (Vn-)} / Z0 =(an – bn)/ Z0 ----------------(5)

Subs eqn (4) & (5) in eqn (1)

[V+] + [V-+ = *Z+ (⅟ Z0) (*V++ – [V-]) -----------------------(6)

= [Z1] ([V+] – [V-]) -----------------------(7)

Where Z1= [Z] / Z0

EQUATION (6) can be written as

{ ( [Z1] + [U] ) [V-] } = { ( [Z1] – [U] ) [V+] }

since (Z1 + U)(V-) = (Z1 – U)(V+)

Where [U] is the unity matrix .

[V-] = [V+] { ( [Z1] – [U] ) / ( [Z1] + [U]) }

[V-] = ( [Z1] – [U] ) { 1/ ([Z1] + [U]) } [V+]

[V-] = [S] [V+] ---------------------------------------------(8)

WHERE,

[S] = ( [ Z1] – [U] ) {1/ ( [Z1] + [U] ) }

Writing transpose of [S] ,

[St] = ( [Z1] – [U] )t {1/([Z1] + [U]) }t

As [Z1] and [U] are symmetrical matrix,

{ 1/([Z1] + [U]) }t = 1/( [Z1]+ [U] )

( [Z1] – [U] )t = ( [Z1] – [U] )

Hence [S] = [St].

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This indicates that Scattering matrix [S] is SYMMETRICAL.

(3.) Unitary property for a lossless junction :

For any lossless network the sum of the products of each term of any one row or of any

column of the S matrix multiplied by its complex conjugate

For lossless n port device , the total power rating

N port must be equal to the total power input to these ports. The mathematical statement

for this power conservation condition is,

∑ | | = ∑ | |

2 ---------------------- (1)

The relationship between bn and an for two port network may be return as

[b] =[S] [a]

Uisng above relations

bn = ∑ ------------------------------------------(2)

sub (2) in (1)

∑ |∑ | ∑ | |

If only i th port is executed and all other ports are matched termianated ,all an=0 except ai

∑ | | = ∑ | |

∑ | | = 1

ie ∑

the above equation states that for a lossless network the product of any column of the scattering

matrix with the conjugate of this column equals UNITY.

If all an=0 except ai & ak

∑ ; for i≠k

This equation states that the product of any column of the scattering matrix with the complex

conjugate of any other column is zero.

In matrix notation ,the relations are expressed as ,

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

[S*] =

[U]= Unit matrix. A matrix [S]for lossless network which satisfies the above three conditions is called

unitary matrx.

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Shifting of reference planes in two port network

(b.) Phase shift property:-

Complex S-parameters of a network are defined with respect to the

position of the port (or) reference planes

For a two port network with unprimed reference planes 1 and 2

(

) = (

) . (

)

Where a1 a2 are incident waves & b1,b2 are out going waves

S-matrix for any network when the reference plane for one of its ports is shifted away along the

transmission line is given by (in fig, shifted reference is mentioned as )

(

) =

) . (

) (for lossless network)

(

) = (

) . (

)

Where l1 l2 = path length.

β 1 β 2 =phase constant.

This property is valid for any number of ports and is called the phase shift property applicable to

shift of reference planes. The resultant MATRIX is

( ) = (

) . (S) . (

)

(4.) zero property of S matrix:-

The sum of products of each term of any column (or row) multiplied by the complex

conjugate of the corresponding terms of any other column(or row) as zero and as

S11 + S21 = 0

b 1 = S11 a1 + S12 a2

b 2 = S21 a1 + S22 a2

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WAVEGUIDE TEES: SCATTERING MATRIX FOR THREE PORT NETWORK :-

*Here a piece of wave guide is attached to the broad wall of the wave guide.

*Port 3 is usually the input port. The port 1 and 2 are in series and hence it is called a series tee

junction.

*Tee can be matched by the screw Iris or inductive or capacitive windows at the junction.

*If the E-plane tee is perfectly matched ,then the diagonal components S11,S22,S33 are zero

because there will be no reflection.

