Chapter 9: Capacitance and the Capacitor

Two conductors close together, with an insulator between them

A capacitor is two conductors, the plates, held close together with an insulator, the dielectric, between them. A resistive layer would let a steady current flow from plate to plate, so the part would be a resistor.

Pumping electrons from one plate, through itself, to the other

The battery acts as an electron pump: electrons leave one plate, pass through the battery and collect on the other, so one plate turns positive and the other negative. None cross the gap between the plates.

It falls as the capacitor's voltage rises, and then stops

The capacitor's growing voltage opposes the battery, so the current falls, and when the two voltages are equal it stops: the capacitor is fully charged. It cannot rise, as more charge means more opposition.

The lamp glows briefly, fading as the capacitor discharges

Current flows from the positive plate through the lamp to the negative one. As the charges even out, the voltage falls, so the current and the glow die away; nothing keeps the lamp lit once the energy is used.

A flexible diaphragm across the pipe

A diaphragm stretches under pressure, storing energy, and pushes the fluid back when the pressure goes. None passes through it, just as no electrons cross the insulator. A narrow section passes a steady flow, like a resistor.

Two parallel lines of equal length

A capacitor is drawn as two parallel lines of equal length, its plates, with a gap for the insulator between them. A long and a short line make a cell, the long one being its positive terminal.

The dielectric

The material between the plates is the dielectric: air, plastic or a liquid, for instance. An electrode is a conductor through which current enters or leaves, like the plates themselves, not the layer between.

It increases

A higher dielectric constant gives more capacitance from the same plates and spacing. The capacitance would fall only for a material with a lower dielectric constant, and air, close to 1, is about the lowest.

1

A vacuum has a dielectric constant of exactly 1, and air is very close to 1; other materials have higher values. It is not 0: plates with a vacuum between them still have capacitance.

How well a part stores energy in an electric field

Capacitance measures how much energy can be held in the electric field between two conductors. Storing energy in a magnetic field is what a coil, an inductor, does; turning energy into heat is a resistor's job.

In the electric field between its plates

The energy of a charged capacitor is held in the electric field between its plates, and comes back out when it discharges. A magnetic field holds the energy of a coil carrying a current, not of a capacitor.

It increases

More plate area gives more capacitance, so doubling the area raises it, to about twice. It would decrease only if the plates were made smaller or moved further apart.

It decreases

Capacitance falls as the gap between the plates widens. It rises when the plates are brought closer together, not when they move apart.

The working voltage marked on it

Capacitance depends on the plate area, the plate spacing and the material's dielectric constant. The working voltage is only the highest voltage the capacitor can safely take; it does not change the capacitance.

It has a high capacitance

A large capacitor is one with a high capacitance, so it can hold much energy; its case need not be big. A big case may simply mean a high working voltage.

Farad

Capacitance is measured in the farad, symbol F, so 10 µF is ten microfarads and 68 pF is 68 picofarads. The capacitor stores energy in a field, but the F names the unit, not the field.

A capacitance of 0,47 µF

MFD is an old marking for µF, used because makers had trouble printing the µ, so 0,47 MFD is 0,47 µF of capacitance. It is not milli: 0,47 mF would be a thousand times more.

Much like the resistor colour code

Capacitor colour codes are similar to the resistor colour code of chapter 5, so the same reading skills apply; other capacitors simply have their values printed on them. It is not a wholly separate system.

4,7 nF

1 nF = 1000 pF, so 4700 pF ÷ 1000 = 4,7 nF. 47 nF would be 47 000 pF, ten times too much; each step from pF to nF to µF is a factor of 1000.

680 nF

1 µF = 1000 nF, so 0,68 µF × 1000 = 680 nF. 68 nF is ten times too small: it would be 0,068 µF.

15 000 pF

1 µF = 1 000 000 pF, so 0,015 µF × 1 000 000 = 15 000 pF, or 15 nF. 1500 pF has one zero too few, a slip of ten.

It is an AC source

The '~' sign, a small sine wave, marks an AC (alternating) source. In arithmetic '~' can mean 'about', but on a source it says nothing about how exact the voltage is.

