228 BELL SYSTEM TECHNICAL JOURNAL 



2A2PQ COS l(pt - e) - (qt - <p)], 

 3AiP'Q cos l(pt - e) - {qt - if)'], 

 ZAiPQ' cos l{pt - d) - {qt - (p)], 

 l/2^3<3'cos (3g/ - 3,^), 

 AiPQ^ cos [_{pt - d) - {3qt - 3<p)]. 



In addition to the desired term in the second order modulation of fre- 

 quency {pt — qt)l2ir there are two other terms of the same frequency 

 in the fourth order modulation. These have respectively the co- 

 efficients ZAiP^Q and ZA^PQ^. With a given value of carrier input P, 

 the first of these is proportional to Q and it will add to the second order 

 term but will cause no serious trouble. The second term, however, is 

 proportional to Q^ and, therefore, would cause the input to the tuned 

 circuit to depart from the desired linear relationship with respect to Q. 

 However, the ratio of the cofficient of this fourth order term to that of 

 the second order term is SAiC^jlA^ which is proportional to Q^ and, 

 therefore, the effect of the fourth order term will fall off rapidly as Q 

 is reduced. In the third order modulation, there is a component of 

 frequency 3qt/2T which would be passed by the tuned circuit if there 

 were a component of 1/3 the resonant frequency of this circuit in the 

 wave to be analyzed. Considering the extreme sharpness of the tuned 

 circuit, it is rather improbable that such a condition will occur. More- 

 over, this component is proportional to Q^ and, therefore, will fall off 

 rapidly as Q is reduced. In the fourth order modulation, there is also a 

 component AiPQ^ cos \^{pt — d) — {3qt — 3(j))'] of frequency {pt — 3qt)/2Tr. 

 This component would indicate a third harmonic in the unknown wave 

 although it actually contained no other frequency than that of qt/2T. 

 However, the ratio of its coefficient to the desired second order term is 

 AiQ^/3A2 and, therefore, the false indications of a third harmonic can 

 be reduced to any desired extent by reducing Q. 



It is evident, therefore, that in an analyzer of this type it is desirable 

 to keep the magnitude of the input of the unknown wave as low as is 

 consistent with obtaining satisfactory meter readings. In the two 

 analyzers now in operation, false indications by the introduction of 

 extraneous frequency components due to the third, fourth and higher 

 orders of modulation are negligibly small. Measurements on an essen- 

 tially pure frequency within the frequency limits of the apparatus show 

 harmonics less than 0.1 per cent of the fundamental. With an 

 indicated harmonic of not more than this magnitude, it is difficult to 

 tell whether such a harmonic is actually present in the wave or a false 

 indication. At any rate, the harmonic is small enough as to be of no 

 significance. It is, of course, difficult to build the modulator so that it 



