Ch 31: Alternating Current

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 14th Edition

Ch 31: Alternating Current

문제 25a

31장, 문제 25a

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?

검증된 단계별 안내

Understand the concept of power factor in an AC circuit. The power factor is defined as the cosine of the phase angle (φ) between the total voltage and the total current in the circuit. It is given by the formula:

\cos (φ)

Calculate the inductive reactance (X_L) using the formula:

X_{L} = ω L

, where

ω

is the angular frequency and

L

is the inductance.

Calculate the capacitive reactance (X_C) using the formula:

X_{C} = \frac{1}{ω}

, where

C

is the capacitance.

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

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

, where

R

is the resistance.

Calculate the power factor using the formula:

\cos (φ) = \frac{R}{Z}

, where

R

is the resistance and

Z

is the total impedance.

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주요 개념

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Impedance in AC Circuits

Impedance is the total opposition a circuit offers to the flow of alternating current, combining resistance, inductive reactance, and capacitive reactance. In a series AC circuit, impedance (Z) is calculated using Z = √(R² + (X_L - X_C)²), where R is resistance, X_L is inductive reactance, and X_C is capacitive reactance. Understanding impedance is crucial for determining the circuit's behavior at different frequencies.

Reactance

Reactance is the opposition to the change in current flow in an AC circuit due to inductors and capacitors. Inductive reactance (X_L) is given by X_L = ωL, and capacitive reactance (X_C) is given by X_C = 1/(ωC), where ω is the angular frequency, L is inductance, and C is capacitance. Reactance affects the phase difference between voltage and current, influencing the power factor.

Power Factor

The power factor in an AC circuit is the ratio of real power used to do work to apparent power flowing in the circuit, expressed as cos(φ), where φ is the phase angle between voltage and current. It indicates how effectively electrical power is converted into useful work, with a power factor of 1 being ideal. Calculating the power factor involves understanding the phase relationship between circuit elements and their reactances.