Inductance

The inductance of an electrical circuit is a coefficient which translates the fact that a current the beam creates a magnetic field through the section surrounded by this circuit. It results a Flux from it from the magnetic field through the section limited by this circuit.

The inductance is equal to the quotient of the flow of this magnetic field by the intensity of the current crossing the circuit. The unit of inductance is the Henry (H).

By extension, one indicates by inductance all Electrical circuit or electric Dipôle which by its construction has a certain value of inductance (physical size). These dipoles are generally reels, often called inductances or coil by abuse language (or Métonymie) as for resistance.

Clean inductance

In English coil inductance who gave the word coil
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 Flux of the magnetic field (called flow of induction formerly) $\backslash \; scriptstyle\; \{\backslash \; Phi\}$ . Inductance L of the electrical circuit is then defined like the relationship between the flow embraced by the circuit and the current:

$L\; =\; \backslash \; frac\; \{\backslash \; Phi\}\; \{I\}\; \backslash ,$

It is important to specify that flow $\backslash \; scriptstyle\; \{\backslash \; Phi\}$ in question is that produced by the current I and not that coming from another source (running, magnet, etc).

In spite of its popularity, this definition presents two disadvantages. First is that the definition of inductance is given according to the flow $\backslash \; scriptstyle\; \{\backslash \; Phi\}$ which is an inaccessible physical size directly. There does not exist means of measuring the magnetic flux without varying it according to time. The second disadvantage is that 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”). ' A second definition which does not present these disadvantages is:

$v=L\; \{di\; \backslash \; over\; dt\}\; \backslash ,$

where:

• $\backslash \; scriptstyle\; \{L\}$ is the clean inductance of the circuit or component.
• $\backslash \; scriptstyle\; \{v\}$ is the terminal voltage of the circuit.
• $\backslash \; scriptstyle\; \{di\; \backslash \; over\; dt\}$ is the variation of the current which crosses the circuit with time (measured in amps/second).
• $\backslash \; scriptstyle\; \{v\}$ and $\backslash \; scriptstyle\; \{I\}$ are instantaneous values.

the signs are such as the terminal of the circuit or composing by which the current which returns increases is positive compared to the other. This sentence can be read in the other direction: when one applies a tension to an inductance, the current which returns by the positive end increases with time. (to be translated into good French!)

It is only by using this definition that one could measure the value of the inductance of a circuit and, from there, to determine the magnetic flux are equivalent which crosses “circumscribed surface” equivalent but it would be necessary for that which the terminal voltage of this portion of circuit depends only on magnetic phenomena. Unfortunately this one depends on a great number of very diverse physical effects (of which the Joule effect), which prevents any possible measurement of the inductance of a portion of circuit

Moreover, this definition is not valid only for portions of circuit presenting of non-linearities (EP: inductances with ferromagnetic core). The value of inductance will depend then on the value of the current and its history (Hystérésis)).

Part of the flow produces by the current crosses the cable itself. It is thus advisable to distinguish the external inductance and the inductance interns of 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 kilohertzs and inductance does not vary any more.

Mutual inductance

When a circuit 1 crossed by a current noted $i\_1\; \backslash ,$, produces a magnetic field through a circuit 2, one can write:

$M\_\; \{1/2\}\; =\; \backslash \; frac\; \{\backslash \; Phi\_2\}\; \{i\_1\}\; \backslash ,$

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

The dipole “Inductance”, or winds

Its symbol in the diagrams is L . An inductance L is a dipole such as:

$U\; =\; L\; \backslash \; frac\; \{di\}\; \{dt\}\; \backslash ,$

This relation comes from the expression of the flow of the magnetic field and the Faraday's law which will be seen into magnetostatic:

$U\; =\; \backslash \; frac\; \{D\; \backslash \; Phi\}\; \{dt\}\; \backslash ,$ and of $\backslash \; Phi=\; L\; \backslash \; cdot\; I\; \backslash ,$

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. The term stored power 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.

The instantaneous power provided to inductance is equal to:

$P\; =\; U\; \backslash \; cdot\; I\; =\; L\; \backslash \; frac\; \{di\}\; \{dt\}\; \backslash \; cdot\; I\; \backslash ,$

By using the following mathematical transformation:

$\backslash \; frac\; \{D\; (i^2)\}\{dt\}\; =i\; \backslash \; cdot\; \backslash \; frac\; \{D\; (I)\}\{dt\}\; +\; \backslash \; frac\; \{D\; (I)\}\{dt\}\; \backslash \; cdot\; I\; =\; 2\; \backslash \; frac\; \{D\; (I)\}\{dt\}\; \backslash \; cdot\; I\; \backslash ,$

the relation is obtained:

$P\; =\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; cdot\; L\; \backslash \; frac\; \{D\; (i^2)\}\{dt\}\; \backslash ,$

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:

$W\; =\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; cdot\; L\; (i^2\_\; \{tf\}\; -\; i^2\_\; \{Ti\})\; \backslash ,$

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 sound inductance will be large. This property is often used to remove small nondesired current fluctuations.

