Power Factor in AC circuit

The power factor is determined by the cosine of the phase angle between voltage and current. In AC circuits, the phase angle between voltage and current is aligned, or in other words, zero. But, practically there exists some phase difference between voltage and current. The value of the power factor always lies between 0 and 1. For a purely capacitive circuit, it is 0 and for the purely resistive circuit, it is 1. whereas, there is no power factor in DC circuits due to zero frequency. It is used to measure how effectively the incoming power is used in an electrical system(energy efficiency).

Power Factor

Power factor is defined as the ratio of real power to the apparent power.

The formula for the power factor is given below :

Power Factor = cosФ = Real power/Apparent power

Real Power – The capacity of the electricity for performing work. It is also known as true power, active power, and working power. It is measured in kilowatts(kW).

Apparent Power – The combination of real power and reactive power. It is also known as total power. It is measured in kilovolt amperes(kVA).

Different Types of Power in AC Circuit

Instantaneous Power: It is defined as the product of voltage across the element and instantaneous current through the element at any instant of time. It is denoted by the letter P. The rate at which an element absorbs energy (in watts).

P=V x I

Power Factor Graph

where, V and I are the instantaneous voltage and current. Since

    V=V msinωt
    I=I m sin(ωt-Ф).
Then the instantaneous power at any time t can be expressed as
    P=V m I msinωtsin(ωt-Ф)
By using trigonometric identity,
     sin(ωt-Ф)=sinωtcosФ-cosωtsinФ
The power becomes:
      P instantaneous = V m I m cosФsin2ωt - V m I m sinФsinωt cosωt 
Averaging this power over a complete cycle gives the average power.

Average Power: It can be calculated by calculating the instantaneous power of the circuit. As the instantaneous power continuously fluctuates over time, calculating the average requires integration. By averaging over a single period (T) of the sinusoidal function, the average power can be determined. The second term in the expression averages to zero. It is an odd function of t. The average of the first term is given by

P average = V m I m cosФ ∫T0 sin2ωtdt  ∕  T
               =V m I m cosФ ∕  2 
Since the rms voltage and current are given by 
V = V m  ∕  √2  and I = I m  ∕  √2 
The average power can be expressed as 
P average = VI cosФ

Apparent Power: The product of the rms voltage and rms current is called apparent power. The apparent power is the same as the real power when the impedance is a pure resistance. The apparent power is greater than real power when reactance exists. It is measured in kilovolt amperes(kVA). It is denoted by letter S.

Reactive Power: The power that corresponds to storage and retrieval of energy rather than consumption. It is measured in kilovolt-Ampere reactive (k VAR).It is denoted by letter Q.

Power triangle represents the real power, reactive power and apparent power of the AC circuit in the right-angle triangle. It shows the relationship between all three powers.

Power triangle

Real power, P = VI cosФ

Reactive power, Q = VI sinФ

Apparent power, S = VI [S² = P² + Q² ]

Complex Power: It is the vector sum of real power and reactive power.

S = P + Q j

Problem 1. Given a circuit with an apparent power of 600VA and a real power of 300W, calculate the power factor.

Given,
Real power = 300W
Apparent power = 600VA
To find power factor ,
Power Factor =  Real power/Apparent power
                         = 300 / 600
                         = 0.5

Problem 2. In a circuit, the real power is 12W and the reactive power is 5VAR. A Calculate the apparent power and power factor.

Given,
Real power (P) = 12W
Reactive power (Q) = 5VAR
To find the Apparent power(S), by using the relationship between real power, reactive power and apparent power.
S2 = P2 + Q2 
S2 = 122 + 52
S2 = 169
S = 13VA[Apparent power]
Now, to find the power factor:
Power Factor = Real power/Apparent power
                         = 12 / 13 
                         = 0.923
So, the Apparent power is 13VA and the Power factor is 0.923.

Problem 3. In a circuit, the apparent power is 5VA and the reactive power is 4VAR. Calculate the power factor of the circuit.

