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Inductors in Series and Parallel | Energy Stored in Inductor

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In this section, we will calculate an equivalent inductance of inductors in series and inductors in parallel.

Inductors in Series

The method for determining the total inductance of the following circuit is similar to that used for series resistors. According to Kirchhoff’s voltage law, we can write following equation for the following figure:
inductors in series

                  ${{V}_{T}}={{V}_{1}}+{{V}_{2}}+{{V}_{3}}\text{    }\cdots \text{    (a)}$

The current flowing in the circuit is iT. The rate of change of current flowing in the circuit is $\frac{d{{i}_{t}}}{dt}$.

Dividing both sides of equation (a) by $\frac{d{{i}_{t}}}{dt}$, we obtain following expression:

          $\frac{{{V}_{T}}}{{}^{d{{i}_{t}}}/{}_{dt}}=\frac{{{V}_{1}}}{{}^{d{{i}_{t}}}/{}_{dt}}+~\frac{{{V}_{2}}}{{}^{d{{i}_{t}}}/{}_{dt}}+\frac{{{V}_{3}}}{{}^{d{{i}_{t}}}/{}_{dt}}\text{      }\cdots \text{    (b)}$

The left side of equation (b) is the total voltage divided by the rate of change of current. This term gives the total inductance LT. Each term on the right side of equation (b) gives the value of an individual inductance:

 

$\frac{{{V}_{1}}}{{}^{d{{i}_{t}}}/{}_{dt}}={{L}_{1}}$

$\frac{{{V}_{2}}}{{}^{d{{i}_{t}}}/{}_{dt}}={{L}_{2}}$

$\frac{{{V}_{3}}}{{}^{d{{i}_{t}}}/{}_{dt}}={{L}_{3}}$

So, the total inductance would be,

                  ${{L}_{T}}={{L}_{1}}+{{L}_{2}}+{{L}_{3}}\text{      }\cdots \text{     (c)}$

Equation (c) states that when inductors are connected in series, the total inductance is the sum of the individual inductances.

If there are two or more equal value inductors in series, the total inductance may be found by;

${{L}_{T}}=NL$

Where N is the number of equal inductors and L is the value of the single inductor.

Inductors in Parallel

By Appling Kirchhoff’s current law to the following figure, we can determine how inductors in parallel combine;
inductors in parallel

                  $~{{i}_{T}}={{i}_{1}}+{{i}_{2}}+{{i}_{3}}\text{     }\cdots \text{      (d)}$

In order to express above equation as a rate of change of current take derivative on both sides;

          $\frac{d{{i}_{T}}}{dt}=\frac{d{{i}_{1}}}{dt}+\frac{d{{i}_{2}}}{dt}+\frac{d{{i}_{3}}}{dt}\text{      }\cdots \text{      (e)}$

Since voltage across an inductor is

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

 And also since VT is the total voltage across the parallel inductance,

 $\frac{{{V}_{T}}}{{{L}_{T}}}=\frac{{{V}_{T}}}{{{L}_{1}}}+\frac{{{V}_{T}}}{{{L}_{2}}}+\frac{{{V}_{T}}}{{{L}_{3}}}\text{      }\cdots \text{    (f)}$

Dividing both sides of equation (c) by VT would give us following equation;

 $\frac{1}{{{L}_{T}}}=\frac{1}{{{L}_{1}}}+\frac{1}{{{L}_{2}}}+\frac{1}{{{L}_{3}}}\text{      }\cdots \text{     (g)}$

Equation (g) states that reciprocal of the total inductance is equal to the sum of reciprocals of the individual inductances connected in parallel.

If two or more parallel inductors are equal. The total inductance may be determined by dividing the value of one of the inductors by the number of equal inductors.

${{L}_{T}}=\frac{L}{N}$

Whereas L is the value of one of the equal indicators and N is the number of equal inductors.

Energy stored in an inductor

The power entering in an inductor at any instant is;

$P=Vi=Li\frac{di}{dt}$

When the current is constant, the derivative is zero and no additional energy is stored in the inductor. When the current increases, the current derivative has a positive value and the power is positive. The total energy in the inductor at any given time is the integral of the power from minus infinity to that time;

 

${{W}_{L}}=\underset{-\infty }{\overset{t}{\mathop \int }}\,Vi~dt=\underset{-\infty }{\overset{t}{\mathop \int }}\,Li\frac{di}{dt}dt$

As we know that

$i\left( -\infty  \right)=0$

So,

${{W}_{L}}=\underset{0}{\overset{i}{\mathop \int }}\,Li~di$

After integration, we have final energy expression;

${{W}_{L}}=\frac{1}{2}L{{i}^{2}}\text{      }\cdots \text{    (h)}$          

WL= energy stored in the inductor at time t in Joules

i = the current in the inductor at time t in ampere

Equation (e) indicates that total energy in the inductor depends only on the instantaneous value of the current. In order for the energy stored in the inductor, as given by equation (e), to be positive, the current and the voltage must have consistent signs, as shown in the following figure:

energy in inductor

You May Also Read: 

Inductors in Series and Parallel

Capacitors in Series and Parallel

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About Ahmad Faizan

Mr. Ahmed Faizan Sheikh, M.Sc. (USA), Research Fellow (USA), a member of IEEE & CIGRE, is a Fulbright Alumnus and earned his Master’s Degree in Electrical and Power Engineering from Kansas State University, USA.