The transformation $$Ω = tan \beta = tan(\frac{wL}{V_p})$$

Maps he plane, which repeats with a period of . This transformation was introduced by P. Richards to synthesize an LC network using open and short-circuited transmission line stubs. Thus, if we replace the frequency variable with Ω. We can write the reactance of an inductor as

$$jX_L=jΩL=jL tan \beta l$$

And the susceptance of a capacitor as

$$jB_c = jΩC = jC tan \beta l$$

These results indicate that an inductor can be replaced with a short-circuited stub of length and characteristic impedance L, while a capacitor can be replaced with an open-circuited stub of length $\beta l$ and characteristic impedance 1/C. A unity filter impedance is assumed.

Cutoff occurs at unity frequency for a low-pass filter prototype; to obtain the same cut off frequency for the Richards-transformed filter.

$$Ω= 1 = tan \beta l$$

Which gives a stub length of $l=λ/8$ where λ is the wavelength of the line at cutoff frequency . At the frequency $w_0=2w_c$, the lines will be λ/4 long and an attenuation pole will occur. At frequencies away from $w_c$, the impendence of the stubs will no longer match the original lumped-element impedances, and the filter response will differ from the desired prototype response. In addition, the response will be periodic in frequency, repeating every $4w_c$ .

In principle, then Richards transformation allows the inductors and capacitors of a lumped-element filter to be replaced with short circuited and open –circuited transmission line stubs as illustrated in figure Since the electrical lengths of all the stubs are the same (λ/8 at $w_c$) these lines are called commenstrate lines.

The additional transmission line sections are called unit elements and are λ/8 long at the unit elements are thus commensurate with the stubs used to implement the inductors and capacitors of the prototype design. The four Kuroda identities are illustrated in Table where each box represents a unit element, or transmission line, of the indicated characteristics impedance and length (λ/8 at $w_c$). The inductors and capacitors represent short-circuit and open-circuit stubs,