A circuit has a coil of inductance 120 μH and resistance of 15.7 Ω, which is connected in series with a capacitor of 211 pF. Determine (a) resonant frequency, (b) voltage drop across R, L and C at resonance and (c) the quality factor Q of the coil. Assume that the impressed voltage is 0.157 V.

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Accepted Answer

Given L = 120 μH, C = 253.3 pF and R = 15.7 Ω.

(a) The resonant frequency, [latex]f_{O} =\frac{1}{2\pi \sqrt{LC} }= \frac{1}{2\pi \sqrt{120 imes 10^{-6} imes 211 imes 10^{-12} } }= 1 MHz[/latex]

(b) The current at resonant frequency,

[latex]I_{o} =\frac{V}{R}= \frac{0.157}{15.7}= 0.01 A [/latex]

Therefore, voltage across R, [latex]V_{R} = I_{o} × R = 0.01 × 15.7 = 0.157 V[/latex]

Voltage across L, [latex]V_{L} = I_{o} × X_{L} = I_{o} × (2 \pi f_{o} L)[/latex]

[latex]= 0.01 × 2 \pi × 1 × 10^{6} × 120 × 10^{–6} = 7.54 V[/latex]

Voltage across C, [latex]V_{C} = I_{o} × X_{C} = I_{o} ×\frac{1}{2\pi f_{O}C}= \frac{0.01}{2\pi imes 1 imes 10^{6} imes 211 imes 10^{-12} }= 7.54 V [/latex]

(c) The quality factor, [latex]Q =\frac{X_{L} }{R}= \frac{2\pi f_{O}L }{R} = \frac{2\pi imes 1 imes 10^{6} imes 120 imes 10^{-6} }{15.7} = 48.[/latex]

Question 13.4: A circuit has a coil of inductance 120 μH and resistance of ...

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