Power in AC Circuits

Currents are categorised into alternating current (AC) and direct current (DC), based on the direction of currents that flow through a circuit. An electric current is called AC when it periodically flows in its reverse direction. In contrast, when the current flows in one direction, it is known as DC. AC voltage can easily be transformed from higher to lower voltage and vice versa. Considering this property of AC, the voltage power from the power station can be reduced to make it safer for domestic appliances. Only AC can be used with capacitors and inductors to turn the flow of electricity in another direction. These properties of AC help tune the radio station. Thus, AC is mostly preferred for home appliances. 

Power in AC circuits

The AC circuits always exhibit reactance. The average power absorbed by the circuit becomes the total power stored and is returned through a complete cycle. Thus, the average power consumed by the circuit is 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)

Consider the LCR circuit; 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 the voltage

Im – Amplitude of current

ω – Angular frequency

Φ – Phase constant

Current amplitude is given as Im = Vm/Z

Where Z is the impedance of the circuit, calculated by

Z = √[ R2+(XL−XC)2]

Here, XL is the inductor’s impedance, and XC is the capacitor’s impedance. 

tanΦ= (XL−XC)/R

or

Φ=tan-1 (XL− XC)/R

All the above equations can be used to find power consumption in an AC circuit.

We know, P = IV

P = (Vm × sin⍵t) × (Im × sin(⍵t + Φ)) 

To find the average value of power in ac circuits, 

Pavg = (Vm × sin⍵t) × (Im × sin(⍵t + Φ))

Wattless Currents

Wattless current is also known as idle current. The current involved in an AC circuit is known as wattless current as the average power consumed in this circuit corresponds to zero.

Wattless current is generated only in an electric circuit with a single capacitor or inductor because, in inductive and capacitive electrical AC circuits, both the voltage and current differ by a phase angle of 90° or π/2.

There is no formula for calculating wattless current as it involves a single capacitor or inductor. However, the presence of the wattless currents can be verified by a satisfying condition indicating that the total energy consumption in a particular circuit is equal to zero. To confirm the presence of wattless current in a circuit, one must know the place difference between the two sinusoidal waves. 

The formula for wattless current is

P = IVsinӨ

where,

P – power in Watts.

I – current in Amperes.

V – voltage in Volts.

Ө – angular phase difference between the current and voltage 

Quality factor

In a resonant circuit, the tuning or sharpness at resonance is called the Quality factor or Q-factor. In the resonant circuit, the quality factor is the ratio of the voltage across the inductor or capacitor to the voltage applied.

Quality factor = voltage across inductor or capacitor / applied voltage —(1)

We know that the voltage across L = Iꞷ0L —(2)

In the above equation, ꞷ0 is the angular frequency of the ac at resonance. At the resonance, the applied voltage is the potential drop across the resistance. But, across inductance(L), the potential drop is the drop across capacitance. They are 180o out of phase. The potential reductions cancel out, and the only potential drop across resistance will exist.

Applied voltage = IR —(3)

We can substitute equations (2) and (3) in equation (1),

Q = Iꞷ0L / IR

Q = ꞷ0L /R

The quality factor is also defined as the ratio of reactive power of the capacitor or inductor to the average power of the resistor at resonance.

Quality factor = reactive power of capacitor or inductor / average power of the resistor

In a pure inductive circuit, the power is reactance XL at resonance,

Quality factor = I2XL / I2R

= ꞷrL/ R

For the capacitance reactance XC at resonance,

Quality factor = I2XC / I2R

= 1 / ꞷr RC

Power Factor in AC circuit

The ratio of the genuine power flowing through the circuit to the apparent power existing in the circuit is known as the Power Factor in an AC circuit.

It normally falls between -1 and 1 and is dimensionless.

power factor of an ac circuit is equal to = True Power/ Apparent power

Also, cosΦ = R/Z

R- resistance in the circuit

Z- impedance of the circuit.

Important points

• Divide the RMS voltage by the impedance to get Ohm’s law for the RMS value of an alternating current.

• The phase angle affects the average power delivered to an LCR circuit.

Conclusion

There is an opposition to the current flow known as resistance in AC and DC circuits, measured in ohms. The resistance offered to an alternating current circuit is called AC resistance. AC resistance is always more than DC resistance in conductors. Due to this, the effective area of the conductor is reduced as its resistance starts increasing. The resistance that occurs in an AC is known as impedance.

Thus, the AC’s magnitude and direction modify periodically, similar to the alternating voltage. The sinusoidal wave of the AC is essential. It is economical and transmits less waste energy. Based on the principles of mutual induction, AC easily transforms into other types of currents. Thus, it avoids power loss during transmission. Due to this property, AC is mainly used in household appliances as it is safer than other currents.