- Electric Power Definition: Electric power is defined as the rate at which electrical energy is transferred by an electric circuit, measured in watts (W).
- Single Phase Power: Single phase power uses one alternating voltage and current wave, commonly found in homes.
- Three Phase Power: Three phase power uses three alternating currents offset by 120 degrees, providing stable and efficient power, ideal for industries.
- Active Power: Active power is the useful power consumed in a circuit, measured in watts (W), and does real work.
- Reactive Power: Reactive power supports the voltage levels necessary for active power transfer but does no useful work itself, measured in volt-amperes reactive (VAR).
Complex Power
For a sinusoidal single-phase circuit, complex power is defined with RMS phasors. The voltage and current phasors can be written as V.ejα and I.ejβ, where α and β use the same angular reference. Under the passive sign convention, complex power is the voltage phasor multiplied by the complex conjugate of the current phasor:
The difference α − β is the phase angle φ between voltage and current.
The relationship can therefore be written as:
For RMS values, P = VIcosφ and Q = VIsinφ.
The quantity S = P + jQ is complex power.
Its magnitude, |S| = (P2 + Q2)½, is apparent power in volt-amperes (VA). It equals RMS voltage times RMS current. Equipment VA ratings reflect voltage, current and thermal limits, including current-related heating described by Joule’s law of heating.
Under the passive load convention used for complex power, Q is positive when current lags voltage in an inductive load and negative when current leads voltage in a capacitive load.
Single Phase Power
A single-phase electrical transmission system uses one alternating voltage. Single phase power is common in distribution and smaller installations, while a bulk electrical power system normally uses three phases. The circuit behaviour discussed below comes from electrical resistance, inductance and capacitance.
Resistance
Resistance opposes the flow of current and converts electrical energy into heat. In an ideal resistor supplied by a sinusoidal source, current and voltage are in phase. If RMS current I flows through resistance R for t seconds, the converted energy I2Rt is active energy. Its conversion rate is active power.
Inductance
An ideal inductor stores energy in its magnetic field as current changes. For instantaneous current I and inductance L henries, the stored energy is:
In sinusoidal steady state, the repeated exchange of this energy is associated with positive reactive power under the passive convention.
Capacitance
An ideal capacitor stores energy in its electric field. For instantaneous electric potential difference V and capacitance C, the stored energy is:
In sinusoidal steady state, the repeated exchange of this field energy is associated with negative reactive power under the passive convention.
Active Power and Reactive Power
Consider a sinusoidal single phase power circuit in which current lags voltage by φ.
Let the instantaneous voltage be v = Vmsinωt.
The instantaneous current is i = Imsin(ωt – φ).
Here Vm and Im are the peak values.
Multiplying v by i gives the instantaneous power:
Active Power
Resistive Power
For a purely resistive single-phase circuit, the phase angle is φ = 0:
The instantaneous power is never negative because voltage v and current i reverse together. Its average value is VI for RMS quantities, and the resistor converts that energy into heat. The average conversion rate is active power. In a resistive electrical circuit, another name is Resistive Power.
Reactive Power
Inductive Power
For a purely inductive circuit, current lags voltage by φ = +90o. Substituting +90o gives:
Instantaneous power alternates at twice the supply frequency. Its signs for the four quarter cycles, 0o to 90o, 90o to 180o, 180o to 270o and 270o to 360o, show energy moving into the magnetic field and back to the source. An ideal inductor therefore has zero average real power. Its positive sinusoidal Q is reactive power, also called inductive power.
In a purely inductive circuit, magnetic-field energy rises and falls with the square of instantaneous current. Instantaneous power has equal positive and negative areas over a cycle. Its average real power is zero, while the RMS phasor relationship gives positive reactive power, also called inductive power.
Capacitive Power
For a purely capacitive circuit, current leads voltage by 90o, so φ = -90o.
The instantaneous capacitive power changes sign through the intervals 0o to 90o, 90o to 180o, 180o to 270o and 270o to 360o. Energy enters and leaves the electric field at twice the supply frequency. An ideal capacitor has zero average real power and negative reactive power under the passive convention.
Active Component and Reactive Component of Power
The instantaneous power equation can be separated into average and oscillating parts:
The term containing VmImcosφ has the nonzero average that defines real power.
This active component, also called real or true power, represents net energy transferred to the load over a complete cycle.
The term containing VmImsinφ alternates with zero average. It represents energy exchanged with electric and magnetic fields and is associated with reactive power.
