# The physical meaning of inductance

What is the the physical meaning of inductance? We know that the resistance "R" of a conductor is how easily the electrons flow through it, etc., but what about the inductance "X"?

• How many magnetic feild lines can be made per amp of current. Jun 16 '16 at 2:06
• Feb 18 at 14:09

In addition to the other answers, the symbol for inductance is L, not X. The symbol for inductive reactance is $X_{L}$, and for capacitive reactance is $X_{C}$.

What is the the physical meaning of inductance?

Hugh Young, in his textbook University Physics , states the following: for a coil with $N$ turns of wire carrying current $i$, the current creates a magnetic flux $\Phi_{B}$ that passes through each turn. The coil's inductance $L$ (a.k.a. self-inductance) is given by

$$L=\frac{N\, \Phi_{B}}{i}$$

Inductors store energy in a field of magnetic flux $\Phi_{B}$ whose magnitude is a function of the current flowing through the inductor:

$$\Phi_{B}=\frac{L\, i}{N}$$

If an electric current flowing through an inductor changes with time $\frac{\mathrm{d} i}{\mathrm{d} t} \neq 0$, this changing current produces a changing magnetic field $\frac{\mathrm{d}\Phi_{B}}{\mathrm{d} t} \neq 0$, that in turn produces a non-zero electromotive force (emf) $\varepsilon$ across the inductor, measured in units of Volts, whose polarity opposes ("resists") the change in current :

$$\varepsilon = -L\frac{\mathrm{d}i}{\mathrm{d} t} = -N\frac{\mathrm{d}\Phi_{B}}{\mathrm{d} t}$$

This opposition to the change in current flow is the inductive reactance.

If an electric current flowing through an inductor is constant (does not change with time, a.k.a., direct current) $\frac{\mathrm{d} i}{\mathrm{d} t}=0$, the field of magnetic flux $\Phi_{B}$ is not changing with time (its magnitude is constant) $\frac{\mathrm{d} \Phi_{B}}{\mathrm{d} t}=0$, then no emf is produced, $\varepsilon = 0$. Therefore, an ideal inductor (zero resistance) does not oppose direct current flowing through it.

When an ideal inductor having inductance $L$ is driven with a sinusoidal current having radian frequency $\omega$ (or frequency f), the magnitude of the inductor's steady-state inductive reactance $X_{L}$ is given by

$$X_{L}=j\omega L = j2\pi fL$$

where $j=\sqrt{-1}$. This equation is derived from a phasor transformation of the first order differential equation

$$v_{L}(t)=L\, \frac{\mathrm{d}}{\mathrm{d} t}i_{L}(t)$$

where $v_{L}(t)$ is the instantaneous voltage across the inductor, and $i_{L}(t)$ is the instantaneous sinusoidal current through the inductor at time t:

$$i_{L}(t) = I_{m} cos(\omega t + \phi)$$

References

 H. Young. "Inductance," in University Physics, 8th ed. Reading, Massachusetts: Addison-Wesley, 1992, ch. 31, pp. 869-870.

• Up you go, this is the best explaination of the inductance. Jun 17 '16 at 13:30

When we use the term 'inductance', we're usually shortening what should actually be called 'self-inductance'.

When a current flows through a wire, a magnetic field is produced around that wire, with its field strength in proportion to the current strength, changing as the current changes.

We also know that a changing magnetic field will induce a voltage over a wire in proportion to the strength of the field and the rate at which it changes.

So, if we have a wire with a changing current flowing through it (think AC or DC with some ripple), then it will produce a changing magnetic field around itself. This changing magnetic field will then induce a voltage back across that same wire, but this induced voltage will be of the opposite polarity to the voltage of the original applied current.

Since the induced voltage opposes the applied voltage/current, the apparent effect is an increased impedance (complex resistance, or resistance with a phase angle), and since the strength of the induced voltage increases in proportion with the frequency of the applied voltage/current (as frequency is the rate of change), the inductor's impedance increases with frequency.

The inductance (with the unit Henrys) of an inductor is a measure of this self-inductance effect.

(Self)inductance is increased by coiling the wire around so that the coils all contribute to and share the same magnetic field.
Wrapping the coils around a magnetically permeable core also increases (self)inductance by 'concentrating' the magnetic field.

• Voltage is induced not current and it is the induced voltage that counteracts the current that directly created the mag field. Jun 17 '16 at 8:03
• Both the inductor and capacitor oppose the input (exciting) voltage by an opposite voltage but in a different manner - the inductor opposition is maximum in the beginning while the capacitor opposition is maximum at the end (I mean the case of a DC input voltage). Feb 18 at 16:46

If you expect an answer of the type

"R" of a conductor is how easily the electrons flow through it

(actually it should say how "difficult" instead of "easy")

I'd say

"L" is how difficult it is to change to current through the component.

