Chapter 10: Inductance and the Inductor

It opposes the change in current

A changing current changes the coil's magnetic field, and that induces a voltage which always acts against the change. If it helped the change along, a current switched on would leap up at once instead of growing gradually.

Self inductance

It is self inductance: the voltage appears in the very coil whose current is changing. Mutual inductance is the effect between two coils, where one coil's changing current induces a voltage in the other.

It builds up gradually to its full value

As the current starts to rise, its magnetic field grows and induces a voltage that opposes the rise, so the current climbs gradually. It does flow: the wire's resistance is low, and an inductor does not block DC.

The voltage induced in it when its current changes

Back EMF is the voltage a coil induces in itself when its current changes; it pushes back against the change. With a steady current nothing is induced: only the small drop across the wire's resistance remains.

A voltage far above the battery's appears, and a spark jumps

With the switch open the current has no low-resistance path, so the collapsing field drives the voltage up until a spark crosses the gap. This back EMF can be many times the battery voltage, not just equal to it.

They switch the coil's current on and off

The ignition coil is the inductor and the points are its switch: when they open, the coil's current is cut off and it produces a high back EMF. The spark itself jumps across the gap of the spark plug.

Thousands of volts

Cutting off the coil's current makes it produce several kilovolts, far above the 12 V supply and enough to jump the plug's gap; a few hundred volts would not. Real ignition coils add a second, many-turn winding.

Like an electromagnet: it makes a magnetic field

Current in the turns sets up a magnetic field around the coil, exactly as in an electromagnet. Holding charge, with its energy in an electric field, is what a capacitor does.

In its magnetic field

While the current builds up, the inductor takes energy from the circuit and stores it in its magnetic field. An electric field is where a capacitor keeps its energy; heat in the wire is energy lost, not stored.

It is given back to the circuit

As the current falls, the inductor gives its stored energy back to the circuit, for example as the spark across a gap. An ideal inductor turns none of it into heat: it only borrows energy and returns it.

Electromotive force

EMF is electromotive force, an early name for voltage, so a back EMF is a voltage. Electromagnetic force is a different idea: the force between charged particles, not the push that drives a current.

Henry

Inductance, written L (I is current), is measured in the henry, H, after Joseph Henry; most coils are µH or mH. Oliver Heaviside coined the word inductance but no unit is named after him; hertz is Hz.

Winding more turns on it

More turns give more inductance. The thickness of the wire is not one of the factors that set it (turns, diameter, spacing and core), so thicker wire alone does not raise it; nor do the voltage or the colour.

The one with the larger diameter

With the turns and the length equal, the coil of larger diameter has the greater inductance, as each turn encloses more of the magnetic field. A smaller diameter gives less, not more.

It decreases

Closer spacing gives more inductance, so spreading the same turns over a longer coil lowers it. It would increase only if the turns were pushed closer together.

Its permeability

Permeability sets how strong a magnetic field a given current produces; ferrite's is far higher than air's, so a ferrite core raises the inductance. The dielectric constant is the property that matters in a capacitor.

Wind it on a ferrite core

Ferrite's permeability is far higher than air's, so the same winding on a ferrite core can have a hundred times the inductance. A plastic tube acts like air and changes almost nothing; wider spacing lowers the inductance.

From stiff wire, or wound on a plastic former

An air-cored coil, drawn as a row of loops, is wound from stiff wire that supports itself or on a plastic former, which acts like air. A brass tube would be a shorted turn, lowering the inductance and wasting power.

An air-cored coil

A ferrite core raises the inductance but adds losses, which are worst at radio frequencies, so for RF power the air-cored coil, self-supporting or on a plastic former, loses least. Resistance wire or shorted turns waste power.

It stores more energy

The inductance says how much energy a coil stores in its magnetic field at a given current, so the larger one stores more and opposes changes more strongly. Less energy would mean less inductance.

470 µH

1 mH = 1000 µH, so 0,47 mH × 1000 = 470 µH. 47 µH is ten times too small: it would be 0,047 mH.

