Handbook of Microwave Component Measurements. Joel P. Dunsmore

      The conversions between the S‐matrix and the Y‐matrix are

      (2.36)

      where Δ_YS = (1 + S₁₁)(1 + S₂₂) − S₂₁S₁₂

      (2.37)

      where ΔY = (Y₀ + Y₁₁)(Y₀ + Y₂₂) − Y₂₁Y₁₂

       2.6.3 ABCD Parameters

      Just as the T‐parameters provide for easy concatenation of devices using a and b waves, a similar matrix representation can be used when the terminal characteristics are defined in terms of voltage and current. These are sometimes called transfer parameters (reminiscent of the T‐parameters) or chain parameters as networks can be chained together in a matrix multiplication manner.

      The functional definition of ABCD parameters is found in at least two different forms, one of which is

(2.38)

      The second form replaces the minus sign with a plus sign with resulting changes in the derived values.

From Eq. 2.38 the values for ABCD parameters can be defined as

      (2.39)

      The transformations between the ABCD‐matrix and the S‐matrix are

      (2.40)

      (2.41)

      where

       2.6.4 H‐Parameters or Hybrid Parameters

      Because of its intrinsic transfer function, as a voltage‐controlled current source, transistor performance has often been described using hybrid parameters. Their functional definition is

      (2.42)

      From this, the definition of individual H‐parameters is

      (2.43)

      The H‐matrix is most simply defined in terms of the other impedance matrixes as

      (2.44)

       2.6.5 Complex Conversions and Non‐equal Reference Impedances

      It is important to note that all the conversions described in the previous sections are valid only for the case of Z₀₁ = Z₀₂ = Z₀ and that Z₀ be pure‐real. The transformations for cases where the port impedances are not equal, and not real, have been computed and are available in papers by Marks and Williams (1992) and with using somewhat different wave definitions, by D.A. Frickey (1994). While the case of complex termination impedances is unusual, the case for differing reference impedances on ports is more common. Since the network elements do not change when reference impedance of the system is changed, Y‐, Z‐, and H‐related parameters also do not change with reference impedance. S‐ and T‐ parameter values do change, and thus it is critical to know the reference impedance for each case.

References

      1 Frickey, D.A. (1994). Conversions between S, Z, Y, H, ABCD, and T parameters which are valid for complex source and load impedances. IEEE Transactions on Microwave Theory and Techniques 42 (2): 205–211.

      2 Hong, J.‐S. and Lancaster, M.J. (2001). Microstrip Filters for RF/Microwave Applications. New York: Wiley. Print.

      3 Keysight Technologies (n.d.‐a) S‐Parameters... Circuit Analysis and Design [Online]. Available at: http://literature.cdn.keysight.com/litweb/pdf/5952-0918.pdf (Accessed 20 January 2020).

      4 Keysight Technologies (n.d.‐b) S‐Parameter Design [Online], Available at: https://literature.cdn.keysight.com/litweb/pdf/5952-1087.pdf?id=1000001947:epsg:apn (Accessed 28 December 2018).

      5 Marks, R.B. and Williams, D.F. (1992). A general waveguide circuit theory. Journal of Research of the National Institute of Standards and Technology 97: 533–561.

      6 Mavaddat, R. (1996). Network Scattering Parameters. Singapore: World Scientific. Print.

      7 Smith, P.H. (1944). An improved transmission line calculator. Electronics 17: 130–133.

      8 Tippet, J.C. and Speciale, R.A. (1982). A rigorous technique for measuring the scattering matrix of a multiport device with a 2‐port network analyzer. IEEE Transactions on Microwave Theory and Techniques 30 (5): 661–666.