*When the waves fed in side arm (port1),the waves appearing port 1 and 2 with equal magnitude

but in opposite phase.

*The electric field in E- plane shown in the following fig.

Scattering matrix for E plane Tee:

In E-plane tee junction, power in port 3 is the difference of the two signals

entering at 1 and 2 simultaneously. It is a three port junction and its S MATRIX is given by

When waves are fed into port 3, the waves appearing at port1 and port 2 of the

collinear arm will be in opposite phase and in the same magnitude.

S13 = -S23 -------------(1)

-ve sign indicates that S13 and S23 have opposite signs.

* for matched junction , the S MATRIX s given by 0.

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S = (

) ------------------------------- (2)

*FOR A SYMMETRY PROPERTY of S matrix ,the symmetric terms in above equations are equal and

they are

S12 = S21 , S13 =S31 ,S23 =S32 ----------------------(3)

*FOR THE ZERO property of S =MATRIX , the sum of the products of each term of any column(or

row)

Multiplied by the complex conjugate of the corresponding terms of any other ( or row) is zero

S11. +S21. +S31. = 0 -----------------------------------(4)

Hence ( S31 = S13)

This means that either S13 or S23 or both shown ie zero.

*From the unity property of S –matrix , the sum of the products of each term of any one row (or

column) multiplied by its complex conjugate is unity,ie

Using eqn (2)

S12. + S13. = 1 -------------------------(6)

S21. + S23. =1 --------------------------(7)

S31. + S32. =1 ---------------------------(8)

Using eqn (3)

S12 = S21

| | + | | = 1

USING eqn (6) | | = 1 - | |

USING eqn (7) | | = 1 - | |

If s12 = s21 , then

| | = 1 - | | = 1 - | | --------------------------------(9)

Equation (8) and (9) are contradictor , for if S13 = 0 (using eqn (5))

Then S23 is also zero , thus eqn (8) is false . this inconsistency proves the statement that the tee

junction cannot be matched to the three arms . In other words , the diagonal elements of the S-

matrix of the tee junction are not all zeros.

In general , when an E-plane tee is constructed of an empty wave guide , it is poorly matched at

tee junction.

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Sij ≠ 0 if i=j.

However , since the collinear arm is usually symmetric about the side arm ,

|S13| = |S23| and S11=S22

Therefore the S-matrix can be simplified in to

S = (

) and

[S] ] = 1 (ACTUALLY unitary property is ([S] )t = 1 since it satisfies

symmetry property , we can write [S] ] = 1 )

S11. + S12. + S13. = 1 ---------------------------(10)

S12. + S11. + S13. =1 ---------------------------(11)

S13. + S13. + S33. =1 ----------------------------(12)

For a matched 3 is matched , s33=0

From eqn (12) | | + | | = 1

2| | = 1

S13 = 1/√ .

eqn (10) is

| | + | | + | | = 1

If | | = ¼ , then | | & | | = 1/4

Therefore S11 = S22 = 1/2

S12 = S22 = ½

S = (

√

√ √

√ )

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H-PLANE TEE: (SHUNT TEE)

A piece of waveguide is attached perpendicular to the narrow wall of a waveguide.

Port 3 is usually the input port.

The arm containing port 1 and 2 are in shunt and hence it is also called a shunt tee.

If two input waves are fed into port 1 and port 2 of the collinear arm, the output wave at

port 3 will be in phase and additive.

On the other hand, if the input is fed into port 3, the wave will split equally into port 1 and

port 2 in phase and in the same magnitude.