The effect of inductance vis-a-vis the variations of the current is similar in mechanics to the effect of the mass vis-a-vis 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 an intensity, it is a little to stop a car by sending it against a wall.

Precaution for use

One should not exceed in instantaneous value the maximum value of the intensity prescribed by the manufacturer. In the event of going beyond, even very in short, one is likely “to saturate” the Magnetic circuit, which causes a reduction in the value of inductance being able to involve an overcurrent.

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.

Impedance

At every moment

$\backslash \; frac\; \{di\}\; \{dt\}\; =\; \backslash \; frac\; \{U\}\; \{L\}$.

There is $u\; (T)\; =\; U\; \backslash \; sqrt\; \{2\}\; \backslash \; sin\; (\backslash \; Omega\; T)$ and $i\; (T)\; =\; I\; \backslash \; sqrt\; \{2\}\; \backslash \; sin\; (\backslash \; Omega\; T\; -\; \backslash \; varphi)$.

$\backslash \; frac\; \{di\}\; \{dt\}\; =\; \backslash \; frac\; \{U\}\; \{L\}\; \backslash \; sqrt\; \{2\}\; \backslash \; sin\; (\backslash \; Omega\; T)$

Thus $i\; (T)\; =\; \backslash \; left\; (\backslash \; frac\; \{U\}\; \{L\}\; \backslash \; sqrt\; \{2\}\; \backslash \; right)\; \backslash \; left\; (-\; \backslash \; frac\; \{1\}\; \{\backslash \; Omega\}\; \backslash \; cos\; (\backslash \; Omega\; T)\; \backslash \; right)\; =\; \backslash \; frac\; \{U\}\; \{L\; \backslash \; Omega\}\; \backslash \; sqrt\; \{2\}\; (-\; \backslash \; cos\; (\backslash \; Omega\; T))$

One obtains finally: $i\; (T)\; =\; \backslash \; frac\; \{U\}\; \{L\; \backslash \; Omega\}\; \backslash \; sqrt\; \{2\}\; \backslash \; sin\; \backslash \; left\; (\backslash \; Omega\; T\; -\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; \backslash \; right)\; =\; I\; \backslash \; sqrt\; \{2\}\; \backslash \; sin\; (\backslash \; Omega\; T\; -\; \backslash \; varphi)$. Thus: $I\; =\; \backslash \; frac\; \{U\}\; \{L\; \backslash \; Omega\}$.

• Law of Ohm in effective values: $U\; =\; L\; \backslash \; Omega\; I\; =\; ZI\; \backslash \; Leftrightarrow\; Z\; =\; L\; \backslash \; Omega\; =\; 2\; \backslash \; pi\; Lf$ with $Z$ in Ohms, $L$ in Henrys, $\backslash \; omega$ in rad/s and $f$ in Hz.

• Uninterrupted, $f\; =\; 0$: a perfect reel behaves like a short-circuit (indeed: $Z\; =\; 0\; \backslash \; Rightarrow\; U\; =\; 0\; \backslash \; cdot\; I\; =\; 0$).

In complexes

$\backslash \; underline\; \{U\}\; =\; \backslash \; underline\; \{Z\}\; \backslash ,\; \backslash \; underline\; \{I\}$ with

• $\backslash \; underline\; \{U\}\; =\; 0$
• $\backslash \; underline\; \{I\}\; =\; =\; \backslash \; frac\; \{U\}\; \{L\; \backslash \; Omega\},\; -\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}\; rad$

From where: $|\backslash \; underline\; \{Z\}|\; =\; Z\; =\; \backslash \; frac\; \{U\}\; \{I\}\; =\; L\; \backslash \; omega$; and $Arg\; (\backslash \; underline\; \{Z\})\; =\; \backslash \; varphi\; =\; \backslash \; frac\; \{\backslash \; pi\}\; \{2\}$

One from of deduced that $\backslash \; underline\; \{Z\}\; =\; \backslash \; mathbf\; \{J\}\; L\; \backslash \; omega$ with $\backslash \; underline\; \{Z\}$ imaginary pure from the form $\backslash \; underline\; \{Z\}\; =\; \backslash \; mathbf\; \{J\}\; X$ and $X\; =\; L\; \backslash \; Omega\; >\; 0$.

