The circuit that fights its own change

A circuit carrying a current is threaded by the magnetic field that current makes. Change the current and the flux through the circuit changes, and a changing flux produces an emf. So the circuit produces an emf in itself, in the direction that opposes what is being done to it — and it does so whether or not anything else is present, whether or not the circuit was designed with a coil in it, and whether or not anybody wanted it to.

That single sentence turns the equation of a circuit from an algebraic one into a differential one. Ohm’s law says the current is the voltage divided by the resistance, and it is an answer: give the voltage, get the current. With self-inductance the equation contains the rate of change of the current as well as the current, and the answer is no longer a number but a history.

The number a coil actually has #

The quantity is defined by the flux a circuit’s own current puts through it: $\Phi = LI$, and the constant $L$ depends only on the shape. For a long solenoid every turn is threaded by the full interior field $\mu_0 n I$, there are $N$ turns of area $A$, and $L = \mu_0 N^2 A/\ell$ follows in two lines. The trouble is that the two lines used a limit and the result is quoted as a formula.

The reason for the shortfall is visible in the derivation’s own assumption. Every turn is supposed to be threaded by the full interior field, and near the ends of a short coil the field has already begun to spread out — so the end turns link less flux than the middle ones, and the total is less than $N$ times the middle turn’s. The ratio is Nagaoka’s coefficient, tabulated in 1909 and still the standard correction, and the computation here reproduces it without being told it.

That is worth a moment because it is a common shape of mistake. A limit taken for a good reason becomes a formula, the formula acquires a name, and the condition under which it was derived stops being quoted. The result is not a small error: a coil built as a squat pancake, which is what a great many are, has a third of the inductance the formula predicts. The same shape of mistake is at work in the field outside a solenoid, which the same limit says is zero and which is not.

That the two curves coincide is the useful part. The inductance itself changes by a large factor between the two coils — the number of turns enters squared and the area linearly — but the fraction of the ideal value does not change at all, because it is a statement about the shape of the field and the field’s shape does not know how many turns are making it. So the correction can be tabulated once, against one dimensionless number, and applied to any coil. Whenever a correction factor turns out to depend on a single ratio, something has been learned about what the answer can possibly depend on, and that is usually worth more than the factor.

Note also what the sum-over-pairs computation says about the definition. Self-inductance is the sum of every turn’s mutual inductance with every other, plus each turn’s with itself. There is no separate physics in “self” — it is the same mutual inductance the two-circuit case uses, applied to one circuit and its own parts.

What it does to a circuit #

Close a switch on a coil and a resistor across a supply and the current does not appear. It grows, exponentially, with a time constant $L/R$ that has nothing to do with the supply and everything to do with the coil.

Two things in that figure are worth stating carefully because they are usually stated wrongly.

The first is what the coil opposes. At the instant of closing, the voltage across the coil is the whole supply and the current is zero; at the end, the current is at its full value and the voltage across the coil is zero. A coil carrying a steady current is a piece of wire. It does not oppose current at all — it opposes change, which is why the whole of its behaviour lives in a derivative and why a circuit’s response to a step tells more about it than its response to anything steady.

The second is the energy. The area under the field’s curve is $\tfrac12 LI^2$, which is the standard result, and the standard result usually arrives as a formula rather than as a story. The story is that the supply is pushing charge against an emf the circuit is generating in opposition, and the work done against that emf is exactly what ends up in the field. Integrating it numerically here gives $\tfrac12 LI^2$ to four parts in ten thousand, which is a check rather than a restatement, because the integrand is a power and the closed form is not used to compute it.

There is a third thing the figure shows that is easy to miss. Over the whole rise, the resistor dissipates exactly as much as the field stores. That is not a coincidence of these numbers: for a step of voltage into an inductor and resistor in series it is exact, and it means that charging a magnetic field through a resistance is at best fifty per cent efficient — the same fifty per cent, and for the same reason, as charging a capacitor through a resistor.