S13 = S23

For symmetry property,

S12 = S21

S23 = S32

S13 = S31

For a matched junction S33 =0

Using these values, S becomes

S11 S12 S13

[ S ] = S12 S22 S13

S13 S13 0

According to unitary property

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[S][S]* =[I]

Ґ S11 S12 S13 Ґ S11* S12* S13* 1 0 0

S12 S22 S13 S12* S22* S13* = 0 1 0

S13 S13 0 ] S13* S13* 0 ] 0 0 1

Taking product on LHS, we get

S11 S 11* +S12 S 12 * + S13 S13* =1

Out S11 S11*=I S11 I^2

S12 S12*=I S12 I^2

S13 S13*=I S12 I^2

I S11 I^2 + IS12 I^2 + I S13 I^2 = 1

I S12 I^2 + IS22 I^2 + I S13 I^2 = 1

IS13 I^2 + I S13 I^2 = 1

S13 S 11 * + S13 S12* =0

Solving the above equations,we get

S13 =1/√2, S11 =S22, S11 =S- S12, S12=1/2= S22

Using above values S becomes

1/2 -1/2 1/√2

-1/2 1/2 1/√2

1/√2 1/√2 0

CIRCULATORS: (THREE PORT NETWORK )

A circulator is a microwave passive multiport device in which the incident wave at port 1 is

coupled to port 2 only. Incident wave at port 2 is coupled to port 3 only and so on.

It is usually made up of ferrite.

The circulator containing three or four ports is most common.

A circulator consists of two non-reciprocal phase shifter waveguide sectional mounted side

by side with two slots in the walls.

Circulator can handle 100 or mire kilowatts of average power.

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The ideal circulator is matched device. It means when all the ports are terminated in a

matched load except one port, the input impedance of the other port is equal to

characteristics impedance of input line.

APPLICATIONS:

1. It is used to separate the input and output in negative resistance application.

2. It is used to couple a transmitter and receiver to a common antenna.

S-matrix:

It matrix is given by

S11 S12 S13

[ S ] = S21 S22 S23

S31 S32 S33 --------(1)

As its properly matched function,

S11 = S22 = S33 = 0

The circulator is a non-reciprocal device and hence it is not symmetrical. It means

Sij ≠ Sji

but [S] is unitary

[ S ] [ S*]=1

0 S12 S13 0 S12* S13*

[ S ] = S21 0 S23 , [S*]= S21* 0 S23*

S31 S32 0 S31* S32* 0

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This gives

IS12 I^2 + I S13 I^2 = 1 --→ I S13 I^2 = 1- IS12 I^2 -------(2)

IS21 I^2 + I S23 I^2 = 1 --→ I S23 I^2 = 1- IS21 I^2 -------(3)

IS31 I^2 + I S32 I^2 = 1 --→ I S32 I^2 = 1- IS31 I^2 -------(4)

Using zero property of S matrix,

S13 S23* = 0 , S12 S32* = 0 , S21 S31* = 0 and using zero property,

S23= 0 S12 = 0 S31 = 0

IS12 I=1 .`. S23= 0

I S32 I=1 .`. S31 = 0

I S13 I=1 .`. S12 = 0

0 0 S13

[ S ] = S21 0 0

0 S32 0

0 0 1

[ S ] = 1 0 0

0 1 0

COMPARISON OF S,Z AND Y-MATRICS:

PARAMETERS S-MATRIX Z-MATRIX Y-MATRIX

1. Type of matrix Square Square Square

2. Information

given by the

matrix

It relates the incident

and reflected waves at

terminals

It relates voltages and

currents of junction.

It relates voltages and

currents of junction.

3. Example S11 S12

S21 S22

Z11 Z 12

Z 21 Z 22

Y11 Y12

Y21 Y22

4. Properties It is symmetric and

unitary. For a

reciprocal network

S12 = S21. For

symmetric network

For a reciprocal

network Z12 = Z21.

For symmetric

network Z11 = Z22.

For a reciprocal

network Y12 = Y21.

For symmetric network

Y11 = Y22 .

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S11 = S22

5. Units of

element

No unit ohms mhos

6. No. of matrix

element

Equal Equal Equal

H,Y,A AND Y- PARAMETERS CANNOT BE USED IN MICROWAVE FREQUENCIES FOR THE FOLLOWING

REASONS:

1. Equipment is not readily available to measure total voltage and total current at the ports of

the network.

