Commensurate line circuit

In some technologies, including the widely used microstrip, series connections are difficult or impossible to implement.

Nevertheless, the commensurate line theory remains the basis for many of these more advanced filter designs.

Very wide lines, on the other hand, allow the possibility of undesirable transverse resonant modes to form.

[3] Electrical length can also be expressed as the phase change between the start and the end of the line.

The fundamental problem is that using more than one length generally requires more than one frequency variable.

A well developed theory exists for synthesising lumped-element circuits from a given frequency function.

[7] 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).

This multiple passband result is a general feature of all distributed-element circuits, not just those arrived at through Richards' transformation.

[9] A UE connected in cascade is a two-port network that has no exactly corresponding circuit in lumped elements.

There are lumped-element circuits that can approximate a fixed delay such as the Bessel filter, but they only work within a prescribed passband, even with ideal components.

[1] Two or more UEs connected in cascade with the same Z0 are equivalent to a single, longer, transmission line.

Some circuits can be implemented entirely as a cascade of UEs: impedance matching networks, for instance, can be done this way, as can most filters.

[1] Kuroda's identities are a set of four equivalent circuits that overcome certain difficulties with applying Richards' transformations directly.

Here the symbols for capacitors and inductors are used to represent open-circuit and short-circuit stubs.

Filter circuits frequently use a ladder topology with alternating series and shunt elements.

However, they have the disadvantage of requiring the addition of an ideal transformer with a turns ratio equal to the scaling factor.

[14] In the decade after Richards' publication, advances in the theory of distributed circuits took place mostly in Japan.

[16] One of the major applications of commensurate line theory is to design distributed-element filters.

However, a single length of transmission line can only be precisely λ/4 long at its resonant frequency and there is consequently a limit to the bandwidth over which it will work.

Applying Kuroda's identities will convert these to all shunt capacitors, which are open circuit stubs.

However, this is not the case in other technologies such as coaxial line, or twin-lead where the short circuit may actually be helpful for mechanical support of the structure.

Example commensurate line design for a 4 GHz, 50 Ω, third order 3 dB Chebyshev low-pass filter . A. Prototype filter in lumped elements, ω=1, Z 0 =1. B. Filter frequency and impedance scaled to 4 GHz and 50 Ω; these component values are too small to easily implement as discrete components. C. The prototype circuit transformed to open-wire commensurate lines by Richards' transformation. D. Applying Kuroda's identities to prototype to eliminate the series inductors. E. Impedance scaling for 50 Ω working, frequency scaling is achieved by setting the line lengths to λ/8. F. Implementation in microstrip .
Frequency response of a fifth order Chebyshev filter (top), and the same filter after applying Richards' transformation
Kuroda's identities