Almost none: a tiny leakage at most

Once charged to the source voltage, the capacitor balances the source, so no current flows except a very small leakage through the imperfect insulator. A steady current would need charge to cross the insulator.

The capacitor charges, discharges and recharges the other way, in turn

As the source reverses, the capacitor charges, discharges and charges again with the opposite polarity, so current flows to and fro in the wires. No charge ever crosses the insulator between the plates.

It increases

At a higher frequency the current flows each way for a shorter time, so the capacitor charges less and builds up less opposing voltage, and more current flows. At a low frequency it charges a lot, and little current flows.

It dissipates no power: energy taken in is given back

A resistance turns electrical energy into heat. A capacitor only borrows energy while it charges and returns it while it discharges, so no power is dissipated in it at all; its opposition, XC, is called capacitive reactance.

XC = 1 ÷ (2πfC)

XC = 1 ÷ (2πfC), with f in hertz and C in farad. Frequency and capacitance are both under the line, so a rise in either lowers the reactance; 2πfC is upside down, and would rise with frequency.

It falls

XC = 1 ÷ (2πfC): the frequency is under the line, so the higher the frequency, the lower the reactance. A capacitor opposes high-frequency AC less, not more; a rising reactance is how a coil behaves.

It falls to a third

XC = 1 ÷ (2πfC) is inversely proportional to f: three times the frequency gives a third of the reactance. Three times the reactance would mean it rose with frequency, which is how a coil behaves.

It halves

XC = 1 ÷ (2πfC) has C under the line, so twice the capacitance gives half the reactance: a bigger capacitor passes AC more easily. Doubling would need C on top, as in the upside-down 2πfC.

It stays the same as before the change

Capacitance is fixed by the plate area, the spacing and the dielectric constant, none of which depends on frequency. What changes with frequency is the reactance, XC = 1 ÷ (2πfC).

Hz and F

XC = 1 ÷ (2πfC) takes f in hertz and C in farad to give ohms. MHz with pF gives a number a million times too small, the answer in megohms, and kHz with µF gives kilohms.

10 Ω

XC = 1 ÷ (2πfC) = 1 ÷ (2π × 1000 × 15 × 10⁻⁶) = 1 ÷ 0,0942 = 10,6 Ω, nearest 10 Ω. Reading µF as nF gives 10,6 kΩ; the 25 V rating plays no part.

227 Ω

XC = 1 ÷ (2πfC) = 1 ÷ (2π × 7 × 10⁶ × 100 × 10⁻¹²) = 1 ÷ 0,0044 = 227 Ω. Reading 100 pF as 10 pF gives ten times too much, 2,27 kΩ.

45,5 Ω

XC = 1 ÷ (2πfC) = 1 ÷ (2π × 3,5 × 10⁶ × 1 × 10⁻⁹) = 1 ÷ 0,022 = 45,5 Ω. Reading 3,5 MHz as 35 MHz gives ten times too little, 4,55 Ω.

23 Ω

XC = 1 ÷ (2πfC) = 1 ÷ (2π × 145 × 10⁶ × 47 × 10⁻¹²) = 1 ÷ 0,0428 = 23,4 Ω, nearest 23 Ω. Reading 47 pF as 470 pF gives about 2,3 Ω.

8,8 Ω

XC = 1 ÷ (2πfC) = 1 ÷ (2π × 1,8 × 10⁶ × 0,01 × 10⁻⁶) = 1 ÷ 0,113 = 8,8 Ω. Reading 0,01 µF as 0,1 µF gives ten times too little, 0,88 Ω.

34 Ω

XC = 1 ÷ (2πfC) = 1 ÷ (2π × 21 × 10⁶ × 220 × 10⁻¹²) = 1 ÷ 0,029 = 34 Ω. Reading 21 MHz as 2,1 MHz gives ten times too much, about 340 Ω.

46,8 kΩ

XC = 1 ÷ (2πfC) = 1 ÷ (2π × 50 × 68 × 10⁻⁹) = 1 ÷ 0,000 021 36 = 46 800 Ω = 46,8 kΩ. Reading 68 nF as 68 pF gives a thousand times too much, 46,8 MΩ.