3,9 mH

1 H = 1000 mH, so 0,0039 H × 1000 = 3,9 mH. 39 mH would be 0,039 H, ten times too much, and 3900 mH is 3,9 H.

0,03 H

In millihenry, 0,03 H = 30 mH, which beats 25 mH; 900 µH is only 0,9 mH and 2700 µH is 2,7 mH. Reading 0,03 H as 3 mH would make 25 mH look the largest.

47 × 10⁻⁶ H

Micro means millionths, so 47 µH = 47 × 10⁻⁶ H = 0,000 047 H. 47 × 10⁻³ H is milli, 47 mH, a thousand times too much.

All the time, throughout every cycle

An AC supply changes the current continually, not just when its voltage reverses, so a voltage is induced throughout the cycle. Only a changing current induces one: a steady DC current, once built up, induces none.

It reverses and opposes the reduction

While the supply raises the current, the induced voltage opposes the rise; once it starts to reduce the current, the induced voltage reverses so that it still opposes the change, now trying to keep the current up.

It can change, but it takes some time to reach its new value

The induced voltage opposes every change in the current, so a change takes time; it never happens in an instant. Opposing a change does not prevent it: the current does reach its new value.

In the 50 mH inductor

A larger inductance opposes a change in current more strongly, so with the same voltage the current in the 50 mH coil changes ten times more slowly. Its rate of change is set by the voltage and the inductance.

It dissipates no power: it gives the energy back

A resistance turns electrical energy into heat. An inductor only borrows energy while the current rises and returns it as the current falls, so no power is dissipated in it; its opposition, XL, is called inductive reactance.

XL = 2πfL

XL = 2πfL, with f in hertz and L in henry: the reactance rises in step with the frequency and with the inductance. 1 ÷ (2πfL) is upside down, like a capacitor's formula, and would fall as the frequency rose.

It falls

XL = 2πfL is proportional to frequency: halve f and XL halves. At a lower frequency the current changes more slowly, so the inductor opposes it less. A reactance that rises as the frequency falls is a capacitor's.

It halves

XL = 2πfL is proportional to L, so half the inductance gives half the reactance. Doubling would need L under the line, as C is in a capacitor's formula, where a smaller capacitor gives more reactance.

240 Ω

XL is proportional to frequency, and 7,2 MHz is twice 3,6 MHz, so the reactance doubles: 2 × 120 = 240 Ω. Halving it to 60 Ω is how a capacitor's reactance would change.

4 MHz

XL is proportional to frequency, and 200 Ω is four times 50 Ω, so the frequency must be four times 1 MHz: 4 MHz. A quarter of the frequency, 250 kHz, is what a capacitor would need.

Hz and H

XL = 2πfL takes f in hertz and L in henry to give ohms. Entering microhenry without converting gives a number a million times too big; kHz with H, or MHz with mH, gives the answer in kilohms.

440 Ω

XL = 2πfL = 2π × 14 × 10⁶ × 5 × 10⁻⁶ = 440 Ω. Reading 5 µH as 5 mH gives a thousand times too much, 440 kΩ.

100 Ω

XL = 2πfL = 2π × 7 × 10⁶ × 2,2 × 10⁻⁶ = 96,8 Ω, nearest 100 Ω. Reading 2,2 µH as 22 µH gives ten times too much, about 1 kΩ.

137 Ω

XL = 2πfL = 2π × 145 × 10⁶ × 0,15 × 10⁻⁶ = 137 Ω. Reading 145 MHz as 145 kHz gives a thousand times too little, 137 mΩ.

943 Ω

XL = 2πfL = 2π × 455 × 10³ × 330 × 10⁻⁶ = 943 Ω. Reading 330 µH as 33 µH, a zero dropped, gives ten times too little, 94,3 Ω.

31,4 Ω

XL = 2πfL = 2π × 50 × 0,1 = 31,4 Ω. Reading 100 mH as 100 H gives a thousand times too much, 31,4 kΩ; leaving out the 2π gives 5 Ω.

130 Ω

XL = 2πfL = 2π × 1000 × 0,02 = 126 Ω, nearest 130 Ω. Reading 1 kHz as 1 Hz gives a thousand times too little, about 130 mΩ.

