Radio broadcast .. (1922-30)

Mathematical Discussion of the Fada Circuit BAND-PASS FILTER DESIGN By E. A. UEHLING Engineering Department, F. A. D. Andrea, Inc. SOME NOTES are given in this article on the design of the band selector used in the Fada receiver described by the writer in July, 1929, RADIO BROADCAST. In this receiver there is a signal selector of band-pass characteristics E receding the first radio-frequency ampli- er tube. In most filters already used the band of frequencies transmitted is narrow at long waves and very wide at short waves. The selectivity at the short wavelengths is usually not very good, even in the best receivers, because of the increased resist- ance of the radio-frequency circuits at the higher frequencies. It is obvious that if the width of the band transmitted by a band-pass filter increases as the wave- length decreases, the tendency toward broad tuning at the shorter wavelengths will be even more pronounced. The Band-Pass Circuit A simplified circuit of a band-pass filter having more desirable characteristics is shown in Fig. 1. It will be noted first of all that no magnetic coupling exists between the two circuits. There are two principal advantages of coupling these circuits as shown and these advantages will be de- scribed as follows: We are interested in the width of the transmission band which de- pends on the value of the quantity Ji, » — RiRj) where Ri and R 2 are the circuit resistances and M 2 is the absolute value of square of the coupling impedance. It will be seen that this coupling impedance should vary as the product RiR2 varies with frequency, so that the quantity (M 2 — RiR2) is as nearly constant with frequency as it can be made. When the coupling between the circuits is magnetic the variation of the mutual reactance can be expressed as: = L d o) If the coupling between the circuits is capacitive, the variation of the mutual reactance with frequency can be expressed as: d - d a) J_ 'u'C Suppose we decide on 4 per cent, coupling as the value which gives the desired width of band at the longest wavelengths. With 230-microhenry tuning coils the mutual inductance will then have to be 9.6 micro- henries and the variation of the mutual reactance with frequency, as expressed by the first formula, will be 9.6 X 10 - «. Now, if the coupling between the circuits is capacitive and a coupling of 4 per cent, is again chosen, the coupling capacity will be about 10,000 micromicrofarads, and at 550 meters the variation of reactance with frequency, as determined by the second of the two formulas, will be 10 X 10 - 6 . For either type of coupling the variation of reactance with frequency at 550 meters when the coupling percentage is adjusted for the same width of band is the same. But in the case of magnetic coupling this variation is constant regardless of fre- quency, and in the case of the capacity coupling the variation in reactance with change in frequency decreases as the frequency increases. Thus we find at 200 meters the variation of capacitive react- ance with frequency is equal to only 1X10 - 6 as compared with 10 X 10 - 6 at the same frequency for magnetic coupling. So in the broadcast range capacity coupling gives a more nearly uniform width of band than magnetic coupling, provided the width of the band is made the same for both types of coupling at 550 meters. That is, the actual arithmetic variation in band width is less for capacity coupling than for in- ductive coupling. The second of the two principal ad- vantages of this type of band-pass filter is that whatever variation in band width there is, it is in the most desirable direction as already stated. As the receiver tuning dial is turned to the shorter wavelengths, r— 'h ngAnt Long Ant Short Ant Fig. 1 2 megohms To Input of / first Tube the coupling percentage is reduced con- stantly, This reduction in percentage of coupling is slightly more than is required to give constant width of band with the result that there is a slight decrease in band width at the lower wavelengths. The use of a band-pass filter is not with- out some loss in voltage amplification as compared with other methods of signal selection. There is a voltage gain in this band-pass filter of about 2 at 550 kilo- cycles and about 4 or 5 at 1500 kilocycles. A comparison is given below of the voltage amplification obtained with the ordinary tuned antenna circuit and the circuit as used in this receiver will be shown. The ratio of the voltage E 3 impressed on the grid of the first amplifier tube to the voltage EI impressed in the antenna is Ei 'A where (referring to Fig. 1) mi is the mutual impedance between the first two circuits, m 2 the mutual impedance between the last two circuits, L the inductance of the tuning coils in the band-pass filter, and Z/', Z 2 ', and Za the impedance of the three circuits respectively as influenced by the reaction of the following circuits. The ordinary antenna circuit differs from the circuit of Fig. 1 only in the ab- sence of the third circuit. In this case the ratio of the voltage E» impressed on the grid of the first amplifier tube to the voltage EI impressed in the antenna is Ea _ mitoL Ei" zTzT where mi and L are the same quantities as before, and Zi' and Z» are the impedances of the two circuits which are in general of different value than Z-" and Z/ given above. But we are interested in the ratio =r, Va the ratio of the voltage impressed on the grid when the band-pass filter is used to that impressed on the grid when the ordi- nary antenna circuit is used. We get this ratio by dividing one equation by the other. Ej Ez Zi'Z; Zi"Zz'Z3 mid)L ' Zi'Zi The resonant frequency of the antenna circuit is so far above the highest frequency of the broadcast band that its reactance to broadcast frequencies is very high, and therefore Zi, the impedance of the antenna circuit, is very high compared with the impedance of the other circuits. Since Zi is very large, its coupling to the following circuits changes its value very little. Therefore, then Zi = Zi" = Zi' approximately Ej Now the value of Zs is ttl Substituting this value of 2$ in the equa- tion above Ej ~ The value of Zj' is Zt ^ _ Zt Zaf 22' = VRj'2 R, + Zi 2R: then = Ej 2Rs But Z2 is the impedance of circuit II with circuit III removed, then Zj = R! hence E, E, = 5. NOVEMBER 1929 • 57