126 Impedance Matchingthe impedance looking into the transmission line is periodic with distance. It can be shown from (5.40) that for each distance l traveled down the transmission line, the impedance makes two clockwise rotations about the center of the Smith chart. Transmission lines can also be used to synthesize reactive impedances. Note that if ZL is either an open or short circuit, then by making the transmission line an appropriate length, any purely reactive impedance can be realized. These types of transmission lines are usually referred to as open- and short-circuit stubs. This topic will not be addressed further because the size of transmission line matching elements over much of the frequency range of RFICs is very large compared with the relatively small size of lumped inductors, capacitors, and transformers.5.13 S, Y, and Z ParametersS, Y, and Z parameters (scattering, admittance, and impedance parameters, respec-tively) are widely used in the analysis of RF circuits. For RF measurements (for example, with a network analyzer), typically S-parameters are used. These may be later converted to Y or Z parameters in order to perform certain analyses. In this section, S-parameters and conversions to other parameters will be described. S-parameters are a way of calculating a two-port network in terms of incident and reflected (or scattered) power. Referring to Figure 5.33, assume that port 1 is the input, a1 is the input wave, b1 is the reflected wave, and b2 is the transmitted wave. We note that if a transmission line is terminated in its characteristic imped-ance, then the load absorbs all incident power traveling along the transmission line, and there is no reflection.The S-parameters can be used to describe the relationship between these waves as follows: 111 112 2bS aS a=+ (5.41) 222 221 1bS aS a=+ (5.42)This can also be written in a matrix as: 111121221222[ ] [ ][ ]bSSabS abSSaéùéù é ù===êúêú ê úëû ë ûëû (5.43)Thus, S-parameters are reflection or transmission coefficients and are usually nor-malized to a particular impedance. S11, S22, S21, and S12 will now each be defined. 211110abSa== (5.44)Figure 5.33 General two-port system with incident and reflected waves.