AC Voltage Inductor Explained for Students

This article gives a clear knowledge of the AC Voltage and Inductor when an AC voltage is applied to the inductor. But first thing is first. We need to know about an inductor. In the presence of the electric current, an inductor works as most of the power electronic circuit that stores energy in the form of magnetic energy.

An Inductor is a coil of wire that sets an altering magnetic field around it when an altering current flows through it. Due to this inductance,a coil is subjected to an alternating current and a back emf is induced in the coil. The current across the inductor changes to equalize the current passing through it. The other names of inductors are choke, reactor, or just coil.

The inductor voltage can measure the amount of electromotive force (voltage) generated for a given rate of change of current.  Consider that an inductor produces an EMF of 1 volt when current passes through the inductor. This inductor possesses an inductance of 1 Henry and changes at the rate of 1 ampere per second.

These are the symbols used for the inductor:

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AC Voltage Applied to an Inductor

An inductor can oppose or block the passage of current through it. There are two cases to do so. One case is for the DC circuit, and another is for the AC circuit. The first inductor is connected to a DC supply, and the second one is connected to the AC supply. In the DC circuit, a constant current flows through this inductor, and the bulb connected to it glows brightly. But in the AC circuit, the bulb does not lighten up as brightly as the first one; this happens because the inductor opposes the flow of alternating current.

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AC Voltage Applied to an Inductor Derivation 

Here is a derivation of AC Voltage Applied to an Inductor Class 12:By using Lenz’s law, the flow of AC current through an indicator can be examined. We know that alternating current changes with magnitude and direction. When the current rises from zero to top value, then the magnetic field, as well as the current of the coil, increases. That is why we find an induced EMF and its direction as stated by the Lenz’s Law:

\[E = - \frac{d\phi}{dt} \]

In certain cases, when the current declines from peak value to zero, the magnetic field of the coil also becomes zero. Therefore, the relation of the induced EMF is:

\[E = -L \frac{di}{dt} \]

Supplied voltage must be equal to the reverse Emf to maintain the current.We know that 

\[\phi  = L \times i \]

The amount of voltage that the AC source requires is given by:

\[V = L \times \frac{di}{dt}\]

Across the coil, the relation between current and voltage is:

\[V = L \times \frac{di}{dt}\]

\[\frac{di}{dt} = \frac{V}{L} \]

\[\frac{di}{dt} = \frac{V_m Sin \omega t}{L}\]........... 1

Here, 

Vm = Maximum voltage

ω = Angular frequency.

After integrating equation 1

\[i = \int \frac{V_m sin \omega t}{L} dt\]

\[i = \frac{-V_m cos \omega t}{\omega L} dt\]

\[i = \frac{V_m sin (\omega t - \frac{\pi}{2})}{L} dt\]

\[i = i_m sin (\omega t - \frac{\pi}{2})\]

Here, im = Maximum value of current 

So, the relation for inductive reactance V = Vmsin⁡ωt

\[V = V_m sin \omega t\]

\[V = i_m sin \omega t\]

\[V = i_m sin (\omega t - \frac{\pi}{2})\]

\[i = -i_m cos \omega t\]

\[i_m = \frac{V_m}{\omega L}\]

\[i_m = \frac{V_m}{X_L}\]

In the above expression, Inductive reactance = XL

AC Voltage Applied to an Inductor Derivation with Graph 

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AC Voltage Applied to an Inductor and Power Consumed

An AC voltage is applied to the inductor to know its function in an electric circuit. The circuit diagram given in Fig.1 shows the presence of an inductor and an A Voltage V, which is represented by the symbol “~.”

The expression for the AC voltage, V = Vmsin⁡ωt

The expression for the amplitude of the current \[i_m = \frac{V_m}{L \omega} \] 

As we know, the symbol for inductive resistance is XL.  XL = ωL

Hence, the amplitude of the current in this circuit can be written as:

\[i_m = \frac{V_m}{X_L} \]  

The above equation’s dimensional formula for inductance signifies the same dimensional formula for resistance. Ohm is the SI unit of capacitance. The purpose of the capacitive resistance is to block the current flow in a purely inductive circuit. This exactly happens when resistance delays the current flow in a purely resistive circuit.After knowing this, we can clearly infer that the inductive resistance is directly proportional to the frequency and the inductance. A conclusion is found from the above expressions that the current in an inductive circuit is π/2 behind the voltage across the capacitor.

Do you know?

Inductors work on the principle of magnetism that stores the energy from the current in the form of a magnetic field. A magnetic field is developed in the inductor when a voltage is applied across the terminals of an inductor.We can determine the value of the current inductor flow by its own self-induced EMF or back EMF value because the progress of the current passing through an inductor is not on the spot.

When current is at its peak value, a steady-state current is flowing through the coil, and there is no back EMF induced to block the current passage. That is why the coil behaves as a short circuit, which allows maximum current passage through it in the case of a DC circuit. 

Conclusion:

This article focuses on Inductor Voltage and its application. You can go through the article for a better understanding of the topic.