Every circuit you will ever build is assembled from a tiny vocabulary of passive parts — wire, batteries, switches, resistors, capacitors, inductors, transformers, fuses. Each is governed by one or two relationships you can read off a label or a ring of colored bands. This chapter is the alphabet; the rest of the book is the language.
You decide to light an LED. You grab "a resistor" out of a drawer — it has four colored stripes: brown, black, red, gold. You wire it in series with the LED, clip on a fresh 9 V battery, and it glows. Good. But three questions immediately start biting.
First: what value is that resistor, and is it about to cook? The part is a tiny quarter-watt (¼ W) film resistor. If too much power flows through it, the film overheats, the value drifts, and eventually it chars and opens. Those four stripes are the only label it has — you must decode them.
Second: how long will the battery actually last? The label says "9V" but says nothing about how much energy is inside. A coin cell and a car battery can both read "1.5 V"-ish per cell, yet one runs a watch for years and the other cranks an engine. Voltage is not capacity.
Third: later you add a relay to switch a small motor, and suddenly the circuit glitches, the microcontroller resets, and an inline fuse keeps blowing. Why does adding one electromechanical switch wreak havoc on an otherwise calm circuit?
All three mysteries dissolve once you know the components. A resistor's bands encode R = (10·d₁ + d₂) × 10mult. A battery's runtime is capacity ÷ load current. A relay coil is an inductor that kicks back a voltage spike when switched off, and a fuse is rated about 50% above the normal current. Let's decode the mystery resistor right now.
Set the supply voltage across our brown-black-red-gold resistor. The decoder reads the bands to a value, then computes the power dissipated. Watch the bar turn red when it exceeds the ¼ W rating — that is the moment the resistor starts to cook.
Drag the multiplier band down to red ×1 (10 Ω) and crank the voltage — the power explodes and the resistor "fries." Drag it up and the same voltage is harmless. This single picture is the entire problem of power rating, which we solve properly in Tab 8.
The humblest component is wire, and we treat it as a perfect zero-resistance connection — right up until it isn't. Run 20 A through a thin lamp cord over a long distance and it gets warm, drops voltage, and in the worst case starts a fire. Wire has resistance, and that resistance depends on its thickness.
In North America thickness is specified by the American Wire Gauge (AWG), and the counterintuitive rule is: bigger gauge number means thinner wire. The numbering comes from how many times the wire was drawn through successively smaller dies. A 10 AWG wire is a fat 2.59 mm in diameter; a 22 AWG wire is a skinny 0.64 mm. Each step of 3 in gauge roughly halves or doubles the cross-sectional area.
Resistance per unit length follows directly: thinner wire has less copper to carry current, so it has more resistance. A 10 AWG wire is about 1 Ω per 1000 ft; a 22 AWG wire is about 16.2 Ω per 1000 ft — sixteen times worse. Pick wire too thin for the current and you waste energy as heat and droop the voltage your circuit sees.
At DC, current spreads evenly across the whole cross-section. But as frequency rises, the magnetic field inside the conductor pushes current toward the outer surface — the skin effect. The current "crowds" into a thin shell near the surface, so the effective cross-section shrinks and the AC resistance climbs well above the DC value. At radio frequencies the center of a thick wire carries almost nothing, which is why RF conductors are often hollow tubes or silver-plated (current rides the surface anyway).
You run 2 A of motor current down 50 ft of 22 AWG wire and back (100 ft round-trip). Resistance ≈ 16.2 Ω/1000 ft × 0.1 kft = 1.62 Ω. Voltage lost in the wire: V = I·R = 2 A × 1.62 Ω = 3.24 V. On a 12 V supply you have thrown away over a quarter of your voltage in the wire alone! Swap to 10 AWG (1 Ω/1000 ft → 0.1 Ω round trip) and the drop falls to 2 A × 0.1 Ω = 0.2 V. Thickness matters.
Pick a gauge and a length, set the current. The widget shows the wire diameter, total resistance, and how much of your supply voltage is lost as heat in the wire. Notice that increasing the gauge number shrinks the wire and worsens the drop.
A wire is fine for DC and low frequencies. But when signals get fast — video, radio, digital edges — a pair of conductors behaves like a continuous chain of tiny inductors and capacitors. This distributed structure has a property called characteristic impedance, written Z₀, and getting it wrong causes reflections that smear and ring your signals.
