page_title: Commensurate line circuit
section_title: Omega-domain
hierarchical_section_title: Commensurate line circuit / Omega-domain
context_page_description: Commensurate line circuits are electrical circuits composed of transmission lines that are all the same length; commonly one-eighth of a wavelength. Lumped element circuits can be directly converted to distributed-element circuits of this form by the use of Richards' transformation. This transformation has a particularly simple result; inductors are replaced with transmission lines terminated in short-circuits and capacitors are replaced with lines terminated in open-circuits. Commensurate line theory is particularly useful for designing distributed-element filters for use at microwave frequencies.
  It is usually necessary to carry out a further transformation of the circuit using Kuroda's identities. There are several reasons for applying one of the Kuroda transformations; the principal reason is usually to eliminate series connected components. In some technologies, including the widely used microstrip, series connections are difficult or impossible to implement.
  The frequency response of commensurate line circuits, like all distributed-element circuits, will periodically repeat, limiting the frequency range over which they are effective.
context_section_description: Richards' transformation can be viewed as transforming from a s-domain representation to a new domain called the Ω-domain where,
  If Ω is normalised so that Ω=1 when ω=ωc, then it is required that,
  and the length in distance units becomes,
  Any circuit composed of discrete, linear, lumped components will have a transfer function H(s) that is a rational function in s.  A circuit composed of transmission line UEs derived from the lumped circuit by Richards' transformation will have a transfer function H(jΩ) that is a rational function of precisely the same form as H(s).  That is, the shape of the frequency response of the lumped circuit against the s frequency variable will be precisely the same as the shape of the frequency response of the transmission line circuit against the jΩ frequency variable and the circuit will be functionally the same.
  However, infinity in the Ω domain is transformed to ω=π/4k in the s domain.  The entire frequency response is squeezed down to this finite interval.  Above this frequency, the same response is repeated in the same intervals, alternately in reverse.  This is a consequence of the periodic nature of the tangent function.  This multiple passband result is a general feature of all distributed-element circuits, not just those arrived at through Richards' transformation.
caption: Frequency response of a fifth order Chebyshev filter (top), and the same filter after applying Richards' transformation