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Capacitors in AC Circuits: Capacitive Reactance and Phasor Diagrams
The voltage across a capacitor lags the current. Due to the phase difference, it is useful to introduce phasors to describe these circuits.
Learning Objective

Explain the benefits of using a phasor representation
Key Points
 When a capacitor is connected to an alternating voltage, the maximum voltage is proportional to the maximum current, but the maximum voltage does not occur at the same time as the maximum current.
 If the AC supply is connected to a resistor, then the current and voltage will be proportional to each other. This means that the current and voltage will "peak" at the same time.
 The rms current in the circuit containing only a capacitor C is given by another version of Ohm's law to be
$I_{rms} = \frac{V_{rms}}{X_C}$ , where Xc is the capacitive reactance.
Term

rms
Root mean square: a statistical measure of the magnitude of a varying quantity.
Full Text
In the previous Atom on "Resistors in AC Circuits", we introduced an AC power source and studied how resistors behave in AC circuits. There, we used the Ohm's law (V=IR) to derive the relationship between voltage and current in AC circuits. In this and following Atoms, we will generalize the Ohm's law so that we can use it even when we have capacitors and inductors in the circuit. To get there, we will first introduce a very general, pictorial way of representing a sinusoidal wave, using phasor.
Phasor
The key idea in the phasor representation is that a complex, timevarying signal may be represented as the product of a complex number (that is independent of time) and a complex signal (that is dependent on time). Phasors separate the dependencies on A (amplitude),
Fig 3
A phasor can be seen as a vector rotating about the origin in a complex plane. The cosine function is the projection of the vector onto the real axis. Its amplitude is the modulus of the vector, and its argument is the total phase \omega t+\theta. The phase constant \theta represents the angle that the vector forms with the real axis at t = 0.
Capacitors in AC circuits
Earlier in a previous Atom, we studied how the voltage and the current varied with time. If the AC supply is connected to a resistor, then the current and voltage will be proportional to each other. This means that the current and voltage will "peak" at the same time. We say that the current and voltage are in phase.
When a capacitor is connected to an alternating voltage, the maximum voltage is proportional to the maximum current, but the maximum voltage does not occur at the same time as the maximum current. The current has its maximum (it peaks) one quarter of a cycle before the voltage peaks. Engineers say that the "current leads the voltage by 90^{∘}". This is shown in .
Fig 2
The current peaks (has its maximum) one quarter of a wave before the voltage when a capacitor is connected to an alternating voltage.
For a circuit with a capacitor, the instantaneous value of V/I is not constant. However, the value of V_{max}/I_{max} is useful, and is called the capacitive reactance (X_{C}) of the component. Because it is still a voltage divided by a current (like resistance), its unit is the ohm. The value of X_{C} (C standing for capacitor) depends on its capacitance (C) and the frequency (f) of the alternating current.
The capacitor is affecting the current, having the ability to stop it altogether when fully charged. Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor. This is considered to be an effective resistance of the capacitor to AC, and so the rms current I_{rms} in the circuit containing only a capacitor C is given by another version of Ohm's law to be
Phase representation
Since the voltage across a capacitor lags the current, the phasor representing the current and voltage would be give as in . In the diagram, the arrows rotate in counterclockwise direction at a frequency
Fig 4
Phasor diagram for an AC circuit with a capacitor
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Key Term Reference
 AC
 Appears in these related concepts: Driven Oscillations and Resonance, Power, and Safety Precautions in the Household
 Component
 Appears in these related concepts: Adding and Subtracting Vectors Using Components, Cathode Ray Tube, TV and Computer Monitors, and the Oscilloscope, and Position, Velocity, and Acceleration as a Function of Time
 DC
 Appears in these related concepts: Resistors and Capacitors in Series, Impedance, and Resistors in AC Circuits
 Law
 Appears in these related concepts: TwoComponent Forces, Physics and Other Fields, and Models, Theories, and Laws
 Ohm's law
 Appears in these related concepts: Energy Usage and Ohm's Law
 alternating current
 Appears in these related concepts: Education and the Professions, Different Types of Currents, and The Inventions of the Telephone and Electricity
 atom
 Appears in these related concepts: Overview of Atomic Structure, Description of the Hydrogen Atom, and Stable Isotopes
 capacitance
 Appears in these related concepts: Capacitors with Dielectrics, Combinations of Capacitors: Series and Parallel, and Capacitance
 capacitor
 Appears in these related concepts: Introduction and Importance, ParallelPlate Capacitor, and ParallelPlate Capacitor
 circuit
 Appears in these related concepts: Electric Potential Energy and Potential Difference, Microwaves, and Maxwell's Equations
 complex numbers
 Appears in these related concepts: The Wave Function, Addition, Subtraction, and Multiplication, and Imaginary Numbers
 current
 Appears in these related concepts: Reporting LongTerm Liabilities, The Battery, and Magnetic Force Between Two Parallel Conductors
 diagram
 Appears in these related concepts: Motion Diagrams, Bohr Orbits, and B.4 Chapter 4
 frequency
 Appears in these related concepts: Frequency of Sound Waves, Characteristics of Sound, and Sound
 inductor
 Appears in these related concepts: Energy in a Magnetic Field, Energy Stored in a Magnetic Field, and Inductance
 instantaneous
 Appears in these related concepts: Rotational Kinetic Energy: Work, Energy, and Power, Polarization, and Instananeous Velocity: A Graphical Interpretation
 magnitude
 Appears in these related concepts: Multiplying Vectors by a Scalar, Roundoff Error, and Components of a Vector
 ohm
 Appears in these related concepts: Poiseuille's Equation and Viscosity, Phase Angle and Power Factor, and Null Measurements
 phase
 Appears in these related concepts: The Kinetic Molecular Theory of Matter, The Phase of Orbitals, and The Production of Electromagnetic Waves
 phasor
 Appears in these related concepts: Resonance in RLC Circuits, Inductors in AC Circuits: Inductive Reactive and Phasor Diagrams, and Phasors
 plane
 Appears in these related concepts: Shape and Volume, Shape, and Introduction to The Four Fundamental Spaces
 power
 Appears in these related concepts: Force of Muscle Contraction, What is Power?, and Authority
 reactance
 resistance
 Appears in these related concepts: Resistors in Parallel, Resisitors in Series, and Introduction to Blood Flow, Pressure, and Resistance
 resistor
 Appears in these related concepts: Charging a Battery: EMFs in Series and Parallel, The Loop Rule, and Current and Voltage Measurements in Circuits
 rms current
 Appears in this related concept: RLC Series Circuit: At Large and Small Frequencies; Phasor Diagram
 rms voltage
 sinusoidal
 Appears in these related concepts: Sinusoidal Nature of Simple Harmonic Motion, Root Mean Square Values, and Introduction to The Sampling Theorem
 voltage
 Appears in these related concepts: The Nernst Equation, Electric Potential Due to a Point Charge, and Principles of Electricity
 wave
 Appears in these related concepts: Properties of Waves and Light, Waves, and Atomic Structure
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Source: Boundless. “Capacitors in AC Circuits: Capacitive Reactance and Phasor Diagrams.” Boundless Physics. Boundless, 26 May. 2016. Retrieved 28 Oct. 2016 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/inductionaccircuitsandelectricaltechnologies22/accircuits162/capacitorsinaccircuitscapacitivereactanceandphasordiagrams5846271/