# Formula for Calculating Wattless Current in AC

## What is the Formula Used For Calculating Wattless Current in AC?

The electric current in an AC circuit is said to be Wattless Current when the average power consumed or dissipated in the given electrical circuit is equal to zero or null. The wattless current is also known as the idle current. The generation of wattless current essentially happens in purely inductive or capacitive electrical circuits only. In other words, the wattless current is produced only in an electrical circuit with a single capacitor or inductor.

The reason behind this is that, in both inductive and capacitive electrical AC circuits, the voltage and current differ by a phase angle of 900 or π/2. We know that the average power depends upon the cosine component of the AC signal, which eventually becomes zero as the angle between them is ninety.

## Wattless Current

Let us have a look at the wattless current and the definition of wattless current. When in any electrical circuit, to be more precisely in any AC electrical circuit only an inductor or capacitor is connected, then the total power consumption will be zero. The current flowing in such an electrical circuit in which the power consumption is zero is known as the wattless current. Such electrical current will not consume any energy from the circuit.

Theoretically, we can say that the generation of wattless current occurs as it flows along the direction of voltage or sometimes completely against it, causing net work done zero. As the net work done approaches zero, the power will also start approaching zero, further which will end up resulting in a Wattless current.

### What is Wattless Current?

Then, what is the formula used for calculating wattless current in AC circuits? And what is wattless current?

There is no particular formula for the wattless current. The wattless current is a condition or special case of any AC circuit with only a capacitor or inductor. But we can verify the presence of wattless current by satisfying the condition, which says the total power consumption must be equal to zero. This condition can be verified, which requires knowledge of the phase difference between the two sinusoidal waves. Let us have a look at the derivation of the wattless current with the help of the power equation.

### Derivation:

The current in an electrical AC circuit is said to be Wattles Current when in any electrical circuit, to be more precisely in any AC electrical circuit consisting of only an inductor or capacitor, and the total power consumption is zero. Wattles’s current is also known as Idle Current. Or in other words, in any electrical AC circuit containing either a capacitor or inductor, then the current flowing in the circuit is said to wattles if the average power consumed in the circuit is zero. The formula used for calculating the wattless current is given by:

P= V I cos ϕ……..(1)

Where,

V-The voltage applied to the circuit

I-The current flowing in the circuit

ϕ-The phase difference between the voltage and current

When the electrical circuit is consisting of only a capacitor or only an inductor then the phase difference between the voltage and current is π/2.Then equation (1) becomes:

P= V I cos  π/2

P=V I (0)

P=0…..(2)

From equation (2) it is clear that the power consumed in an AC circuit with only an inductor or capacitor is zero. This condition is satisfied when the phase difference between the voltage and current is ninety. The current flowing in such a circuit is known as the wattless current.

Here, one important point to note is that when an AC circuit consists of only a capacitor it will lead by π/2 phase angle and at the same time when an AC circuit consists of only an inductor it will lag by π/2 phase angle.

### Examples:

1. The RMS Current In An AC Circuit is 2A. If The Wattless Current is √3A,Then Calculate The Corresponding Power Factor of The Circuit?

Sol:

Given,

The RMS current in the AC circuit = Irms = 2 A

The wattless current in the circuit = Iwatt = √3 A

We are asked to determine the power factor of the circuit. We know that in order to attain the wattless current the phase angle between the current and voltage must be equal to π/2. Here we know that the given current is wattless current, but we want to find the power factor.

The Irms current can be split into two components as Irms cos ϕand the Irms sin ϕ.The cosine component of the Irms is known as the power factor. Mathematically, we write:

Power factor = cos ϕ

P. F =Cos ϕ

Now, the Irms sin ϕcomponent is perpendicular to the voltage, then the sine component of the Irms is the wattless current. Therefore we write:

Iw.l = I sin ϕ

√3 = 2 sin ϕ

sin ϕ = √3/2

ϕ = π/3 or 600

Therefore, the power factor is given by:

P. F = Cos 60 = 1/2

Hence, the power factor in a given AC circuit is 0.5.

2. What is Meant By The Wattless Current? Show That an Ideal Inductor Does Not Consume Any Power In An AC Circuit.

Sol:

The wattless current is defined as the current flowing in an electrical AC circuit such that the circuit consists of either a capacitor or inductor, then the current flowing in the circuit is said to be wattless if the average power consumed in the circuit is zero. The formula used for calculating the wattless current or wattless current formula is given by:

P= V I cos ϕ……..(1)

Where,

V-The voltage applied to the circuit

I-The current flowing in the circuit

ϕ-The phase difference between the voltage and current

When the electrical circuit is consisting of only an ideal inductor then the phase difference between the voltage and Irms current is π/2. The Irms current can be split into two components as Irms cos ϕ and the Irms sin ϕ. Both the components of the RMS current play individual roles according to the need. For now, we assume that the cosine component of the RMS only contributes to the power, thus it is considered the only cosine as the wattless component of current. Then equation (1) becomes:

P= V I cos π/2

P=V I (0)

P=0…..(2)

From equation (2) it is clear that the power consumed in an AC circuit with only an ideal inductor is zero. Hence proved that an ideal inductor does not consume any power in an AC circuit.