Low-pass filter

• N : filter order
• ε: parameter governing ripple in the pass-band
• T_N : Tchebycheff polynomial T_N(x)=2x.T_N-1(x)-T_N-2(x) with T₀(x)=1 et T₁(x)=x
• w_C: cutoff radian frequency in rad.s-1 rad.s^-1 with w =2π.f   ; f : fréquency in Hertz.

Chebyshev low-pass filters responses: :

Normalised low-pass prototype filter

Synthesis of the normalised Chebyshev low-pass prototype filter:

Normalised impedance: R’₀=1W (G’₀=1W^-1, g₀=1)
Normalised cut-off radian frequency: w’_C=1rad.s^-1
L_Ar : ripple expressed in dB.

hen it is necessary to denormalize the low_pass prototype filter by a frequency transformation (normalised low-pass => low-pass) and an impedance (1ohm => 50ohm) transformation. This leads to a ladder circuit (parallel capacities, Ci; series inductances, Li) with a cut-off frequency meeting the specification and functioning within a 50 Ohm environment.

The capacities and the inductances of the ladder network that synthesises the specified frequency response consist of line sections of short length with respect to the wavelength at the filter cut-off frequency. These components are denoted as semi-lumped elements. The parallel capacities consist of line sections with a low characteristic impedance, Zcapa, (whose value is usually about 25), which means line sections of width, Wcapa. For a given filter, as the line sections synthesizing the capacities have all the same width, Wcapa. it is only the respective length of each section that sets the value of the synthesised capacity.
To achieve a parallel capacity of value Ci, the line section length, lcapa, is determined as follows:

εreff capa: effective relative dielectric permittivity for a line of width, Wcapa, engraved on the chosen substrate.
The procedure is alike for the series inductances, which consist of line sections with a high characteristic impedance, Zind, (usually around 100 ohm). These inductive line sections are, thus, very thin, and their width, Wind, is close to the technological limit allowed by the selected substrate.
To achieve a series inductance of value, Li , the line section length, lind, is determined as follows:

εreff ind: effective relative dielectric permittivity for a line of width, Wind, engraved on the chosen substrate.
The conductor circuit below shows how to cascade the parallel capacities and series selfs to achieve, in microstrip technology, the low-pass filtering function modelled from the ladder filter:

Conductor circuit of a low-pass microstrip filter (N=5)

W0: width of access line with a characteristic impedance, Z0, usually around 50 ohm.
Remark: the line section lengths have to be slightly adjusted to compensate for the effects induced by width discontinuities