Draft:Reactive Power

• Comment: Need to add more independent sources. Kirokeer (talk) 19:34, 16 October 2025 (UTC)

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Reactive power is a quantity considered in the context of AC power system analysis. It is useful in describing requirements for managing power flow through AC electrical systems during the process of reactive power management. Reactive power is considered when a system contains electrical loads with nonzero electrical reactance. Such loads are almost always present in AC power systems because they arise from capacitance and inductance within the system. Two example situations in which reactive power is considered are (1) determining the reactive power required to operate an electric motor (an inductive load) and (2) determining the amount of reactive power that an electrical generator is able to produce during operation.

Mathematical Definitions

Given sinusoidal voltage and current with RMS voltage $V$, RMS current $I$, and phase difference $\varphi$ between the voltage and the current, the reactive power is $$Q=VI\sin(\varphi )$$ More explicitly, the above formula for reactive power applies when the voltage across the system as a function of time is $v(t)={\sqrt {2}}V\cos(\omega t+\phi _{v})$ and the current through the system as a function of time is $i(t)={\sqrt {2}}I\cos(\omega t+\phi _{i})$. Then the phase difference between voltage and current is $\varphi =\phi _{v}-\phi _{i}$.

Reactive power is also often considered in relation to the complex power $S$. The relationship between reactive and complex power is $Q=\mathrm {Im} (S)$.

To understand reactive power, it is necessary to contrast it with real power $P$, also known as active power. Real power corresponds to the average power delivered to a system over time and it is given by $$P=VI\cos(\varphi )$$ It can be considered “more real” than reactive power because in operation of devices such as electric motors, the mechanical work done by the motor corresponds to the the real power $P$ multiplied by the time that it is operated. In contrast, reactive power describes the amplitude of power that sinusoidally flows into and out of an electrical system at twice the frequency of the ac voltage. Thus, reactive power is only transiently stored in devices (e.g. in the magnetic field of an inductor) before being returned to the power supply. This statement is made mathematically precise by considering the instantaneous power flow $p(t)$ into an electrical system. This instantaneous power can we written as $$p(t)=P[1+\cos(2\omega t+2\phi _{v})]+Q\sin(2\omega t+2\phi _{v})$$ The term $P[1+\cos(2\omega t+2\phi _{v})]$ can be interpreted as the real power flow, as would be delivered to a purely resistive load, which averaged over time corresponds to net power delivery $P$. The term $Q\sin(2\omega t+2\phi _{v})$ can be interpreted as the reactive power flow, that would be delivered to a purely inductive or capacitive load, which delivers no net power when averaged over time.

Consuming and Generating Reactive Power

In the context of reactive power management, it is said that inductive loads “consume reactive power” ($Q>0$) and that capacitive loads “generate reactive power” ($Q<0$). This is useful terminology in the context of electrical grid operation where the typical concern is supplying power to large inductive loads such as motors. Capacitors added to the power system tend to reduce the requirements on power stations to “generate reactive power.” Physically, this occurs because the transient energy stored in the electric field of capacitive loads is converted to energy stored in the magnetic fields of inductive loads and vice versa; this is the same phenomenon that occurs in LC circuits. However, it should be noted that the terminology "consuming" and "generating" reactive power is a product of convention. For this reason, it is important to note that powering large capacitive loads is fundamentally just as difficult as powering large reactive loads. In other words, “generating a lot of reactive power” is essentially an equivalent problem to “supplying a lot of reactive power.” This is because even when $Q<0$ is a large negative number, it still means that there is a large oscillation in instantaneous power.

In a power system, the sum of the reactive power consumed by all elements within the system is equal to the total reactive power consumed by the entire system. This is the case one would hope for based on how the accounting of power consumption works. However, it is important to recognize that passive elements, namely capacitors, are said to continuously generate reactive power. This is a possible point of confusion because it is physically impossible for passive elements to generate power continuously, and yet they are said to continuously produce reactive power. This is solely a consequence of terminology. When it is recognized that reactive power indicates a power oscillation amplitude rather than net power flow, it is clear that no real power is actually generated by a capacitor.

Reactive Power as an Equivalent Description of Current and Phase

It is often the case that loads in an electrical system are supplied with power at a nominal and well regulated voltage. For example, this is the situation for power outlets within buildings. Given the formulas $Q=VI\sin(\varphi )$ and $P=VI\cos(\varphi )$, one perspective in this context is that $Q$ and $P$ are simply alternative variables for describing the amplitude and phase of the current. Considering this may be helpful in giving physical meaning to the notion of reactive power. However, one should also recognize that it is useful to work with real power and reactive power rather than currents and phases. Doing so simplifies problems of power generation to considering devices as “needing a certain amount of reactive and real power to operate.”