Power in A.C Circuit

The average power dissipated in an A.C. circuit per cycle is given as:

P=12I0V0

But, I0=2(Irms)−−−−−−√ and

V0=2(Vrms)−−−−−−√∴P=12(2(Irms)−−−−−−√×2(Vrms)−−−−−−√)∴P=2IrmsVrms2∴P=IrmsVrms

According to Ohm’s law, V = IR

∴P=I2R=V2R

Power is always dissipated in a resistor. This because the current and the voltage are in phase.  In an inductor, power dissipation is zero due to the fact that power is positive when energy is  stored in the magnetic field of the coil and negative when the energy is given back to the voltage  supply on the next part of the cycle.

In a capacitor, power dissipation is also zero due to the fact that power is positive when energy is  stored in the electric field between the capacitor’s plates and negative when the energy is given back to the voltage supply.

In essence, power dissipation is zero both in an inductor and a capacitor because the voltage and  current are out of phase by90°.

Therefore, in an a.c. circuit containing inductor, capacitor and a resistor in series, power is only  dissipated in the resistor.

Hence, P=IV=I2R=V2R

Example 5:

A circuit consists of a resistor 500ohms and a capacitor of 5μF connected in series. If  an alternating voltage of 10v and frequency 50 Hz is applied across the series circuit, calculate

(a) the reactance of the capacitor

(b) the current flowing in the circuit

(c) the voltage across the capacitor

(d) If the capacitor is replaced with an inductor of 150mH, calculate the impedance and voltage across the inductor

Solution:

(a) XC=12πfL=12π×50×5.0×10−6=636.62ohms

(b) Z=R2+X2C−−−−−−−√ since there is no L

Z=5002+636.622−−−−−−−−−−−−√=809.5ohmsI=VZ=10809.5A=12.35×10−3A

(c) VC=IXC=12.35×10−3×636.62=7.86V

(d) XL=ωL=2πfL=2×π×50×10−3ohms=47.12ohmsZ=R2+X2L−−−−−−−√=5002+47.122−−−−−−−−−−−√=502.2ohmsI=VZ=10502.2=19.9×10−3AV=IXL=19.9×10−3×47.12=938×10−3V

EVALUATION (POST YOUR ANSWERS USING THE QUESTION BOX BELOW FOR EVALUATION AND DISCUSSION):

1. Why is power only dissipated in the resistor in a series circuit containing resistor, capacitor and an inductor?
2. State the formula for power.

GENERAL EVALUATION (POST YOUR ANSWERS USING THE QUESTION BOX BELOW FOR EVALUATION AND DISCUSSION):

1. Tell the meaning of the following denotations: V₀, V_rms, I₀, I
2. In what component of the series circuit is voltage and current in step with each other?
3. Critically examine the passage of an a.c through an inductor.
4. Define the reactance and the resistance of a capacitor and a resistor respectively.
5. Discuss why power is not dissipated in an inductor when current passes through it.