Question #193613

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?

Expert's answer

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"

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