664 14. S-Parameters b1 S11 S12 a1 S11 S12 14 = ,S= (scattering matrix) (14.1.3) b2 S21 S22 a2 S21 S22 S-Parameters The matrix elements S11,S12,S21,S22 are referred to as the scattering parameters or the S-parameters. The parameters S11, S22 have the meaning of reflection coefficients, and S21, S12, the meaning of transmission coefficients. The many properties and uses of the S-parameters in applications are discussed in [1135–1174]. One particularly nice overview is the HP application note AN-95-1 by Anderson [1150] and is available on the web [1847]. We have already seen several examples of transfer, impedance, and scattering ma- trices. Eq. (11.7.6) or (11.7.7) is an example of a transfer matrix and (11.8.1) 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. 14.1 Scattering Parameters The traveling wave variables a1,b1 at port 1 and a2,b2 at port 2 are defined in terms of V1,I1 and V2,I2 and a real-valued positive reference impedance Z0 as follows: 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, V1 Z0I1 V2 Z0I2 a1 = a2 = and scattering matrix. Fig. 14.1.1 shows a typical two-port network. 2 Z0 2 Z0 (traveling waves) (14.1.4) − + V1 Z0I1 V2 Z0I2 b1 = b2 = 2 Z0 2 Z0 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: − + − V2 Z0I2 V2 Z0( I2) a2 = = Fig. 14.1.1 Two-port network. 2 Z0 2 Z0 V + Z I V − Z (−I ) = 2 0 2 = 2 0 2 The transfer matrix, also known as the ABCD matrix, relates the voltage and current b2 2 Z0 2 Z0 at port 1 to those at port 2, whereas the impedance matrix relates the two voltages † The term traveling waves is justified below. Eqs. (14.1.4) may be inverted to express V1,V2 to the two currents I1,I2: the voltages and currents in terms of the wave variables: V1 AB V2 = (transfer matrix) I CD I 1 2 = + = + (14.1.1) V1 Z0(a1 b1) V2 Z0(a2 b2) (14.1.5) V1 Z11 Z12 I1 1 1 = (impedance matrix) = − = − V2 Z21 Z22 −I2 I1 (a1 b1) I2 (b2 a2) Z0 Z0 × Thus, the transfer and impedance matrices are the 2 2 matrices: = In practice, the reference impedance is chosen to be Z0 50 ohm. At lower fre- AB Z11 Z12 quencies the transfer and impedance matrices are commonly used, but at microwave T = ,Z= (14.1.2) CD Z21 Z22 frequencies they become difficult to measure and therefore, the scattering matrix de- scription is preferred. −1 The admittance matrix is simply the inverse of the impedance matrix, Y = Z . The The S-parameters can be measured by embedding the two-port network (the device- scattering matrix relates the outgoing waves b1,b2 to the incoming waves a1,a2 that under-test, or, DUT) in a transmission line whose ends are connected to a network ana- are incident on the two-port: lyzer. Fig. 14.1.2 shows the experimental setup. † A typical network analyzer can measure S-parameters over a large frequency range, 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. for example, the HP 8720D vector network analyzer covers the range from 50 MHz to 14.1. Scattering Parameters 665 666 14. S-Parameters 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. −jδ1 −jδ2 a1 e 0 a1 a2 e 0 a2 = , = (14.1.7) jδ1 jδ2 b1 0 e b1 b2 0 e b2 where δ1 = βll and δ2 = βl2 are the phase lengths of the segments. Eqs. (14.1.7) can be rearranged into the forms: jδ1 b1 b1 a1 a1 e 0 = D , = D ,D= jδ2 b2 b2 a2 a2 0 e The network analyzer measures the corresponding S-parameters of the primed vari- ables, that is, Fig. 14.1.2 Device under test connected to network analyzer. b S S a S S 1 = 11 12 1 = 11 12 Fig. 14.1.3 shows more details of the connection. The generator and load impedances ,S (measured S-matrix) (14.1.8) b2 S21 S22 a2 S21 S22 are configured by the network analyzer. The connections can be reversed, with the generator connected to port 2 and the load to port 1. The S-matrix of the two-port can be obtained then from: b1 b1 a1 a1 = D = DS = DS D ⇒ S = DS D b2 b2 a2 a2 or, more explicitly, jδ1 jδ1 S11 S12 e 0 S11 S12 e 0 = jδ2 jδ2 S21 S22 0 e S21 S22 0 e (14.1.9) + S e2jδ1 S ej(δ1 δ2) = 11 12 j(δ1+δ2) 2jδ2 S21e S22e Fig. 14.1.3 Two-port network under test. Thus, changing the