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?

Verified step by step guidance

1. First, calculate the inductive reactance (X_L) using the formula: $X{L}_{}=\omega L$, where $\omega$ is the angular frequency given by $\omega =2\pi f$. Substitute the values for L and f to find X_L.
2. Next, calculate the capacitive reactance (X_C) using the formula: $X{C}_{}=\frac{1}{\omega }$. Substitute the values for C and f to find X_C.
3. Determine the impedance (Z) of the circuit using the formula: $Z=\sqrt{R^2+{(}^X}$. Substitute the values for R, X_L, and X_C to find Z.
4. Calculate the phase angle (φ) using the formula: $\phi =\tan -1\left(\frac{X}{L_{}}\right)$. Substitute the values for X_L, X_C, and R to find φ.
5. Finally, determine the power factor (pf) using the formula: $\mathrm{pf}=\cos \left(\phi \right)$. Use the phase angle calculated in the previous step to find the power factor.

Key Concepts

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.