Capacitors and Inductors

Resistors only ever turn electrical energy into heat. Capacitors and inductors are different: they store energy, a capacitor in an electric field and an inductor in a magnetic field, and give it back later. That one difference makes time appear in your circuits. A capacitor takes time to fill, an inductor takes time to build its current, and a pair of simple resistor-and-capacitor circuits can delay a signal, smooth a supply, filter out noise or clean up a bouncing switch. It also explains why a motor or relay coil can destroy the transistor that switches it, which Basic Electronics 5, Basic Electronics 6 and Basic Electronics 9 will solve. This chapter builds on Ohm's law (Basic Electronics 1), series and parallel rules (Basic Electronics 2) and the divider (Basic Electronics 3).

1. Capacitors

What a capacitor is

A capacitor is two conductors (plates) separated by an insulator (the dielectric). No current flows through it, but when a voltage is applied, charge piles up on one plate and an equal charge leaves the other. The more charge it holds per volt, the bigger its capacitance C, measured in farads (F):

Energy stored

Moving a small extra charge dq onto a capacitor that already holds voltage v = q/C takes energy dE = v × dq. Adding that up from empty (0) to full charge Q:

Capacitors in parallel and series

Capacitors combine the opposite way round to resistors: parallel adds, series reduces. Because the voltage shares unevenly, putting capacitors in series to survive a higher voltage is risky unless you add balancing resistors; the sensible fix is one capacitor with a higher voltage rating.

Types you will meet

Reading the value

Voltage rating, polarity and tolerance

• Voltage rating is the most the capacitor may see continuously. Choose at least 1.5 to 2 × the working voltage for an electrolytic (a 10 V part on a 5 V rail, a 25 V part on 12 V). Ceramics also lose capacitance as the DC voltage across them rises ("DC bias derating"): a 10 µF X5R ceramic rated 6.3 V can deliver well under half of 10 µF at 5 V, depending on the part, so read the datasheet curve for a smoothing capacitor.
• Polarity. An electrolytic connected backwards leaks heavily, heats up and can vent or burst with a pop and a puff of electrolyte. Check the stripe before you power up. Ceramic and film capacitors have no polarity.
• Tolerance is wide: a 100 µF ±20 % electrolytic is anywhere from 80 to 120 µF, so a timing circuit built with it is only accurate to about that. Use film or C0G where the time or frequency really matters.

2. Charging and discharging through a resistor

Connect a resistor R in series with a capacitor C and a supply V_s through a switch. The moment the switch closes the capacitor is empty, so it looks like a short circuit and the current is V_s/R. As charge builds, the capacitor's voltage rises, the voltage left across the resistor falls, and the current dwindles. It is the same shape as water filling a bucket through a narrow pipe: fast at first, ever slower as it fills.

Derivation

The 63 % and 99 % points

Worked example: 10 kΩ and 220 µF from 5 V

3. Capacitors and AC: reactance and RC filters

Capacitive reactance

A capacitor passes a changing voltage but not a steady one. The faster the voltage changes, the more current it passes. Starting from i = C dv/dt with a sine wave v = V_p sin(ωt), where ω = 2πf:

A capacitor in series with a resistor is a frequency-dependent voltage divider. Because the capacitor's current leads its voltage by a quarter of a cycle (90°), the reactance and the resistance do not add directly; they add at right angles, like the sides of a triangle:

The RC low-pass filter

The RC high-pass filter

Swap the two parts and the capacitor is in the signal path: it passes fast changes and blocks slow ones, including DC. Taking the output across the resistor,

4. Decoupling and smoothing

Decoupling: a tiny local reservoir

A microcontroller or motor driver draws current in sudden bursts as its transistors switch, for example 50 mA in 10 ns. The supply wires back to the battery are not perfect: they have inductance (roughly 1 nH per millimetre, so about 100 nH for 10 cm). A rapidly changing current through an inductance produces a voltage, v = L di/dt (section 5):

• 100 nF ceramic as close as possible to each IC's power pins, with short leads (that is the purpose of the "104" caps on every board).
• 10 to 100 µF electrolytic (bulk) near a motor driver, regulator or the point where power enters a board: slower, bigger bursts.
• A small and a large capacitor in parallel is useful: each covers a different range of speeds.

Smoothing a rectified supply

A rectifier (Basic Electronics 5) turns AC into a series of humps. A big capacitor fills at each peak and feeds the load between peaks. While feeding the load with a steady current I for one gap of T = 1/f seconds, the voltage falls by dV = I × T / C, so:

5. Inductors

An inductor is a coil of wire, often wound on an iron or ferrite core. Current through it makes a magnetic field; if the current changes, the field changes, and a changing field induces a voltage in the coil that opposes the change (Lenz's law). The inductance L is measured in henries (H; practical sizes are µH to mH, up to a few H for relay and motor coils).