19,5 µH

L = XL ÷ (2πf) = 220 ÷ (2π × 1,8 × 10⁶) = 19,5 × 10⁻⁶ H = 19,5 µH. Reading 1,8 MHz as 18 MHz gives ten times too little, 1,95 µH.

6 MHz

f = XL ÷ (2πL) = 1000 ÷ (2π × 27 × 10⁻⁶) = 5,89 MHz, nearest 6 MHz. Taking 1 kΩ as 1 Ω gives a thousand times less, about 6 kHz.

3 kΩ

XL = 2πfL = 2π × 100 × 5 = 3142 Ω, about 3,1 kΩ, nearest 3 kΩ. Reading 5 H as 5 mH gives a thousand times too little, about 3 Ω.

738 Ω

XL = 2πfL = 2π × 25 × 10³ × 4,7 × 10⁻³ = 738 Ω. Reading 25 kHz as 25 MHz gives a thousand times too much, 738 kΩ.

8,53 µH

L = XL ÷ (2πf) = 1500 ÷ (2π × 28 × 10⁶) = 8,53 × 10⁻⁶ H = 8,53 µH. Reading 1,5 kΩ as 15 kΩ gives ten times too much, 85,3 µH.

The magnitude of its reactance, |X|

Ohm's Law works with the size of the reactance in place of R: V = I × |X|, I = V ÷ |X| and |X| = V ÷ I. An ohmmeter reads only the wire's small DC resistance, far below the reactance at radio frequencies.

6 V

V = I × |X| = 0,04 × 150 = 6 V. Multiplying 40 by 150 without turning the milliamperes into amperes gives 6000 V, a thousand times too much.

300 Ω

|X| = V ÷ I = 9 ÷ 0,03 = 300 Ω. Dividing 9 by 30 without changing the milliamperes into amperes gives 0,3 Ω, a thousand times too small.

No: both are in ohms, but they do not add

Reactance is measured in ohms but is not resistance, so the two cannot simply be added; they combine into an impedance, met in the next chapter, here 100 Ω, not 140 Ω. Sharing a unit does not make them add.

15,9 mA

XL = 2π × 2 × 10⁶ × 50 × 10⁻⁶ = 628 Ω, so I = V ÷ |XL| = 10 ÷ 628 = 0,0159 A = 15,9 mA. Writing 0,0159 A as 0,0159 mA gives a thousand times too little.

26,5 mA

XL = 2π × 10 000 × 0,003 = 188,5 Ω, so I = 5 ÷ 188,5 = 0,0265 A = 26,5 mA. Reading 3 mH as 3 µH gives a thousand times too much, 26,5 A.

88 V

XL = 2π × 3,5 × 10⁶ × 16 × 10⁻⁶ = 352 Ω, so V = I × |XL| = 0,25 × 352 = 88 V. Reading 0,25 A as 2,5 A gives ten times too much, 880 V.

150 mH

|X| = V ÷ I = 4,8 ÷ 0,1 = 48 Ω, so L = |X| ÷ (2πf) = 48 ÷ 314 = 0,153 H = 153 mH, nearest 150 mH. Reading 4,8 V as 48 V gives ten times too much, about 1,5 H.

370 mA

XL = 2π × 50 × 2 = 628 Ω, and I = V ÷ |XL| = 230 ÷ 628 = 0,366 A = 366 mA, nearest 370 mA. Reading 2 H as 2 mH gives a thousand times too much, about 370 A.

26 V

XL = 2π × 40 × 10³ × 6,8 × 10⁻³ = 1709 Ω, so V = I × |XL| = 0,015 × 1709 = 25,6 V, nearest 26 V. Reading 15 mA as 15 µA gives a thousand times too little, about 26 mV.

16,5 V

XL = 2π × 50 × 10³ × 1,5 × 10⁻³ = 471 Ω, so V = I × |XL| = 0,035 × 471 = 16,5 V. Reading 50 kHz as 5 kHz, a zero dropped, gives ten times too little, 1,65 V.

90°

The voltage and current of a perfect inductor are 90° apart, a quarter of a cycle. At 180° they would be exact opposites, one at its positive peak while the other is at its negative peak.