Two geometries dominate. Coaxial cable (coax) is a center conductor wrapped by a tubular shield, with dielectric between — the shield blocks interference, which is why it carries antenna and video signals. Twin-lead is two parallel wires held a fixed distance apart, common in old TV antenna feeds. The geometry and the dielectric between the conductors set the inductance L and capacitance C per unit length.
The remarkable thing about Z₀ is that it does not depend on the cable's length — only on its cross-section and materials. It is the ratio of voltage to current of a wave traveling down the line. Typical values: coax runs 50–100 Ω (RG-8 ≈ 53 Ω, RG-58 ≈ 50 Ω, RG-11 and RG-59 ≈ 73–75 Ω), while twin-lead is about 300 Ω.
If a cable of impedance Z₀ is terminated by a load resistance equal to Z₀, the wave is fully absorbed — matched. If the load differs, part of the wave bounces back as a reflection, producing standing waves, ghosting on video, and lost power in RF. This is why you terminate a 50 Ω line with a 50 Ω load, and why old TVs had 75 Ω and 300 Ω terminals with a balun between them.
The insulating dielectric between conductors stores electric energy and sets C. Its dielectric constant κ (relative permittivity) ranges from air ≈ 1.0, to Teflon ≈ 2.1, to polyethylene ≈ 2.3. A higher κ raises C, which lowers Z₀ and slows the wave. Designers trade these to hit a target impedance with a manufacturable geometry.
RG-11 has roughly C = 21 pF/ft and L = 0.112 µH/ft. Then:
That matches the spec sheet's 75 Ω rating — this is genuinely how cable impedance is engineered. Raise C (denser dielectric) and Z₀ drops; raise L (thinner center conductor) and Z₀ rises.
Adjust the per-foot inductance and capacitance. The widget computes Z₀ and flags whether it lands near a standard cable (50, 75, or 300 Ω). See how denser dielectric (higher C) pulls Z₀ down.
Components are useless without a way to join them. Connectors are the standardized plugs and jacks that make connections repeatable, and each is chosen for its electrical and mechanical job. Getting the connector right matters: a video signal in a connector designed for audio will reflect, hum, or simply not fit.
Three families you will meet constantly: the BNC connector is a bayonet-locking coax connector used for 50 Ω and 75 Ω RF and oscilloscope probes — it maintains the cable's impedance through the junction. The RCA (phono) connector is the cheap push-on plug for consumer audio and composite video. The banana plug is a single springy pin for bench power supplies and multimeter leads, where you just need a fat, easy, high-current connection.
BNC — bayonet lock, impedance-controlled, RF & scopes. Twist a quarter turn to lock.
RCA — push-on, color-coded (red/white audio, yellow video), cheap and ubiquitous.
Banana — single pin, high current, stackable, bench gear and meters.
Terminal block / header — screw or pin connections for prototyping and signal breakouts.
A schematic is a diagram that uses standardized symbols to represent components and lines to represent connections (not physical wires). Reading schematics fluently is the single most important literacy skill in electronics — the symbol tells you the component, and crossing/jumping lines tell you what connects to what. A dot at an intersection means a junction; no dot (or a hop) means the wires merely cross.
The standard symbols for the components in this chapter, drawn on the canvas. These are exactly what you will see in every circuit diagram from here on. Click "Next page" to cycle through the set.
A battery converts chemical energy into electrical energy. A primary cell is single-use: once the chemistry is spent, you throw it out. The defining number on the label is the nominal voltage — the "typical" terminal voltage during most of its life — but that single number hides a whole discharge story.
The common chemistries form a ladder of quality. Carbon-zinc is the cheapest, with a sloping discharge and poor performance under heavy load. Alkaline improves capacity and high-current behavior at modest cost — it is the default AA/AAA you buy. Lithium primaries are the premium tier: light, long shelf life, wide temperature range, and a famously flat discharge curve that holds voltage nearly constant until it suddenly drops at the end.
Most single cells sit near 1.5 V nominal (alkaline and carbon-zinc), while lithium primaries are often ~3 V per cell. A "9 V" battery is six 1.5 V cells stacked in series inside one case. Lead-acid cells (rechargeable, covered next tab) are 2 V each, ganged into 6 V and 12 V packs.
No real battery holds its nominal voltage forever. As it drains, terminal voltage sags — gently for lithium, steeply for carbon-zinc. Worse, every battery has an internal resistance Rint: under load, some voltage is dropped inside the cell, so the terminal voltage you actually get is Vterm = Vemf − I·Rint. A weak old battery has high Rint, which is why it reads fine with no load but collapses the instant you draw current.