14 MHz

f = 1 ÷ (2π × XC × C) = 1 ÷ (2π × 75 × 150 × 10⁻¹²) = 14,1 MHz, nearest 14 MHz. Reading pF as nF gives a thousand times less, about 14 kHz.

398 nF

C = 1 ÷ (2π × f × XC) = 1 ÷ (2π × 10 × 10³ × 40) = 398 × 10⁻⁹ F = 398 nF. Taking 10 kHz as 10 MHz gives a thousand times less, 398 pF.

69 mA

XC = 1 ÷ (2π × 100 000 × 2,2 × 10⁻⁹) = 723 Ω, and I = V ÷ |XC| = 50 ÷ 723 = 0,069 A = 69 mA. Reading 2,2 nF as 22 nF gives ten times too much, 690 mA.

140 mA

XC = 1 ÷ (2π × 400 × 4,7 × 10⁻⁶) = 84,7 Ω, and I = 12 ÷ 84,7 = 0,142 A = 142 mA, nearest 140 mA. Reading µF as nF gives a thousand times less, 140 µA.

37,3 mA

XC = 1 ÷ (2π × 3,6 × 10⁶ × 330 × 10⁻¹²) = 134 Ω, so I = V ÷ |XC| = 5 ÷ 134 = 0,0373 A = 37,3 mA. Reading 3,6 MHz as 36 MHz gives ten times too much, 373 mA.

9,95 V

XC = 1 ÷ (2π × 1600 × 1 × 10⁻⁶) = 99,5 Ω, so V = I × |XC| = 0,1 × 99,5 = 9,95 V. Taking 1 µF as 1 mF gives a thousand times less, 9,95 mV.

400 Ω

Ohm's Law works with reactance: |XC| = V ÷ I = 20 ÷ 0,05 = 400 Ω. Dividing 20 by 50 without changing the milliamperes into amperes gives 0,4 Ω, a thousand times too small.

6 V

V = I × |XC| = 0,03 × 200 = 6 V. Multiplying 30 by 200 without turning the milliamperes into amperes gives 6000 V, a thousand times too much.

The size of X, with any minus sign left off

|X| is the magnitude of X: its size without any minus sign. Ohm's Law uses it for a reactance, as I = V ÷ |XC|, so nothing is made negative.

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

Resistance and reactance share the ohm, but they combine in another way, shown later in the course: here about 141 Ω, not 200 Ω. Being in the same unit does not make them simply add.

90°

In a capacitor the current and voltage are 90° apart at all times, 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

The current and voltage of a capacitor 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°.

0,25 µs

The period is t = 1 ÷ f = 1 ÷ 1 000 000 = 1 µs, and the 90° shift is a quarter of it: 0,25 µs. 0,5 µs, half the period, would be a 180° shift.

It stays at 90°

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

The current leads the voltage

In a capacitor the current leads the voltage, by 90°: the current flows first, and the voltage builds up as the charge collects on the plates. A voltage leading the current is how a coil behaves.

It lags behind the current

Because the current in a capacitor leads, the voltage lags behind it, peaking a quarter of a cycle later. Saying the voltage leads puts the two the wrong way round.

CTotal = C1 + C2 + …

In parallel the capacitances simply add, CTotal = C1 + C2 + …, like resistors in series. The sum of reciprocals is the rule for capacitors in series, the other way round from resistors.

1/CTotal = 1/C1 + 1/C2 + …

In series the reciprocals add, 1/CTotal = 1/C1 + 1/C2 + …, as for resistors in parallel, so the total is below the smallest capacitor. Simply adding is the rule for capacitors in parallel.

Resistors in series

Parallel capacitances add, CTotal = C1 + C2 + …, just as resistances in series do. Capacitors in series follow the rule for resistors in parallel, the reciprocals adding: the two rules are swapped.

188 nF

In parallel the capacitances add: 4 × 47 = 188 nF, or 0,188 µF. 11,75 nF, 47 ÷ 4, comes from the rule for capacitors in series used by mistake.

