Power in AC Circuit

What is an Alternating Current?

Based on the direction of current flowing through the circuit, it is differentiated in two types. One is Alternating current and another is Direct current. When an electric current, reverses its direction periodically while flowing through an electric circuit is called Alternating Current (AC). On the other hand, when current flows in only one direction is known as Direct current (DC).

The major advantage of alternating current is that AC voltages can be easily transformed from higher to lower voltage levels and vice-versa. Due to this virtue, high voltage power from power stations can be reduced to a safer voltage for domestic use. Only Alternating current is compatible with capacitors and inductors. By using them within the Alternating current circuits, the flow of electricity can be turned. This property helps tune the radio stations. Because of these reasons, AC electricity is most preferred for home appliances.


Power Consumed in an AC circuit

An electric circuit produces power which is given by the expression, P= I V.

Where, I – the current flowing through the circuit and

V- Voltage across it.

AC circuits always offer reactance, therefore there are two components of power, power component because of the magnetic field and another is because of the electric field. The average power absorbed by the circuit becomes the sum of power stored and returned through complete one cycle. Thus, the average power consumed by the circuit will be the instantaneous power within one cycle.

As the current flowing through the circuit and voltage are dependent on time, the instantaneous power is also dependent on time t. It is given by,

 p (t) = I (t) x v (t). 

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Consider the LCR circuit as shown in the figure, voltage V is applied to the circuit. The voltage V is given by,

V=Vm× sinωt

The current in this case is written by:

I=Im × sin(ωt+Φ )

Where,   Vm - amplitude of Voltage

               Im -  Amplitude of current

ω- Angular frequency.

Φ - Phase constant

Current amplitude is given as, Im=Vm/Z

Where, Z- impedance of the circuit and it is calculated by 

Z = \[\sqrt{(X_{L} - X_{C})^{2} + R^{2}}\]

\[Tan\Phi\] = \[\frac{X_{L} - X_{C}}{R}\] 

\[\Phi = arctan\frac{x_{L} - x_{C}}{R}\] 

Using all above equations to find the consumption of power in an AC circuit,

We know, P = IV

P = (\[V_{m}\]  × sin⍵t) × (\[1_{m}\] × sin(⍵t + \[\Phi\])) 

To find the average value of power

\[P_{avg}\] = (\[V_{m}\]  × sin⍵t) × (\[1_{m}\] × sin(⍵t + \[\Phi\])) 

Using The Trigonometric Formula,

2 sin A sin B = cos (A- B) - cos (A +B)

We get,

\[P_{avg} = Average\: of\: \frac{V_{m} I_{m}[cos\Phi\: -\: cos(2\omega t + \Phi)] }{2}\]

\[P_{avg} = \frac{V_{m} I_{m}cos\Phi}{2}\]

\[Average\: of\:  \frac{V_{m} I_{m} cos(2\omega t + \Phi) }{2}\] = 0

\[P_{avg} = \frac{V_{m} I_{m}}{2} cos\Phi\]


\[P_{avg} = V_{rms} ×  I_{rms} × cos\Phi\] 

For a resistive circuit, Φ=0 , which results in cosΦ = 1

Hence, Pavg = Vrms x Irms

For inductive circuit, Φ= 900 , (Voltage across the inductor leads the current by 900)

so, cosΦ = 0

Hence, Pavg = 0

For a capacitive circuit,Φ = -900. It implies, cosΦ= 0 (It means, in case of capacitors, voltage lags the current by 900)

Hence, Pavg= 0

The average power of an AC circuit is called the true power of the circuit.


Power Factor

  • The power factor of an alternating current is defined as the ratio of the true power flowing through the circuit to the apparent power present in the circuit. 

  • It is usually in the interval of -1 to 1 and is dimensionless.

Power Factor = True Power/ Apparent power

Also, cosΦ = R/Z

R- resistance in the circuit

Z- impedance of the circuit.


Fun Facts

  • Ohm’s law for the RMS value of an alternating current is calculated by dividing the RMS voltage by the impedance.

  • The average power delivered to a LCR circuit varies with the phase angle.


Solved Examples

1. An AC generator whose emf is given by, v(t)= [4.00V] sin[(1.00 x 104 rad/s) t],

 is connected to a LCR circuit for which L=2.00 x 10-3H, C= 4.00 x 10-6 F and R= 5Ω

  1. What is the RMS voltage across the generator?

  2. What is the impedance of the LCR circuit?

  3. What is the average power output of the generator?


Solution-

The RMS voltage across the generator is 1/√2 times the amplitude of voltage.

Hence, Vrms = 1/√2 x (4.00V) = 2.83 V

The impedance of circuit is given by,

 \[Z = \sqrt{r^{2} + (z_{r} - z_{c})^{2}}\]

= 52+ {[1 x 104 x 2 x 10-3] - [1/ (1 x 104) x (4 x 10-6)]}

= 7.07Ω

The average power output of the generator is given as

\[P_{avg} = \frac{V_{rms}^{2} R}{Z^{2}}\] = \[\frac{2.83^{2} × 5}{7.07^{2}}\] = 0.801W

2. Two loads of magnitude 10KW each, are operating with a power factor 0.8 (each of them is lagging). What is the combined power factor for both loads?


Solution- We know, Apparent power = True power / power factor

For load 1

Apparent power = 10 KW/ 0.8

Apparent power =  12.5 KVA

For load 2

Apparent power = 12.5 KVA

Combined Power factor =  Total true power / total Apparent power

= (10+10) / (12.5+12.5)

Combined power factor = 0.8 (Lags)

FAQ (Frequently Asked Questions)

1. How is an Alternating current produced?

  • Ans- Various sources of electricity such as electromechanical generators, produce current. 

  • The current is produced with voltages such that it alternates in polarity, reversing between positive and negative over time. 

  • Many times, an alternator is used to generate alternating current.

  • When a magnetic field generates in an alternator, a loop of wire is rapidly inserted to produce an electric current. 

  • As the wire starts spinning, it attains polarity, and consequently voltage and current get induced. The generated current can change the direction. 

  • As the current reverses its direction, the voltage too changes the direction accordingly. 

  • If an AC circuit is connected to an oscilloscope and the behaviour is plotted against the time, then several waveforms (sine, triangle and square) are observed. Most buildings are wired with an oscillating voltage in the sine wave form 

2. What are the applications of an Alternating current?

  • Ans- Generation and transportation of alternating current over a long distance is comparatively easy, that is why Alternating current is mostly used in mains-wired buildings (offices, residential complexes).

  • At high voltages less energy is utilized in power transmission. At high voltages, lower currents are produced resulting in less heat in transmission lines as they have comparatively less resistance. 

  • Alternating currents can be easily converted to high voltage from low voltage and vice-versa by using transformers.

  • Alternating current is of great use in electric motors. These motors are used in home appliances such as refrigerators, washing machines, air conditioners.