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4. Volume 03

3.1 Capacitors store charge

A capacitor is two metal plates with an insulator between them. Push charge onto it and it stores the charge; the more charge it holds, the higher its voltage. Its capacitance says how much charge each volt stores.

Two plates and a gap

Picture two metal plates facing each other, very close but not touching. Connect them to a battery. Electrons pile onto the plate joined to -, and leave the plate joined to +. No charge crosses the gap, because the gap is an insulator. But the plates now hold separated charge, and a voltage between them.

Disconnect the battery and the charge stays put. The capacitor holds its voltage, like a tiny rechargeable battery. Real capacitors roll up long, thin plates to fit a lot of area into a small part.

The symbol is just that: two short parallel lines, for the two plates.

Charge and capacitance

The charge a capacitor stores is its capacitance times its voltage:

Q = C x V

Capacitance is measured in farads (F). A farad is enormous, so real parts are microfarads (µF, millionths), nanofarads (nF, billionths) or picofarads (pF, a thousandth of a nanofarad):

100 µF at 5 V holds 500 µC
1 nF at 5 V holds 5 nC

A charged capacitor also stores energy, which it can give back quickly - the flash in a camera works this way:

energy = C x V x V / 2: 100 µF at 5 V stores 1.25 mJ

Kinds of capacitor

An electrolytic capacitor must go the right way round. Reversed, it can overheat and burst. Every capacitor also has a voltage rating, and must never be used above it.

Combining capacitors

Capacitors combine the opposite way to resistors. In parallel, the plate areas add up, so the capacitances add. In series, they combine like resistors in parallel:

100 nF and 100 nF in parallel: 200 nF
100 nF and 100 nF in series: 50 nF

3.2 Charging and discharging: the RC time constant

Through a resistor, a capacitor charges quickly at first and then more and more slowly. The time constant, R × C, sets the pace: after one time constant it is 63% charged, and after five it is as good as full.

Charging through a resistor

At the moment the switch closes, the capacitor is empty and has no voltage across it. So the whole 5 V is across R1, and the capacitor behaves like a plain wire. Much later, the capacitor is full at 5 V and nothing flows:

circuit rc-charge: at the first instant the capacitor acts like a wire: 500 µA flows
after a long time: the capacitor holds 5 V and 0 A flows

In between, the current falls away smoothly. As the capacitor fills, its voltage rises, so less of the 5 V is left across R1 to push current through it. Less current means slower filling. So the capacitor charges fast at first, and ever more slowly after that.

The time constant

The resistance times the capacitance gives a time, in seconds, that sets the pace:

time constant = R x C = 10 kΩ x 100 µF = 1 s

The shares are the same for any R and C. After one time constant, any RC circuit is 63% of the way there; after five, it is within 1% and counts as finished.

Discharging

Take the battery away and join the full capacitor's leads through the same resistor, and it empties along the mirror-image curve. After one time constant, 37% of the voltage is left.

Vc = Vs x (1 - e^(-t / RC))

a step-by-step charge in 200000 small steps agrees with the formula to within 0.001 V

3.3 RC filters

Because a capacitor takes time to charge, an RC circuit smooths fast changes away and lets slow ones through. That is a low-pass filter, and its cut-off frequency is 1 / (2π × R × C).

Frequency

Many signals go up and down over and over again. The number of times a signal repeats each second is its frequency, measured in hertz (Hz). A signal that repeats a thousand times a second is 1 kHz. Sound you can hear runs from about 20 Hz to 20 kHz.

The low-pass filter

Put a resistor in series with a signal, and a capacitor from its far end to ground:

The circle with a wave in it is a signal source: a voltage that keeps changing. When the input changes slowly, the capacitor has plenty of time to follow, and the output matches the input. When it changes fast, the capacitor cannot keep up. It only half-charges before the input turns round, so the output swings much less.

The cut-off frequency marks the change-over:

fc = 1 / (2 x pi x R x C)
circuit lowpass: R1 = 1 kΩ and C1 = 100 nF, so fc = 1.59 kHz

Here pi is the number π, about 3.14. For this filter, the output compares with the input like this:

Well below the cut-off, almost all of the signal gets through. At the cut-off, 0.707 of it does. Ten times above, only about a tenth does, and each further tenfold rise cuts it by ten again.

The high-pass filter

Swap the resistor and the capacitor, and the filter works the other way round. That is a high-pass filter. A steady voltage is blocked completely, because a charged capacitor passes no current. Fast changes pass straight through. It has the same cut-off formula.

