S- Parameters

Impedance and Admittance matrices For n ports network we can relate the voltages and currents by impedance and admittance matrices Impedance matrix Admittance matrix where

Reciprocal and Lossless Networks Reciprocal networks usually contain nonreciprocal media such as ferrites or plasma, or active devices. We can show that the impedance and admittance matrices are symmetrical, so that. Lossless networks can be shown that Zij or Yij are imaginary Refer to text book Pozar pg193-195

Example Find the Z parameters of the two-port T –network as shown below I1 I2 V1 V2 Solution Similarly we can show that Port 2 open-circuited This is an example of reciprocal network!! Port 1 open-circuited

S-parameters Input signal reflected signal transmitted signal Port 1 Microwave device Port 2 Vi2 Vi1 Input signal Vr2 Vr1 reflected signal transmitted signal Vt1 Vt2 Transmission and reflection coefficients

S-parameters Let Vb1= b1 , Vi1=a1 , Vi2=a2 , Then we can rewrite Voltage of traveling wave away from port 1 is Voltage of Reflected wave From port 1 Voltage of Transmitted wave From port 2 Voltage of transmitted wave away from port 2 is Let Vb1= b1 , Vi1=a1 , Vi2=a2 , Then we can rewrite

S-parameters S11and S22 are a measure of reflected signal at port 1 and port 2 respectively S21 is a measure of gain or loss of a signal from port 1 to port 2. S12 ia a measure of gain or loss of a signal from port 2 to port 1. Hence In matrix form Logarithmic form S11=20 log(r1) S22=20 log(r2) S12=20 log(t12) S21=20 log(t21) S-matrix

S-parameters Vr2=0 means port 2 is matched

Multi-port network Port 5 network Port 1 Port 4 Port 2 Port 3

Example Below is a matched 3 dB attenuator. Find the S-parameter of the circuit. Z1=Z2= 8.56 W and Z3= 141.8 W Solution By assuming the output port is terminated by Zo = 50 W, then Because of symmetry , then S22=0

Continue V1 Vo V2 From the fact that S11=S22=0 , we know that Vr1=0 when port 2 is matched, and that Vi2=0. Therefore Vi1= V1 and Vt2=V2 Therefore S12 = S21 = 0.707

Lossless network For lossless n-network , total input power = total output power. Thus Where a and b are the amplitude of the signal Putting in matrix form at a* = bt b* =at St S* a* Note that bt=atSt and b*=S*a* Called unitary matrix Thus at (I – St S* )a* =0 This implies that St S* =I In summation form

Conversion of Z to S and S to Z where

Reciprocal and symmetrical network Since the [U] is diagonal , thus For reciprocal network Since [Z] is symmetry Thus it can be shown that

Example A certain two-port network is measured and the following scattering matrix is obtained: From the data , determine whether the network is reciprocal or lossless. If a short circuit is placed on port 2, what will be the resulting return loss at port 1? Solution Since [S] is symmetry, the network is reciprocal. To be lossless, the S parameters must satisfy For i=j |S11|2 + |S12|2 = (0.1)2 + (0.8)2 = 0.65 Since the summation is not equal to 1, thus it is not a lossless network.

continue Reflected power at port 1 when port 2 is shorted can be calculated as follow and the fact that a2= -b2 for port 2 being short circuited, thus (1) b1=S11a1 + S12a2 = S11a1 - S12b2 Short at port 2 a2 (2) b2=S21a1 + S22a2 = S21a1 - S22b2 -a2=b2 From (2) we have (3) Dividing (1) by a1 and substitute the result in (3) ,we have Return loss

ABCD parameters Voltages and currents in a general circuit Network V2 V1 Voltages and currents in a general circuit In matrix form Given V1 and I1, V2 and I2 can be determined if ABDC matrix is known. This can be written as Or A –ve sign is included in the definition of D

Cascaded network However V2a=V1b and –I2a=I1b then The main use of ABCD matrices are for chaining circuit elements together Or just convert to one matrix Where

Determination of ABCD parameters Because A is independent of B, to determine A put I2 equal to zero and determine the voltage gain V1/V2=A of the circuit. In this case port 2 must be open circuit. for port 2 open circuit for port 2 short circuit for port 2 short circuit for port 2 open circuit

ABCD matrix for series impedance Z V2 V1 for port 2 open circuit for port 2 short circuit V1= - I2 Z hence B= Z V1= V2 hence A=1 for port 2 short circuit for port 2 open circuit I1 = - I2 = 0 hence C= 0 I1 = - I2 hence D= 1 The full ABCD matrix can be written

ABCD for T impedance network Z1 Z2 V1 Z3 V2 for port 2 open circuit therefore then

Continue for port 2 short circuit Solving for voltage in Z2 Hence But VZ2 Solving for voltage in Z2 Z2 Z3 Hence But

Continue Z1 I1 I2 for port 2 open circuit V2 Z3 Analysis Therefore

Continue for port 2 short circuit I1 is divided into Z2 and Z3, thus VZ2 Z2 Z3 I1 is divided into Z2 and Z3, thus Full matrix Hence

ABCD for transmission line Input V1 Transmission line Zo g V2 z =0 z = - For transmission line f and b represent forward and backward propagation voltage and current Amplitudes. The time varying term can be dropped in further analysis.

continue At the input z = - At the output z = 0 (2) (1) At the output z = 0 (4) (3) Now find A,B,C and D using the above 4 equations for port 2 open circuit For I2 =0 Eq.( 4 ) gives Vf= Vb=Vo giving

continue Note that From Eq. (1) and (3) we have for port 2 short circuit For V2 = 0 , Eq. (3) implies –Vf= Vb = Vo . From Eq. (1) and (4) we have

continue for port 2 open circuit For I2=0 , Eq. (4) implies Vf = Vb = Vo . From Eq.(2) and (3) we have for port 2 short circuit For V2=0 , Eq. (3) implies Vf = -Vb = Vo . From Eq.(2) and (4) we have

continue Note that The complete matrix is therefore Where a= attenuation k=wave propagation constant When the transmission line is lossless this reduces to Lossless line a = 0

Table of ABCD network Transmission line Z Series impedance Shunt impedance

Table of ABCD network Ideal transformer n:1 Z1 Z2 T-network Z3 Z3 Z1 p-network Ideal transformer n:1

Conversion S to ABCD For conversion of ABCD to S-parameter For conversion of S to ABCD-parameter Zo is a characteristic impedance of the transmission line connected to the ABCD network, usually 50 ohm.