### Table of Contents

### Introduction

In alternating current (AC) circuits, the transmission of electrical energy and the behaviour of electric machines is often easier to understand by working with power rather than dealing with voltages and currents. Power can be defined in various ways, the applicability of the definition depending on the type of circuit.

The most common definition of power is: **energy spent per unit of time [J/s]**.

P [W] = E [J] / time [s]

In AC circuits the notion of power is a bit different, in the sense that there are three types of power:

- active power
- reactive power
- apparent power

### Active power

For a better understanding of the concept of active power, we are going to use a simple AC circuit which consists of a sinusoidal voltage source and a resistor.

In this simple circuit we can see that the current is flowing through the resistor R = 2 Ω, alternating its direction. If we assume that the AC voltage is E [V] and the current is I [A], the power through the resistor will be P [W] = EI.

When a resistor is put in an AC circuit, it will not change the phase (θ) between the voltage (E) and current (I) passing through it. With other words, the **voltage and current passing through a resistor are in phase with each other**. If we represent voltage and current passing through the resistor as vectors (phasors), they will overlap.

For a better visualisation of this concept, let’s create a simple simulation model of the circuit above using Simetrix.

In this simulation model we have a voltage source with the amplitude of 162 V and frequency of 60 Hz. The electric current will flow through a resistor of 2 Ω. The voltage, current and power on the resistor are measured and the results are plotted in the image below.

As we can see in the upper plot, the voltage and current are in phase, there is no delay between the signals. Also, the power is always positive, being the product of voltage and current, which have the same sign, either positive or negative. The power that we measure at the resistor is **active power** because it always flows from the source (voltage) to the load (resistor). In a purely resistive circuit all power through the circuit is active power.

In this case, the power is converted from one form to another, for example: from electricity to heat, from mechanical to electrical, etc. This is the definition of power commonly used in physics and it’s also known as **active power** or **true power** or **real power**.

In conclusion, **active power P**, also known as **true/real power**, is the power which is converted from one form in another (e.g. electricity to heat) and it’s measured in **Watt [W]**. The power in a purely resistive electric circuit is entirely active power.

#### Example on how to calculate active power

Calculate the active power dissipated on a resistor R = 2 Ω in an AC circuit, with the peak voltage E_{peak} = 162 V and the frequency f = 60 Hz. Visualize the voltage, current and power function of time t [s] on a plot, together with their peak and RMS values.

**Step 1**. Calculate E_{rms} = E_{peak} / √2 = 162 / √2 = 114.55 V

**Step 2**. Calculate I_{peak} = E_{peak} / R = 162 / 2 = 81 A

**Step 3**. Calculate I_{rms} = I_{peak} / √2 = 81 / √2 = 57.28 A

**Step 4**. Calculate active power P = E_{rms} ⋅ I_{rms} = 114.55 ⋅ 57.28 = 6.56 kW

**Step 5**. Visualize the voltage, current and power function of time t [s] on a plot.

### Reactive power

When an AC electric circuit contains an inductor or capacitor, additional to the resistor, the power contained in this circuit is not entirely active/true/real. This is because the inductor and capacitor can store energy as magnetic or electric fields and the release it back in the circuit as electric energy.

In the circuit below we are linking in series an AC voltage source, a resistor and an inductor.

The current is still traveling through the components, alternating its direction, but in this case there is a phase delay of 90º between the voltage and current. This phase delay is due to the inductor which is converting energy from electric to magnetic form and reverse. If we represent the voltage and current phasor for the circuit above, we get this:

In this case we can see that the **voltage is leading the current** with a phase delay of 90º.

To explain the concept of reactive power, we are going to use a simple AC electric circuit which contains an AC voltage source with the amplitude of 120 V and a frequency of 60 Hz, a 1 μΩ resistor and 5 mH inductor (see image below). The resistance is set very low in order to have a “purely” inductive circuit.

The power is measured at the inductor’s terminal and the voltage across it. The current measurement is done before the resistor but it’s the same for both components, being connected in series. This circuit is simulated in Simetrix and the results shown below.

