  # Power and Power Factor in AC Circuits

By BYJU'S Exam Prep

Updated on: September 25th, 2023 Power and power factor are fundamental concepts in AC circuits that play a crucial role in understanding and analyzing electrical systems. Power represents the rate at which energy is transferred or consumed in an electrical circuit, while power factor quantifies the efficiency of power utilization. By comprehending these concepts, engineers, and electricians can optimize energy usage, improve system performance, and ensure the smooth operation of electrical equipment.

The power factor is a measure of how effectively the circuit utilizes real power. It is defined as the ratio of real power to the product of the voltage and current in an AC circuit. The power factor is expressed as a value between 0 and 1, where a power factor of 1 indicates efficient utilization of real power, while a power factor closer to 0 implies a significant presence of reactive power and inefficient power consumption. In this article, Byjus Exam Prep will discuss deeper into the concepts of power and power factor in AC circuits, exploring their significance, calculation methods, and practical implications.

Table of content ## What Does Mean Power and Power Factor in AC Circuits?

Understanding power and power factor is crucial in various applications, such as power transmission, industrial machinery, and residential electrical systems. By analyzing and optimizing power factors, engineers can ensure stable voltage levels, prevent equipment damage, and minimize energy costs. A high power factor is desirable in electrical systems as it minimizes losses, reduces energy wastage, and improves the overall efficiency of the circuit. Power factor correction techniques are often employed to compensate for reactive power, leading to improved power factors and enhanced system performance.

In AC circuits, power has two components: real power (P) and reactive power (Q). Real power is responsible for performing useful work, such as providing mechanical energy or producing light, while reactive power is associated with the exchange of energy between capacitive and inductive elements within the circuit. Both real and reactive power contribute to the total power consumed by a load.

## Different Types of Power in AC Circuit

### Instantaneous Power

It is used for the special case of steady-state sinusoidal signals.

Instantaneous power supplied to Impedance: p(t) = v(t) i(t) = i2(t) R • v(t) = Vm sin(ωt), where the radian frequency in radians per second ω = 2πf and the frequency in Hertz f = 1 T and T is the period of the sine wave.

### Average Power

Power absorbed or supplied during one cycle. For sinusoidal (and other periodic signals) we can compute averages over one period. If voltage and current are in phase:

If voltage and current are in quadrature:

### Apparent Power

The product of rms voltage (V) and rms current (I) is called apparent power and is denoted by S. It is measured in volt-ampere (VA).

S=VI

### Reactive Power

The product of applied voltage and the reactive component of current (I sin φ) is called reactive power and is denoted by Q. It is measured in reactive volt-ampere.

Q=VI sin φ

### Active Power

While the power is called active power, true power or real power. The unit of Active or Real power is Watt where 1W = 1VA. Active power is denoted by P.

P = VI Cos φ

### Complex Power

It helps in measure of power using phasors.

###  Power Factor (cos φ)

It is defined as the ratio of true power to apparent power. It is a measure of the angle between current and voltage phasors.  Apparent Power is measured in VA, Reactive Power is measured in VAR, and True Power is measured in Watts. These three types of power are trigonometrically related to one another. In the above right triangle, P = adjacent length, Q = opposite length, and S = hypotenuse length. The opposite angle is equal to the circuit’s impedance (Z) phase angle. The numerical value of the cosine of the phase angle between the applied voltage and the current drawn from the supply voltage gives the power factor.
Cos φ = R/Z
• The nature of the power factor is always determined by the position of the current with respect to the voltage.
• For the pure inductor, the power factor is cos 90o (zero lagging)
• For pure capacitors, the power factor is cos 90o (zero but leading).
• For a purely resistive circuit voltage and current are in phase i.e., φ = 0 [cos 0 = 1]. Such a circuit is called a unity power factor circuit.

## Examples Related to Power Factor in AC Circuits

Example-1: What is the resistance of a light bulb that uses an average power of 75.0 W when connected to a 60 Hz power source with a peak voltage of 170 V? Solution:

PAvg = (VRMS) 2 / R = (VPeak) 2 / (2R)
R = (VPeak) 2 / (2 PAvg) = (170)2 / (2 * 75) = 193 Ohms

Example-2: An inductor has a 54.0 Ohm reactance at 60 Hz. What will be the maximum current if this inductor is connected to a 50 Hz source that produces 100 V rms.
Solution:

XL = ω L
XL 50 Hz  = 2 π (50) L
XL 60 Hz  = 2 π (60) L
XL 50 Hz / XL 60 Hz =  2 π (50) L / ( 2 π (60) L ) = 5/6
XL 50 Hz=  5/6 (XL 60 Hz) = 5/6 (54) = 45 Ohms
I Peak 50 Hz  = VPeak / XL 50 Hz = (1.41) (100) /45 =3.13 A

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