AC Voltage applied to an Inductor

Introduction

AC voltage is caused by an alternating current. The change in magnetic flux, when a current-carrying conductor is allowed to rotate in a magnetic field, induces an alternating voltage in the coil. This is known as induced electromotive force. The inductor is a coil of wire in which back emf is induced when an AC passes through it. This back emf produces a current which opposes the AC input.

AC Voltage

An alternating current (AC) is defined as an electric current that changes its direction as well as its magnitude regularly. Thus, AC voltage is the determination of voltage difference between terminals where AC current flows. Mathematically an AC voltage and AC be represented as

$$\mathrm{V(t)=V_0\:sin(\omega t)}$$

$$\mathrm{I(t)= I_0\:sin(\omega t)}$$

Inductor

An inductor is a passive electronic component that is used for storing energy by converting electrical energy into magnetic energy. The inductor is a coil of insulated wire that changes its magnetic field when the current flowing through it changes. Thereby, it induces an electromotive force. This induced emf opposes the flow of current in the circuit. Lenz’s law states that an induced emf produced in the circuit always opposes the current flowing through it. According to this, the induced emf is,

$$\mathrm{V =- L\frac{di}{dt}}$$

Where i=source current and the L is inductance of the coil

Inductance(L) and their Types

The ratio between the emf induced in the circuit and the change of current for time is known as the coil's inductance. There are classified into two and they are

• Self-induction

• Mutual Induction

Self-Induction

If the current passing through the coil changes with time, then the magnetic flux associated with the coil changes and so it induces an electromotive force in the circuit itself. This is called self-induction.

Mutual-Induction

Suppose the current passing through the primary coil changes then the magnetic flux associated with the coil changes. This change makes the change in the magnetic component of the secondary coil thereby it induces an emf in the secondary circuit. This is known as mutual inductance. Both occur due to a change in magnetic flux associated with the coil as the current passing through it changes. Henry (H) is the unit for inductance.

Inductor with AC Biasing

It can be easily described by a simple circuit. In an AC circuit an inductor, a bulb, and a switch are connected in series. Whenever the switch is connected AC flows through it but the bulb does not glow suddenly. It slowly rises. Because during this time the induced current opposes the growth of the source current. When the switch is disconnected the bulb does not suddenly turn off. Because the induced emf produced in the circuit opposes the decay of source current. In a DC circuit, this opposing factor is defined as resistance. But in AC circuits it is noted as impedance.

AC Voltage applied to an Inductor

If an AC voltage applied to the inductor, then the magnetic lines of force change which in turn magnetic flux associated with the coil changes thereby it induces an emf which opposes the current flow through the inductor. This opposition to the current flow is due to the AC resistance of the coil. This is also known as inductive reactance $\mathrm{X_L}$

$$\mathrm{X_L = \omega L}$$

$$\mathrm{X_L = 2\pi fL}$$

https://lizenzhinweisgenerator.de/?lang=en&url=/wiki/File:Sydney_(AU),_Coast,_New_Zealand_sea_lions_--_2019_--_3489.jpg

Generally, ac voltage is represented by,

$$\mathrm{V(t) = V_0\:sin(\omega t)}$$

$$\mathrm{\omega= Angular\:frequency}$$

$$\mathrm{V_0= maximum\:voltage}$$

As per the definition of electromagnetic induction by Faraday, the induced emf is denoted as,

$$\mathrm{V=-L\frac{di}{dt}}$$

When current passes through the inductor the induced emf opposes the current. The source voltage must be equal to the back emf to maintain the current balance.

$$\mathrm{L\frac{di}{dt}=V_0 \:sin(\omega t)}$$

$$\mathrm{\frac{di}{dt}=\frac{V_0 sin(\omega t)}{L}}$$

$$\mathrm{\int \frac{di}{dt}=\frac{V_0}{L}\int sin(\omega t)}$$

(As V0 and L are constant they come out the integral)

Integrating this equation we get,

$$\mathrm{i=\frac{V_0}{L}(−cos(\omega t)/ \omega)}$$

$$\mathrm{i =\frac{V_0}{L\omega}(−cos \:\omega t) + C}$$

To find the value of C,

If $\mathrm{V_0=0}$,then i=0 which means C=0 thus

$$\mathrm{i =\frac{V_0}{L\omega}(−cos \:\omega t)}$$

$\mathrm{cos\omega t}$ can be written as $\mathrm{sin(90^{\circ}-\omega t)}$

By inserting the negative in -cosωt will become $\mathrm{sin\:(\omega t- 90^{\circ})}$

This can be modified as,

$$\mathrm{i =\frac{V_0}{L\omega}(sin(\omega t−90^{\circ}))}$$

$$\mathrm{i =\frac{V_0}{X_L}(sin(\omega t−90^{\circ}))}$$

$$\mathrm{i =i_0(sin(\omega t−90^{\circ}))\:\:\:\:\:(as\:\:\frac{V_0}{X_L}=i_0)}$$

$$\mathrm{X_L = L\omega}$$

$\mathrm{X_L}$ = Reactive inductance of the coil

The S.I unit for reactive inductance is ohm.

