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 average power dissipated?

검증된 단계별 안내

Step 1: Calculate the angular frequency (ω) of the AC power source using the formula ω = 2πf, where f is the frequency of the power line. For this problem, f = 60 Hz.

Step 2: Determine the inductive reactance (X_L) using the formula X_L = ωL, where L is the inductance of the inductor (0.12 H).

Step 3: 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).

Step 4: Compute the total impedance (Z) of the circuit using the formula Z = √(R² + (X_L - X_C)²), where R is the resistance (75 Ω).

Step 5: Calculate the average power dissipated using the formula P_avg = (V_rms² / Z) × (R / Z), where V_rms is the root mean square voltage of the power source (120 V).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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 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). Understanding impedance is crucial for calculating current and power in AC circuits.

Average Power in AC Circuits

The average power dissipated in an AC circuit is calculated using the formula P = VIcos(φ), where V is the root mean square (RMS) voltage, I is the RMS current, and φ is the phase angle between the voltage and current. The phase angle is determined by the impedance of the circuit and affects how much of the power is actually used for work versus being stored in the reactive components. This concept is essential for determining how much power is effectively consumed by the circuit.

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 circuit can draw maximum power from the source, and the phase angle φ becomes zero, meaning the voltage and current are in phase. While this concept is not directly needed for calculating average power, it provides insight into the behavior of the circuit under different frequencies, which can affect power dissipation.