INDUCTANCES, CAPACITANCES AND RESISTANCES 



on 7 by 90 degrees, and Vq lags on / by 90 degrees. It follows that Vj^ and Vq are 

 in anti-phase with one another, that is, they pass though maxima and minima 

 in step with each other, but with opposite polarity, as suggested in Figure 5.2. 



I slnQjt 



Figure 5.2 



The instantaneous potential difference between A and B is y^ minus Vq, and 

 the modulus of the reactance of the combination is \vi^ — vdJI which equals 

 \vi\ll — \vq\II. However \vjjll = Xj^ and \vq\II ^= Xq so the reactance is 

 Xj^ — Xq. This is the reason for taking capacitive reactance as negative and 

 inductive reactance as positive. Common sense suggests that if a number of 

 reactances are connected in series, the effective reactance ought to be obtained 

 by adding them all up. And so it can be, for 



The impedance of the arrangement between C and D is therefore 



R+J(Xr.-Xa) 



= " +i('"^ - ^) 



When ft) is small, coL is smaller than l/a>C and the circuit behaves as if it 

 contained only the resistance and a capacitance C'such that XjcoC = l/o^C — 

 oiL. When ft> is large, mL is larger than IfoiC and the circuit behaves as if 

 it contained only the resistance and the inductance L' such that coL' = coL — 

 l/ft)C. When oy has the critical value \l(LCy- the reactances cancel, and it 

 is as if only the resistance was there. This is the basis of series resonance. 



P 

 Vv^A/ 1 







Figure 5.3 

 Series resonance 



If series L, C, and R are connected to a constant voltage alternator (Figure 

 5.3), the current is given by 



E 



1 = 



''+jh-^) 



74 



