A capacitor in an AC circuit exhibits capacitive reactance \(X_C\), which opposes changes in voltage across its plates. Unlike inductive reactance, capacitive reactance decreases with increasing frequency, making capacitors high-pass components. The current through a capacitor leads the voltage by exactly 90° because the capacitor responds to voltage changes (dv/dt), not voltage itself. When voltage tries to change, the capacitor must charge or discharge, causing current to flow. This current reaches its maximum when voltage is changing most rapidly (at voltage zero crossing) and is zero when voltage change is slowest (at voltage peaks). Like inductors, capacitors consume no real power; they store energy in their electric field during one quarter cycle and return it during the next quarter cycle.
Capacitive reactance represents the capacitor's opposition to changing voltage through charge storage and release. The 90° phase relationship occurs because maximum current flows when voltage is changing most rapidly (at voltage zero crossing), while zero current flows when voltage change is slowest (at voltage peaks). The inverse frequency dependence arises because higher frequencies mean faster voltage changes, requiring more current flow for the same voltage amplitude, effectively reducing the opposition (reactance).
In an alternating current (AC) circuit containing only a capacitor, the capacitor offers opposition to the flow of current, known as capacitive reactance. This opposition is frequency-dependent, and there is a distinct phase difference between the voltage across the capacitor and the current flowing through it.
| Property | Details |
|---|---|
| Scalar/Vector Nature | Instantaneous voltage and current are scalars. In AC analysis, they are often represented as phasors, which have magnitude and phase angle, to simplify calculations involving their sinusoidal nature. |
| SI Units | Capacitance (C) in Farads (F), Voltage (V) in Volts (V), Current (I) in Amperes (A), and Capacitive Reactance (Xc) in Ohms (Ω). |
| Magnitude | The magnitude of the current is given by I = V / Xc, where Xc = 1 / (2πfC). The capacitive reactance is inversely proportional to both the AC frequency (f) and the capacitance (C). |
| Phase Relationship | The current through a pure capacitor leads the voltage across it by 90 degrees (π/2 radians). This means the current reaches its maximum value a quarter of a cycle before the voltage does. |
| Energy & Power | An ideal capacitor does not dissipate energy. It stores energy in its electric field as voltage increases and returns this energy to the circuit as voltage decreases. The average power consumed over a full AC cycle is zero. |
| Dimensional Formula | Capacitance: [M⁻¹ L⁻² T⁴ I²]. Capacitive Reactance: [M L² T⁻³ I⁻²]. |
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| \(X_C\) | Capacitive Reactance | Ohm (Ω) | Frequency-dependent opposition to AC current. |
| \(C\) | Capacitance | Farad (F) | Measure of charge storage capability. |
| \(\omega\) | Angular Frequency | rad/s | Rate of change of phase angle, \( \omega = 2\pi f \). |
| \(f\) | Frequency | Hertz (Hz) | Number of cycles per second of the AC waveform. |
| \(V_C, U_C\) | Voltage | Volt (V) | Voltage across the capacitor, lags current by 90°. |
| \(I, i_C\) | Current | Ampere (A) | Current through the capacitor, leads voltage by 90°. |
| \(Z_C\) | Complex Impedance | Ohm (Ω) | Complex representation of reactance, \(Z_C = -jX_C\). |
| \(Q_C\) | Reactive Power | VAR | Power stored and returned by the electric field (negative by convention). |
| \(W_C\) | Stored Energy | Joule (J) | Energy stored in the capacitor's electric field. |
| \(\phi\) | Phase Angle | degrees or rad | Phase difference between voltage and current (\(-90°\) for a capacitor). |
The derivation starts from the fundamental relationship between current and the rate of change of voltage for a capacitor.
Assuming a sinusoidal voltage source, where \( v(t) = V_0 \cos(\omega t) \), we can find the current by taking the derivative.
Using the trigonometric identity \( -\sin(x) = \cos(x + 90°) \), we can express the current as a cosine function to compare its phase with the voltage.
This shows that the current amplitude is \( I_0 = \omega C V_0 \) and that the current leads the voltage by 90°. Capacitive reactance \(X_C\) is defined as the ratio of voltage amplitude to current amplitude, similar to Ohm's Law.