3.4 Inductors

An inductor is a coil of wire. It stores energy in a magnetic field and fights any change in its current. In a settled circuit it is just a wire - but switch its current off suddenly, and it makes a very large voltage.

A coil of wire

Current through a wire makes a magnetic field around it. Wind the wire into a coil, and the fields of all the turns add up to a strong one. Building that field takes energy, and the field gives the energy back when the current falls. The result is a part that resists any change in its current.

Inductance is measured in henries (H). Real inductors are usually millihenries (mH) or microhenries (µH). The symbol is a row of loops, like a coil.

Steady current: just a wire

Once the current has stopped changing, an inductor is only a coil of copper, with almost no resistance:

circuit inductor: once settled, the current is 50 mA - the inductor is just a wire

When the switch first closes, though, the current cannot jump straight to 50 mA. The inductor holds it back, and it grows with a time constant of L / R:

time constant = L / R = 10 mH / 100 Ω = 100 µs
after one time constant the current has reached 31.6 mA

That is the capacitor's story turned round. A capacitor's voltage cannot change instantly; an inductor's current cannot either.

Switching an inductor off

The voltage across an inductor depends on how fast its current changes:

V = L x change in current / time

Open the switch, and the current tries to fall to zero at once. The inductor fights it by making whatever voltage it takes to keep the current going:

stopping 50 mA in 1 µs: V = 10 mH x 50 mA / 1 µs = 500 V

Five hundred volts, from a 5 V battery. The spike jumps across the opening switch contacts as a spark, and it can destroy a transistor. Motors, relays and loudspeakers are all coils, so they all do this. Volume 05 shows the simple diode that makes them safe.

3.5 Debouncing a switch with RC

A switch's contacts bounce for a few milliseconds when pressed, making a burst of on-off pulses. An RC filter smooths the burst into one slow rise, which a chip reads as a single press.

Switch bounce

A push button is two springy metal contacts. When they meet, they bounce apart and together a few times before they settle. That switch bounce lasts a few milliseconds. People never notice, but a chip does: it can read every bounce as a new press.

Here is a button feeding a chip's input, with an RC filter in between:

circuit debounce: button held down: the input settles at 4.55 V; released: 0 V
R1 x C1 = 10 kΩ x 1 µF = 10 ms

One press, with and without the filter

In this press, the contacts touch three times in the first 3.5 ms before they settle. The chip treats anything above 2.5 V as a 1:

switch alone: the input rises through 2.5 V 4 times, so the chip sees 4 presses
with the RC filter: it rises through 2.5 V once, 10.8 ms after the first touch

The filter costs a delay of about 11 ms, which nobody can feel. A time constant of 5 ms to 20 ms suits most buttons.

Finishing the job

A slowly rising voltage has one more problem. While it creeps through the switching point, a little electrical noise can tip an ordinary input back and forth. A Schmitt trigger input fixes this. It switches to 1 at a higher voltage than it switches back to 0, so a slow edge crosses cleanly, once. Many chips offer one on their input pins.

Debouncing can also be done by the chip itself: a program, or a small digital circuit, ignores the input until it has been steady for a few milliseconds. State Machines, Volume 07 builds that digital debouncer.

What you learned

• A capacitor stores charge: Q = C × V. Capacitance is in farads, usually µF, nF or pF.
• Capacitors add in parallel; electrolytic capacitors have a + and a - lead.
• Through a resistor, a capacitor charges 63% in one time constant, R × C, and is full after about five.
• An RC low-pass filter passes slow signals and smooths fast ones; its cut-off frequency is 1 / (2π × R × C).
• An inductor fights changes in current; steady, it is a wire; its time constant is L / R.
• Switching an inductor off suddenly makes a large voltage spike: V = L × change in current / time.
• An RC filter, ideally with a Schmitt trigger input, turns a bouncing press into one clean press.

Key words from this volume

Every word below has a plain-English entry in the glossary.

• Capacitor
• Capacitance
• Farad (F)
• Electrolytic capacitor
• Time constant
• Low-pass filter
• Cut-off frequency
• Frequency
• High-pass filter
• Inductor
• Henry (H)
• Switch bounce
• Schmitt trigger

Practice

47 kΩ x 10 µF = 470 ms; about full after 5 time constants: 2.35 s

10 µF at 12 V holds 120 µC and stores 720 µJ

10 kΩ and 10 nF: fc = 1.59 kHz

10 µF and 22 µF: 32 µF in parallel, 6.88 µF in series

20 mH, 100 mA stopped in 2 µs: V = 1 kV

Interview corner

Volume 04 meets the diode, which lets current flow one way only, and the LED, a diode that gives out light.