There are few thing to observe from the simulation results. First, we can see that the current is lagging behind the voltage, with a phase delay of 90º. Also, the power at the inductor changes sign, being either positive or negative. The change in sign means that the power if flowing from the voltage source to the inductor (positive power) and from the inductor towards the voltage source (negative power). The power displayed in the plot is purely **reactive**, which means that it’s not dissipated as heat at all. Reactive power is also know as **imaginary power**.

This change in power sign is due to the behaviour of the inductor, which charges with energy from the circuit and then discharges the same energy back in the circuit. You can say that the power is “recycled” and is not used to produce heat or mechanical work.

The circuit above helps explaining how the inductor behaves and how the power is converted from electric to magnetic form and back. The circuit is divided into two parts, a charging circuit (left) and discharging circuit (right). The activation/deactivation of each circuit is done through a switch S. The charging/discharging action is related to the inductor. When the inductor is charging, the power from the voltage source (12 V) is split into two parts: an active/real power, which is being dissipated by the resistor R_{1} = 140 Ω and a reactive/imaginary power which is being stored in the inductor L = 3H.

When the switch S is opening the charging circuit, it is also closing the discharging circuit. In this state, the energy stored in the inductor is discharged through the resistor R_{2} = 140 Ω. In the discharge phase all the electrical power in the discharge circuit is active power due to the fact that it passes through the resistor and the inductor is loosing its stored energy.

The power which flows back and forth between the source and the load is called **reactive or imaginary power**. The symbol for reactive power is **Q** and it’s also calculated by the product between voltage and current, but the unit of measurement is **Volt Ampere Reactive [VAR]**.

Due to the increase and reduction of magnetic field (inductor) or electric field (capacitor), reactive power (Q) takes power away from the AC circuit, which makes it more difficult for the active power (P) to be supplied directly to a circuit or load.

#### Example on how to calculate reactive power

Calculate the reactive power in an inductor L = 5 mH in an AC circuit, with the peak voltage E_{peak} = 162 V and the frequency f = 60 Hz.

**Step 1**. Calculate E_{rms} = E_{peak} / √2 = 162 / √2 = 114.55 V

**Step 2**. Calculate the inductive reactance X_{L} = 2 ⋅ π ⋅ f ⋅ L = 2 ⋅ π ⋅ 60 ⋅ 5⋅10^{-3} = 1.885 Ω

**Step 3**. Calculate I_{peak} = E_{peak} / X_{L} = 162 / 1.885 = 85.994 A

**Step 4**. Calculate I_{rms} = I_{peak} / √2 = 85.994 / √2 = 60.771 A

**Step 5**. Calculate reactive power Q = E_{rms} ⋅ I_{rms} = 114.55 ⋅ 60.771 = 6.9614 VAR

### Summary

The main differences between active and reactive power are summarized in the table below.

Active Power | Reactive Power |

It’s the real power used/consumed/dissipated by the circuit/load | It’s the power flowing back and forward from the source into the circuit, without doing any useful work |

It’s always positive, does not change direction | Can be positive or negative, changes direction periodically |

Flows only from source to load/circuit | Flows from source to load or reverse, from load to source |

Symbolized by the letter P and measured in Watts, P [W] | Symbolized by the letter Q and measured in Volt-Ampere-Reactive, Q [VAR] |

Measured in a real circuit with a Wattmeter | Measured in a real circuit with a VARmeter |

Generates useful work like, mechanical power, heat or light | Does not generate useful work, only oscillates back and forth between source and load/circuit |

Depends on the dissipative elements of the circuit (resistance) | Depends on the inductive or capacitive elements of the circuit (reactance) |

It’s maximum in a purely resistive circuit | It’s maximum in a purely inductive or capacitive circuit |

### References

[1] Theodore Wildi, Electrical Machines, Drives and Power Systems, 6th Edition, Pearson, 2005.

[2] Stan Gibilisco, Teach Yourself Electricity and Electronics, 3rd Edition, McGraw-Hill, 2001.