The Voltage across the Inductor: Energy Stored in Inductor

An inductor is a passive circuit element that stores energy in the form of a magnetic field. Inductors are made of wrapped conducting wires or coil, to enhance the effectiveness of the inductor number of turns is increased. The inductance of an inductor increases as with increasing the number of turns. An inductor is not anticipated to dissipate energy, it only stores energy and then delivers it to the circuit when required. If the current is passed through it the voltage across the inductor will change gradually.

inductor symbols
Various symbols of inductors

The inductance of an inductor

In the 1800’s Oersted, showed that a current-carrying conductor produces a magnetic field around it. When he was conducting some experiments, a compass present nearby the current-carrying conductor deflected, he notices.

After few years, Ampere showed by taking some careful experiments that the magnitude of magnetic flux is directly related to the amount of current flowing through it.

Michal Faraday and Joseph Henry discovered that a changing magnetic field can produce a voltage across its neighboring circuits. They also showed that the magnitude of voltage is dependent on the rate of change of flux.

$v \propto \frac{d\Phi }{dt} \quad$ OR
 $ v= L\frac{d\Phi }{dt} \quad
\dots(1)$

The letter “L” is symbol for inductance of an inductor.

Inductance is the measure of the property of an inductor that opposes the change in current flow.

If a current is passed through an inductor, the voltage across the inductor changes gradually rather than abruptly. It can be represented mathematically

$v=L\frac{di}{dt} \quad \dots(2)$

From equation (2) it can be implied that if there is a constant current flowing through the inductor, the voltage across the inductor will be zero. It means that an inductor will act as a short circuit for the DC power supply.

An inductor acts like a short circuit to a dc source.

According to equation (2), discontinuous change in inductor current need infinite voltage across the inductor, which is practically impossible. So inductor opposes a change in current either positive change or negative change explained by the Lenz Law.

Lenz Law:

The Lenz Law state that the direction of the induced voltage is always such that it opposes its cause which produces it. As discussed earlier that the magnetic flux is directly related to the current flowing through it. So, when the current increases the flux also increases, this change in flux causes a voltage to induce in such a direction that opposes the existing current. Similarly, when current and flux decreases, it causes the voltage to induce in such a direction that opposes the existing current. This is “choking” and that’s why inductors are named chock.

The inductance of an Inductor :

The inductance of an inductor can be found using the following formula:

$L=\frac{N^{2}\mu A}{l} \quad \dots (3)$

Where N is the number of turns in the coil, A is the cross-sessional area of the wire, l is length and μ is the permeability of the core, where magnetic flux passes. The permeability depends upon the material used and varies from material to material. Just like capacitors and resistors, the inductor is also available in the market from a few micro-Henry to tens of Henry, which may be fixed or variable.

Energy Stored in Inductor:

The inductor works like a capacitor and doesn’t dissipate energy. It stores electric energy in the form of the magnetic field during the charging phase and releases the same energy to the circuit in the decay phase. Energy stored in the inductor is the multiplication of current through the inductor and voltage across the inductor.

The inductor absorb power is

$p(t)=v(t)i(t)$

From equation (2) we know that

$v(t)=L\frac{di}{dt} $

Putting it to the above equation

$p(t)=(L\frac{di}{dt}) i(t)$

Where stored energy can be find by integrating both sides up to charging time

$\int_{t_{0}}^{t}{p (t)dt}=L\int_{t_{0}}^{t}{i(t) \frac{di}{dt’}} dt’$

$W_{Stored}=L\int_{i(t_{0})}^{i(t)}{i ‘(t)} di^{‘}$

$W_{Stored}=L\frac{1}{2}\lbrack i(t)^{2}-i(t_{0})^{2}\rbrack $

If the inductor is initially not charged or left unconnected for a long time, so the initial current  $i(t_{0})$ will be zero. Thus the following formula is applicable for finding stored energy.

$W_{Stored}=\frac{1}{2}Li^{2}$

The energy stored is represented by the graph below.

Energy Stored in inductors
What is the Maximum Energy Stored in an Inductor?

Look at the above graph and you understand the maximum energy storage in an inductor. The graph has current, voltage, and power lines. Where it has also told us about the energy stored in an inductor by the shaded area. The energy stored in the area under the power curve. And this could be maximum if the power of the inductor goes to zero. Or the current or voltage of the inductor goes to zero. As the exponential decay or rise it theoretically t -> infinity. but practically when it consumes the time of 5$\tau$.

