Power in AC Circuit: Power Factor, Equation, Formulas and Solved Examples

Power in ac circuit is the rate of work or the quantity of energy transferred by a circuit per unit of time.

Alternating Circuit is alternating current sources connected to a load. When an alternating e.m.f. is connected to an electrical circuit it is called an alternating circuit. The emf and current in the AC circuit at the same frequency but phase may or may not differ across the elements used. We will study the AC circuit with resistor inductor and capacitor in this article.

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A.C. Circuit with Pure Resistor

Consider a resistance R connected across an alternating emf. The voltage is given by  \(e=e_osin\omega{t}\\\)

\(\text{if i is the current then,}\\\)

\(e=iR\\\) \(\therefore i={e\over{R}}\\\)

\(i={e_osin\omega{t}\over{R}}\\\)

\(\text{Let  } i_o=i_{max}\\\)

\(i_o={e_o\over{R}}\\\)

\(\text{Hence,} i=i_osin\omega{t}\\\)

\(\text{For }i=i_o, sin\omega{t}=1\\\)

\(\text{For e to be maximum i.e.  } e_o, sin\omega{t}=1\)

The peak value and zero value of both current and EMF occur at the same time. Hence, the current and EMF are said to be in phase.

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Power in Pure Resistor Circuit

The Power in ac circuit of pure resistor is as follows:

The equation of power varies with time. Hence we consider average power. 

A.C. Circuit with Pure Inductor

Consider an inductor L connected across an alternating emf Using Kirchhoff’s law of electricity,

Thus we can see that there is a phase difference of π/2 between current and emf. We can conclude that e.m.f leads current by phase of π/2 or the current lags behind emf by phase of π/2 Further, we can see that,

The term L represents the effective opposition of the inductor to the flow of AC. It is called inductive reactance.

A.C. Circuit with Pure Capacitor

Consider an inductor C connected across an alternating emf. Let q be the charge on any plate of the capacitor at any instant. Hence, we get the potential difference V=q/C. This is equal to the instantaneous value of emf i.e. e=q/C

q=Ce

Thus we can see that there is a phase difference of π/2 between current and emf. We can conclude that currently leads current by phase of π/2 or the emf lags behind emf by phase of π/2 Further, we can see that,

This is called capacitive reactive. Capacitive Reactance is the opposition offered to the AC by the capacitor.

Study about the Conductor in Magnetic Field here.

Power Factor

The average power in ac circuit is given by

The product does not give the actual power and is called apparent power. It gives actual power only when multiplied by factor . The factor is called power factor. of an A.C. circuit.

Inductor, Capacitor, Resistor A.C. Circuit

Now let us consider the combination of all three Inductor, Capacitor, Resistor connected in series with the AC. 

Let , and be the emf across the Inductor, Capacitor, Resistor and let i be the current. We know that exceeds current by π/2 while lags current by π/2. Further, we also know that is in phase with the current. Hence, the phasor diagram will be given by,

From the phasor diagram we get,

This quantity is called an impedance of the circuit. Impedance is defined as the ratio of r.m.s. The voltage of current over to r.m.s. current. The angle between e and i in the phasor diagram is denoted φ and given by,

LR Circuit

If Capacitance is absent, LCR is called as LR Circuit. 

In the LR circuit, Xc=0. Therefore, 

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Resonant Frequency of LR Circuit

The resonant frequency is defined as when the voltage across the resistor and the voltage across the inductor or resistor are equal. 

RC Circuit

If Inductor is absent, LCR is called RC Circuit. 

In RC circuit, XL=0.Therefore, 

Resonant Frequency of RC Circuit

The resonant frequency is defined as when the voltage across the resistor and the voltage across the capacitor or resistor are equal.

Power in LCR AC Circuit

Here, erms.irmsis called apparent power and cosis called power factor. 

Resonant Frequency of LCR Circuit

The resonant frequency is defined as when the voltage across the resistor and the voltage across the inductor, capacitor or resistor are equal.

Check the concept of Electrostatic Coulomb Law here.

Q-Factor of power in ac circuit

The quality factor relates the maximum or peak energy stored in the circuit to the energy dissipated during each cycle of oscillation.

Hence, mathematically it becomes the ratio of resonant frequency to bandwidth.

So LCR circuits, the Q factor can be given by:

Hope this helped you to understand the important concept of A.C. Circuits & Power. Do practice it now on the Testbook App through the free mock tests.

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