2. Short and open circuits are difficult to achieve over a broadband of frequencies.

3. Active devices, such as power transistors and tunnel diodes, frequently will not have stability

for a short or open circuit.

MULTIPORT (OR) FOUR PORT NETWORK:

(1) MAGIC TEE:

1. It is a four port device and it is also called magic tee because of its unusual

characteristics.

2. Its four arms are two side arms, shunt arm and series arm. The shunt arm is called H-

arm and series arm is called E-arm. The side arms are called collinear arms. It is used to

produce sum and difference signal simultaneously.

3. The electric field in shunt and series arms are perpendicular to each other and hence

they are said to be cross polarized.

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CHARACTERISTICS:

1. If two inphase waves of equal magnitude are fed into ports 1 and 2, the output at port 3 is

additive and output at port 4 is subtractive. Hence port 3 is called the sum and port 4 is

called the difference.

2. A wave incident at port 4 ( E arm ) divides equally between port 1 and 2 but opposite in

phase with no coupling to port 3 (H arm ) . Thus

S24 = - S14 ( -ve sign indicates opposite phase) ----------------------(1)

3. A wave incident at port 3 (H arm) divides equally between port 4 and 2 in phase with no

coupling to port 4 (E arm). Thus

S13 = S23 ---------------------(2)

4. Collinear ports 1 and 2 are isolated from each other.

S12 = S21 = 0 --------------(3)

5. Port 3 and port 4 are isolated from each other i.e no coupling between shunt and series

arm.

S34 = S43 = 0 --------------(4)

SCATTERING MATRIX FOR MAGIC TEE:

Magic tee is a four port junction. S matrix is given by

S11 S12 S13 S14

[ S ] = S21 S22 S23 S24

S31 S32 S33 S34 ------------------------------------(5)

S41 S42 S43 S44

Due to symmetry property

S12 = S21

S31 = S31

S14 = S41

S23 = S32 } ---------(6)

S24 = S42

S34 = S43

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The S – matrix for a magic tee , matched port 3 and port 4 is given by i.e S33 & S44 is equal

to zero.

S34 = S44=0 ----------(7)

Using equation (7),(6),(4)

S11 S12 S13 S14

[ S ] = S21 S22 S13 -S14

S13 S13 0 0 ------------------------------------(8)

S14 -S14 0 0

Using unitary property,

[S][S]*=[U] when symmetry property is satisfied.

[S][S]*t=[U] when symmetry property is not satisfied.

Here symmetry property is satisfied

[S][S]*=[U]

S11 S12 S13 S14 S11* S12* S13* S14 *

S21 S22 S13 -S14 S21* S22* S13* -S14*

S13 S13 0 0 * S13* S13* 0 0 = [ 0 ]

S14 -S14 0 0 S14* -S14* 0 0

I S11 I^2 + IS12 I^2 + I S13 I^2+ I S14 I^2 = 1 ------------------------------------(9)

I S12 I^2 + IS22 I^2 + I S23 I^2+ I S24 I^2 = 1 ------------------------------------(10)

I S13 I^2 +I S13 I^2 = 1 ------------------------------------(11)

I S14 I^2 + I S14 I^2 = 1 ------------------------------------(12)

Using equation (11)

2IS13I^2 = 1/2

S13= 1/√2

lllrly, S14= 1/√2

subtracting eqn. (9) and (10)

I S11 I^2 - IS22 I^2 = 0

15

I S11 I = IS22 I

Substitute IS13I^2 = ½ and IS14I^2 = 1/2 in eqn.(9)

I S11 I^2 + IS12 I^2 + 1/2+1/2= 1

I S11 I^2 + IS12 I^2 =0

From equ(3) S12 =0 .`. S11 must be equal to zero.