Active power and reactive power have the same physical dimension, but active power is stated in watts (W) and reactive power in volt-amperes reactive (var).
A single-phase source produces one alternating voltage and current. The next sections apply the equation to circuits containing electrical resistance, capacitance, inductance or combinations of them to develop the single phase power equation.
Single Phase Power Equation for Purely Resistive Circuit
For a purely resistive single phase power calculation, connect resistance R across a sinusoidal voltage source:
V(t) is instantaneous voltage.
Vm is peak voltage.
ω is angular frequency in radians per second.
By Ohm’s law:
Substituting V(t) gives:
V(t) and IR are in phase. The voltages and current therefore have zero phase difference.
The instantaneous power is:
This single phase power equation contains a constant average term:
and a twice-frequency term:
The oscillating term averages to zero over a complete cycle. The average resistor power is shown below.
Single Phase Power Equation for Purely Inductive Circuit
An ideal inductor opposes changes in current through its induced emf. The applied voltage drives that changing current. The diagram shows an ideal inductor connected to a sinusoidal source Vrms.
The inductor voltage is:
The single phase power equation shows that I lags V by π/2, as the phasor diagram illustrates.
Instantaneous power is:
This single phase power formula contains only a twice-frequency alternating term. Its average over a full cycle is zero.
Single Phase Power Equation for Purely Capacitive Circuit
An ideal capacitor charges and discharges as its voltage changes:
The single phase power calculation shows that I(t) leads V(t) by π/2.
Capacitor power is a twice-frequency alternating term whose average over a full cycle is zero.
Single Phase Power Equation for RL Circuit
A resistor and ideal inductor are connected in series across a voltage source V. Their RMS drops are VR = IR and VL = IXL.
The phasor voltage drops form a right triangle. OA represents IR, AD represents IXL, and OD is the resultant of VR and VL.
This is the impedance magnitude of the RL circuit.
The vector diagram shows V leading I. The phase angle φ is:
Instantaneous power contains an average term, 0.5 VmImcosφ, and a twice-frequency term containing 0.5 VmIm whose average over a cycle is zero.
The average term is the real power transferred to the load.
Using RMS values, P = VI cos Φ watts.
For sinusoidal voltage and current, cosφ is the displacement power factor:
Current resolves into Icosφ in phase with V and Isinφ in quadrature. VIcosφ is active power, and VIsinφ is reactive power. The older term wattless current can mislead because quadrature current still contributes to conductor losses.
Single Phase Power Equation for RC Circuit
In a series RC circuit, current is in phase with resistor voltage and leads capacitor voltage. The supply current therefore leads supply voltage by φ. If V = Vmsinωt, then I = Imsin(ωt + φ).
The active-power calculation follows the same form as for an R-L circuit, but an RC load has leading electrical power factor.
Three Phase Power Definition
A balanced three phase power source has three sinusoidal phase voltages of equal magnitude and frequency. In a balanced three-phase electric power circuit, the voltage waveforms are separated by 120o. Equal phase loads produce current waveforms separated by 120o as well. A phase is 120o from the next phase in the selected sequence. The Three phase power definition does not require three isolated circuits: star and delta connections can share conductors and connection points. Each complete cycle spans 360o, while adjacent balanced phases remain 120o apart.
A three phase system is unbalanced when phase magnitudes, phase loads or the nominal 120o displacement do not form a balanced set.
Advantages of Three Phase System
Three-phase systems offer practical benefits for bulk power transfer and rotating machinery.
- For a sinusoidal load, instantaneous single phase power is:
It contains a twice-frequency term. For a balanced sinusoidal load, the total three phase power equation is:
The three oscillating terms cancel, so total instantaneous power is constant. This produces smoother torque in three-phase motors. - For a given power and line voltage, a balanced three-phase system can transfer power with less conductor material than an equivalent single-phase arrangement. Machine rating depends on design, cooling and operating limits rather than a universal 1.5 factor.
- A conventional Single phase induction motor needs an auxiliary starting method because one stator winding does not create starting torque. A correctly supplied three-phase induction motor creates a rotating field and is normally self-starting.
- Three-phase equipment often delivers steadier power and suits high-power loads. Power factor and efficiency depend on load and machine design, so neither is automatically higher in every installation.
Three Phase Power Equation
To derive the three phase power equation for a balanced sinusoidal load, use the same assumptions for the three phase power calculation: equal phase-voltage magnitudes and equal phase currents, with each phase displaced by 120o. Let φ be the same impedance angle between voltage and current in every phase.
