296 



BELL SYSTEM TECHNICAL JOURNAL 



If the terms in powers of co higher than the first are neglected, the 

 resistance is 



1 Z 



G coC 



(4) 



where Z is the resistance of a one farad condenser at l/2x cycles 

 per second. 



In terms of the rate of change of induction, B, in the sample and 

 the output voltage e of the integrator, the differential equation of the 

 circuit in Fig. 3 is 





(5) 



where 



/ = 



1 



RCo 



R RCco 

 '^ r''^ Z 



Let us now assume that the magnetizing field is sinusoidal, 



H = III cos co/ ; 



that 



and that 



B = fxH, fx being considered constant; 



e = a cos Oil -\- h sin co/. 



Making these substitutions in equation (5) and solving for a and b 

 by equating coefficients of like terms, we get 



NS ^, 



1 



Rc^'^^'T^n 



- [cos wt — J sin oif\. 



(6) 



The value of / is less than .01 in this apparatus so that J"^ can be 

 neglected compared with unity. The negative sign is merely a matter 

 of convention, since the curve can be turned by reversing one pair 

 of terminals. The sign of the cos co/ term will therefore be taken 

 positive henceforth, and we have 



NS 

 RC 



B 



cos CO 



1 /, , i?\ . 1 . 



/ — -^-TT- 1 + — sm co/ — - sm co/ 

 RCw \ r / Z 



(7) 



The value of e therefore departs from that for a perfect integrator 

 by two terms, one depending upon the time constant of the integrator, 

 the frequency, and the value of the grid leak resistance of the first 

 amplifier stage; the other term depending only upon the conductance 

 of the integrating condenser. 



