Stub Filter (Richards' Transformation)
Richards' transformation Ω = tan(βℓ) maps a lumped lowpass prototype onto commensurate transmission-line stubs. Choosing each stub to be λ/8 long at the cutoff frequency makes tan(βℓ)=1 there, so a prototype inductor becomes a short-circuited (series) stub and a prototype capacitor becomes an open-circuited (shunt) stub. This tool gives the characteristic impedance of each stub directly from the g-values — the starting point before applying Kuroda's identities to make everything shunt and physically separated.
Equations & Parameters ▸
Richards: \(\Omega=\tan\beta\ell\), stubs \(\ell=\lambda/8\) at \(f_c\) so \(\tan\beta\ell=1\).
series inductor \(g_k\to\) short-circuit series stub, \(Z_0^{stub}=g_k Z_0\)
shunt capacitor \(g_k\to\) open-circuit shunt stub, \(Z_0^{stub}=Z_0/g_k\)
stub physical length \(\ell=\dfrac{v_p}{8f_c}=\dfrac{\lambda_g}{8}\)
series inductor \(g_k\to\) short-circuit series stub, \(Z_0^{stub}=g_k Z_0\)
shunt capacitor \(g_k\to\) open-circuit shunt stub, \(Z_0^{stub}=Z_0/g_k\)
stub physical length \(\ell=\dfrac{v_p}{8f_c}=\dfrac{\lambda_g}{8}\)
| N | Filter order (1–10). |
| Response | Butterworth or Chebyshev prototype g-values. |
| LAr | Chebyshev passband ripple (dB). |
| Z0, fc | System impedance and cutoff; f_c and ε_eff set the λ/8 stub length. |
| εeff | Effective permittivity of the line (1 for air). |
Reference: D. M. Pozar, Microwave Engineering, 4th ed., §8.4.
Inputs
1 – 10
Shape
dB
ChebyshevΩ
SystemGHz
λ/8 lengthLine ε_eff
Stub impedances (λ/8)
Stub realisation