$\mathrm{i_0}$ =maximum current

Thus, the current in the inductor circuit is 90 degrees behind the voltage.

Power in ac Circuit

Power in a circuit change continuously as voltage and current changes in the circuit. Power in the inductor circuit is

$$\mathrm{P = VI}$$

If there is a phase difference between voltage and current then,

$$\mathrm{V(t) = V_0 \:sin(\omega t)}$$

$$\mathrm{I(t) = I_0 \:sin(\omega t-\Phi)}$$

Thus power

$$\mathrm{P = V_0 I_0 \:sin(\omega t-\Phi) sin(\omega t)}$$

Since

$$\mathrm{sinA\:sinB=\frac{1}{2}cos(A-B)-cos(A+B)}$$

$$\mathrm{P=\frac{V_0 I_0}{2}(cos \Phi - cos(2\omega t- \Phi))}$$

Average power

$$\mathrm{P_{avg}=\frac{V_0 I_0}{2}(\int\: cos⁡\:\Phi- cos(2\omega t-\Phi) )}$$

$$\mathrm{P_{avg}=\frac{V_0 I_0}{2}cos\: \Phi\:\:\:(By\:substituting\:limits\: cos(2\omega t-\Phi)=0)}$$

$$\mathrm{P_{avg}=\frac{V I}{2}cos\:\Phi}$$

For pure inductor phase difference if

$$\mathrm{\Phi=\frac{\pi}{2}\:\:then\:\:cos\frac{\pi}{2}=0}$$

Thus power $\mathrm{P=0}$

Power Factor

Power factor in an AC circuit is the ratio between apparent and actual power.

$$\mathrm{Power\:factor=\frac{Apparent \:power}{True\:power}}$$

Conclusion

The magnitude and direction of the alternating current changes regularly. This change in alternating current changes the magnetic flux associated with the coil which is connected to the source, thereby it induces an emf in the opposite direction in the circuit.

In an inductor circuit applied by AC voltage the current lags 90 degrees behind the voltage. It depends upon both voltage and frequency of the applied voltage. For a pure inductor the power consumed is zero. Because the phase difference between the voltage and current is nil.

FAQs

Q1. The Inductance of the Inductor is 1.5H which is operating at the Frequency of 50Hz. Calculate its reactive inductance of the Inductor?

Ans. The reactive inductance is $\mathrm{X_L=2\Pi fL}$

$$\mathrm{X_L = 2 \times 3.14 \times 50 \times 1.5}$$

$$\mathrm{X_L = 471\Omega}$$

Thus, the reactive inductance of the inductor is $\mathrm{471\Omega}$.

Q2. What is the relation between Voltage and Current for a pure Inductor Circuit?

Ans. In a purely inductive circuit the current falls behind the voltage by 90 degrees.

Q3. Express AC current and AC Voltage Mathematically.

Ans. Alternating current is defined as an electric current which changes its direction as well as its magnitude regularly. Thus, AC voltage is the determination of voltage difference between terminals where a.c current flows. Mathematically an AC voltage and AC current be represented as,

$$\mathrm{V(t)= V_0\:sin(\omega t)}$$

$$\mathrm{I(t)= I_0 sin(\omega t)}$$

Q4. What happens in an Inductor and what is its Voltage?

Ans. Inductor is a coil of insulated wire that changes its magnetic field when current passes through it changes thereby it induces an electromotive force in the direction opposite to that of the current flowing through it. This induced emf always opposes the current flowing through the circuit. The inductor voltage is

$$\mathrm{V=L\frac{di}{dt}}$$

Q5. Define Inductance and what is its Unit?

Ans. The ratio between the emf induced in the circuit and the change of current with respect to time is known as the inductance of the coil. The unit of inductance is Henry (H).

Q6. What is Electromagnetic Induction?

Ans. An Inductor is a coil of insulated wire that changes its magnetic field when there is a change in current flowing through it. thereby it induces an electromotive force. This induced emf opposes the current pass through the circuit. This is called electromagnetic induction and it is well defined by Faraday.

Q7. What is Lenz's Law?

Ans. The emf induced due to the change in AC current passing through the circuit always opposes the current flowing through it.