INDUCTANCES, CAPACITANCES AND RESISTANCES 



Parallel resonance 



Suppose the parallel L-C-R circuit be connected to a constant voltage 

 alternator (Figure 5.7) and suppose 7? = 0. Then the impedance of the 

 circuit is {jcoL . — jlcoL}l{jcoL — j/mC}. When w = 1/(LC)^'- the denomi- 

 nator becomes zero and the impedance goes to infinity (contrast this with 

 the series circuit, in which the impedance at resonance is a minimum). 



I Zp is pure 

 is equivalent to A\ < resistive 

 at resonance \J < and equal 



iLC-T2) ' ' c/? 



% 



Figure 5.7 



With a practical circuit R ^0 and at resonance the impedance does not 

 become infinite. We have 



(R+j(oL).- ^ 



coC 



z. = - 



which simplifies and rationalizes to 



R ./ L R^ (olJ\ 



z =^ 



Parallel resonance is defined as occurring at that frequency at which Z^ 

 becomes purely resistive. Thus the j term in the above equation must dis- 

 appear, i.e. 



L R^ m.V' ^ * 



— — — ^ — == 



C^cOg CcOg C 



where w^ is the parallel resonant frequency. The solution is 



/ 1 i?2w/2 



Substituting back, it emerges eventually that, at ojg, the impedance of the 

 circuit is simply 



z =A 



" CR 

 LJCR is called the 'dynamic resistance' of the circuit. 



Parallel L-C-R connected to constant-current alternator 



In Figure 5.8, if the parallel circuit is resonant at the alternator frequency, 

 then it may be replaced by a resistance LjCR and the potential difference 

 across the circuit is V — I . LJiCR). 



The capacitance current is F/(— y/a>C) —JmCV. 



78 



