An L-R-C series circuit with L = 0.120 H, R = 240 Ω, and C = 7.30 μF carries an rms current of 0.450 A with a frequency of 400 Hz. What are the phase angle and power factor for this circuit?

검증된 단계별 안내

First, calculate the inductive reactance (X_L) using the formula:

X L = ω L

, where

ω

is the angular frequency given by

ω = 2 π f

. Substitute the values for L and f to find X_L.

Next, calculate the capacitive reactance (X_C) using the formula:

X C = \frac{1}{ω}

. Substitute the values for C and f to find X_C.

Determine the impedance (Z) of the circuit using the formula:

Z = \sqrt{R^{2} + (^{X L - X C) 2}}

. Substitute the values for R, X_L, and X_C to find Z.

Calculate the phase angle (φ) using the formula:

φ = \tan - 1 (\frac{X}{L})

. Substitute the values for X_L, X_C, and R to find φ.

Finally, determine the power factor (pf) using the formula:

pf = \cos (φ)

. Use the phase angle calculated in the previous step to find the power factor.

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주요 개념

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

Impedance in L-R-C Circuits

Impedance in an L-R-C circuit is the total opposition to the flow of alternating current, combining resistance (R), inductive reactance (X_L), and capacitive reactance (X_C). It is calculated using Z = √(R² + (X_L - X_C)²), where X_L = 2πfL and X_C = 1/(2πfC). Understanding impedance is crucial for determining the phase angle and power factor.

Phase Angle

The phase angle in an L-R-C circuit indicates the phase difference between the voltage across the circuit and the current through it. It is determined by the arctangent of the reactance difference over resistance: φ = arctan((X_L - X_C)/R). A positive phase angle suggests the circuit is inductive, while a negative angle indicates it is capacitive.

Power Factor

The power factor in an L-R-C circuit is the cosine of the phase angle, representing the ratio of real power used in the circuit to the apparent power. It is calculated as cos(φ), where φ is the phase angle. A power factor close to 1 indicates efficient power usage, while a lower value suggests more reactive power is present.