Constant k filter

From Wikipedia, the free encyclopedia

Linear analog electronic filters
Network synthesis filters
Image impedance filters

Constant k filter

m-derived filter

Zobel network (constant R) filter

Composite image filter

Simple filters
edit

Constant k filters are a type of electronic filter designed using the image method. They were invented by George Campbell who published his work in 1922[1] but had clearly invented them some time before[2] as his colleague at AT&T Co, Otto Zobel, was already making improvements to the design at this time. Campbell's filters were far superior to the simpler single element circuits that had previously been used. The term "constant k" was not, however, used by Campbell. It was coined by Zobel[3] to distinguish them from his m-derived filter design.

The great advantage Campbell's filters had over the RL circuit and other simple filters of the time was that he could design filters for any desired degree of stop band rejection or steepness of transition between pass band and stop band. It was only necessary to add more filter sections until the desired response was obtained.[4]

The filters were designed by Campbell for the purpose of separating multiplexed telephone channels on transmission lines but their subsequent use has been much more widespread than that. The design techniques used by Campbell have largely been superseded. However, the ladder topology used by Campbell with the constant k is still in use today with implementations of modern filter designs such as the Tchebyscheff filter. Campbell gave constant k designs for low-pass, high-pass and band-pass filters. Band-stop and multiple band filters are also possible.[5]

Contents
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.

[edit] Derivation

Constant k low-pass filter half section. L = Ck2
Constant k low-pass filter half section. L = Ck2
Constant k band-pass filter half section. L1 = C2k2 and L2 = C1k2
Constant k band-pass filter half section. L1 = C2k2 and L2 = C1k2
Constant k prototype low-pass filter ZiT image impedance
Constant k prototype low-pass filter ZiT image impedance
Constant k prototype low-pass filter transfer function for a single half-section
Constant k prototype low-pass filter transfer function for a single half-section

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

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 Zi as the size of the section (in terms of component values) approaches zero. 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.

[edit] Image impedance

See also Image impedance#Derivation

The image impedances of the section are given by;

Z_{iT}^2=Z^2 + k^2

and,

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

It can be seen that the image impedance in the pass band of the filter is purely real and in the stop band it is purely imaginary. For the low-pass half section pictured for instance,

Z_{iT}^2=-(\omega L)^2 + \frac{L}{C}

The transition occurs at a cut-off frequency given by,

\omega_c=\frac{1}{\sqrt{LC}}

Below this frequency the image impedance is real,

Z_{iT}=L\sqrt{\omega_c^2-\omega^2}

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

Z_{iT}=iL\sqrt{\omega^2-\omega_c^2}

[edit] Transmission parameters

See also Image impedance#Transfer function

For a constant k section in general the transmission parameters for a half-section are given by;

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

And for n half-sections;

\gamma_n=n\gamma\,\!

Again, for the low-pass L section example, below the cut-off frequency the transmission parameters are given by;

\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;

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

[edit] Prototype transformations

The plots shown of image impedance, attenuation and phase change are the plots of a low-pass prototype filter section. The prototype has a cut-off frequency of ωc=1 rad/s and a nominal impedance k=1Ω. This is produced by a filter half-section where L=1 henry and 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.

[edit] Cascading sections

Several L 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 half-sections. Where ZiT faces ZiT, the section is called a Π section. Where ZiΠ faces ZiΠ the section so formed is a T section. Further additions of half-sections to either of these forms a ladder network which may start and end with series or shunt elements.

It should be born 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.

Image filter sections
Unbalanced L Half section T Section Π Section
Ladder network
Balanced C Half section I Section Box Section
Ladder network
Textbooks and design drawings usually show the unbalanced implementations, but in telecoms it is often required to convert the design to the balanced implementation when used with balanced lines. edit

[edit] See also

[edit] References

  1. ^ Campbell, G A, "Physical Theory of the Electric Wave-Filter", Bell System Tech J, November 1922, vol 1, no 2, pp 1-32.
  2. ^ Bray gives 1910 as the start of Campbell's work on filters
  3. ^ White, G, "The Past", Journal BT Technology, Vol 18, No 1, pp107-132, January 2000, Springer Netherlands.
  4. ^ Bray, J, Innovation and the Communications Revolution, Institute of Electrical Engineers
  5. ^ Zobel, O J, Multiple-band wave filter, US patent 1 509 184, filed 30 April 1920, issued 23 Sept 1924.
  • Mathaei, Young, Jones Microwave Filters, Impedance-Matching Networks, and Coupling Structures McGraw-Hill 1964