Wave Filter Principles

In preceding sections dealing with transformer frequency response, means for extending frequency range have been considered. In broadcast transmitters this is a vital problem. But in other applications amplifiers are used over a limited frequency range. It is sometimes desirable to allow certain frequencies which are present to pass through the amplifier at full amplitude but to suppress as nearly as possible certain other frequencies. The means usually employed to accomplish this result is a wave filter. In any such filter, the band of frequencies which it is desired to transmit is known as the transmission band, and that which it is desired to suppress is known as the attenuation band. At some frequency, known as the cut-off frequency, the filter starts to attenuate. Transition between attenuation and transmission bands may be gradual or sharp; the filter is said to have gradual or sharp cut-off accordingly. When a filter is used in conjunction with a transformer-coupled amplifier, the frequency response of both filter and amplifier must be coordinated. In a later section it will be shown how transformer response may be improved through the use of wave filter principles.

To avoid introducing losses and attenuation in the transmission bands, reactances as nearly pure as practicable are used in the elements of a wave filter.

Fig. 135. Low-pass filter sections.

For example, in the "low-pass" filter T-section of Fig. 135, the inductance arms shown as L/2 and the capacitance C are made with losses as low as possible. Capacitors ordinarily used in filters have low losses, but it is a problem to make inductors which have low losses. Values of inductor Q ranging from 10 to 200 are common, depending upon the value of inductance and the frequency of transmission. Therefore in wave filters the loss is mostly in the inductors. It can be shown⁽¹⁾ that for pure reactance arms the values of reactance are such that in the transmission band

[83]

where Z₁ is the reactance of the series arm and Z₂ is the reactance of the shunt arm. In the T-section of Fig. 135, Z₁ is 2πf[ (L/2) + (L/2) ]

= 2πfL and Z₂ is the reactance of C. The attenuation for sections of filter like Fig. 135 is shown in Fig. 136, for a pure reactance network starting at the cut-off frequency.

Fig. 136. Attenuation per section with pure reactance arms.

The attenuation is shown in decibels, and the abscissas are one-fourth of the ratio of series to shunt reactance in a full section.

It is important, in the transmission band, to terminate the sections of filter in the proper impedance. Like a transmission line, a wave filter will deliver its full energy only into an impedance which is equal to its characteristic impedance. Many wave filters are composed of several sections which simulate transmission lines. A properly constructed filter exhibits the same impedance at either end when terminated at the opposite end with an impedance equal to its characteristic impedance. The impedance seen at any one point in the filter is called its image impedance; it will be the same in either direction provided that the source and terminating impedances are equal. In general, however, the image impedance will not be the same for all points in the filter. For example, the impedance looking into the left or T-sec-tion of Fig. 135 (if it is assumed to be terminated properly) will not be the same as that seen across the capacitor C. For that reason, another half-series arm is added between C and the termination to keep equal input and output impedances. The terminating sections at both the sending and receiving ends of a filter network are half-sections, whereas the intermediate sections are full sections. A full T-section of the type shown in Fig. 135 includes an inductance L equal to L/2 + L/2. The image impedance seen at the input terminals of the T-section of Fig. 135 is known as the mid-series impedance, and that seen across capacitor C is known as the mid-shunt impedance.

Likewise, in the pi-section shown at the right in Fig. 135, the mid-shunt image impedance is seen at the input or output terminals. The mid-series impedance is seen at a point in the middle of coil L. This section terminates properly in its characteristic impedance at either end. Note that adjacent sections have C/2 for the shunt arm, so that a full section would again be composed of a capacitor C and an inductance L. The choice of T- or pi-sections is determined by convenience in termination, or by the kind of image impedance variation with frequency that is desired.

If these precautions are not observed, wave reflections are likely to cause a loss of power transfer in the transmission band.

(1)	See Transmission Networks and Wave Filters, by T. E. Shea, D. Van Nostrand Co., New York, 1929, p. 187.