The RL time constant

Back-EMF and why coils need protection

The law v = L di/dt has a dangerous consequence. When current through a coil is switched off, di/dt is large and negative, so the coil produces a large voltage in the direction that tries to keep the current flowing. This is the back-EMF (or inductive kick). Think of a heavy wheel: you cannot stop it instantly, and the faster you try, the more force you feel.

A spinning motor adds a second, related effect: it also generates a voltage proportional to its speed, the motor's own back-EMF, which limits its speed and current. Basic Electronics 9 covers both kinds.

6. A switch debounce circuit

A mechanical switch does not close cleanly. For a millisecond or several, the springy contacts bounce, making and breaking the circuit dozens of times. A microcontroller reads every bounce as a separate press, so one press can count as five. The fix is to make the signal change slowly enough that the bounces are smoothed away. An RC circuit does it with two parts. The pull-up resistor from Basic Electronics 3 charges a capacitor on the input pin.

7. Build it: charge, filter, coil

Three stages. Stages A and B use only a multimeter and a stopwatch (a phone is fine) on a breadboard. Stage C, the filters and the inductor, shows things that need an oscilloscope on a real bench (a scope reads signals changing in microseconds and milliseconds; a multimeter cannot follow them), so do those in a simulator. If you own a signal generator and scope, build the filter from the parts list and compare with the table.

Parts list

Stage A: charge and discharge, measured

1. With J1 out, discharge C1 with the 100 Ω resistor and confirm the meter reads about 0 V. Check the supply voltage too.
2. Charge: push J1 into the + rail and start the stopwatch at the same moment. Write down the voltage at 1.5 s, 2.2 s, 4.4 s, 6.6 s and 10 s (a phone video of the meter and stopwatch makes this easy).
3. Discharge: once at full voltage, move J1 from the + rail to the − rail, restart the stopwatch, and record the voltage at the same times.
4. Estimate the time constant: it is the time when the charge reading reaches 63 % of the supply (3.16 V on 5 V). Divide by R and you have your capacitor's real value.

Stage B: debounce circuit in slow motion

Build the debounce schematic from section 6 with the 100 µF capacitor, which stretches the 10 ms time constant to 1 s so the meter can follow it. The real-speed version (1 µF) is in the simulator.

1. Power up and wait about 5 s: the pin node reads the supply (about 5 V).
2. Press and hold the button: the reading drops at once to about 0.05 V (5 × 100 / 10,100).
3. Release and start the stopwatch: the reading climbs back, reaching 63 % (3.16 V) at 1 s and 70 % (3.5 V) at 1.2 s. That 70 % point is where a 5 V logic input would see a "1".

Stage C: simulator only (filters and coil)

Falstad Circuit Simulator (falstad.com/circuit). Use Draw to add parts, double-click a part to edit its value, and right-click a node to send it to the scope.

• RC charging (compare with Stage A): a 5 V DC source, a switch, a 10 kΩ resistor and a 220 µF capacitor to ground. Scope the capacitor node and click the switch. The 63 % mark must fall at 2.2 s (you may need to slow or speed the simulation with its speed slider).
• Low-pass and high-pass: an AC voltage source (set the amplitude to 1 V, frequency to 100 Hz, 1 kHz, then 10 kHz), a 1.6 kΩ resistor and a 100 nF capacitor. Scope the source and the output together and read the output amplitude. Then swap R and C for the high-pass.
• Coil current: a 12 V DC source, a 240 Ω resistor and a 240 mH inductor (set the inductance in its edit box) in series, then a switch to ground. Right-click the resistor and choose the scope to show its current. Close the switch: the current should climb towards 50 mA with a 1 ms time constant.
• Flyback: in the same circuit, open the switch and look at the voltage at the node between the coil and the switch. Then add a diode with its anode at ground and its cathode at the +12 V end of the coil (across the coil, as in the section 5 schematic), and repeat. Without the diode the node jumps to a very large voltage; with it, the node rises only to about 12.7 V.

Tinkercad Circuits (tinkercad.com): the breadboard, resistor, electrolytic capacitor (it shows its polarity), push button, power supply and multimeter cover stages A and B with the same hole lists. The function generator and oscilloscope components give the filters (set the generator to a sine wave of 1 V amplitude and use the scope's cursors to read the output). Tinkercad has no ideal inductor in its basic parts, so use Falstad for stage C's coil.

Expected readings

The real readings spread because of tolerance: the capacitor is ±20 % (and electrolytics change with age) and the resistor is ±5 %. The numbers that do not depend on tolerance are the fractions: the reading is always 63 % of the supply after one measured time constant, and 99 % after 4.6.

8. Challenges

Next, Basic Electronics 5 introduces diodes and LEDs: one-way current, forward voltage, sizing an LED resistor, the flyback and protection diodes promised above, and rectifier basics.