Ch 31: Alternating Current

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 31: Alternating Current

Problem 23a

Chapter 31, Problem 23a

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

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Textbook Question

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A series ac circuit contains a 250-Ω resistor, a 15-mH inductor, a 3.5-μF capacitor, and an ac power source of voltage amplitude 45 V operating at an angular frequency of 360 rad/s.What is the power factor of this circuit?

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Textbook Question

An L-R-C series circuit is connected to a 120-Hz ac source that has V_rms = 80.0 V. The circuit has a resistance of 75.0 Ω and an impedance at this frequency of 105 Ω. What average power is delivered to the circuit by the source?

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A resistor with R = 300 Ω and an inductor are connected in series across an ac source that has voltage amplitude 500 V. The rate at which electrical energy is dissipated in the resistor is 286 W. What is (a) the impedance Z of the circuit; (b) the amplitude of the voltage across the inductor; (c) the power factor?

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