Power of a circuit is defined as the rate of consumption of electric energy in that circuit.

**POWER IN AC CIRCUITS**

**Power of a circuit is
defined as the rate of consumption of electric energy in that circuit. **It is given by the
product of the voltage** **and current. In an AC circuit, the voltage and
current vary continuously with time. Let us first calculate the power at an
instant and then it is averaged over a complete cycle.

The alternating voltage
and alternating current in the series *RLC* circuit at an instant are
given by

*v*=*V _{m}* sin ω

where *ϕ* is the
phase angle between *υ* and *i*. The instantaneous power is then
written as

Here the average of sin2 ω*t* over a cycle is 1/2 and
that of sin ω *t* cos ω*t* is zero. Substituting
these values, we obtain average power over a cycle.

where *V _{RMS}*

**Special Cases**

(i) For a purely
resistive circuit, the phase angle between voltage and current is zero and cos *ϕ*
= 1.

∴* P _{av} *=

(ii) For a purely
inductive or capacitive circuit, the phase angle is ± π/2 and cos(± π/2)=0.

∴ P_{av}
= 0

(iii) For series RLC circuit, the phase angle

(iv) For series RLC circuit at
resonance, the phase angle is zero and
cos . ϕ =1

∴ P_{av} = V_{RMS} I_{RMS}

Consider an AC circuit
in which there is a phase angle of *ϕ* between *V _{RMS}* and

Now, *I _{RMS}*
is resolved into two perpendicular components namely,

(i) The component of
current ( *I* * _{RMS}*
cos φ) which is in phase with
the voltage is called active component. The power consumed by this current

(ii) The other component
( *I* * _{RMS}*
sin φ) which has a phase angle
of π

The current in an AC
circuit is said to be wattless current if the power consumed by it is zero.
This wattless current happens in a purely inductive or capacitive circuit.

The power factor of a
circuit is defined in one of the following ways:

(i)** Power factor =
cos ϕ = cosine of the angle of lead or lag**

(ii) **Power factor =**
** R/Z** =
Impedance / Resistance

(iii) **Power factor =**
*VI* cos φ / VI

= True power / Apparent power

Some examples for power
factors:

(i) Power factor = cos
0° = 1 for a pure resistive circuit because the phase angle *ϕ* between
voltage and current is zero.

(ii) Power factor =
cos(±π /2 )= 0 for a purely inductive or capacitive circuit because the phase
angle *ϕ* between voltage and current is ±π /2 .

(iii) Power factor lies
between 0 and 1 for a circuit having *R*, *L* and *C* in varying
proportions.

There are many
advantages and disadvantages of AC system over DC system.

(i) The generation of AC
is cheaper than that of DC.

(ii) When AC is supplied
at higher voltages, the transmission losses are small compared to DC
transmission.

(iii) AC can easily be
converted into DC with the help of rectifiers.

(i) Alternating voltages
cannot be used for certain applications e.g. charging of batteries,
electroplating, electric traction etc.

(ii) At high voltages,
it is more dangerous to work with AC than DC.

**EXAMPLE 4.26**

A series RLC circuit
which resonates at 400 kHz has 80 μH inductor, 2000 pF capacitor and 50 Ω
resistor. Calculate (i) Q-factor of the circuit (ii) the new value of
capacitance when the value of inductance is doubled and (iii) the new Q-factor.

**Solution**

L = 80 × 10^{-6}H;
C = 2000 × 10^{-12} F

R = 50 Ω; *f*_{r} = 400 × 10^{3}Hz

**EXAMPLE 4.27**

A capacitor of
capacitance 10^{-4} /π F, an inductor of inductance 2/ π H and a
resistor of resistance 100 Ω are connected to form a series RLC circuit. When
an AC supply of 220 V, 50 Hz is applied to the circuit, determine (i) the
impedance of the circuit (ii) the peak value of current flowing in the circuit
(iii) the power factor of the circuit and (iv) the power factor of the circuit
at resonance.

**Solution**

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12th Physics : Electromagnetic Induction and Alternating Current : Power In AC Circuits |

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