13 S-Parameters 13.1 Scattering Parameters Linear two-port (and multi-port) networks are characterized by a number of equivalent circuit parameters, such as their transfer matrix, impedance matrix, admittance matrix, and scattering matrix. Fig. 13.1.1 shows a typical two-port network. Fig. 13.1.1 Two-port network. Thetransfer matrix , also known as the ABCD matrix, relates the voltage and current at port 1 to those at port 2, whereas theimpedance matrixrelates the two voltages V1,V2 to the two 1currentsI,I 2: ��� V1 I 1 = A B C D V2 I2 (transfer matrix) V1 V2 = Z11 Z12 Z21 Z22 I 1 ���I 2 (impedance matrix) (13.1.1) Thus, the transfer and impedance matrices are the 2��2 matrices: T= A B C D , Z= Z11 Z12 Z21 Z22 (13.1.2) Theadmittance matrixis simply the inverse of the impedance matrix,Y=Z���1. The scattering matrix relates theoutgoing wavesb1,b2 to theincoming wavesa1,a2 that areincidenton the two-port: ���In the figure,I2 flows out of port 2, and hence���I2 flows into it. In the usual convention, both currents I1,I2 are taken to flow into their respective ports. 526 13. S-Parameters b1 b2 = S11 S12 S21 S22 a1 a2 , S= S11 S12 S21 S22 (scattering matrix) (13.1.3) The matrix elementsS11,S12,S21,S22 are referred to as thescattering parametersor the S-parameters . The parametersS11,S22 have the meaning of reflection coefficients, andS21,S12, the meaning of transmission coefficients. The many properties and uses of theS-parameters in applications are discussed in [980���1019]. One particularly nice overview is the HP application note AN-95-1 by Anderson [995] and is available on the web [1354]. We have already seen several examples of transfer, impedance, and scattering ma- trices. Eq. (10.7.6) or (10.7.7) is an example of a transfer matrix and (10.8.1) is the corresponding impedance matrix. The transfer and scattering matrices of multilayer structures, Eqs. (6.6.23) and (6.6.37), are more complicated examples. The traveling wave variablesa1,b1 at port 1 anda2,b2 at port 2 are defined in terms ofV1,I 1 andV2,I 2 and a real-valued positive reference impedanceZ0 as follows: a1= V1+Z0I1 2 Z0 b1= V1���Z0I1 2 Z0 a2 = V2���Z0I2 2 Z0 b2 = V2+Z0I2 2 Z0 (traveling waves) (13.1.4) The definitions at port 2 appear different from those at port 1, but they are really the same if expressed in terms of the incoming current���I2: a2= V2���Z0I 2 2 Z0 = 2V2+Z0(���I) 2 Z0 b2= V2+Z0I 2 2 Z0 = 2V2���Z0(���I) 2 Z0 The term traveling wavesis justified below. Eqs. (13.1.4) may be inverted to express the voltages and currents in terms of the wave variables: V1= Z0(a1+b1) I1= 1 Z0 (a1���b1) V2= Z0(a2+b2) 2I= 1 Z0 (b2���a2) (13.1.5) In practice, the reference impedance is chosen to beZ0 = 50 ohm. At lower fre- quencies the transfer and impedance matrices are commonly used, but at microwave frequencies they become difficult to measure and therefore, the scattering matrix de- scription is preferred. TheS-parameters can be measured by embedding the two-port network (the device- under-test, or, DUT) in a transmission line whose ends are connected to a network ana- lyzer. Fig. 13.1.2 shows the experimental setup. A typical network analyzer can measureS-parameters over a large frequency range, for example, the HP 8720D vector network analyzer covers the range from 50 MHz to

13.1. Scattering Parameters 527 40 GHz. Frequency resolution is typically 1 Hz and the results can be displayed either on a Smith chart or as a conventional gain versus frequency graph. Fig. 13.1.2 Device under test connected to network analyzer. Fig. 13.1.3 shows more details of the connection. The generator and load impedances are configured by the network analyzer. The connections can be reversed, with the generator connected to port 2 and the load to port 1. Fig. 13.1.3 Two-port network under test. The two line segments of 1lengthsl,l 2 are assumed to have characteristic impedance equal to the reference impedance Z0. Then, the wave variablesa1,b1 and a2,b2 are recognized as normalized versions of forward and backwardtraveling waves . Indeed, according to Eq. (10.7.8), we have: a1 = V1+Z0I1 2 Z0 = 1 Z0 V1+ b1 = V1���Z0I1 2 Z0 = 1 Z0 V1��� a2= V2���Z0I2 2 Z0 = 1 Z0 V2��� b2= V2+Z0I2 2 Z0 = 1 Z0 V2+ (13.1.6) Thus,a1is essentially the incident wave at port 1 andb1the corresponding reflected wave. Similarly,a2 is incident from the right onto port 2 and b2 is the reflected wave from port 2. The network analyzer measures the wavesa1,b1 and a2,b2 at the generator and load ends of the line segments, as shown in Fig. 13.1.3. From these, the waves at the inputs of the two-port can be determined. Assuming lossless segments and using the propagation matrices (10.7.7), we have: 528 13. S-Parameters a1 b1 = e���j��1 0 0 ej��1 a1 b1 , a2 b2 = e���j��2 0 0 ej��2 a2 b2 (13.1.7) where��1 l=��land��2=��l 2 are the phase lengths of the segments. Eqs. (13.1.7) can be rearranged into the forms: b1 b2 =D b1 b2 , a1 a2 =D a1 a2 , D= ej��1 0 0 ej��2 The network analyzer measures the correspondingS-parameters of the primed vari- ables, that is, b1 b2 = S11 S12 S21 S22 a1 a2 , S = S11 S12 S21 S22 (measuredS-matrix) (13.1.8) TheS-matrix of the two-port can be obtained then from: b1 b2 =D b1 b2 =DS a1 a2 =DSD a1 a2 ��� S=DSD or, more explicitly, S11 S12 S21 S22 = ej��1 0 0 ej��2 S11 S12 S21 S22 ej��1 0 0 ej��2 = S11e2j��1 S12ej(��1+��2) S21ej(��1+��2) S22e2j��2 (13.1.9) Thus, changing the points along the transmission lines at which theS-parameters are measured introduces only phase changes in the parameters. Without loss of generality, we may replace the extended circuit of Fig. 13.1.3 with the one shown in Fig. 13.1.4 with the understanding that either we are using the extended two-port parametersS, or, equivalently, the generator and segmentl 1 have been re- placed by their Th�� evenin equivalents, and the load impedance has been replaced by its propagated version to distance 2l. Fig. 13.1.4 Two-port network connected to generator and load.