Opening of the circuit

Let us examine the practical behavior of an inductance when one stops the circuit which feeds it. In the diagram of right-hand side we represented an inductance which takes care through a resistance and a switch. The condenser drawn in dotted lines represents the stray capacities of inductance. Although this condenser is drawn separates, actually, it forms share of inductance because it represents the stray capacity between the turns of winding. Any winding has stray capacities, even those especially conceived to decrease them, like winding in “honeycomb”.
At one moment $\backslash \; scriptstyle\; \{t\_\; \backslash \; circ\}$ the switch opens. By looking at the definition of inductance:

$v\; =\; L\; \{di\; \backslash \; over\; dt\}$

we see that, so that the current which crosses an inductance stops instantaneously, one would need the appearance of an infinite tension on its terminals, which is impossible. What makes the current? Well, it continues to circulate. By or? The current “manages”. At the beginning, the only way available is through the stray capacities. The current continues to circulate by negatively charging the high point with the condenser in the drawing. We find ourselves with a circuit LLC which will oscillate with a pulsation:

$\backslash \; textstyle\; \{\backslash \; Omega\; =\; \{1\; \backslash \; over\; \backslash \; sqrt\; \{LLC\}\}\}$

where $\backslash \; scriptstyle\; \{C\}$ is the equivalent value of the stray capacities. If the electric insulation and the electric rigidity of winding are sufficient to resist the high voltages and if the switch stops the circuit well, the oscillation will continue with a decreasing amplitude because of the dielectric Pertes of the stray capacities and resistive Pertes of winding. If, moreover, the reel comprises a core Ferromagnétique there will be also losses due to the Hystérésis. It should be noted that the maximum tension of the oscillation can be very high. This is worth not Surtension to him. This comes owing to the fact that the maximum of tension corresponds to the moment when all the energy of inductance $\backslash \; scriptstyle$ was transferred to the stray capacities $\backslash \; scriptstyle$. If these last are small, the tension can be very large and of the electric arcs can occur between turns of winding or the open contacts of the switch. Although the electric arcs are generally harmful and dangerous, in certain applications they useful and are wished. This case of the arc cutting, gas-discharge lamps, Light-arc furnaces used out of metallurgy, etc In the case of the arc cutting, the switch of our diagram is formed by contact between metal to weld and the electrode.
Notice that there is many overpressures, but not surcourants. Immediately after the opening the current remains the same one then it decreases. What arrives when the arc appears depends on the electric characteristics of the arc. And the electric characteristics of the arc depend especially on the current which circulates there and the length. When the current is large (tens of amps) the arc is formed by a thick way of atoms and ionized molecules which have a low resistance to the passage of the current. These thick arcs have a raised thermal inertia and remain conducting during several milliseconds without passage of the current. They do not die out in the passing by zero of the alternative course. One can weld with the arc with current not rectified. These arcs dissipate hundreds of Watts and can melt of metals and create fires. If the arc occurs between the contacts of the switch, the circuit is not really opened and the current continues to circulate.
The nondesired arcs constitute a problem serious and difficult to solve when high voltages and great powers are concerned.
When the currents are small, the arc cools quickly and stops leading the current.
In the drawing of right-hand side we illustrated a particular case which can occur, but which is only one of the possible cases. We increased the time scale around the opening of the switch and the formation of the arc.
After the opening of the switch, the terminal voltage of inductance increases (with the reversed sign). At the moment $\backslash \; scriptstyle\; \{t\_1\}$ the tension is sufficient to create an arc between two towers of the reel. The arc has a low electrical resistance and discharges the stray capacities quickly. The power, instead of charging the stray capacities, starts to be on by the arc. We drew a case in which the terminal voltage of the arc remains with little close constant. As the tension is negative, the current in the reel decreases until, at the moment $\backslash \; scriptstyle\; \{t\_2\}$, it is not sufficient to maintain the arc and this one dies out and stops leading. The power starts again to be on by the stray capacities and, this time, the oscillation continues while diminishing without creating new arcs because the tension will not reach values too high.
Let us recall that it is only about one possible case. Other cases can occur.
One can explain why it happens that one receives an electric shock by measuring the electrical resistance of a winding with simple a Ohmmètre, whereas this one delivers only a few milliamperes under a few volts. The reason is that when one disconnects the test probes of the ohmmeter, if one continues to touch the terminals of winding, the milliamperes which circulated, continue to do it, but while passing by the fingers.
The rule is that, to avoid the arcs and overpressures, it is necessary to protect the circuits by envisaging a way for the current from inductance when the circuit stops. In the diagram of right-hand side one finds the example of a transistor which controls the current in a reel (that of a relay, for example). When the transistor is blocked, the current which circulates in the reel charges the stray capacities and the tension of the collector increases and can easily exceed the tension collector-bases authorized and destroy the transistor. While placing a diode, as in the diagram, with the cut, the current will circulate by the diode and the tension with the collector will not exceed the supply voltage plus 0,6 V of the diode. The functional price of this protection is that the current met longer to be decreased, which, in certain cases, can be a disadvantage. If it is the case, one can decrease this time by replacing the rectifier diode not a Diode Zener or a Diode Transil.
It should not be forgotten that the safety device will have to be able to almost absorb all the energy stored in inductance.

External bonds

• Calculation of the coaxial reels

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