AC power


In an electric circuit, instantaneous power is the time rate of flow of energy past a given point of the circuit. In alternating current circuits, energy storage elements such as inductors and capacitors may result in periodic reversals of the direction of energy flow. Its SI unit is the watt.
The portion of instantaneous power that, averaged over a complete cycle of the AC waveform, results in net transfer of energy in one direction is known as instantaneous active power, and its time average is known as active power or real power. The portion of instantaneous power that results in no net transfer of energy but instead oscillates between the source and load in each cycle due to stored energy is known as instantaneous reactive power, and its amplitude is the absolute value of reactive power.

Active, reactive, apparent, and complex power in sinusoidal steady-state

For a simple alternating current circuit in steady-state; consisting of a source and a linear time-invariant load, both the current and voltage are sinusoidal at the same fixed frequency, given by:
with and the RMS, and the phasors and the phase shift between the voltage and current. The instantaneous power is given by the product:
If the load is purely resistive, the two quantities reverse their polarity at the same time. Hence, the instantaneous power is always positive, such that the direction of energy flow does not reverse and always is toward the resistor. In this case, only active power is transferred.
If the load is purely reactive, then the voltage and current are 90 degrees out of phase. For two quarters of each cycle, the product of voltage and current is positive, but for the other two quarters, the product is negative, indicating that on average, exactly as much energy flows into the load as flows back out. There is no net energy flow over each half cycle. In this case, only reactive power flows: There is no net transfer of energy to the load; however, electrical power does flow along the wires and returns by flowing in reverse along the same wires. The current required for this reactive power flow dissipates energy in the line resistance, even if the ideal load device consumes no energy itself. Practical loads have resistance as well as inductance, or capacitance, so both active and reactive powers will flow to normal loads.
Apparent power is the product of the RMS values of voltage and current. Apparent power is taken into account when designing and operating power systems, because although the current associated with reactive power does no work at the load, it still must be supplied by the power source. Conductors, transformers and generators must be sized to carry the total current, not just the current that does useful work. Insufficient reactive power can depress voltage levels on an electrical grid and, under certain operating conditions, collapse the network. Another consequence is that adding the apparent power for two loads will not accurately give the total power unless they have the same phase difference between current and voltage.
Conventionally, capacitors are treated as if they generate reactive power, and inductors are treated as if they consume it. If a capacitor and an inductor are placed in parallel, then the currents flowing through the capacitor and the inductor tend to cancel rather than add. This is the fundamental mechanism for controlling the power factor in electric power transmission; capacitors are inserted in a circuit to partially compensate for reactive power 'consumed' by the load. Purely capacitive circuits supply reactive power with the current waveform leading the voltage waveform by 90 degrees, while purely inductive circuits absorb reactive power with the current waveform lagging the voltage waveform by 90 degrees. The result of this is that capacitive and inductive circuit elements tend to cancel each other out.
Engineers use the following terms to describe energy flow in a system :
  • Active power, P, or real power: watt ;
  • Reactive power, Q: volt-ampere reactive ;
  • Complex power, S: volt-ampere ;
  • Apparent power, |S|: the magnitude of complex power S: volt-ampere ;
  • Phase of voltage relative to current, φ: the angle of difference between current and voltage;. Current lagging voltage, current leading voltage.
These are all denoted in the adjacent diagram.
In the diagram, P is the active power, Q is the reactive power, S is the complex power and the length of S is the apparent power. Reactive power does not do any work, so it is represented as the imaginary axis of the vector diagram. Active power does do work, so it is the real axis.
The unit for power is the watt. Apparent power is often expressed in volt-amperes since it is the product of RMS voltage and RMS current. The unit for reactive power is var, which stands for volt-ampere reactive. Since reactive power transfers no net energy to the load, it is sometimes called "wattless" power. It does, however, serve an important function in electrical grids and its lack has been cited as a significant factor in the Northeast blackout of 2003. Understanding the relationship among these three quantities lies at the heart of understanding power engineering. The mathematical relationship among them can be represented by vectors or expressed using complex numbers, S = P + j Q.

Calculations and equations in sinusoidal steady-state

The formula for complex power in phasor form is:
where and are the complex valued voltage and current, respectively, written in phasor form with the amplitude as RMS. Also by convention, the complex conjugate of I is used, which is denoted , rather than I itself. This is done because otherwise using the product V I to define S would result in a quantity that depends on the reference angle chosen for V or I, but defining S as V I* results in a quantity that doesn't depend on the reference angle and allows to relate S to P and Q.
Other forms of complex power are derived from Z, the load impedance.
Consequentially, with reference to the power triangle, real power is derived as:
For a purely resistive load, real power can be simplified to:
R denotes resistance of the load.
Reactive power is derived as:
For a purely reactive load, reactive power can be simplified to:
where X denotes reactance of the load.
Combining, the complex power is back-derived as
and the apparent power as
These are simplified diagrammatically by the power triangle.

Power factor

The ratio of active power to apparent power in a circuit is called the power factor. For two systems transmitting the same amount of active power, the system with the lower power factor will have higher circulating currents due to energy that returns to the source from energy storage in the load. These higher currents produce higher losses and reduce overall transmission efficiency. A lower power factor circuit will have a higher apparent power and higher losses for the same amount of active power. The power factor is 1.0 when the voltage and current are in phase. It is zero when the current leads or lags the voltage by 90 degrees. When the voltage and current are 180 degrees out of phase, the power factor is negative one, and the load is feeding energy into the source. Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle of current with respect to voltage. Voltage is designated as the base to which current angle is compared, meaning that current is thought of as either "leading" or "lagging" voltage. Where the waveforms are purely sinusoidal, the power factor is the cosine of the phase angle between the current and voltage sinusoidal waveforms. Equipment data sheets and nameplates will often abbreviate power factor as "" for this reason.
Example: The active power is and the phase angle between voltage and current is 45.6°. The power factor is. The apparent power is then:.

Reactive power

In a direct current circuit, the power flowing to the load is proportional to the product of the current through the load and the potential drop across the load. The power that happens because of a capacitor or inductor is called reactive power. It happens because of the AC nature of elements like inductors and capacitors. Energy flows in one direction from the source to the load. In AC power, the voltage and current both vary approximately sinusoidally. When there is inductance or capacitance in the circuit, the voltage and current waveforms do not line up perfectly. The power flow has two components – one component flows from source to load and can perform work at the load; the other portion, known as "reactive power", is due to the delay between voltage and current, known as phase angle, and cannot do useful work at the load. It can be thought of as current that is arriving at the wrong time. To distinguish reactive power from active power, it is measured in units of "volt-amperes reactive", or var. These units can simplify to watts but are left as var to denote that they represent no actual work output.
Energy stored in capacitive or inductive elements of the network gives rise to reactive power flow. Reactive power flow strongly influences the voltage levels across the network. Voltage levels and reactive power flow must be carefully controlled to allow a power system to be operated within acceptable limits. A technique known as reactive compensation is used to reduce apparent power flow to a load by reducing reactive power supplied from transmission lines and providing it locally. For example, to compensate an inductive load, a shunt capacitor is installed close to the load itself. This allows all reactive power needed by the load to be supplied by the capacitor and not have to be transferred over the transmission lines. This practice saves energy because it reduces the amount of energy that is required to be produced by the utility to do the same amount of work. Additionally, it allows for more efficient transmission line designs using smaller conductors or fewer bundled conductors and optimizing the design of transmission towers.