The behavior of a purely capacitive AC circuit is critically dependent on the frequency of the AC source. Different frequency regimes define its primary application and characteristics.
| Type / Case | Description | When to Use |
|---|---|---|
| Low-Frequency Limit (DC) | As frequency approaches zero, capacitive reactance (Xc) approaches infinity. The capacitor effectively becomes an open circuit, blocking the flow of steady current. | Used in DC blocking capacitors to separate AC signals from DC bias in electronic circuits. |
| High-Frequency Limit | As frequency becomes very high, capacitive reactance (Xc) approaches zero. The capacitor acts almost like a short circuit or a simple wire, offering very little opposition to current flow. | Used in bypass or decoupling capacitors to short high-frequency noise to ground, and in high-pass filters. |
| Intermediate Frequencies | At frequencies between the extremes, the capacitor exhibits a finite, non-zero reactance. It impedes the current to a degree determined by the specific frequency and capacitance value. | This is the general operating principle for using capacitors in filtering and timing circuits to control signal flow based on frequency. |
Filtering and Coupling
Capacitors are used to create high-pass filters and for AC coupling between amplifier stages. They pass high-frequency signals while blocking DC and attenuating low frequencies.
Power Factor Correction
In industrial settings, large inductive loads (like motors) cause a lagging power factor. Banks of capacitors are used to provide leading reactive power, compensating for the inductive load and improving the overall power factor to reduce energy costs.
Motor Starting and Running
Single-phase induction motors use starting and running capacitors to create a phase-shifted current in an auxiliary winding. This generates a rotating magnetic field necessary to start the motor and improve its running efficiency.
Energy Storage and Pulse Power
Capacitors store electrical energy and can release it very quickly. This is used in applications like electronic photo flashes, defibrillators, and large-scale pulse power systems for research.
Power Factor Correction Banks
In factories with many electric motors, the overall electrical system becomes highly inductive. This causes inefficiency. Large banks of high-voltage capacitors are installed in parallel with the load. Their capacitive nature counteracts the motors' inductance, bringing the power factor closer to unity and reducing electricity costs.
Audio Amplifier Coupling
In a multi-stage audio amplifier, the DC operating voltage of one stage must not affect the next. A coupling capacitor is placed between stages. For the AC audio signal, the capacitor has low reactance and lets the signal pass. For the DC bias voltage, the capacitor has infinite reactance, acting as an open circuit and blocking it.
Single-Phase Motor Starting
Appliances like refrigerators and air conditioners use single-phase induction motors, which cannot start on their own. A 'start capacitor' is used in series with an auxiliary winding to create a second current that is out of phase with the main current. This phase difference creates a rotating magnetic field, providing the initial torque to spin the motor.
RF Bypassing in Electronics
In a radio receiver or transmitter, it's crucial to keep high-frequency signals from traveling along power supply lines. Small capacitors are placed between the power line and ground. To high-frequency RF noise, these capacitors have very low reactance and act as a short circuit, shunting the unwanted noise to ground while letting the DC power pass through unaffected.
| Quantity | Symbol | SI Unit | Dimensional Formula |
|---|---|---|---|
| Capacitance | C | Farad (F) | [M⁻¹ L⁻² T⁴ I²] |
| Reactance / Impedance | \(X_C, Z_C\) | Ohm (Ω) | [M L² T⁻³ I⁻²] |
| Voltage | V | Volt (V) | [M L² T⁻³ I⁻¹] |
| Current | I | Ampere (A) | [I] |
| Angular Frequency | \(\omega\) | rad/s | [T⁻¹] |
| Reactive Power | Q | VAR | [M L² T⁻³] |
| Energy | W | Joule (J) | [M L² T⁻²] |
The formula is \(X_C = 1/(\omega C)\). It calculates capacitive reactance (\(X_C\)), which is the opposition a capacitor presents to the flow of alternating current (AC) in an electrical circuit. This opposition is frequency-dependent and is measured in Ohms (\(\Omega\)).
In the formula, \(X_C\) is the capacitive reactance in Ohms (\(\Omega\)). The variable \(\omega\) (omega) represents the angular frequency of the AC signal in radians per second (rad/s). The variable \(C\) is the capacitance of the capacitor, measured in Farads (F).
This formula is used in AC circuit analysis to determine how much a capacitor will impede the flow of current at a given frequency. To use it, you must know the capacitance \(C\) and the angular frequency \(\omega\) of the AC source. It is fundamental for designing filters, oscillators, and timing circuits.
A frequent error is using the standard frequency \(f\) (in Hertz) directly in the formula instead of the angular frequency \(\omega\). You must first convert frequency to angular frequency using the relationship \(\omega = 2\pi f\). Forgetting this conversion will lead to a result that is incorrect by a factor of \(2\pi\).
Capacitive reactance is a key principle behind audio crossover networks in speakers, where capacitors are used as high-pass filters to direct high-frequency signals to tweeters. It is also essential for AC coupling between amplifier stages, allowing AC signals to pass while blocking DC voltage.
Capacitive reactance acts as the effective 'resistance' for a capacitor in an AC circuit, allowing an AC version of Ohm's Law to be applied: \(V_C = I_C X_C\). Critically, it also defines the phase relationship where the current (\(I_C\)) flowing through the capacitor *leads* the voltage (\(V_C\)) across it by exactly 90 degrees.