A quarter of a cycle

Voltage and current are 90° apart, and 90° is a quarter of the 360° cycle, so their peaks are a quarter of a cycle apart. Half a cycle would be 180°, exact opposites.

0,1 µs

The period is t = 1 ÷ f = 1 ÷ 2 500 000 = 0,4 µs, and the 90° shift is a quarter of it: 0,1 µs. 0,2 µs, half the period, would be a 180° shift.

It stays at 90°

The voltage and current of an inductor are always 90° apart, whatever the frequency; only the reactance changes, doubling here. Doubling the angle to 180° would make them exact opposites, which no change of frequency does.

It lags behind the voltage

In an inductor the current lags the applied voltage, by a quarter of a cycle, as the VIL of CIVIL says: voltage leads current in L. A current that leads its voltage is what a capacitor gives.

In an inductor, voltage leads current

CIVIL splits into CIV, in a capacitor C the current I leads the voltage V, and VIL, in an inductor L the voltage V leads the current I. Current leading voltage is the capacitor's half, the other way round.

Capacitor: current leads voltage; inductor: voltage leads current

CIV: in a capacitor, current leads voltage; VIL: voltage leads current in an inductor. The two behave in opposite ways, and swapping them is the usual slip; neither has current and voltage in step.

The voltage across the coil

The voltage leads, so it peaks first and the current follows a quarter of a cycle later. In a capacitor it is the other way round: the current peaks first.

It returns, later in each cycle, the energy it stores

With voltage and current 90° apart, energy flows into the magnetic field during one quarter cycle and back out during the next, so on average none is used. Storing energy for ever would need an ever-growing current.

In the resistance of its wire

Real inductors are wound from wire with some resistance, which turns a little energy into heat; a ferrite core adds losses of its own. The reactance itself dissipates nothing: the energy it takes in is all returned.

Resistors

Inductances add in series and combine by reciprocals in parallel, just like resistances, as long as the coils do not share their magnetic fields. Capacitors follow the opposite rules, so they are the wrong model.

LTotal = L1 + L2 + …

In series the inductances simply add, LTotal = L1 + L2 + …, as resistances in series do. The sum of reciprocals is the rule for inductors in parallel.

1/LTotal = 1/L1 + 1/L2 + …

In parallel the reciprocals add, 1/LTotal = 1/L1 + 1/L2 + …, so the total is below the smallest inductor. Simply adding is the rule for inductors in series.

38,8 µH

In series the values add: 10 + 22 + 6,8 = 38,8 µH. 3,42 µH, less than the smallest, is what the parallel rule gives, the wrong rule here.

6 µH

1/LTotal = 1/10 + 1/15 = 0,1667, so LTotal = 1 ÷ 0,1667 = 6 µH. 25 µH, simply adding them, is the series total, the wrong rule here.

1,32 mH

1/LTotal = 1/2,2 + 1/3,3 = 0,4545 + 0,3030 = 0,7576, so LTotal = 1 ÷ 0,7576 = 1,32 mH. 5,5 mH is the series sum, the wrong rule here.

10 µH

1/LTotal = 3 × 1/30 = 1/10, so LTotal = 10 µH, a third of one inductor. 90 µH would be the total in series, the wrong rule here.

30 µH

1/L2 = 1/12 − 1/20 = 1/30, so L2 = 30 µH: 30 µH in parallel with 20 µH gives 12 µH. Subtracting, 20 − 12 = 8 µH, uses the series idea, the wrong rule here.

11 µH

In series the values add, so the extra inductor must supply 33 − 22 = 11 µH. 66 µH comes from working the reciprocal rule for parallel inductors, the wrong rule here.

20 µH

The series pair makes 20 + 20 = 40 µH, and 40 µH in parallel with 40 µH gives half of that, 20 µH. Adding all three, 80 µH, treats the parallel coil as if it were in series.

It is less than the smallest

Adding reciprocals makes 1/LTotal bigger than any single 1/L, so the total is below even the smallest inductor. A total above the largest is what the same inductors give in series.