A 1.5 V AA with Rint = 0.15 Ω drives a 2 Ω load. Current I = 1.5 V / (0.15 + 2) = 0.70 A. Terminal voltage = 1.5 − 0.70×0.15 = 1.5 − 0.105 = 1.40 V. Now let the cell age until Rint = 1 Ω: I = 1.5/3 = 0.50 A and Vterm = 1.5 − 0.50×1 = 1.0 V. Same chemistry, same "1.5 V" label, but the tired cell delivers a third less voltage to the load.
Each chemistry discharges differently. Pick a chemistry and toggle internal-resistance sag on/off. Lithium's flat curve is why it powers cameras and sensors that need stable voltage; carbon-zinc's droop is why it fades early under load.
A secondary cell is rechargeable: the chemical reaction runs backwards when you push current in. Lead-acid (cars, UPS), NiMH (AA rechargeables), and lithium-ion (phones, laptops) are the workhorses. But whether a cell is single-use or rechargeable, the question "how long will it last?" comes down to one quantity: capacity.
Capacity is measured in amp-hours (Ah) or milliamp-hours (mAh). A 1 Ah battery can supply 1 A for 1 hour, or 0.5 A for 2 hours, or 2 A for half an hour. It is the product of current and time. The runtime is simply capacity divided by load current:
Typical numbers: a NiMH AA holds about 2300 mAh, a NiMH 9 V about 250 mAh (much less — tiny cells). This is why your "9V smoke-detector battery" dies so much faster than a pile of AAs delivering the same energy.
The C-rating expresses current as a multiple of capacity. "1C" means a current numerically equal to the capacity: a 1000 mAh cell at 1C delivers 1000 mA for 1 hour. At 0.5C it delivers 500 mA for 2 hours; at 2C, 2000 mA for half an hour (ideally). High C-rates stress the cell, heat it, and via internal resistance actually reduce usable capacity — another non-ideal effect (Peukert's law).
A 1800 mAh pack drives a 120 mA load. Runtime t = 1800 mAh / 120 mA = 15 hours. The discharge rate is 120/1800 = 0.067C — a gentle drain, so the ideal estimate is reliable. Now double the load to 240 mA: t = 1800/240 = 7.5 h at 0.13C. A second example: 2000 mAh at 250 mA gives t = 2000/250 = 8 hours.
Set the capacity and load. The gauge drains in real-imagined time; the readout shows runtime t = Capacity/Load and the C-rate. Toggle the internal-resistance penalty to see how heavy loads shave real usable capacity.
A switch makes or breaks a circuit. We describe one by counting its poles (how many separate circuits it switches) and throws (how many positions each pole can connect to). SPST is single-pole single-throw — a plain on/off. SPDT adds a second throw (a changeover, A-or-B). DPDT switches two circuits at once, each with two positions — perfect for reversing a motor.
SPST — one circuit, on/off. The light switch.
SPDT — one circuit, selects between two destinations.
DPST — two circuits switched together, on/off.
DPDT — two circuits, each changeover; reverses motors.
A relay is a switch operated by an electromagnet instead of your finger. A small current through a coil creates a magnetic field that pulls an armature, flipping the contacts. This lets a tiny control signal (a microcontroller pin, a few milliamps) switch a big load (a mains motor, amps at hundreds of volts) with full electrical isolation between the two sides.
Here is the villain from Tab 0. The relay coil is an inductor, and inductors resist sudden changes in current. When you switch the coil off, the collapsing magnetic field tries to keep current flowing, and induces a large reverse voltage spike — inductive kickback, governed by V = L·(di/dt). With current cut abruptly, di/dt is enormous, so the spike can reach hundreds of volts, arcing across contacts and resetting nearby chips.
The fix is a flyback diode placed across the coil, reverse-biased in normal operation. When the field collapses, the diode gives the current a safe loop to die out gently, clamping the spike to about 0.7 V. One cheap diode tames the glitch — this is the standard solution to the "circuit glitches when the relay switches a motor" mystery.
Energize and de-energize the relay coil. Watch the contacts flip and the coil voltage spike at turn-off. Toggle the flyback diode to see the spike get clamped to a safe ~0.7 V.
Resistors are too small to print a number on, so their value is painted as a ring of colored bands. Decoding them is a rite of passage. A standard 4-band resistor has two significant-digit bands, a multiplier band, and a tolerance band. A 5-band resistor adds a third significant digit for precision parts.