Why the field is where the energy is #

The energy could be booked as a property of the circuit — $\tfrac12 LI^2$, a number attached to a current — and for circuit purposes it usually is. There is a second account, in which the energy is spread through space at a density $B^2/2\mu_0$, and for a long solenoid the two agree exactly: the field inside is $\mu_0 nI$, the volume is $A\ell$, and the product is $\tfrac12 \mu_0 n^2 I^2 A\ell = \tfrac12 LI^2$.

The two accounts are indistinguishable as long as nothing leaves. They part company when something does, and the argument for preferring the field account is the one made about electrostatic energy and is the same argument: a changing current somewhere sends energy off at the speed of light, and an account that keeps the energy attached to circuits has to explain where it was in the interval between leaving one and arriving at the other.

For a coil the practical consequence is about geometry rather than philosophy. Two coils of the same inductance can hold their energy in very different volumes, and the volume is what decides whether the field is small enough not to matter to the neighbours and whether the material inside is being asked to hold more energy density than it can. A superconducting magnet’s stored energy is a real hazard, measured in megajoules, and it is stored in the bore — which is why what happens when the material stops being superconducting is the central engineering problem of such a magnet rather than an afterthought.

Opening the switch #

Closing a switch on an inductor is gentle. Opening one is not, and the asymmetry is complete.

The reasoning is worth doing slowly because it is entirely a statement about what is and is not allowed to jump. The energy in the field is $\tfrac12 LI^2$, so a discontinuous change in current would be a discontinuous change in energy, which needs an infinite power. The current therefore cannot jump. The voltage is under no such constraint — it is not the state variable, and no energy is attached to it — so when the current’s path is suddenly made a poor one, the current keeps flowing and the voltage becomes whatever that path requires.

That is why an inductive load destroys switches, why a relay coil needs a diode across it, and why an ignition coil works: the last is the effect used deliberately, with a twelve-volt supply and a break arranged to produce tens of kilovolts. It is also why the eddy currents in a falling magnet’s tube behave so differently from a wire circuit — in a solid the current has no switch to open and always finds a path.

The two figures together make the point that the voltage is not a property of the coil. Its magnitude is the current times whatever resistance the current is forced into, so it is set by the switch, not by the inductance. What the coil sets is the duration — $L$ over the new resistance — and therefore the energy delivered, which is the whole $\tfrac12 LI^2$ however fast it is given up.

The one that is not a coil #

Nothing above requires a coil. Every circuit encloses some area and therefore has an inductance, and the fact matters as soon as anything changes quickly.

A loop of wire ten centimetres square has an inductance of a few hundred nanohenries. That sounds negligible, and it is negligible for anything slow. For a current changing at a hundred amps a microsecond — an ordinary figure inside switching electronics — it is a few tens of volts, generated by nothing but the loop’s own geometry, appearing in series with whatever was supposed to be there. The remedy is geometric rather than electrical: reduce the area the current encloses, which is why fast circuits are laid out with their outward and return paths on top of one another and why a ground plane is not a luxury.

The same reasoning explains a fact about mutual inductance that looks like a coincidence. Two circuits couple equally in both directions because the coefficient is a geometrical integral symmetric in the two loops; self-inductance is that integral with both loops the same, so it is automatically positive and cannot be made zero by any arrangement of a closed circuit. A circuit with no inductance is a circuit enclosing no area, which is a circuit that is not closed.

The lower bound nobody can get under #

There is a limit on how small an inductance can be made, and it is geometric rather than technological.

The energy in the field of a current-carrying loop is the integral of $B^2/2\mu_0$ over all space, and the field near a wire is $\mu_0 I/2\pi r$ however the wire is arranged. So a loop of perimeter $P$ made of wire of radius $a$ has an inductance of roughly $(\mu_0 P/2\pi)\ln(P/a)$, and the logarithm is the only place the wire’s thinness enters. Making the wire fatter helps logarithmically, which is to say hardly; making the loop smaller helps in proportion, which is why the only real remedy is area.

The consequence is a number every designer of fast electronics eventually meets: about one nanohenry per millimetre of loop. A component with leads five millimetres long has ten nanohenries of lead inductance whatever it is made of, and at a current changing at a hundred amps a microsecond that is a volt — appearing in series with a component whose whole purpose may be to hold a voltage steady to a millivolt. No better capacitor fixes it. Only a shorter loop does.

