Energy in a Coupled Electric Circuit

After learning what is the mutual inductance and dot convention, we will move on how to calculate the energy in a coupled electric circuit.

We can call an electric circuit as a coupled circuit if the circuit has a mutual inductance from two coils or inductors.

Make sure to read what is electric circuit first.

Energy in a Coupled Electric Circuit

We saw that the energy stored in an inductor is given by

Energy in a Coupled Electric Circuit
(1)

We now want to determine the energy stored in magnetically coupled coils.

Consider the circuit in Figure.(1).

Energy in a Coupled Electric Circuit
Figure 1. The circuit for deriving energy stored in a coupled circuit.

We assume that currents i1 and i2 are zero initially so that the energy stored in the coils is zero.

If we let i1 increase from zero to I1while maintaining i2 = 0, the power in coil 1 is

Energy in a Coupled Electric Circuit
(2)

and the energy stored in the circuit is

Energy in a Coupled Electric Circuit
(3)

If we now maintain i1 = I1 and increase i2 from zero to I2, the mutual voltage induced in coil 1 is M12 di2/dt, while the mutual voltage induced in coil 2 is zero, since i1 does not change.

The power in the coils is now

Energy in a Coupled Electric Circuit
(4)

and the energy stored in the circuit is

Energy in a Coupled Electric Circuit
(5)

The total energy stored in the coils when both i1 and i2 have reached constant values is

Energy in a Coupled Electric Circuit
(6)

If we reverse the order by which the currents reach their final values, that is, if we first increase i2 from zero to I2 and later increase i1 from zero to I1, the total energy stored in the coils is

Energy in a Coupled Electric Circuit
(7)

Since the total energy stored should be the same regardless of how we reach the final conditions, comparing Equations.(6) and (7) leads us to conclude that

Energy in a Coupled Electric Circuit
(8a)

and

Energy in a Coupled Electric Circuit
(8b)

This equation was derived based on the assumption that the coil currents both entered the dotted terminals.

If one current enters one dotted terminal while the other current leaves the other dotted terminal, the mutual voltage is negative so that the mutual energy MI1I2 is also negative.

In that case,

Energy in a Coupled Electric Circuit
(9)

Also, since I1 and I2 are arbitrary values, they may be replaced by i1 and i2, which gives the instantaneous energy stored in the circuit the general expression

Energy in a Coupled Electric Circuit
(10)

The positive sign is selected for the mutual term if both currents enter or leave the dotted terminals of the coils; the negative sign is selected otherwise.

We will now establish an upper limit for the mutual inductance M.

The energy stored in the circuit cannot be negative because the circuit is passive.

This means that the quantity 1/2L1i12 + 1/2L2i22 − Mi1i2 must be greater than or equal to zero,

Energy in a Coupled Electric Circuit
(11)

To complete the square, we both add and subtract the term i1i2√(L1L2) on the right-hand side of Equation.(11) and obtain

Energy in a Coupled Electric Circuit
(12)

The squared term is never negative; at its least, it is zero. Therefore, the second term on the right-hand side of Equation.(12) must be greater than zero; that is,

Energy in a Coupled Electric Circuit
(13)

Thus, the mutual inductance cannot be greater than the geometric mean of the self-inductances of the coils.

The extent to which the mutual inductance M approaches the upper limit is specified by the coefficient of coupling k, given by

Energy in a Coupled Electric Circuit
(14)

or

Energy in a Coupled Electric Circuit
(15)

where 0 ≤ k ≤ 1 or equivalently 0 ≤ M ≤ √(L1L2). The coupling coefficient is the fraction of the total flux emanating from one coil that links the other coil. For example, in Figure.(2),

energy in coupled circuit
Figure 2. Mutual inductance M21 of coil 2 with respect to coil 1
Energy in a Coupled Electric Circuit
(16)

and in Figure.(3)

Energy in a Coupled Electric Circuit
Figure 3. Mutual inductance M12 of coil 1 with respect to coil 2.
Energy in a Coupled Electric Circuit
(17)
 

If the entire flux produced by one coil links another coil, then k = 1 and we have 100 percent coupling, or the coils are said to be perfectly coupled. Thus,

The coupling coefficient k is a measure of the magnetic coupling between two coils; 0 ≤ k ≤ 1.

For k < 0.5, coils are said to be loosely coupled; and for k > 0.5, they are said to be tightly coupled.

We expect k to depend on the closeness of the two coils, their core, their orientation, and their windings. Figure.(4) shows loosely coupled windings and tightly coupled windings.

Energy in a Coupled Electric Circuit
Figure 4. Windings: (a) loosely coupled, (b) tightly coupled; cutaway view demonstrates both windings.

The air-core transformers used in radio frequency circuits are loosely coupled, whereas iron-core transformers used in power systems are tightly coupled.

Energy in a Coupled Electric Circuit Example

Consider the circuit in Figure.(5). Determine the coupling coefficient. Calculate the energy stored in the coupled inductors at time t = 1 s if v = 60 cos (4 t + 30◦) V.

Energy in a Coupled Electric Circuit
Figure 5

Solution:
The coupling coefficient is

Energy in a Coupled Electric Circuit

indicating that the inductors are tightly coupled. To find the energy stored, we need to obtain the frequency-domain equivalent of the circuit.

Energy in a Coupled Electric Circuit

The frequency-domain equivalent is shown in Figure.(6). We now apply mesh analysis. For mesh 1,

Energy in a Coupled Electric Circuit
(1.1)

For mesh 2

Energy in a Coupled Electric Circuit

or

Energy in a Coupled Electric Circuit
(1.2)

Substituting this to Equation.(1.1) yields

Energy in a Coupled Electric Circuit

and

Energy in a Coupled Electric Circuit

In the time-domain,

Energy in a Coupled Electric Circuit

At time t = 1 s, 4t = 4 rad = 229.2◦, and

Energy in a Coupled Electric Circuit

The total energy stored in the coupled inductors is

Energy in a Coupled Electric Circuit
Energy in a Coupled Electric Circuit
Figure 6

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Reference:  Fundamentals of electric circuits by Charles K. Alexander and Matthew N. O. Sadiku