## What is a complex form?

The complex form is a topic related to power engineering. The complex form of power is the representation of electric power in the form of a complex number. The complex number consists of a real and imaginary part, in the same way, the complex form of power also consists of a real and an imaginary part. The real part of the complex form of power represents the active power, whereas the imaginary part of the complex form of power represents the reactive power. Considering the power as $S$, the active power as $P$, and the reactive power as $Q$, the equation for the complex form of power is written as-

Where $i$ is an imaginary number.

## Fundamentals of power system

A power system is defined as the connection or network of the various components that convert the non-electrical energy into the electric form and supply the electric form of energy from the source to the load. The power system is an important parameter in power engineering and the electrical engineering profession. The powers in the power system are primarily categorized into two types- active power and reactive power.

## Active power

The power which is actually utilized in a circuit is known as active power. Active power is also known as true power, real power, watt-full power, and useful power. The loads that utilize the active power are transformers, induction motors, generators, and so on.

### Active power in direct current (DC) circuits

The active power in the DC circuits is the product of the voltage across the circuit load and the current flowing through it. Hence, the formula for active power in a DC circuit is given as-

Where $P$ is the active power, $V$ is the voltage across the load, and $I$ is the current flowing through the circuit.

### Active power in the alternating current (AC) circuits

The active power in the alternating current circuits is different than that of the direct current circuits because the AC circuits consist of the phase differences between the current and voltage due to the sinusoidal nature of the current flowing through the circuit. Hence, the formula for active power in an AC circuit is given as-

For a three-phase AC system, the formula for active power is given as-

Where $P$ is the active power, $V$ is the line voltage across the load, and $I$ is the line current flowing through the circuit and $\mathrm{cos}\theta$ is the power factor.

## Reactive power

The power which flows back and forth in a circuit between the load and the source is known as reactive power. The reactive power is also known as useless power or wattless power. The reactive power is of no use for the consumer and the circuit and it creates an extra burden on the system supplying the electricity. The reactive power is useful for the reactive components of a circuit such as the inductors and capacitors. The reactive power is absorbed by the inductors and capacitors and utilized to create the magnetic field and electrostatic field. The magnetic field is created by the inductors and the electrostatic field is created by the capacitors. The reactive power is not present in the case of a DC circuit due to the absence of phase angle $\theta$. In the case of AC circuits, the reactive power is present and is given using the following formula-

Where $Q$ is the reactive power.

The value of reactive power is positive in the case of the inductive loads and the value of reactive power is negative in the case of the capacitive loads.

## Apparent power

Apparent power is the total of the active power and reactive power. Mathematically, the apparent power is given as the product of the root mean square values of the voltage and the current. The relationship between active power, reactive power, and apparent power is geometrically given in the form of a phasor diagram known as the power triangle. The voltage is considered as the reference phasor in the phasor diagram. The relationship between active power, reactive power, and apparent power is given as-

Where ${S}^{\text{'}}$ is the apparent power, $P$ is the active power, and $Q$ is the reactive power.

In the case of a purely resistive circuit, the apparent power is equal to the real power or active power. In the case of a purely inductive or purely capacitive circuit, the apparent power is greater than the real power or active power.

## Importance of the complex form of power

Following are the importance of using the complex form of power:

• The complex form of power is distributed in the active power and reactive power as the real part and imaginary part respectively. Hence, it is easier to identify the two powers.
• The power triangle of the system can be easily constructed when the values of all components in the complex form of power are known. The real part ($P$) or the active power will represent the base, the imaginary part ($Q$) or the reactive power will represent the perpendicular, and the hypotenuse ($S$) or the power will represent the apparent power of the system.
• The magnitude of the complex form of power will give the value of apparent power. Hence, the apparent power can be obtained as-

## Context and Applications

The complex form is useful for the students undergoing the following courses:

• Bachelors in Technology (Electrical Engineering)
• Masters in Technology (Electrical Engineering)
• Masters in Technology (Power System and Power Electronics)

## Practice Problems

Q1. What does the real part of the complex form of the power represent?

1. Active power
2. Reactive power
3. Passive power
4. Resistance

Answer: Option a

Explanation: The real part of the complex form of the power represents the Active power.

Q2. What does the imaginary part of the complex form of the power represent?

1. Active power
2. Reactive power
3. Passive power
4. Resistance

Answer: Option b

Explanation: The imaginary part of the complex form of the power represents the reactive power.

Q3. Which of the following is not an alternative name for active power?

1. True power
2. Real power
3. Watt-full power
4. Wattless power

Answer: Option d

Explanation: Wattless power is not an alternative name for active power.

Q4. Which of the following is also known as useless power?

1. Active power
2. Reactive power
3. Real power
4. Watt-full power

Answer: Option b

Explanation: Reactive power is also known as useless power.

Q5. In which of the following cases, the apparent power is equal to the active power?

1. In the case of a purely resistive circuit
2. In the case of a purely inductive circuit
3. In the case of a purely conductive circuit
4. In the case of a purely capacitive circuit

Answer: Option a

Explanation: In the case of a purely resistive circuit, the apparent power is equal to the active power.

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