Inductance

According to the theorem of Amp very current traversing a circuit creates a magnetic field through the section which it surrounds. The inductance of this circuit is the quotient of the flow of this magnetic field by the intensity of the current crossing the circuit. The unit of inductance is Henry (H). In any rigor this term has interest only for the situations in which flow is or perhaps considered as proportional to the current.

By synecdoque, one calls inductance any electronics component intended by his construction to have a certain value of inductance, as one calls resistance of the components used for this property. These dipoles are generally reels, often called coil.

Simple reel

Clean inductance

Symbol of inductance

Symbol of inductance

The most current definition of clean inductance is the following one: The surface circumscribed by an electrical circuit traversed by a current I is crossed by the flow of the magnetic field called formerly flow of induction Φ. inductance L of the electrical circuit is then defined like the relationship between the flow embraced by the circuit and the current : L = Φ ⁄ I

Let us specify that flow Φ is that produced by current I and not that coming from another source.

Visualization of a surface circumscribed by a reel with three whorls.

- This definition presents three disadvantages.
- The definition of inductance is given according to the flow Φ which is an inaccessible physical size directly. There does not exist means of measuring the magnetic flux without varying it according to time.
- The surface circumscribed by the circuit is not always easy to determine and in certain cases, it does not even exist, for example if the circuit makes a node.
- The definition supposes that flow is proportional to the intensity of the current. It is not the case when flow crosses a magnetic material, in this case, one observes a magnetic hysteresis loop.

- A second definition, which presents only the third disadvantage, rises from the Faraday's law which is only the really applicable one in all the situations.
- e (t) = - dΦ(t) ⁄ dt
- if L constant one is deduced from it
- e (t) = - L * di(t) ⁄ dt
- where L is the clean inductance of the circuit or component
- E is the electromotive force at the boundaries of the generating circuit in convention
- di(t) ⁄ dt is the variation of the current which crosses the circuit with time measured in amps ⁄ second.
- E and I are instantaneous values.
- We notice that
- the relation applicable only to the situations in which flow is or perhaps is not regarded as proportional to the current.
- when the current is constant, di ⁄ dt is null and consequently the electromotive force E self-induced is null too.
- The sign (-) indicates that the self-induced electromotive force at the boundaries of inductance is opposed to the variations of the current which crosses it.
- when one applies a constant tension to an inductance, the current which returns by the positive end increases with time.

From this definition, one could measure the value of the inductance of a circuit, then to determine the magnetic flux are equivalent which crosses equivalent circumscribed surface if the terminal voltage of this portion of circuit depended only on magnetic phenomena. Unfortunately, a great number of very diverse physical effects influence this tension. One cannot thus measure the inductance of a portion of circuit.

As already indicated, this definition is not valid for portions of circuit presenting of non-linearities. One can define an inductance which depends on the value of the current and its history : L = ∂Φ ⁄ ∂I

Part of the flow produced by the current crosses the cable itself. It is thus advisable to distinguish external inductance and internal inductance from a circuit. The internal inductance of a cable decreases when the frequency of the current increases because of the skin effect or effect of skin. In practice, the effect of skin is almost complete from one or two tens of kilocycle and inductance does not vary any more.

Mutual inductance

A circuit crossed by a current noted i_{1}, produces a magnetic field through a circuit, one can write : M_{½} = Φ_{2} ⁄ i_{1}

The value of this mutual inductance depends on the two geometrical characteristic, involved circuits, many whorls and of their relative position: distance and orientation.

The dipole Inductance, or winds

symbolization of a perfect reel of inductance L

tension on its terminals U

intensity of the current which crosses it I in convention receiving.

symbolization of a perfect reel of inductance L

tension on its terminals U

intensity of the current which crosses it I in convention receiving.

Its symbol in the diagrams is L. an inductance coil L is a dipole such as the tension on these terminals is proportional to derived from the intensity of the current which crosses it in convention receiver : u = L * di ⁄ dt

This relation comes from the expression of the magnetic flux into magnetostatic : u = dΦ ⁄ dt et de Φ = L * i, consequence of the definition of L.

This equation shows that the intensity of the current crossing an inductance cannot undergo discontinuity, that would correspond indeed to an infinite tension on its terminals, therefore with an infinite power.

Instantaneous power

Note: one can store only energy. Stored the power term is thus an abuse language which actually corresponds to the power that one provides to the inductance and which comes to increase the energy stored in the latter.

In convention receiver the instantaneous power provided to inductance is equal to : P = u * i = L * di ⁄ dt * i

By using the following mathematical transformation : d(i²) ⁄ dt = i * d(i) ⁄ dt + d(i) ⁄ dt * i = 2 * d(i) ⁄ dt * i, one obtains the relation : P = ½ * L(i²_{tf} - i²_{ti})

The instantaneous power provided to an inductance is related to the variation of the square of the intensity which crosses it: if this one increases, inductance stores energy. It restores some in the contrary case.

The energy exchanged between 2 moments Ti and tf is worth : W = ½ * L(i²_{tf} - i²_{ti})

It results from it that it is difficult to vary quickly the current which circulates in a reel and this more especially as the value of its inductance will be large. This property is often used to remove small nondesired current fluctuations.

The effect of inductance towards the variations of the current is similar in mechanics to the effect of the mass towards the variations speed: when one wants to increase speed it is necessary to provide kinetic energy and this more especially as the mass is large. When one wants to slow down, this energy should be recovered. To disconnect a reel traversed by a current, it is a little to stop a car by sending it against a wall.

Power in sinusoidal mode

In sinusoidal mode, an ideal inductance (of which resistance is null) does not consume active power. On the other hand, there are storage or restitution of energy by the reel during the variations of the intensity of the current.

execution time customer :

runtime server : 0.478 seconds