Physics
Textbooks
Boundless Physics
Circuits and Direct Currents
RC Circuits
Physics Textbooks Boundless Physics Circuits and Direct Currents RC Circuits
Physics Textbooks Boundless Physics Circuits and Direct Currents
Physics Textbooks Boundless Physics
Physics Textbooks
Physics
Concept Version 11
Created by Boundless

Phase Angle and Power Factor

In a series RC circuit connected to an AC voltage source, voltage and current maintain a phase difference.

Learning Objective

  • Compare the currents in the resistor and capacitor in a series RC circuit connected to an AC voltage source


Key Points

    • In a series RC circuit connected to an AC voltage source, the currents in the resistor and capacitor are equal and in phase.
    • In a series RC circuit connected to an AC voltage source, the total voltage should be equal to the sum of voltages on the resistor and capacitor.
    • In a series RC circuit connected to an AC voltage source, voltage and current have a phase difference of $\phi$ , where $cos\phi = \frac{R}{\sqrt{R^2+(\frac{1}{\omega C})^2}}$ . cosϕ is called the power factor.

Terms

  • alternating current

    (AC)—An electric current in which the direction of flow of the electrons reverses periodically having an average of zero, with positive and negative values (with a frequency of 50 Hz in Europe, 60 Hz in the US, 400 Hz for airport lighting, and some others); especially such a current produced by a rotating generator or alternator.

  • impedance

    A measure of the opposition to the flow of an alternating current in a circuit; the aggregation of its resistance, inductive and capacitive reactance. Represented by the symbol Z.

  • rms

    Root mean square: a statistical measure of the magnitude of a varying quantity.


Full Text

Phase Angle

Impedance is an AC (alternating current) analogue to resistance in a DC circuit. As we studied in a previously Atom ("Impedance"), current, voltage and impedance in an RC circuit are related by an AC version of Ohm's law: $I = \frac{V}{Z}$, where I and V are peak current and peak voltage respectively, and Z is the impedance of the circuit.

In a series RC circuit connected to an AC voltage source as shown in , conservation of charge requires current be the same in each part of the circuit at all times. Therefore we can say: the currents in the resistor and capacitor are equal and in phase. (We will represent instantaneous current as i(t). )

Series RC Circuit

Series RC circuit.

On the other hand, because the total voltage should be equal to the sum of voltages on the resistor and capacitor, so we have:

$\begin{aligned} v(t)& = v_R(t)+v_C(t) \\ &= i(t)R + i(t)\frac{1}{j \omega C} \\ &= i(t) (R+\frac{1}{j\omega C}) \end{aligned}$,

where $\omega$ is the angular frequency of the AC voltage source and j is the imaginary unit; j2=-1. Since the complex number $Z = R+\frac{1}{j\omega C} = \sqrt{R^2+(\frac{1}{\omega C})^2} e^{j\phi}$ has a phase angle $\phi$ that satisfies $cos\phi = \frac{R}{\sqrt{R^2+(\frac{1}{\omega C})^2}}$,

we notice that voltage $v(t)$ and current $i(t)$ has a phase difference of $\phi$.

For R=0, $\phi = 90^{\circ}$. As learned from the preceding series of Atoms—the voltage across the capacitor VC follows the current by one-fourth of a cycle (or 90º).

Power Factor

Because voltage and current are out of phase, power dissipated by the circuit is not equal to: (peak voltage) times (peak current). The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the average power is IrmsVrmscosϕ, where Irms and Vrms are the root mean square (rms) averages of the current and voltage, respectively. For this reason, cosϕ is called the power factor, which can range from 0 to 1.

[ edit ]
Edit this content
Prev Concept
Impedance
Electric Currents and Magnetic Fields
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.