S11 = S22 =0

Substitute above value in eqn.(8)

0 0 1/√2 1/√2

0 0 1/√2 -1/√2

*S+ = 1/√2 1/√2 0 0

1/√2 -1/√2 0 0

Ґ 0 0 1 1 +

I 0 0 1 -1 I

*S+ = 1/√2 I 1 1 0 0 I

[ 1 -1 0 0 ]

If E arm is considered as port 3 and H arm is considered as port 4 , then [S] matrix changed as

Ґ 0 0 1 1 +

I 0 0 -1 0 I

*S+ = 1/√2 I 1 -1 0 0 I

[ 1 1 0 0 ]

APPLICATIONS:

The magic tee is used for

Mixing

Duplexing

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Producing sum and difference signals.

Impedance measurement.

To couple two transmitted to the antenna without loading.

Example (2) for four port network:

DIRECTIONAL COUPLER:

1. It is a passive four port device. It consists of a primary waveguide 1-2 and a secondary

waveguide 3-4.

2. The guides 1-2 and 3-4 are identical. Any one of them can be used as primary and the other

act as auxiliary guide.

3. The direction coupler is said to be consisting of main arm and an auxiliary arm.

4. The amount of power coupled to the auxiliary arm depends on the no. of holes and their

sized.

5. Matched termination absorb incident power without reflection.

6. The powers at port 4 and 2 have a phase difference of 90⁰. Similarly, the power at port 3

and 1 has a phase difference of 90⁰ when they propagate in reverse direction.

7. The directional coupler are described by coupling factor, directivity and VSWR.

Coupling factor:

It is defined as the ratio of input power and output power at auxiliary arm.

C=10log10 (Pi/Pa) dB

Pi =input power to primary guide.

Pa=power output at auxiliary arm.

C=10log10 (P1/P4)

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Directivity:

It is defined as the ration of power in the auxiliary arm due to power in forward direction

to the power at the same port due to power in the reverse direction.

D(dB) = 10 log 10 (Paf/Par)

Paf = power in the auxiliary arm due to power in forward direction.

Par= power in the auxiliary arm due to power in reverse direction.

D=10log10 (P4/P3)

Transmission loss(dB) =10 log(P1/P2)

Return loss (dB)=10 log (P1/Pr)

Scattering matrix:

Since it is a four port network, its general scattering matrix is given by

S11 S12 S13 S14

[ S ] = S21 S22 S23 S24

S31 S32 S33 S34 ------------------------------------(1)

S41 S42 S43 S44

In a directional coupler all four ports are completely matched. Thus, diagonal elements of

the s-matrix are zeros and

S11 =S22 =S33=S44 =0

As noted , there is no coupling between port 1 and port 3 and between port 2 and port 4.

Thus S13 =S31 =S24=S42 =0

Consequently S matrix of a directional coupler becomes

0 S12 0 S14

[ S ] = S21 0 S23 0

0 S32 0 S34 ------------------------------------(2)

S41 0 S43 0

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Using zero property of the S-matrix

S13 S 14 * + S32 S34* =0 ----------3(a) } -------(3)

S21 S 23 * + S41 S43* =0 ----------3(b) }

Since it is reciprocal network

0 S12 0 S14 0 S12* 0 S14*

[ S ] = S21 0 S23 0 ,[ S *] = S21* 0 S23* 0

0 S32 0 S34 0 S32* 0 S34*

S41 0 S43 0 S41* 0 S43* 0

Using unitary property

[S][S*]=1

I S12 I^2 + I S14 I^2 = 1 ------------------------------------(5)

I S12 I^2 + I S23 I^2 = 1 ------------------------------------(6)

I S23 I^2 + I S34 I^2 = 1 ------------------------------------(7)

I S14 I^2 + I S34 I^2 = 1 ------------------------------------(8)

(5)-(6)

I S14 I^2 - I S23 I^2 = 0

I S14 I^2 = I S23 I^2

(6)-(7)

I S12 I^2= I S34 I^2

By choosing reference plane of port4 with respect to that of port 2 and the reference plane of port 3

with respect to that of 4, we can make S-parameters real.

S12 =S34 =α

Consider 3(b) equation

S21 S23*+ S41 S43*=0

α is +ve and real no.

