Capacitors and Capacitance

Open any transceiver and capacitors outnumber every other component in it. They smooth the power supply, they carry signal from one stage to the next while keeping the DC bias apart, they tune the oscillator, and they short the unwanted radio frequency to earth. The exam asks four or five questions about them, and two of those are traps built specifically to catch someone who learned resistors first. This lesson is written to make you immune to both.

Two plates and a dielectric

A capacitor is the simplest component there is to describe: two conducting plates facing each other, with an insulator between them that never lets current across. That insulator is called the dielectric, and it is usually what gives the capacitor its name — a ceramic capacitor has a ceramic dielectric, a polyester capacitor a thin polyester film.

Connect a battery across it and electrons are pulled off one plate and crowded onto the other. Nothing crosses the gap. The plates end up with equal and opposite charges, and an electric field — an electrostatic field — stretched across the dielectric between them. That field is the store: discharge the capacitor and the field collapses, giving the energy back.

Capacitance, charge and the farad

Capacitance is the measure of how much charge a capacitor holds for each volt you apply. It is the ratio, and nothing more mysterious than that:

One farad is one coulomb of charge stored per volt. That turns out to be an absurdly large amount — a one-farad capacitor built with air between its plates would need plates the size of a cricket ground. So every value you will meet carries a prefix:

1 µF is 1000 nF is 1 000 000 pF. Older Indian circuit diagrams write µF as MFD and picofarads as pf or occasionally mmfd (micro-microfarad) — same thing.

Worked example 1 — charge stored

A 10 µF capacitor is charged to 12 V. How much charge sits on its plates?

Q = C × V = 10 × 10⁻⁶ × 12 = 1.2 × 10⁻⁴ C = 120 µC

What decides the capacitance — and what does not

Only three things set the value of a parallel-plate capacitor:

• Plate area (A) — bigger plates hold more charge at the same voltage. Capacitance is directly proportional to area.
• Plate separation (d) — the closer the plates, the stronger the field for a given voltage and the more charge is held. Capacitance is inversely proportional to separation.
• The dielectric — some insulators polarise more readily than air and let the plates hold more charge for the same voltage. The multiplying factor is the dielectric constant (relative permittivity), written ε_r or K.

What is not in that expression is the item the exam asks about. The thickness of the plates does not affect capacitance at all. The charge lives on the two facing surfaces; whether the metal behind those surfaces is a foil or a slab makes no difference to how much of it there is. Nor does the material the plates are made of, nor the shape of the leads.

Worked example 2 — the effect of the dielectric

Two plates 5 cm square face each other across an air gap of 0.1 mm. Area is 0.05 × 0.05 = 0.0025 m²; separation is 0.0001 m.

C = 8.85 × 10⁻¹² × 1 × 0.0025 ÷ 0.0001 = 2.2 × 10⁻¹⁰ F = 221 pF

Now slide a sheet of mica of the same thickness into the gap (ε_r ≈ 6) and the value rises six-fold, to about 1330 pF, with nothing else changed. Halve the gap instead and the value doubles. That is the whole design trade of the component in two sentences.

Working voltage

Every capacitor carries a second number beside its value: the working voltage, often printed as WV or VDC. A dielectric only withstands so many volts across its thickness. Go past that and the insulation breaks down — the material punctures, the carbonised track that is left behind short-circuits the plates, and the capacitor is finished. Electrolytics fail spectacularly, heating and venting.

Note the tension with everything above: closer plates give more capacitance and a lower working voltage, because the same volts now appear across a thinner insulator. Rate the part above the highest peak the circuit can present — for a 230 V mains-derived rail that peak is about 325 V, not 230 V, as Sine Waves: Peak, RMS and Average explains.

Types you will meet

The electrolytic is the one that needs care. Its dielectric is an oxide film only a fraction of a micron thick, formed by electrolysis on the aluminium foil and maintained by the applied voltage. That is how it packs thousands of microfarads into a small can. Connect it backwards and the film dissolves, the capacitor conducts, the electrolyte boils and the case bursts. The negative lead is marked with a stripe; the longer lead is positive on a radial part. Every other type in that table is non-polarised and may go in either way round.

A variable capacitor changes its value by sliding one set of vanes between another, varying the effective plate area; a trimmer does the same in miniature, for a setting made once at alignment. Two variable capacitors on a common shaft form a gang, so a receiver's RF and oscillator stages tune together.

Energy stored

The voltage is squared, so doubling the voltage on a capacitor stores four times the energy. A 4700 µF reservoir capacitor in a 25 V power supply holds 0.5 × 0.0047 × 25² = 1.47 J. That is enough to give you a real jolt and enough to weld the tip of a screwdriver, which is why a power supply keeps a bleeder resistor across the reservoir and why you check with a meter before putting a hand inside one.

Series and parallel — the reversal

Here is the single most-failed item on this topic. Capacitors combine the opposite way round from resistors.

Do not memorise that as an arbitrary swap — it follows straight from the geometry, and if you hold the reason you will never get it the wrong way round:

• In parallel, the plates sit side by side connected to the same two nodes. You have simply built a capacitor with more plate area, and capacitance rises with area. So the values add.
• In series, the outer plates are further apart than either capacitor's own plates were — you have increased the effective plate separation. Capacitance falls as separation rises. So the total is less than the smallest one present.

The widget below is the fastest way to fix this. Enter the same numbers, then press the Resistors and Capacitors buttons in turn and watch the two answers change places on screen.

Worked example 3 — a mixed network

Two 10 µF capacitors are connected in series, and that pair is connected in parallel with a 5 µF capacitor. What is the total?

The series pair first. Two equal capacitors in series always give half of one of them: 1/C = 1/10 + 1/10 = 2/10, so C = 5 µF.

Now that 5 µF sits in parallel with the other 5 µF, and parallel capacitors add: 5 + 5 = 10 µF.

The RC time constant

Charge a capacitor through a resistor and it does not fill instantly. At the first instant the capacitor is empty, the full supply appears across the resistor, and the current is at its largest. As the plate voltage climbs, less voltage is left across the resistor, so the current falls — and the charging slows. The result is the exponential curve every power supply and every timing circuit lives on.

After one time constant the capacitor has reached 63.2 % of the supply voltage. After each further time constant it covers 63.2 % of what remains. By convention it is treated as fully charged after five time constants, by which point it is at 99.3 %. Discharging follows the same curve downwards: one time constant takes it down to 36.8 % of where it started.

Set R and C in the explorer below and watch the shape stay identical while only the time axis changes — that is the point of a time constant.

Worked example 4 — a timing circuit

A 1 MΩ resistor charges a 1 µF capacitor from a 12 V supply. Find the time constant, the voltage after one time constant, and the time to full charge.

τ = R × C = 10⁶ × 10⁻⁶ = 1 second

After 1 s: 0.632 × 12 = 7.6 V. Fully charged after 5 × 1 = 5 seconds.

What a capacitor does to DC and to AC

This is the behaviour every other use of a capacitor is built on. Apply steady DC and current flows only while the capacitor is charging. Once it is charged, the current stops. A capacitor blocks DC.

Apply AC and the supply reverses many times a second. The capacitor charges one way, discharges, charges the other way, discharges again, and current flows in the wires continuously — even though nothing has ever crossed the dielectric. A capacitor passes AC, and it passes it more readily the higher the frequency, because there is less time between reversals for the plate voltage to build up and oppose the flow.

That frequency-dependent opposition has a name, capacitive reactance, and a formula that belongs to Phase, Reactance, Impedance and Power Factor. For now hold the shape of it: high frequency, easy passage; low frequency, hard; DC, blocked entirely.