Constant k filter

Terminology

Some of the impedance terms and section terms used in this article are pictured in the diagram below. Image theory defines quantities in terms of an infinite cascade of two-port sections, and in the case of the filters being discussed, an infinite ladder network of L-sections. Here "L" should not be confused with the inductance L – in electronic filter topology, "L" refers to the specific filter shape which resembles inverted letter "L".

The sections of the hypothetical infinite filter are made of series elements having impedance 2Z and shunt elements with admittance 2Y. The factor of two is introduced for mathematical convenience, since it is usual to work in terms of half-sections where it disappears. The image impedance of the input and output port of a section will generally not be the same. However, for a mid-series section (that is, a section from halfway through a series element to halfway through the next series element) will have the same image impedance on both ports due to symmetry. This image impedance is designated ZiT due to the "T" topology of a mid-series section. Likewise, the image impedance of a mid-shunt section is designated ZiΠ due to the "Π" topology. Half of such a "T" or "Π" section is called a half-section, which is also an L-section but with half the element values of the full L-section. The image impedance of the half-section is dissimilar on the input and output ports: on the side presenting the series element it is equal to the mid-series ZiT, but on the side presenting the shunt element it is equal to the mid-shunt ZiΠ . There are thus two variant ways of using a half-section.

Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals.

Derivation

The building block of constant k filters is the half-section "L" network, composed of a series impedance Z, and a shunt admittance Y. The "k" in "constant k" is the value given by,

${\displaystyle k^2=\frac{Z}{Y}}$

Thus, k will have units of impedance, that is, ohms. It is readily apparent that in order for k to be constant, Y must be the dual impedance of Z. A physical interpretation of k can be given by observing that k is the limiting value of Z_i as the size of the section (in terms of values of its components, such as inductances, capacitances, etc.) approaches zero, while keeping k at its initial value. Thus, k is the characteristic impedance, Z₀, of the transmission line that would be formed by these infinitesimally small sections. It is also the image impedance of the section at resonance, in the case of band-pass filters, or at ω = 0 in the case of low-pass filters. For example, the pictured low-pass half-section has

${\displaystyle k=\sqrt{\frac{i\omega L}{i\omega C}}=\sqrt{\frac{L}{C}}}$.

Elements L and C can be made arbitrarily small while retaining the same value of k. Z and Y however, are both approaching zero, and from the formulae (below) for image impedances,

${\displaystyle \underset{Z,Y\to 0}{lim}{Z}_{\mathrm{i}}=k}$.

Image impedance

The image impedances of the section are given by

${\displaystyle {Z_{\mathrm{i}\mathrm{T}}}^2=Z^2+k^2}$

and

${\displaystyle \frac{1}{{Z_{\mathrm{i}\mathrm{\Pi }}}^2}={Y_{\mathrm{i}\mathrm{\Pi }}}^2=Y^2+\frac{1}{k^2}}$

Given that the filter does not contain any resistive elements, the image impedance in the pass band of the filter is purely real and in the stop band it is purely imaginary. For example, for the pictured low-pass half-section,

${\displaystyle {Z_{\mathrm{i}\mathrm{T}}}^2=-(\omega L{)}^2+\frac{L}{C}}$

The transition occurs at a cut-off frequency given by

${\displaystyle {\omega }_c=\frac{1}{\sqrt{LC}}}$

Below this frequency, the image impedance is real,

${\displaystyle Z_{\mathrm{i}\mathrm{T}}=L\sqrt{{\omega }_c^2-{\omega }^2}}$

Above the cut-off frequency the image impedance is imaginary,

${\displaystyle Z_{\mathrm{i}\mathrm{T}}=iL\sqrt{{\omega }^2-{\omega }_c^2}}$

Transmission parameters

The transmission parameters for a general constant k half-section are given by

${\displaystyle \gamma ={\sinh }^{-1}\frac{Z}{k}}$

and for a chain of n half-sections

${\displaystyle {\gamma }_n=n\gamma }$

For the low-pass L-shape section, below the cut-off frequency, the transmission parameters are given by

${\displaystyle \gamma =\alpha +i\beta =0+i{\sin }^{-1}\frac{\omega }{{\omega }_c}}$

That is, the transmission is lossless in the pass-band with only the phase of the signal changing. Above the cut-off frequency, the transmission parameters are:

${\displaystyle \gamma =\alpha +i\beta ={\cosh }^{-1}\frac{\omega }{{\omega }_c}+i\frac{\pi }{2}}$

Prototype transformations

The presented plots of image impedance, attenuation and phase change correspond to a low-pass prototype filter section. The prototype has a cut-off frequency of ω_c = 1 rad/s and a nominal impedance k = 1 Ω. This is produced by a filter half-section with inductance L = 1 henry and capacitance C = 1 farad. This prototype can be impedance scaled and frequency scaled to the desired values. The low-pass prototype can also be transformed into high-pass, band-pass or band-stop types by application of suitable frequency transformations.

Cascading sections

Several L-shape half-sections may be cascaded to form a composite filter. Like impedance must always face like in these combinations. There are therefore two circuits that can be formed with two identical L-shaped half-sections. Where a port of image impedance Z_iT faces another Z_iT, the section is called a Π section. Where Z_iΠ faces Z_iΠ the section so formed is a T section. Further additions of half-sections to either of these section forms a ladder network which may start and end with series or shunt elements.

It should be borne in mind that the characteristics of the filter predicted by the image method are only accurate if the section is terminated with its image impedance. This is usually not true of the sections at either end, which are usually terminated with a fixed resistance. The further the section is from the end of the filter, the more accurate the prediction will become, since the effects of the terminating impedances are masked by the intervening sections.