Question

An impedance coil whose resistance and inductance are 18 ohms and 3.82 mH, respectively, is connected in parallel with a series combination of a 5.3 uF capacitor and a variable resistor. For what value of resistance will the circuit be in resonance at 1000 cps?

Answer 1

Let series resistor and inductor combination be $R_L+jX_L$

Let series capacitor and resistor combination be $Rc-jXc$

When these are connected in parallel the total impedance should be resistive, that is the condition for resonance.

i. e imaginary term of parallel combination of $R_L+jX_L \space and\space Rc-jXc$ =0

$i.e (RcX_L - R_LXc) (R_L+Rc) - \newline (X_L-Xc) (R_LRc+X_LXc) =0$

Simplifying we have,

$Rc^2X_L-R_L^2Xc=X_L^2Xc-Xc^2X_L\newline$

Substituting given inductor, capacitor resistor and frequency values we have,

$24Rc^2=5400\newline Rc=15\Omega$