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 peak current I?

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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 (0.12 H).

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

Find the total impedance (Z) of the circuit using the formula: Z = √(R² + (X_L - X_C)²), where R is the resistance (75 Ω).

Calculate the peak current (I) using Ohm's Law for AC circuits: I = V_peak / Z, where V_peak is the peak voltage. Note that V_peak = √2 × V_rms, and V_rms is given as 120 V.

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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.

Ohm's Law for AC Circuits

Ohm's Law states that the current (I) through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the impedance (Z) of the circuit. In AC circuits, this is expressed as I = V/Z, where V is the root mean square (RMS) voltage. The peak current can be found by multiplying the RMS current by √2.

Resonance in RLC Circuits

Resonance occurs in RLC circuits when the inductive reactance equals the capacitive reactance, resulting in maximum current flow at a specific frequency. This frequency is known as the resonant frequency (fr) and is given by fr = 1/(2π√(LC)). While the question does not directly ask for resonance, understanding it helps in analyzing the behavior of the circuit under different frequencies.