Figure shows a series LCR circuit connected to a variable frequen… - Vidyadip

Question

Figure shows a series $\text{LCR}$ circuit connected to a variable frequency $230V$ source. $L = 5.0H, C = 80\ \mu F, R = 40\Omega .$

Determine the source frequency which drives the circuit in resonance
Obtain the impedance of the circuit and the amplitude of current at the resonating frequency.
Determine the rms potential drops across the three elements of the circuit. Show that the potential drop across the $\text{LC}$ combination is zero at the resonating frequency.

✓

Answer

Here, we are given a $\text{LCR}$ circuit.
Inductance, $L = 5.0H$
Resistance, $\text{R}=40\Omega$
Capacitance, $C = 80\ \mu F = 80 \times 10^{-6}F$
Effective voltage,
$E_v = 230$ volt
$\Rightarrow$ Peak voltage,$\text{E}_0=\sqrt{2}\text{E}_{\text{v}}=\sqrt{2}\times230\text{V}$

Resonance angular frequency is given by

$\omega_{\text{r}}=\frac{1}{\sqrt{\text{LC}}}$
$=\frac{1}{\sqrt{5\times80\times10^{-6}}}$
$=\frac{1}{2\times10^{-2}}$
$= 50\ rad/\sec.$

Impedance of the circuit,

$\text{Z}=\sqrt{\text{R}^2+\big(\omega\text{L}-\frac{1}{\omega\text{C}}\big)^2}$
At resonance, $\omega\text{L}=\frac{1}{\omega\text{C}}$
Therefore,
$\text{Z}=\sqrt{\text{R}^2}=\text{R}=40\Omega$
Amplitude of current at resonating frequency
Peak value of current, $\text{I}_0=\frac{\text{E}_0}{\text{z}}=\frac{\sqrt{2}\times230}{40}=8.13\text{A}$
Rms value of current, $\text{I}_{\text{v}}=\frac{\text{I}_0}{\sqrt{2}}=\frac{8.13}{\sqrt{2}}=5.75\text{A}$

Potential drop across $L$

$\text{V}_{\text{L rms}}=\text{I}_\text{v}\omega_{\text{r}}\text{L}$
$= 5.75 \times 50 \times 5.0$
$= 1437.5V$
Potential drop across $R$
$\ce{V_{R rms} = I_v \times R}$
$= 5.75 \times 40$
$= 230$ volts
Potential drop across $C$
$\text{V}_{\text{C rms}}=\text{I}_{\text{v}}\Big(\frac{1}{\omega_{\text{r}}\text{C}}\Big)$
$=5.75\times\frac{1}{50\times80\times10^{-6}}$
$=\frac{5.75}{4}\times10^3$
$= 1437.5V$
Therefore,
Potential drop across $\text{LC}$ circuit
$\ce{V_{LC rms} = V_{L rms} - V_{C rms} = 0}$
Thus, the potential drop across the $\text{LC}$ combination is zero at the resonating frequency.

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