13.2. Power Flow 529 The actual measurements of theS-parameters are made by connecting to a matched load,ZL =Z0. Then, there will be no reflected waves from the load,a2 = 0, and the S-matrix equations will give: b1=S11a1+S12a2=S11a1 ��� S11= b1 a1 ZL=Z0 = reflection coefficient b2=S21a1+S22a2=S21a1 ��� S21= b2 a1 ZL=Z0 = transmission coefficient Reversing the roles of the generator and load, one can measure in the same way the parametersS12 andS22. 13.2 Power Flow Power flow into and out of the two-port is expressed very simply in terms of the traveling wave amplitudes. Using the inverse relationships (13.1.5), we find: 1 2 Re[V1 ���I]= 1 1 2 | a 1|��� 2 1 2 | b 1| 2 ��� 1 2 Re[V2 ���I]= 2 1 2 | a 2|��� 2 1 2 | b 2| 2 (13.2.1) The left-hand sides represent the power flow intoports 1 and 2. The right-hand sides represent the difference between the power incident on a port and the power reflected from it. The quantity Re[V2���I 2 ]/ 2 represents the power transferred to the load. Another way of phrasing these is to say that part of the incident power on a port gets reflected and part enters the port: 1 2 |1 a |= 2 1 2 |1|2+ b 1 2 Re[V1���I1] 1 2 |2 a |= 2 1 2 |2|2+ b 1 2 Re[V2 ���(���I)] 2 (13.2.2) One of the reasons for normalizing the traveling wave amplitudes by Z0 in the definitions (13.1.4) was precisely this simple way of expressing the incident and reflected powers from a port. If the two-port is lossy, the power lost in it will be the difference between the power entering port 1 and the power leaving port 2, that is, Ploss= 1 2 Re[V1 ���I]��� 1 1 2 Re[V2���I2]= 1 2 | a 1 |2+ 1 2 | a 2 |2��� 1 2 | b 1 |2��� 1 2 | b 2| 2 Noting thata���a= | a 12|+|a2| 2 andb���b= | b 12|+|b2|, 2 and writingb���b=a���S���Sa, we may express this relationship in terms of the scattering matrix: Ploss= 1 2 a���a��� 1 2 b���b= 1 2 a���a��� 1 2 a���S���Sa= 1 2 a���(I���S���S)a (13.2.3) 530 13. S-Parameters For a lossy two-port, the power loss is positive, which implies that the matrixI���S���S must be positive definite. If the two-port is lossless,Ploss = 0, theS-matrix will be unitary , that is,S���S=I. We already saw examples of such unitary scattering matrices in the cases of the equal travel-time multilayer dielectric structures and their equivalent quarter wavelength mul- tisection transformers. 13.3 Parameter Conversions It is straightforward to derive the relationships that allow one to pass from one param- eter set to another. For example, starting with the transfer matrix, we have: V1=AV2+BI2 I1=CV2+DI2 ��� V1=A (1 C I1��� D C I 2 ) +BI2= A C 1I��� AD���BC C I 2 V2= 1 C I1��� D C I 2 The coefficients 1ofI,I2are the impedance matrix elements. The steps are reversible, and we summarize the final relationships below: Z= Z11 Z12 Z21 Z22 = 1 C A AD���BC 1 D T= A B C D = 1 Z21 Z11 Z11Z22���Z12Z21 1 Z22 (13.3.1) We note the determinants det(T)= AD���BC and det(Z)=Z11Z22���Z12Z 21 . The relationship between the scattering and impedance matrices is also straightforward to derive. We define the 2��1 vectors: V= V1 V2 , I= I1 ���I2 , a= a1 a2 , b= b1 b2 (13.3.2) Then, the definitions (13.1.4) can be written compactly as: a= 1 2 Z0 (V+Z0I= ) 1 2 Z0 (Z+Z0I)I b= 1 2 Z0 (V���Z0I= ) 1 2 Z0 (Z���Z0I)I (13.3.3) where we used the impedance matrix relationship V = ZIand defined the 2��2 unit matrix I . It follows then, 1 2 Z0 I=(Z+Z0I)���1a ��� b= 1 2 Z0 (Z���Z0I)I=(Z���Z0I)(Z+Z0I)���1a Thus, the scattering matrixSwill be related to the impedance matrixZby S=(Z���Z0I)(Z+Z0I)���1 Z=(I���S)���1(I+S)Z0 (13.3.4)