In an L-R-C series ac circuit, the source has a voltage...

In an $L-R-C$ series ac circuit, the source has a voltage amplitude of $230 \mathrm{~V}, R=100.0 \Omega,$ and the reactance of the inductor is $320 \Omega$. (a) What is the curThe voltage amplitude across the resistor is $115 \mathrm{~V}$. rent amplitude in the circuit? (b) What is the voltage amplitude across the inductor? (c) What two values can the reactance of the capacitor have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance angular frequency? Explain.

In an $L-R-C$ series ac circuit, the source has a voltage amplitude of $230 \mathrm{~V}, R=100.0 \Omega,$ and the reactance of the inductor is $320 \Omega$.
(a) What is the curThe voltage amplitude across the resistor is $115 \mathrm{~V}$. rent amplitude in the circuit? (b) What is the voltage amplitude across the inductor? (c) What two values can the reactance of the capacitor have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance angular frequency? Explain.

Key Concepts

Resonance in RLC Circuits

Resonance occurs in an RLC circuit when the inductive reactance and capacitive reactance are equal in magnitude, leading to a purely resistive total impedance. This condition maximizes the circuit current for a given applied voltage and is pivotal in applications such as filters and oscillators. The analysis involves determining the specific frequency (or angular frequency) at which this cancellation happens.

Inductive Reactance

Inductive reactance quantifies the opposition to current flow provided by an inductor in an AC circuit, and it increases with the frequency of the source. This reactance is a key element in determining the phase shift between the voltage across the inductor and the current through the circuit, directly affecting circuit behavior.

Capacitive Reactance

Capacitive reactance represents the opposition to current flow in a capacitor when subjected to AC voltage, and it inversely depends on the frequency of the applied source. It is significant in determining the overall impedance of the circuit and plays a critical role in tuning and resonance, where its value can counteract that of the inductor.

Series RLC Circuit Analysis

This concept involves analyzing circuits where a resistor, inductor, and capacitor are connected in series. The key idea is that the overall circuit behavior is governed by the sum of the individual impedances of the components. Understanding this arrangement is essential for determining current distribution, voltage drops, and resonance phenomena in AC circuits.

Impedance in AC Circuits

Impedance is a fundamental concept in AC circuit analysis that extends the idea of resistance to include the effects of inductance and capacitance. It is a complex quantity that combines both the real part (resistance) and the imaginary part (reactance) and is crucial for computing the relationship between voltage and current in AC circuits using phasor representations.

Transcript

00:02 Hi, here in this given problem in the given lrc series circuit, the voltage amplitude v0, that is given as 230 volt resistance, the value of resistance in the circuit, 100 ome, inductive reactance, opposition offered by the inductor, 320 om, and voltage amplitude across the resistor, that is 115 volt.

00:52 In the first part of the problem, as in series, the current remains the same.

01:03 So, total circuit current i -o, that will be equal to current through resistor, current amplitude through resistor.

01:23 And using om's law, that is vor, voltage amplitude divided by the resistance, means this is 115 divided by 100.

01:32 So the current amplitude in the circuit is calculated to be equal to 1 .15amper.

01:42 Answer for the first part of this given problem here.

01:47 Then in its second part, now we have to find voltage amplitude across inductor.

02:01 Across inductor, current will remain same.

02:03 So voltage amplitude across inductor vol, that will be equal to i -o -current amplitude multiplied by inductive reactive reactants xl.

02:23 Means this is 1 .15 multiplied by 320.

02:28 So this vol is calculated to be equal to 368 volt.

02:34 Answer for the second part of this given problem here.

02:40 Then the third part of the problem using the concept that total circuit voltage, that is equal to the vector sum of voltage amplitude across inductor and capacitor means that is vol minus voc.

03:02 Suppose vol is more than voc.

03:05 So the net voltage across them that will be vol minus voc square plus vorr square for the resistor.

03:14 So this vol minus voc...

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