The series RLC circuit shown in the figure has R=2.20 Ω, L=9.30 mH, C=2.27 mF, Vm=110 . V, and ω

The series RLC circuit shown in the figure has R=2.20 Ω,...

The series RLC circuit shown in the figure has $R=2.20 \Omega, L=9.30 \mathrm{mH}, C=2.27 \mathrm{mF}$, $V_{\mathrm{m}}=110 . \mathrm{V},$ and $\omega=377 \mathrm{rad} / \mathrm{s}$. a) What is the maximum current, $I_{\mathrm{m}}$, in this b) What is the phase constant, $\phi$, between the voltage and the current? c) The capacitance, $C$, can be varied. What value of $C$ will allow the largest current amplitude oscillations to occur, and what are the magnitudes of this current, $I_{\mathrm{m}}^{\prime}$, and the phase angle, $\phi^{\prime}$, between the current and the voltage?

The series RLC circuit shown in the figure has $R=2.20 \Omega, L=9.30 \mathrm{mH}, C=2.27 \mathrm{mF}$, $V_{\mathrm{m}}=110 . \mathrm{V},$ and $\omega=377 \mathrm{rad} / \mathrm{s}$.
a) What is the maximum current, $I_{\mathrm{m}}$, in this
b) What is the phase constant, $\phi$, between the voltage and the current?
c) The capacitance, $C$, can be varied. What value of $C$ will allow the largest current amplitude oscillations to occur, and what are the magnitudes of this current, $I_{\mathrm{m}}^{\prime}$, and the phase angle, $\phi^{\prime}$, between the current and the voltage?

Key Concepts

Phase Angle in AC Circuits

The phase angle in an AC circuit represents the difference in phase between the voltage and the current. It is influenced by the relative contributions of resistance (which causes no phase shift) and reactance (which causes the current to either lead or lag the voltage). In a series RLC circuit, the phase angle indicates whether the circuit behaves more like an inductor (current lags) or a capacitor (current leads), and it becomes zero at resonance when the inductive and capacitive effects balance each other.

Series RLC Circuit

A series RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor connected in a single loop. It is a fundamental circuit in AC analysis that exhibits unique transient and steady-state behavior depending on the frequency of the applied voltage source. This configuration is used to study energy storage and dissipation, and its behavior is characterized by its total impedance which affects both the magnitude and phase of the current relative to the voltage.

Impedance in AC Circuits

Impedance is the total opposition that a circuit offers to the flow of alternating current (AC) and is a combination of resistance and reactance. In a series RLC circuit, impedance is determined by the resistor, inductor, and capacitor and is generally expressed as a complex quantity, combining both magnitude and phase information. It plays a critical role in determining the current through the circuit given an applied AC voltage and is key to analyzing the circuit’s response at different frequencies.

Resonance in RLC Circuits

Resonance occurs in an RLC circuit when the inductive reactance and capacitive reactance cancel each other out, resulting in the circuit’s impedance being purely resistive. At the resonant frequency, the current amplitude reaches its maximum because the circuit experiences minimal opposition from reactance. This phenomenon is of great importance in tuning circuits and signal processing, as adjusting the circuit parameters to reach resonance can maximize power transfer or signal selectivity.

Transcript

00:01 Have a series rlc circuit having resistance r equals to 2 .20 oom and l equals to 9 .30 millie henry that is 10 to the power minus 3 henry and capacitance c equals to 2 .27 multiplied by 10 to the power minus 3 ferrade and vm is equals to 1110 volt and omega angular frequency is 3 double 7 radian per second.

00:21 Okay.

00:22 So for the part a of the question we have to determine the maximum current in the circuit i am so it will be given by vm by z and z is the impact.

00:30 Impedance so it will be given by vm divided by r square plus x l minus x c whole square and whole underw so first of all calculating value for the x l so xl it will be equals to omega l so omega l is 3 double 7 multiplied by l which is 9 .30 multiplied by 10 to the power minus 3 so from here xl equals to 3 .51 om and now x c it will be 1 by omega c so 1 by omega 3 double 7 but plied by c which is 2 .27 but played by 10 to the power minus 3 ferret so from here xc equals to 1 .17 om so now substituting these value of xl and and xc here so we get i m equals to vm which is equals to 1110 volt divided by r which is 2 .20 whole square plus 3 .51 minus 1 .17 whole square and whole under root okay so from here after solving, we get maximum current in the circuit im equals to 34 .3 ampere.

01:37 So this is the answer for the part a of the problem.

01:40 Now moving to the part b in which we have to determine the phase constant phi, so it is given by 10 inverse of xl minus xc divided by r.

01:53 So we have all the value for the excellent xc.

01:56 So we can write that 10 inverse xl which is equal to the x.

01:59 To 3 .51 minus x c which is 1 .17 divided by r which is 2 .20.

02:06 So from here after solving we get 5 equals to 5 equals to 0 .816 radian or in the degree it is equals to 46 .8 degree.

02:17 So this is the answer for the part b of the problem.

02:21 Okay, this is our part b.

02:22 So now moving to the part c in which we have to determine the value for the capacitance for the largest current amplitude so for the largest current amplitude resonating frequency omega not it will be equals to 1 by under root of l c so omega not here is omega our so from here capacitance c it will be equals to it will be equals to 1 by omega square multiplied by l okay so substituting values we get we can call it as c dash okay so 1 by omega which is 3 double 7 whole square multiplied by l which is 9 .30 multiplied by 10 to the power minus 3 henry...

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