If you want to increase the energy store in an inductor increase the inductance of the inductor and current through it. This can be seen in the energy storage formula as these parameters are directly related.

Inductor Construction :

The basic construction of an inductor contains an insulated (enameled) wire-wound. The winding may be supported by a core or not. In case of not supported by an internal core, is familiar with an air-core inductor. In another case, the winding is supported by an iron core, which is an iron core inductor. Iron is a ferromagnetic high permeable material that provides a low reluctance path for the magnetic flux. It also confines the magnetic flux near the winding and increases the flux linkage. At low frequency, the core is made of thin plates called lamination to reduce eddy current losses. Where for high frequencies, the core is made of soft ferrite because of not producing high eddy current losses.

Eddy Current Losses :

As the core inside the inductor is subjected to the magnetic flux of the inductor. Upon variation in the flux, the iron core behaves like a conductor and voltage is induce inside the core. The induced voltage causes current inside the core, which names as Eddy current. The current increases losses by producing heat.

To counter the eddy current losses, the core is made of laminations, the iron thin plates insulated from each other. Thus the flow of current is limited by increasing the resistance in the current path.

Q Factor of Inductor :

Practically the inductor shows some resistance, which absorbs part of the apparent power and reduces the efficiency of the inductor. Q Factor is the measure of the efficiency of the inductor at the given frequency and equal to the ratio of inductive reactance to the resistance. The higher the Q factor means nearness to the ideal inductor and narrower bandwidth in a resonant circuit. The radio uses the high Q factor inductor with a capacitor to make the circuit resonant.

$Q=\frac{wL}{R}=\frac{2\pi fL}{R}$

The voltage across the Inductor in Series:

Just like resistors and capacitors, inductors are combined in series and/or parallel. We are interested in finding the equivalent inductance of the combination. If there is only one path for current the combination of inductors will be a series combination, as shown in the diagram below.

inductance of an Inductor in series circuit and voltage across the inductor

Applying the KVL to the circuit above

$v=v_{1}+v_{2}+\ldots +v_{n} \ldots (4)$

Voltage across the inductor depends upon change in current

$v=L\frac{di}{dt}$

$v=L_{1}\frac{di}{dt}+L_{2}\frac{di}{dt}+\ldots +L_{n}\frac{di}{dt}
\ldots (5) $

We know the there is only one path for current, so current for all elements are same

$I=I_{1}=I_{2}=\ldots =I_{n}$

Now putting it to the above equation (4)

$v=(L_{1}+L_{2}+\ldots +L_{n})\frac{di}{dt}$

So equivalent inductance in series is

$L_{eq}=L_{1}+L_{2}+\ldots +L_{n}$

Inductors in Series-Parallel Calculator :

Connection type?

First Inductor, L1

Second Inductor, L2

Third Inductor, L3

Resultant Inductance, LT

The voltage across the Inductor in parallel :

If multiple inductors are connected such that there are multiple paths for current to flow, such combination is a parallel combination. Now consider the following parallel combination of inductors shown in the diagram.

inductance of an Inductors in Parallel circuit and voltage across the inductor

Now applying KCL to the above diagram will give us the following equation

$i_{T}=i_{1}+i_{2}+\ldots +i_{n}$

We know that voltages are same in a parallel combination, so putting that into the above equation

$i_{T}=\frac{1}{L_{1}}\int{vdt}+\frac{1}{L_{2}}\int{vdt}+\ldots$
$+\frac{1}{L_{n}}\int{vdt} i=\frac{1}{L}\int{vdt}$

OR

$i_{T}=(\frac{1}{L_{1}}+\frac{1}{L_{2}}+\ldots +\frac{1}{L_{n}}) \int{vdt}$

So equivalent inductance in case of the parallel combination will be

$\frac{1}{L_{eq}}=\frac{1}{L_{1}}+\frac{1}{L_{2}}+\ldots
+\frac{1}{L_{n}}$

Conclusion :

  • Inductors are passive electrical components that store energy in the form of a magnetic field.
  • Inductors resist change in current and they are commonly used in electronic circuits avoiding current surges.
  • The inductance of an inductor increases in series and reduces in parallel.    

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