Three times the value of one

Parallel capacitances add, so three equal ones give C + C + C = 3C, three times one capacitor. A third of the value is what the same three give in series.

130 nF

1/CTotal = 1/220 + 1/330 = 0,004 545 + 0,003 030 = 0,007 576, so CTotal = 1 ÷ 0,007 576 = 132 nF, nearest 130 nF. 550 nF is the parallel sum, the wrong rule here.

120 pF

Parallel capacitances add, so the trimmer supplies the difference: 390 − 270 = 120 pF. Adding the two, 660 pF, forgets that the 270 pF is already part of the 390 pF total.

130 pF

1/CTotal = 3 × 1/390 = 1/130, so CTotal = 130 pF: equal capacitors in series give one value divided by their number. 1170 pF is the parallel sum, the wrong rule here.

It is always less than the smallest

Adding reciprocals makes 1/CTotal bigger than any single 1/C, so CTotal is smaller than even the smallest capacitor. A total between the smallest and largest would be some average, which neither rule gives.

11,6 nF

In parallel the values add: 1,5 + 3,3 + 6,8 = 11,6 nF. About 0,9 nF, less than the smallest of them, is what the rule for capacitors in series gives.

Half the value of one

1/CTotal = 1/C + 1/C = 2/C, so CTotal = C ÷ 2: two 100 pF capacitors in series give 50 pF. Twice the value is what the same pair gives in parallel.

Their tolerance is poor, about ±10%

General-purpose ceramics are cheap and good at RF, but their real value may be some 10% from the marked one, too loose for a part that sets a frequency. Special stable ceramic types are made for that job.

15 kV

Ceramic capacitors are made for voltages of up to about 15 kV. 1,5 kV is ten times too low, and 15 MV is far beyond any practical capacitor.

They are accurate, but quite expensive

Silvered mica capacitors are good at RF and accurate, typically ±1%, but quite expensive, and they come only in small values. Being poor at RF is wrong: they suit RF circuits well.

About 1 pF to 100 nF

Silvered mica capacitors are made only in small values, about 1 pF to 100 nF. About 10 nF to 10 µF is the range of polycarbonate capacitors, the choice for higher values.

When higher values are needed at medium tolerance

Polycarbonate capacitors suit higher values, about 10 nF to 10 µF, at a medium tolerance, typically ±5%. The cheapest choice is the ceramic capacitor, not the polycarbonate one.

Metal foil, a conducting fluid and a thin chemical film

One 'plate' is metal foil, usually aluminium, and the other a conducting fluid; between them, as the dielectric, lies a thin chemical film on the foil. The fluid conducts: it is a plate, not the insulator.

One terminal must always stay the more positive

A polarised capacitor has a positive and a negative terminal, and the positive one must never become the more negative. A voltage that keeps reversing, as AC does, is exactly what it cannot take.

DC circuits such as power supplies

Most electrolytics are polarised, so one terminal must stay positive, which suits DC circuits such as power supplies, where their large values are useful. Frequency-setting parts need small, accurate capacitors instead.

No, but most of them are

Most electrolytic capacitors are polarised and must be connected the right way round, but non-polarised types are also made. Saying every one is polarised goes further than the facts.

Electrolytic

Electrolytic capacitors are made in far higher values than the other types, 1000 µF and more being common. Polycarbonate capacitors, the usual choice for higher values among the rest, stop at about 10 µF.

The rotor

Turning the knob moves the rotor plates in and out of the fixed set, the stator, changing how far they overlap. The stator is the set that stays still.

It changes how far the two sets of plates overlap

Turning the knob swings the rotor plates further into or out of the stator plates, changing the area where they overlap, and capacitance follows that area. The spacing between the plates stays the same.

Tuning radios to different stations

Variable capacitors were once the usual tuning control of radios, setting the frequency received. Volume is set with a potentiometer, a variable resistor, not with a variable capacitor.

That its capacitance can be varied

An arrow drawn through a component's symbol marks it as variable: here the capacitance can be adjusted. A polarised capacitor is marked instead with a + sign at its positive plate.