The same arithmetic run the other way explains why an inductance large enough to be useful is always a coil. Getting to a millihenry with a single turn would need a loop kilometres round; winding a thousand turns into a few centimetres does it, because the flux from each turn threads all the others and the inductance goes as the square of the number rather than in proportion to it. That squaring is the whole reason coils exist, and it is a direct consequence of self-inductance being a double sum over turns.

Where the model stops #

The inductance here is a constant, and it is only constant while the material is linear. A coil wound on an iron core has an inductance that depends on the current, because the iron’s permeability does — and at high enough current the iron saturates and the inductance collapses toward the air-cored value. That is not a small correction: an inductor designed on the linear figure can lose most of its inductance at its rated current, which is why the magnetisation curve of the core is part of the design rather than a property of a material somebody else chose.

The current is assumed to be spread evenly over each wire’s cross-section. At high frequency it is not: the current confines itself to a skin whose depth falls as the inverse square root of frequency, and the internal part of the inductance — the flux inside the wire itself — falls with it. So a coil’s inductance is slightly frequency-dependent even with no core, and its resistance is very much so.

Nothing here radiates. The whole treatment is quasi-static: the field is taken to be the field the present current would make, everywhere, at once. That is excellent for a coil a few centimetres across at kilohertz and wrong at the frequency where the coil is a fair fraction of a wavelength, which for ten centimetres is a few hundred megahertz. Past that a coil is an antenna and the energy going missing is not going into the resistor.

And the capacitance between turns is ignored entirely. Every real coil has some, so every real coil is resonant, and above that self-resonance an inductor behaves as a capacitor. The frequency is often lower than expected — a few megahertz for a large air-cored coil — and a component used above it is not doing what its name says.

The name, and what it hides #

The quantity is called inductance and the phenomenon induction, and both words carry an implication that is worth resisting: that something is being induced in something, by an agent outside it. In the mutual case that reading is fair — one circuit acts on another. In the self case there is no other circuit, and the language of an agent and a patient has nowhere to attach.

What is actually true is a statement about a single object and its own field. A current distribution has a magnetic field; that field has an energy; the energy depends on the current; and a system whose energy depends on a variable resists changing it. The emf is the derivative of that dependence, and Lenz’s law — that the effect opposes the change producing it — is not a separate principle but the statement that the energy is a minimum, positive-definite function of the current. It could not have come out the other way without the field energy being negative.

Reading it that way also makes the sign of the switch-off obvious. Interrupting the current is asking the field to give its energy back on a timescale the circuit did not agree to, and the circuit’s only means of returning it is to drive the current through whatever is available. The spark is the field’s energy arriving somewhere; the four thousand volts is what it costs to deliver it that fast.

What the pictures cannot show #

The transient figures draw a current and a voltage and cannot show what the field is doing between them. During the break, the energy leaves the field and arrives in the arc, and the mechanism is the collapsing field driving a current through whatever will carry one; a picture of the field lines shrinking inward would show the transfer directly and is a much harder thing to draw honestly than a pair of curves.

The inductance figure draws a ratio and hides the absolute numbers, which for the coil computed are a few tens of microhenries. That is deliberate — the ratio is the part that is general and the microhenries are the part that is about one coil — but it means the figure cannot be read for a design. What it can be read for is the correction factor, which is the same for every coil of that shape.

Where the ladder goes next #

The induction ladder began with the field that makes the other, and only while it is changing, went through the magnet that falls slowly, the rule that is two laws wearing one coat and the coupling that is the same both ways. This rung applies the law to a circuit and its own field. The rungs after it: the transformer, where the coupling and the self-inductances are designed together and the leakage is what is left over; the skin effect in earnest, where the current’s own field decides where the current goes; and the transition from a lumped inductance to a transmission line, which is where the quasi-static assumption stops.

The habit worth carrying away is about state. Ask which quantity in a system cannot jump, and the rest of its behaviour follows. In a circuit with an inductor it is the current, and everything about switching — the exponential rise, the impossibility of an instant stop, the voltage spike — is that one constraint applied at different moments.