Each color maps to a digit. The mnemonic-free truth: Black 0, Brown 1, Red 2, Orange 3, Yellow 4, Green 5, Blue 6, Violet 7, Gray 8, White 9. The first two bands give the digits, the third (multiplier) gives the power of ten, and the fourth gives tolerance: gold ±5%, silver ±10%, none ±20%.
Brown = 1, Black = 0, Red = 2 (multiplier), Gold = ±5%. So R = (10×1 + 0) × 10² = 10 × 100 = 1000 Ω = 1 kΩ ±5%. The ±5% window is 950–1050 Ω. Mystery #1 from Tab 0 is solved: it's a 1 kΩ resistor, and at 9 V it dissipates only P = 9²/1000 = 0.081 W — safe for a ¼ W part.
Yellow = 4, Violet = 7, Orange = 3 (multiplier), Gold = ±5%. R = (10×4 + 7) × 10³ = 47 × 1000 = 47 kΩ ±5%. With practice you read these in a second.
Drag each band's slider to choose its color. The decoder reads the bands to a live resistance value and draws the tolerance window. Switch to Encoder mode and type a target value to see which bands produce it. This is the big payoff — the whole color code in your hands.
The color code tells you a resistor's ohms but says nothing about how much power it can safely dissipate. Push too much through and it overheats — the failure mode from Tab 0. Hobby resistors come in standard ratings of ⅛, ¼, ½, 1 W and up. The power a resistor must handle is P = I²R = V²/R.
That last form is handy: given a power rating P and resistance R, the maximum safe voltage across the resistor is V = √(P·R). For a ¼ W, 1 kΩ part: Vmax = √(0.25 × 1000) = √250 ≈ 15.8 V. Our 9 V mystery resistor is comfortably under that. But put 9 V across a 100 Ω ¼ W part: P = 81/100 = 0.81 W — over three times its rating. It fries.
Components rated right at their limit run hot, drift, and die early. The rule of thumb is to derate: pick a resistor rated 2–4× the calculated power. If your resistor must dissipate 0.3 W, do not buy a ½ W part that runs at 60% — buy a 1 W part that loafs at 30%. It will last and stay cool.
You rarely have the exact value you need, so you combine resistors. In series, resistances add: Rtotal = R₁ + R₂ + … (same current through each, voltages add). In parallel, conductances add, so the reciprocal rule applies: 1/Rtotal = 1/R₁ + 1/R₂ + … (same voltage across each, currents add). Parallel resistance is always less than the smallest member.
You need ~750 Ω but only have 1 kΩ resistors. Three 1 kΩ in parallel give 1000/3 = 333 Ω (too low). Instead put a 1 kΩ in parallel with a 3 kΩ: R = (1000×3000)/(1000+3000) = 3&,000&,000/4000 = 750 Ω. If 12 V sits across this combo, total P = 12²/750 = 0.192 W, and because the 1 kΩ branch carries more current it dissipates more — check each branch and derate accordingly.
Set two resistor values and choose series or parallel. The widget computes the equivalent resistance with the correct rule and shows the power each resistor dissipates at the chosen voltage — so you can spot which one needs a beefier rating.
A capacitor stores energy in an electric field between two plates separated by a dielectric. It opposes changes in voltage: it resists charging up instantly, and once charged it holds that voltage. This makes capacitors the great smoothers, couplers, and filters of electronics. Capacitance C (in farads) is the charge stored per volt: Q = C·V.
Dielectric choice defines the type. Ceramic caps are tiny, cheap, great at high frequency (bypass/decoupling). Electrolytic caps pack huge capacitance into a small can but are polarized and leaky — the bulk smoothers in power supplies. Film caps are stable and precise for signal work. Each trades capacitance, voltage rating, stability, and size.
Capacitors combine opposite to resistors. In parallel, capacitances add: C = C₁ + C₂ + … (more plate area, more storage). In series, the reciprocals add: 1/C = 1/C₁ + 1/C₂ + … (like thicker dielectric, less storage). This is the mirror image of the resistor rules — a frequent exam trap.
Three jobs dominate. Coupling: a series cap passes AC signal but blocks DC bias between stages. Bypass/decoupling: a cap from a supply rail to ground shunts noise to ground, keeping the rail clean. Filtering: in a power supply, a big smoothing capacitor after a rectifier fills in the valleys of pulsing DC, leaving a small residual ripple. Bigger C (or lighter load) means smaller ripple. With an RC network the corner is set by the cutoff frequency:
A full-wave rectifier feeds a smoothing cap C, with load current I, ripple frequency f. The peak-to-peak ripple is approximately Vripple ≈ I / (f·C). For I = 0.5 A, f = 120 Hz (full-wave from 60 Hz mains), C = 1000 µF: Vripple ≈ 0.5 / (120 × 1000×10−6) = 0.5 / 0.12 = 4.2 V. Quadruple the cap to 4700 µF and ripple drops to ~0.89 V. Bigger cap, smoother DC.
