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Answered: Wees | bartleby

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AN GRG cIRcuiT i5 ARwe 3Y A TIME VARYING VOLTAGE
Su PPLY WITH A VOLTA QG Of N Lt) = 100V.cos (ut+4).
Aw LRC CIRCUIT IS PRIVEN BY A TIME VARYIN be VOLTAGG
FindS
Wees
1f La ZmlH, C=Amf, R=2e
FIND ALL VALUES FOe waWRO:
%3D
工
R

Expert Solution

Step 1

Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and specify the other subparts (up to 3) you’d like answered.

A Series LCR Circuit

The figure above shows a series LCR circuit. The input voltage is given by $V = V_{0} \sin (ω t + ϕ)$ where $V_{0}$ is the amplitude of the voltage.

The current amplitude is given by

$I_{0} = \frac{V_{0}}{\sqrt{R^{2} + {(ω L - \frac{1}{ω C})}^{2}}}$

The denominator is known as the impedance of the circuit. At resonance, the inductive reactance becomes equal to the capacitive reactance

$ω_{r e s} L = \frac{1}{ω_{r e s} C} \Rightarrow ω_{r e s} = \sqrt{\frac{1}{L C}}$

This is the resonant frequency. At resonance, the impedance becomes only resistive and hence the peak current at resonance becomes

$I_{0} = \frac{V_{0}}{R}$

The potential difference across the resistor is

$V_{R} = I_{0} R = V_{0}$

The potential difference across the inductor

$V_{L} = ω_{r e s} L I_{0}$

The potential difference across the capacitor

$V_{C} = \frac{I_{0}}{ω_{0} C}$

Step 2

In the given question the resistance is $R = 2 Ω$

the inductance is $L = 2 m H = 2 \times 10^{- 3} H$

the capacitance is $C = 0.5 m F = 0.5 \times 10^{- 3} F$

the input voltage $V = 100 \cos (ω t + ϕ) V$

a) The resonant frequency is given by

$ω_{r e s} = \sqrt{\frac{1}{L C}} = \sqrt{\frac{1}{2 \times 10^{- 3} \times 0.5 \times 10^{- 3}}} = 1000 r a d / s$

b) The peak value of current flowing in the circuit at resonance

$I_{0} = \frac{V_{0}}{R} = \frac{100}{2} A = 50 A$

The peak value of the potential difference across the resistor is $V_{R} = V_{0} = 100 V$ .

c) The peak value of potential difference across the inductor is

$V_{L} = ω_{r e s} L I_{0} = 1000 \times 2 \times 10^{- 3} \times 50 V = 100 V$

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