A series R L C circuit has power factor 0.764 and impedance 182 Ωat 442 Hz. (a) What's the resi

step 1 Let's do part A. We know that cosine of phi is equal to R divided by Z. And from here, R equals

A series R L C circuit has power factor 0.764 and impedance...

A series $R L C$ circuit has power factor $0.764$ and impedance $182 \Omega$ at $442 \mathrm{~Hz}$. (a) What's the resistance? (b) If the inductance is $25.0 \mathrm{mH}$, what's the resonant frequency?

Key Concepts

Impedance

Impedance is the measure of the opposition to the flow of alternating current and is composed of resistance (the real part) and reactance (the imaginary part). In a series RLC circuit, the impedance is found by combining the resistor value with the difference between the inductive and capacitive reactances using the Pythagorean formula.

Resonance in RLC Circuits

Resonance in an RLC circuit occurs when the inductive reactance equals the capacitive reactance, resulting in their cancellation. At this resonant point, the circuit behaves purely resistively, and the current reaches its maximum value for a given voltage. The phenomenon is central to tuning and filter applications.

Series RLC Circuit

A series RLC circuit consists of a resistor, an inductor, and a capacitor connected in a single loop. The circuit's behavior—such as its total impedance, energy storage, and dissipative properties—is determined by the interplay between these three components, which leads to frequency-dependent responses.

Power Factor

The power factor in an AC circuit is the cosine of the phase angle between the voltage and current waveforms. It reflects the ratio of the real power (that does useful work) to the apparent power and is directly related to the relationship between the resistance and the overall impedance of the circuit.

Resonant Frequency Calculation

The resonant frequency of a series RLC circuit is determined by both the inductance and the capacitance via the formula f0 = 1/(2??(LC)). This frequency indicates the point at which the energy oscillates most efficiently between the inductor and capacitor, leading to minimal impedance and maximum circuit response.

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