Question 4.14: A series L–C–R circuit comprises an inductor of 20 mH, a cap......
A series L–C–R circuit comprises an inductor of 20 mH, a capacitor of 10 nF, and a resistor of 100 Ω. If the circuit is supplied with a sinusoidal signal of 1.5 V at a frequency of 2 kHz, determine the current supplied and the voltage developed across the resistor.
Step-by-Step
First we need to determine the values of inductive reactance, X_L, and capacitive reactance X_C:
\begin{array}{l}{{X_{\mathrm{{L}}}=2\pi f L=6.28\times2\times10^{3}\times20\times10^{-3}=251\ \Omega}}\\ {{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}\\ {{X_{C}=\frac{1}{2\pi f C}=\frac{1}{6.28\times2\times10^{3}\times100\times10^{-9}}=796.2\ \Omega}}\end{array}The impedance of the series circuit can now be calculated:
Z={\sqrt{R^{2}+\left(X_{\mathrm{{L}}}-X_{\mathrm{{C}}}\right)^{2}}}={\sqrt{100^{2}+\left(251.2-796.2\right)^{2}}}From which:
Z={\sqrt{10,000+297,025}}={\sqrt{307,025}}=554 \OmegaThe current flowing in the series circuit will be given by:
I={\frac{V}{Z}}={\frac{1.5}{54}}=0.0027=2.7{\mathrm{~mA}}The voltage developed across the resistor can now be calculated using:
V = IR = 2.7 mA × 100 F = 270 mV
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