α S23*+ S41 α=0

α (S23*+ S41) =0

S14 =-S23*= S23=β (say)

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.`.S matrix is

I 0 α 0 β I

I α 0 β 0 I

I 0 β 0 α I

I β 0 α 0 I

β is +ve and real and α^2 + β^2 =1.

α = transmission factor. Β = coupling factor.

CONVERSION BETWEEN ABCD AND S-PARAMETERS:

Z and Y parameters are defined by using independent and dependent variable. Another useful

representation is obtained by taking current and voltage at the input port as dependent variable and

voltage and current at the output port as independent variables. These are used particularlyfor

cascading network. If V2 and I2,voltage and current at the output port are the independent

variables, the relationship between them can be expressed as

V1=AV2 –BV2

I1 =CV2 - DI2 (for two port network)

In matrix form,

Ґ V1 + = Ґ A B I Ґ V2 I

I I1 ] = I C D ] I - I2 ]

Two port S-parametric conversion from ABCD parameters

The coefficients A B C D are defined as

A= V1 / V2 ] I2=0 C= I1 / V2 ] I2=0

B= -V1 / I2 ] V2=0 D= - I1 / I2 ] V2=0

The coefficient A and B are dimensionless ,B has the dimensions of impedance and C has the

dimensions of admittance. These coefficients are ABCD parameters.

The scattering matrix represents the relationship between parameter an ‘s

(incident wave amplitude) bn’s (reflected wave amplitude). For a two port network, the

relationship between bn’s and an’s may be written as

b1=S11a1+S12a2

b2=S21a1+S22a2

In the matrix form,

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Ґ b1 I = Ґ S11 S12 | Ґ a1 |

I b2 ] = | S21 S22 ] | a2 ]

The coefficients S11 ,S12 , S21 & S22 are defined as

S11= b1/ a1 | a2 =0 S12= b1/ a2 | a1 =0

S21= b2/ a1 | a2 =0 S22= b2/ a2 | a1 =0

These coefficients (S11 ,S12 , S21 & S22 in terms of A,B,C,D coefficients are)

S11= A+B-C-D

A+B+C+D

S12= 2(AD-BC)

A+B+C+D

S21= . 2 .

A+B+C+D

S11= -A+B-C+D

A+B+C+D

The coefficients A,B,C,D in terms of S11 ,S12 , S21 & S22 are

A= (1+ S11)(1- S22)+ S12 S21

2 S21

B= (1+ S11)(1+ S22)- S12 S21

2 S21

C= (1- S11)(1- S22)- S12 S21

2 S21

D= (1-S11)(1+S22)+ S12 S21

2 S21

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EXAMPLE FOR TWO PORT NETWORK- ISOLATORS:

*An isolator is a two port non-reciprocal device which produces a minimum attenuation to wave

propagation in one direction and very high attenuation in the opposite direction.

* When inserted between a signal source and loud almost all the signal power can be transmitted to

the load and any reflected power from the load is not fed back to the generator output port. This

eliminates variation of source power output and frequency pulling due to changing loads.

* The attenuation is ferrite for negative clockwise circular polarization is very small where as for

positive/counter clockwise circular polarization is very large.

* since the reverse power is absorbed in the ferrite and dissipated as heat, the maximum power

hanling capability of an isolator is limited.

For an ideal lossless matched isolator

I S21 I=1 , IS12 I = IS11 I= I S22 I=0

[S]= 0 0

1 0

ADVANTAGES OF [S] OVER [Z] OR [Y]:

1. In microwave techniques, the source remains ideally constant in power, regards of circuit

changes besides frequency measurement the only other possible measurement parameters

are VSWR,power and phase. These essentially direct correspondence is not possible with [Z]

&[Y] representation.

2. The unitary property of [S] helps a quick check of power balance for lossless structures. No

such immediate check is possible with [Z] or[Y].

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3. [S] is defined for a given set of reference plane only. If the reference planes are changed the

S coefficients vary only in phase. This is not the case in [Z] or[Y],because voltage and current

functions of complex impedance & both magnitude and phase change in [Z] &[Y].

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