A series RLC circuit consists of a 75 Ω resistor, a 0.12 H inductor, and a 30 μF capacitor. It is attached to a 120 V/60 Hz power line. What is the phase angle ϕ?

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Determine the angular frequency (ω) of the AC source using the formula ω = 2πf, where f is the frequency of the power line (60 Hz).

Calculate the inductive reactance (X_L) using the formula X_L = ωL, where L is the inductance of the inductor (0.12 H).

Calculate the capacitive reactance (X_C) using the formula X_C = 1 / (ωC), where C is the capacitance of the capacitor (30 μF or 30 × 10⁻⁶ F).

Find the net reactance (X) of the circuit by subtracting the capacitive reactance from the inductive reactance: X = X_L - X_C.

Calculate the phase angle (ϕ) using the formula tan(ϕ) = X / R, where R is the resistance (75 Ω). Then, take the arctangent of the result to find ϕ in radians or degrees, depending on the desired unit.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Impedance in RLC Circuits

Impedance is the total opposition that a circuit offers to the flow of alternating current (AC) and is represented as a complex number. In a series RLC circuit, the impedance (Z) combines the resistance (R), inductive reactance (XL), and capacitive reactance (XC). The formula for impedance is Z = √(R² + (XL - XC)²), where XL = 2πfL and XC = 1/(2πfC), with f being the frequency.

Phase Angle in AC Circuits

The phase angle (ϕ) in an AC circuit indicates the phase difference between the voltage and the current. It is calculated using the formula ϕ = arctan((XL - XC) / R). A positive phase angle indicates that the circuit is inductive, while a negative angle indicates a capacitive circuit. The phase angle is crucial for understanding power factor and energy consumption in AC systems.

Resonance in RLC Circuits

Resonance occurs in RLC circuits when the inductive reactance equals the capacitive reactance (XL = XC), resulting in maximum current flow and minimal impedance. At resonance, the phase angle is zero, meaning the voltage and current are in phase. Understanding resonance is important for analyzing circuit behavior at different frequencies and optimizing performance in AC applications.