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in Alternating Current by (61.1k points)

Establish a relation for power in an alternating current circuit. What will be effect in formula, when there is no any reactance and resistance in the circuit? Define power factor.

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Average Power in A.C. Circuit:

The power of a circuit is defined as rate of energy transfer between the flowing charge and the circuit. In dc circuits, power is given by the product of voltage and current.

However in an ac circuit both voltage and current keep on varying with time. Thus for an ac circuit, the instantaneous power is the product of the instantaneous voltage and instantaneous current.

For an ac circuit the voltage and current at any instant is given by
V = Vm sinωt
I = Im sin(ωt + ϕ)
when ϕ is phase between the voltage and the current
∴ The instantaneous power is given by
P = VI = Vm sinωt × [Im sin(ωt + ϕ)]
= VmIm sinωt [sinωt cosϕ> + cosωt sinϕ]
= VmIm [sin2 ωt cosϕ + sin 2ωt cos cot sinϕ]
\(\frac{V_{m} I_{m}}{2}\) [(1 – cos2ωt) cosϕ + sin2ωt sinϕ]

If the instantaneous power is assumed to remain constant for a small time dt then find out the work done during this time,
dW = pdt
∴ Work done over a complete cycle

Thus the average power dissipated depends upon the voltage current and cosine of the phase angle between them.

1. Resistive circuit: If the circuit contains only pure R, voltage and current are in same phase i.e., ϕ = 0 or cosϕ= 1. The power dissipated is maximum.
P = Vrms Irms = \(\frac{V_{r m s}^{2}}{R}\)

2. Pure inductive circuit: If the circuit contains only an inductor, then the voltage leads the current in phase by \(\frac{\pi}{2}\) i.e., ϕ = \(\frac{\pi}{2}\)
∴ cos ϕ = 0
P = Vrms Irms\(\cos \frac{\pi}{2}\) = 0
∴ No power is dissipated even though a current is flowing in the circuit.

3. Purely capacitive circuit: If the circuit contains only a capacitor, then the current leads the voltage by \(\frac{\pi}{2}\) i.e., ϕ = \(\frac{\pi}{2}\) or cosϕ = 0
∴ P = Vrms Irms\(\cos \frac{\pi}{2}\) = 0
Thus the average power consumed over a complete cycle is zero here.

4. LCR series circuit: The power dissipated is given by P = Vrms Irms cos ϕ where ϕ = \(\tan ^{-1} \frac{X_{L}-X_{C}}{R}\)
So, ϕ may have non-zero values in a LR, CR or LCR circuit. Even then the power is dissipated only across the resistor.

5. Power dissipated at resonance in LCR circuit: At resonance XC – XL = 0 and ϕ = 0.
Thus cosϕ = 1 and P = I2Z = I2R. Thus the power dissipated is maximum (through R) at resonance.

Power Factor
The average power for an ac circuit is given by
P = Vrms Irmscos ϕ
Vrms Irms is called the apparent or virtual power,
P is called the true power.
cos ϕ is called the power factor of an ac circuit.
∴ True power = Apparent power × Power factor
Thus the power factor is defined as the ratio of
the true power and apparent power for an ac circuit.

Let us calculate its value for different cases.
1. Purely resistive circuit, ϕ = 0°.
∴ Power factor = 1 (= maximum value)
2. Purely inductive circuit, ϕ = 90°.
∴ Power factor = cos 90° = 0 (minimum value)
3. Purely capacitive circuit ϕ = 90°.
∴ Power factor = cos 90° = 0 (minimum)
4. For a series LCR circuit, the power factor is

From the above discussion it becomes clear that the power dissipated in a circuit depends upon the power factor. Larger the value of power factor, greater is the power dissipated.

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