# Reluctance and inductance are opposites, but both store magnetic energy?

Is there a (relatively) simple way to explain this? Inductance is the creation of stored magnetic energy that resists a change in current direction, but reluctance resists the creation of a magnetic field? But also stores magnetic energy?

• Opposites in what sense? Inductance lives in the electric circuit context while reluctance lives in the magnetic circuit context. Also, for goodness sakes, please stop yelling in all caps. If you want to emphasize a word, enclose it with asterisks. – Alfred Centauri Jun 24 '18 at 2:56

The fact that some entity resists a force (or a field), does not mean that it cannot store energy associated with the action of that force. In fact, this is often (if not always) the case, since a force typically has to perform some work on an entity, presumably, overcoming some resistance, in order to energize it.

For instance, a spring resists a compression force, but ends up storing energy created by that very force.

Now let's try to apply this rule to magnetic field.

Magnetic field is created by current.

As you've correctly noted, the inductance resists the increase of the current but we can (loosely) say that the inductor ends up storing magnetic field and, therefore, magnetic energy. This clearly follows the rule outlined above.

At any given current level, the reluctance decreases the magnetic flux and therefore the inductance, which is the ratio of the flux and the current. Therefore, a magnetic circuit with higher reluctance should have lower resistance to the growing current and will end up producing a weaker magnetic field with less magnetic energy. This is, again, consistent with the above rule.

• Nevertheless, V.F., several articles on reluctance say the phenomenon can store magnetic energy, like inductance, despite being the inverse/reciprocal of inductance (they use reciprocal units). – Kurt Hikes Jun 25 '18 at 3:55
• @KurtHikes Could you give me a reference to one of these articles? Reluctance is a ratio between magnetomotive force and magnetic flux, so strictly speaking, it cannot store anything. Although, in some discussions such language (i.e., "reluctance stores magnetic energy") can be used - it should not be taken literally. Reluctance reduces inductance and, therefore, reduces the ability of inductor to store magnetic field. You can make a similar statement about a gap of a capacitor, but you don't ask why the gap is inverse to capacitance, while both store magnetic field. – V.F. Jun 25 '18 at 11:36

Reluctance and inductance are opposites

Inductance $L$ relates a certain magnetic flux linkage $\lambda$ to an electric current $i$ through an $N$ turn coiled conductor:

$$\lambda(i) = N\Phi(i) = Li$$

Reluctance $\mathcal{R}$, on the other hand, relates a certain magnetomotive force (mmf) $\mathcal{F}$ to a magnetic flux through a magnetic core.

$$\mathcal{F}(\Phi) = \mathcal{R}\Phi$$

Are reluctance and inductance opposites? Well, let's see...

If $N$ turns of a conductor are wound around a portion of a magnetic core and the conductor has a current $i$ through, the mmf is just the product of the number of turns and the current:

$$\mathcal{F}(i) = Ni\,[\mathrm{ampere-turns}]$$

The magnetic flux is then

$$\Phi(i) = \frac{\mathcal{F}(i)}{\mathcal{R}} = \frac{N}{\mathcal{R}}i = \frac{L}{N}i$$

Thus,

$$\mathcal{R} = \frac{N^2}{L}$$

Define the permeance $\mathcal{P}$ as

$$\mathcal{P} = \frac{1}{\mathcal{R}}$$

and then

$$\mathcal{P} = \frac{L}{N^2}$$

That is, the permeance is proportional to the inductance. Now, the energy $W$ stored in the magnetic field of an inductance $L$ due to a current $i$ is given by

$$W = \frac{1}{2}Li^2 = \frac{1}{2}\mathcal{P}N^2i^2 = \frac{1}{2}\mathcal{P}\mathcal{F}^2$$

In summary:

(1) from the electric circuit perspective, the inductance relates the energy stored in the magnetic field to the square of the current.

(2) from the magnetic circuit perspective, the permeance relates the energy stored in the magnetic field to the square of the mmf.