Given,
Apparent power(S)= 5VA
Reactive power(Q)= 4VAR
To find Power factor,
Power Factor = Real power/Apparent power
we need the real power to calculate power factor. By using relationship between real power, reactive power and apparent power.
P2 = S2 - Q2
P2 = 52 - 42
P2 = 9
P = 3W[Real power]
Now , we can find power factor
Power Factor = Real power/Apparent power
                         = 3 / 5
                         = 0.6
The power factor of the circuit is 0.6.

Mathematical Analysis

Suppose a voltage V is applied to a LCR circuit, where V is given by :

V = V m sinωt
The current in this case is written by:
I = I m sin(ωt + Ф)
where, 
V m = Voltage Amplitude
I m  = Current Amplitude
 ω   = Angular Frequency
 Ф   = Phase Constant
Now, Current Amplitude is related to voltage Amplitude as:
I m = V m  ∕  Z
where , Z = Impedance of circuit and given by:
Z = √(X L - X  C )2 + R2 

tanФ = (X L - X  C ) ∕  R

Ф = tan-1(X L - X C ) ∕  R
where,
XL is the inductive reactance(2ЛfL)
XC is the capacitive reactance(1/2ЛfC)
f is the frequency of the AC source.

Problem : In a LCR circuit, the inductance(L) is 0.2 Henry, capacitance(C) is 200 microfarads and the resistance(R) is 30 ohms. The AC source has a frequency of 50Hz.

Given,
Inductance(L) = 0.2 henry
Capacitance(C) = 200 microfarads
Resistance(R) = 30 ohms
Frequency(f) = 50Hz

1. Calculate inductive reactance(XL),
XL= 2ЛfL
XL= (2Л)(50)(0.2)
XL≈ 62.83Ω

2. Calculate capacitance reactance(XC),
XC =1/2ЛfC
XC =1/(2Л)(50)(200x10-6)
XC ≈ 15.91Ω

3. Determine the phase angle(Ф),
Ф = tan-1(X L - X C ) ∕  R
Ф = tan-1(62.83 - 15.91 ) ∕  30
Ф ≈ 1.0019 radians

4. Calculate the power factor(PF),
PF = cos(Ф)
PF = cos(1.0019)
PF ≈ 0.999

So, the power factor of the LCR circuit is approximately 0.999.

Power Consumption in AC Circuit

An electric circuit produces power which is P = V I.

where,

V is voltage across it and

I is the current flowing through the circuit

The Alternating Current(AC) circuits always offer reactance, The power in a circuit has two components, one due to the electric field and the other due to the magnetic field. The average power absorbed by the circuit results from the sum of power stored and returned throughout a complete cycle. Consequently, the average power consumed by the circuit is equivalent to the instantaneous power within a single cycle.

Conclusion

Electrical power is the rate at which energy is consumed inside a circuit. All electrical and electronic devices have a maximum amount of electrical power they can safely handle. In AC circuits, three categories of abilities are there Real power, Reactive power and Apparent power.

Real power is denoted by the letter p and calculated in kilowatts. Whereas reactive power is denoted by letter Q and calculated in kilovolt – ampere reactive. Apparent power is denoted by letter S and calculated in kilovolt – ampere.

A power triangle is the relationship between real power, reactive power and apparent power.

FAQs on Power Factor

What is the difference between real power and reactive power?

Real power is the portion of power in an AC circuit that is converted into useful work, such as generating heat, light, or mechanical energy. Reactive power is the power oscillating between the source and reactive components(capacitors or inductors) in the circuit.

How can power factor be improved?

It can be improved by adding power factors correction devices such as capacitors, to the electrical system. These devices counteract the reactive power in the circuit, bringing the power factor closer to unity.

What is a Power Triangle?

The Power triangle shows the relationship between Real power, Reactive power and Apparent power and there are denoted by the base, perpendicular and hypotenuse of the right – angled triangle.