Synthesis of a 5-pole LC Lowpass Filter
with element extraction by pole removal at infinity

By D. Pelz (dpelz@rfcurrent.com)

The filter response function (characteristic function) of a 5-pole lowpass filter has been found by a preceding approximation process. This approximation could be done manually by adjusting the two reflection zeroes in the filters passband and a constant factor in such a way that the desired passband ripple attenuation exists at the passband edge frequencies and at the reflection peak between the two edges. The lowpass filter is required to have a passband from 0.78 to 1.0. Therefore, it makes sense that its reflection zeroes lie within these boundaries. The resulting lowpass filter response is also known as 'constrained passband lowpass filter'.
Passband edge frequencies:

Reflection zeroes:

Constant factor:

Characteristic Function (Reflection Function):
p = sigma + j x omega (complex frequency variable)

Filter response function in dB

K(p) can be converted from its product form into a polynomial by simply multiplying out the product form. Mathcad does this conventiently and produces the polynomial coefficients:

The transfer function polynomial is found from the relationship between mismatch reflection and attenuation (Feldtkeller equation). We only need to find the roots of the polynomial 1 + KF(p):

The Hurwitz condition requires that the roots of the transfer function lie in the left half of the complex number plane. Therefore, we need to select only the left half roots for the transfer function polynomial:

We can now define the transfer function polynomial from its roots:

In order to check the correctness of the found transfer function, we can plot the filter response graph again 
from the roots of the transfer function polynomial:

Now we have all required information for finding the normalised circuit elements of the lowpass filter.
Characteristic function polynomial coefficients:

Transfer function polynomial coefficients:

The short-circuit input impedance function Z1sc(p) is found from the even- and odd parts of the characteristic 
function and the transfer function polynomials. We create these polynomials first:

We can now build the numerator and denominator polynomials of the short-circuit input impedance
function. The numerator is the sum of the odd part polynomials (the characteristic function is itself and
odd function). The denominator is the even part of the transfer function polynomial:

The short-circuit input impedance rational function can now be defined as:

Given that the lowpass filter's transfer function is a polynomial, all its transmission zeroes lie at infinity. That implies that the network only contains series-Ls and shunt-Cs - each of these producing one transmission zero at infinity. The first element of such a 5-th degree network can start with either a series-L or a shunt-C. We chose the series-L. At omega = infinity the shunt-C following the first series-L has a zero impedance. Therefore, the input impedance at omega = infinity must simply be p x L1. We find its normalised value from
Z1sc (p) = p x L1  @ p = infinity by:

Mathcad does not perform well in subsequent pole removal calculations and therefore the element extraction is done by a small subroutine. The algorithm is based on calculating the elements from the highest coefficients of the short-circuit input impedance polynomials and subsequent calculation of the remainder input impedance and so on, until all circuit elements are extracted.

Normalised series inductance values:

Normalised shunt capacitance values:

The odd-degree lowpass filter network has symmetry: L1 = L5, C2 = C4

In order to check the correctness of the found circuit elements, let's analyse the entire circuit. The easiest way to analyse ladder type circuits is probably by calculating the input impedance by successive inversion, starting with the series impedance of the normalised terminating resistance ( 1 ) and the right-most inductance L5, ie.  Zin_1 = 1 + p x L5.

Proof of correctness of extracted circuit elements

Q. E. D.

Denormalisation

In the above calculations, normalised frequencies and normalised circuit elements were used. For lowpass filters the normalisation frequency is usually the filter cutoff frequency. Denormalistion can be carried out in the following way. 
Filter cutoff frequency:

Port impedance: