Stored Energy

Energy stored in a magnetic field, also considering the case of no magnetic saturation

View other versions (2)

Key Facts

Gyroscopic Couple: The rate of change of angular momentum ( $\inline \tau$ ) = $\inline I\omega\Omega$ (In the limit).

$\inline I$ = Moment of Inertia.
$\inline \omega$ = Angular velocity
$\inline \Omega$ = Angular velocity of precession.

Overview

Key facts

The energy stored in a magnetic field is given by:

$E_{stored} = V \int H dB$

where $\inline V$ is the volume, $\inline H$ the magnetic field strength, and $\inline B$ the magnetic flux density.

In the particular case of no magnetic saturation, the energy stored becomes:

$E_{stored} = V \frac{B^2}{2 \mu_0 \mu_r}$

where $\inline \mu_0$ is the magnetic permeability of free space, and $\inline \mu_r$ the relative magnetic permeability.

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Constants

$\mu_0 = 4 \pi \cdot 10^{-7} \; \frac{N}{A^2}$

If we are to neglect the resistance of the circuit wire, then there would be no energy loss in maintaining a magnetic field. However, energy is required to establish the field, and it can then be recovered when the field is destroyed.

For a toroid, the induced voltage $\inline e$ at any instant is:

$e = N \frac{d\Phi}{dt}$

(2)

where $\inline N$ is the number of turns, and $\inline \Phi$ the magnetic flux.

If the current at any instant is $\inline i$ , then the instantaneous power ( $\inline Watts$ ) is:

$P = e \cdot i$

(3)

The energy ( $\inline Joules$ ) released from the coil in a time $\inline dt$ is:

$E = e \cdot i \cdot dt$

(4)

or, by considering (2):

$E = N \cdot i \cdot d\Phi$

(5)

The total energy stored in the coil then becomes:

$E_{stored} = \int_0^{\Phi_{Max}} N \cdot i \cdot d\Phi$

(6)

In order to further define the energy stored in a magnetic field, consider a magnetic circuit of length $\inline l$ and cross-sectional area $\inline A$ , as diagramed in Figure 1.

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We know that the magnetic flux density $\inline B$ can be defined as:

$B = \frac{\Phi}{A}$

(7)

which leads to:

$\Phi = B \cdot A$

(8)

from which:

$d\Phi = A \cdot dB$

(9)

Taking into account equations (9) and (6), we obtain the energy stored in the magnetic circuit:

$E_{stored} = \int N \cdot i \cdot A \cdot dB$

(10)

which can also be written as:

$E_{stored} = \int \frac{N \cdot i}{l} \cdot A \cdot l \cdot dB$

(11)

We know that if the magnetic field strength $\inline H$ is uniform, then:

$H = \frac{N i}{l}$

(12)

Taking into account (12), equation (11) becomes:

$E_{stored} = V \int H dB$

(13)

where $\inline V$ ( $\inline =A l$ ) is the volume. Although this equation was proved for a toroid, it can in fact be demonstrated for all magnetic circuits.

For a $\inline BH$ curve as the one diagramed in Figure 2, $\inline \int H dB$ is the blue shaded area:

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It can be noted that, if there is no magnetic saturation (i.e. the $\inline BH$ curve is straight), then:

$\int H dB = \frac{1}{2} H B$

(14)

We also know that the magnetic field strength $\inline H$ is related to the magnetic flux density $\inline B$ with the equation:

$B = \mu_0 \mu_r H$

(15)

or:

$H = \frac{B}{\mu_0 \mu_r}$

(16)

where $\inline \mu_0$ is the magnetic permeability of free space, and $\inline \mu_r$ the relative magnetic permeability.

Taking into account equations (16), (14), and (13), the energy stored in this particular case becomes:

$E_{stored} = V \frac{B^2}{2 \mu_0 \mu_r}$

(17)