points along the transmission lines at which the S-parameters are measured introduces only phase changes in the parameters. The two line segments of lengths l1,l2 are assumed to have characteristic impedance Without loss of generality, we may replace the extended circuit of Fig. 14.1.3 with the equal to the reference impedance Z0. Then, the wave variables a1,b1 and a2,b2 are one shown in Fig. 14.1.4 with the understanding that either we are using the extended recognized as normalized versions of forward and backward traveling waves. Indeed, two-port parameters S , or, equivalently, the generator and segment l1 have been re- according to Eq. (11.7.8), we have: placed by their Th´evenin equivalents, and the load impedance has been replaced by its propagated version to distance l . + − 2 V1 Z0I1 1 V2 Z0I2 1 a1 = = V1+ a2 = = V2− 2 Z0 Z0 2 Z0 Z0 (14.1.6) − + V1 Z0I1 1 V2 Z0I2 1 b1 = = V1− b2 = = V2+ 2 Z0 Z0 2 Z0 Z0 Thus, a1 is essentially the incident wave at port 1 and b1 the 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 waves a1,b1 and a2,b2 at the generator and load ends of the line segments, as shown in Fig. 14.1.3. From these, the waves at the Fig. 14.1.4 Two-port network connected to generator and load. inputs of the two-port can be determined. Assuming lossless segments and using the propagation matrices (11.7.7), we have: 14.2. Power Flow 667 668 14. S-Parameters The actual measurements of the S-parameters are made by connecting to a matched For a lossy two-port, the power loss is positive, which implies that the matrix I −S†S load, ZL = Z0. Then, there will be no reflected waves from the load, a2 = 0, and the must be positive definite. If the two-port is lossless, Ploss = 0, the S-matrix will be S-matrix equations will give: unitary, that is, S†S = I. We already saw examples of such unitary scattering matrices in the cases of the equal b1 b1 = S11a1 + S12a2 = S11a1 ⇒ S11 = = reflection coefficient travel-time multilayer dielectric structures and their equivalent quarter wavelength mul- a = 1 ZL Z0 tisection transformers. b2 b2 = S21a1 + S22a2 = S21a1 ⇒ S21 = = transmission coefficient a = 1 ZL Z0 14.3 Parameter Conversions Reversing the roles of the generator and load, one can measure in the same way the It is straightforward to derive the relationships that allow one to pass from one param- parameters S12 and S22. eter set to another. For example, starting with the transfer matrix, we have: 1 D A AD − BC 14.2 Power Flow = + V1 = A I1 − I2 + BI2 = I1 − I2 V1 AV2 BI2 C C C C ⇒ Power flow into and out of the two-port is expressed very simply in terms of the traveling I = CV + DI 1 D 1 2 2 V = I − I wave amplitudes. Using the inverse relationships (14.1.5), we find: 2 C 1 C 2 The coefficients of I ,I are the impedance matrix elements. The steps are reversible, 1 ∗ 1 2 1 2 1 2 Re[V I1] = |a1| − |b1| 2 1 2 2 and we summarize the final relationships below: (14.2.1) 1 ∗ 1 2 1 2 − Re[V I2] = |a2| − |b2| Z11 Z12 1 AAD− BC 2 2 2 2 Z = = Z21 Z22 C 1 D The left-hand sides represent the power flow into ports 1 and 2. The right-hand sides (14.3.1) represent the difference between the power incident on a port and the power reflected AB 1 Z11 Z11Z22 − Z12Z21 ∗ T = = from it. The quantity Re[V2 I2]/2 represents the power transferred to the load. CD Z21 1 Z22 Another way of phrasing these is to say that part of the incident power on a port We note the determinants det(T)= AD − BC and det(Z)= Z11Z22 − Z12Z21. The gets reflected and part enters the port: relationship between the scattering and impedance matrices is also straightforward to derive. We define the 2×1 vectors: 1 2 1 2 1 ∗ |a1| = |b1| + Re[V I1] 2 2 2 1 (14.2.2) V1 I1 a1 b1 V = , I = , a = , b = (14.3.2) 1 2 1 2 1 ∗ − |a2| = |b2| + Re[V (−I2)] V2 I2 a2 b2 2 2 2 2 Then, the definitions (14.1.4) can be written compactly as: One of the reasons for normalizing the traveling wave amplitudes by Z0 in the definitions (14.1.4) was precisely this simple way of expressing the incident and reflected 1 1 = + = + powers from a port.