A series circuit consists of an ac source of variable frequency, a 115-Ωresistor, a 1.25-μF capa

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A series circuit consists of an ac source of variable...

A series circuit consists of an ac source of variable frequency, a $115-\Omega$ resistor, a $1.25-\mu F$ capacitor, and a $4.50-\mathrm{mH}$ inductor. Find the impedance of this circuit when the angular frequency of the ac source is adjusted to (a) the resonance angular frequency; (b) twice the resonance angular frequency; (c) half the resonance angular frequency.

Key Concepts

Series RLC Circuit

A series RLC circuit is a configuration where a resistor, an inductor, and a capacitor are connected in a single path for current flow. The circuit's overall behavior depends significantly on the frequency of the applied alternating current (AC), since both the inductor and capacitor have frequency-dependent reactances which influence the total impedance of the circuit.

Impedance

Impedance is a measure of the total opposition a circuit presents to the flow of alternating current. It combines both the resistance, which does not vary with frequency, and the reactance, which does. In a series RLC circuit, the impedance is a complex number that reflects the phase difference between voltage and current, and its value changes with the frequency of the AC source.

Reactance

Reactance is the component of impedance that arises from the presence of inductors and capacitors in a circuit. Inductive reactance increases with frequency, while capacitive reactance decreases with frequency. The net reactance in a series RLC circuit is determined by subtracting the capacitive reactance from the inductive reactance, playing a crucial role in the overall impedance.

Resonance in RLC Circuits

Resonance in an RLC circuit occurs when the inductive and capacitive reactances are equal in magnitude but opposite in phase, resulting in their cancellation. At the resonant frequency, the circuit's impedance is purely resistive, meaning that the current and voltage are in phase, and the energy exchange between the inductor and capacitor is maximized.

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