– Abdu
Jun 15 '16 at 23:51
• V = R x I; V = L x di/dt. As L is bigger the voltage 'opposition' of the inductance to changes in current gets also bigger. Jun 16 '16 at 0:36

Inductance is just "electron inertia."

Adding inductance is like connecting a flywheel to the flowing charges. Or, it's like hanging larger weights on each moving electron.

But also that's a bit of a simplification; it's not just inertia. For example, we can increase the inertia by making the wires longer. That's like increasing the water-inertia of plumbing, by adding longer pipes. Longer pipes, more fluid mass. But with conductors, if two conductors are placed side by side, they magnetically couple together, and their "inertia" increases! For this reason, whenever we double the number of turns in a simple hoop-coil of wire, the "inertia" (inductance) does not just double as one might expect. Instead it goes up 4x. The "flywheel mass" is not just a matter of electron-quantity within the conductors.

For the ultra-simplified version which ignores these EM effects, just pretend that the wires are actually glass pipes full of filthy water, so you can see the suspended dirt moving inside the pipes. What do we get if we wind a coil of clear pipes? Then, whenever we pump a flow, the entire mass of water in the coil will all flow forward as a unit. The coil full of dirty water starts turning, as if the water was in a barrel. We've made a hydraulic flywheel: hard to get it moving, hard to make it stop, and it can store energy if we first get it spinning, then connect the hose-ends together to form a closed loop. (This reveals an important rule: to store energy in a coil, the terminals must be shorted, while storing energy in capacitors requires that the terminals be NOT-shorted.) If our spiral of transparent pipe was a one-turn shorted-out coil, it's definitely an "electron flywheel." But if it's multi-turns and shorted, then it's still a flywheel: one that has been sliced into a weird spiral-loop device. Notice that the "amp-turns" concept becomes visible, where gallons/sec flow rate through the loop is analogous to amperes.

Then, add the induction concept: if we shove a single magnet pole into the center of the fluid-charges flywheel, the magnetic field gives the flywheel a spin, and the whole thing will keep turning until slowed by electric resistance. (Or, it just keeps spinning, if it's a superconductor ring.) When it slows and stops, then yank the magnet pole back out again, and the resistive "flywheel" will briefly spin in the opposite direction.

The most common component that you would refer to as an "inductor" would be some sort of coil of wire. When you put a current through a coil, it creates a magnetic field. As the current increases, the field strength increases. There are a few ways of looking at it, but you could say that an inductor stores energy in its magnetic field, like a capacitor stores energy in its electric field.

If you put a voltage across an inductor, at the beginning there will be a very small current. This will generate an increasingly large magnetic field. This changing field will impede the flow of current, slowing down the increase of the field strength, impeding the flow less and allowing faster change. This forms a kind of equilibrium, allowing the current to increase at a certain rate until some sort of limit is reached, either from the resistance of the inductor or the limits of the power supply.

Equally, when you try to reduce the current, the magnetic field collapses, transferring the magnetic energy back into electrical energy, increasing the voltage. This forms another equilibrium until the current goes to 0.

Inductance is a measure of how much voltage a change in current produces. More inductance means a bigger voltage change for a given current change.

• Inductance is the relation between the flux created and the current you insert... Voltage appear in the picture because of an other electromagnetic concept: Faraday's law. Jun 17 '16 at 13:33
• Yeah, I originally said current, but then someone edited it Jun 17 '16 at 13:44

It may sound bizarre but I regard "inductance" as being defined by space (area within the coil) and shape (coil shape). This is for a single turn inductor in free-space (for simplicity) i.e. no interaction with other objects.

It's easy to define inductance in terms of electrical terms such as the total flux produced per amp and there's nothing wrong in that. That is the usual definition. My version kind of tells me how inductance is increased by maximizing area (for a given wire length per turn) i.e. making a circle rather than making a square or rectangle.

Hence I think of inductance in terms of area and shape.

Resistance is the ratio of the voltage across a component to the direct current through the component.

Inductance is the ratio of the voltage across a component to the rate of change of current through the component. (Here I am referring to the portion of the voltage that is produced by the rate of change of the current.)

In general, most real physical components exhibit some of each effect. Resistors are designed to emphasize and have a well-defined value of resistance, while often having minimal inductance. On the other hand, inductors are designed to emphasize and have a well-defined value of inductance while generally having minimal resistance.

Inductance is like a mass in the mechanical analogy, look into a frequency resonator to understand the coil. A whistle of a police officer at the train station is a good example for a resonator. Inside this whistle, there is a ball (weight) that moves inside a cavity, due to this weight the sound is produced. This weight is the L (coil) and the cavity is the C in an LC resonator circuit.

• Exactly! The inductor stores something like kinetic energy while the capacitor stores something like potential energy. This intuitive notion about these energy-storing elements is sufficient to understand circuits. For example, see my Wikibooks story about LC tank. Feb 18 at 16:35