The voltage and current of the red phase are:
respectively.
The voltage and current of the yellow phase are:
respectively.
The voltage and current of the blue phase are:
respectively.
The red-phase instantaneous power is:
The yellow-phase instantaneous power is:
The blue-phase instantaneous power is:
Total instantaneous power is their sum:
For a balanced sinusoidal load, the twice-frequency terms cancel and total instantaneous power is constant at three times the average real power per phase. Three-phase reactive power can still be nonzero because the complex reactive powers add across the phases.
Reactive power quantifies sinusoidal energy exchange associated with electric and magnetic fields in an electric circuit. Its unit is the volt-ampere reactive (var). An ideal reactive element transfers zero net energy over a cycle, but the associated current affects losses, equipment loading and voltage. Stored field energy can later be dissipated after switching in an electrical DC circuit. In an AC system, reactive power is therefore an operating quantity rather than unused power.
Power factor is the ratio of active power P to apparent power S. For sinusoidal voltage and current, the phase angle also relates to reactive power:
Here θ is the angle between voltage and current, and cosθ is displacement power factor.
With nonsinusoidal waveforms, total power factor remains P/S but is not generally equal to cosθ. Domestic loads can be resistive, inductive, capacitive or electronic. Under the passive convention, inductive current lags voltage and gives positive Q, while capacitive current leads voltage and gives negative Q.
In a sinusoidal RL circuit or RC circuit, instantaneous energy moves into and out of the reactive element during parts of every cycle. The following derivation separates average and alternating power for an RL load:
V = Vmsinωt , I = Imsin(ωt − θ)
The Q1sin2ωt term has zero average. It describes field-energy exchange rather than net energy consumption, although the associated current still affects conductors and equipment.
Use of Reactive Power
AC machines need electric or magnetic fields for energy conversion. An induction electrical motor draws magnetising current. An electrical transformer draws excitation current to establish core flux. These currents contribute to reactive demand. Other loads may operate near unity power factor or supply capacitive vars, so demand depends on the equipment.
Reactive Power in Transmission Lines
Reactive power and voltage are closely coupled in an AC electrical power transmission line. The exact response depends on network topology, line parameters, loading and controls.
For the simplified lossless two-bus model used here, receiving-end reactive power is:
Here θ is power angle, Xl is line reactance, Vs is sending-end voltage, and Vr is receiving-end voltage.
With a small-angle approximation, Qr becomes:
Rearranging gives:
The corresponding positive-voltage solution is:
This result applies to the stated model rather than every transmission network.
The positive root represents the normal positive operating-voltage solution, so Vr stays physically meaningful when Qr is zero.
If Q1 denotes receiving-end load demand and Q2 denotes reactive support under this sign convention, then Qr = Q1 – Q2.
Case – 1
In the simplified model, Q2 = Q1 gives Vs = Vr. In a real network, resistance, shunt capacitance, transformer taps and other buses can still make the voltage differ between the two ends.
Case – 2
When reactive demand exceeds support under this convention, Qr changes accordingly and receiving-end voltage tends to fall. Operators can increase local support or adjust other controls to keep voltage within range.
Case – 3
When reactive support exceeds local demand in this model, Qr changes in the opposite direction and receiving-end voltage tends to rise. Excess voltage can exceed equipment limits.
Actual load patterns vary by network and time. Transmission operators use voltage schedules, generator excitation and compensating devices rather than trying to force Q1 = Q2 at every location.
Reactive Power Compensation
Reactive-power compensation must match the network study and operating range. A shunt reactor absorbs vars, while a shunt capacitor supplies vars under the passive convention.
For a lagging load, correctly sized compensation can improve power factor. A capacitor bank can reduce source current by supplying vars near the load. Other devices include a switched shunt capacitor, shunt reactor, synchronous condenser and power-electronic compensator. An on-load tap-changing transformer regulates voltage by changing ratio; it does not directly create reactive power. An overexcited synchronous motor operating without mechanical load can serve as a synchronous condenser. Transformer turns ratio changes must respect current and voltage limits because a voltage difference affects network power flow.
For active power P, the capacitor rating needed to change displacement power factor from cosθ1 to cosθ2 is:
P is the load’s real-power demand in watts.
If inductive compensation moves the displacement power factor from cosθ2 to cosθ1, the reactor rating is:
The required capacitance or inductance follows from:
Practical selection must account for switching steps, voltage variation, harmonics, resonance and overcompensation.