Rectified DC feeds a smoothing capacitor. Increase C to fill in the valleys and flatten the ripple; increase the load current to drag the ripple back up. The readout shows peak-to-peak ripple and the RC cutoff frequency.
An inductor is a coil of wire that stores energy in a magnetic field. It is the dual of the capacitor: where a cap opposes changes in voltage, an inductor opposes changes in current (V = L·di/dt — the same law behind relay kickback in Tab 6). Inductance L grows with turns, coil area, and especially with a magnetic core (iron or ferrite) that concentrates the field.
Inductors combine like resistors (and opposite to capacitors). In series, L adds: L = L₁ + L₂ + …. In parallel, reciprocals add: 1/L = 1/L₁ + 1/L₂ + … (assuming no mutual coupling).
Wind two coils on the same core and you have a transformer. AC in the primary (Np turns) creates a changing flux that induces a voltage in the secondary (Ns turns). The voltage scales with the turns ratio:
More secondary turns than primary → step-up (higher voltage, lower current). Fewer → step-down. Power is (nearly) conserved, so voltage up means current down. Impedance transforms by the square of the turns ratio, which makes transformers superb impedance matchers:
A center tap is a connection at the electrical midpoint of a winding, giving two equal half-voltages of opposite phase — the basis of full-wave rectifiers and push-pull amplifiers. Real transformers are only 65–99% efficient: copper losses (winding resistance) and core losses (hysteresis, eddy currents) eat the rest.
Np = 200, Ns = 1200, Vp = 120 V. Turns ratio Ns/Np = 6, so Vs = 120 × 6 = 720 V (step-up). Drive the same transformer backwards (120 V into the 1200-turn winding): Vs = 120 × (200/1200) = 120/6 = 20 V (step-down). And if the secondary delivers 0.5 A, the primary draws Ip = 0.5 × 6 = 3 A — current trades inversely with voltage.
Set primary turns, secondary turns, and primary voltage. The coils redraw with the right turn density; the readout gives secondary voltage, the current ratio, and the impedance ratio (turns squared). Cross Nₛ below Nₙ for step-down, above for step-up.
The last passive component is the one that saves the rest. A fuse is a deliberately weak link — a thin metal element that melts and opens the circuit when current exceeds a safe value, protecting wiring and parts from a fault. A circuit breaker does the same job with a resettable bimetallic strip that bends and trips on overcurrent, then can be switched back on.
Fuses come in two speeds. A fast-blow fuse opens almost instantly on overcurrent — for sensitive electronics that must not see a surge. A slow-blow (time-delay) fuse tolerates a brief overload before blowing — essential for inductive loads like motors and transformers that draw a large inrush current at startup but settle to a modest running current. Use slow-blow there, or every power-on trips the fuse (the blown-fuse mystery from Tab 0).
Rate a fuse about 50% above the normal operating current, so normal operation never trips it but a real fault does. If a circuit normally draws 2 A, choose roughly a 3 A fuse (Ifuse ≈ 1.5 × Inormal). And place the fuse in the hot (line) side, before the load, so a fault is disconnected from the source — fusing the return side leaves the circuit live.
A motor draws 2 A running but 8 A inrush for a fraction of a second at startup. A 2 A fast-blow fuse blows on every start. Solution: a slow-blow fuse rated Ifuse ≈ 1.5 × 2 A = 3 A, time-delay type. It ignores the brief 8 A inrush but still opens on a sustained fault — exactly the behavior you want.
Set the normal current and pick a fuse rating. The widget marks the recommended ~1.5× rating, shows the inrush spike, and indicates whether a fast-blow or slow-blow fuse survives startup.
| Component | Key relationship | Non-ideal gotcha |
|---|---|---|
| Wire (AWG) | R = ρL/A; bigger #=thinner | Resistance & skin effect at HF |
| Cable | Z₀ = √(L/C) | Reflections if unmatched |
| Primary battery | Vnominal; Vterm=Vemf−I·Rint | Internal resistance, droop |
| Battery capacity | t = Capacity(Ah)/Load(A) | High-rate & cold loss (Peukert) |
| Switch / Relay | poles × throws; coil = inductor | Inductive kickback (use diode) |
| Resistor (color) | R = (10d₁+d₂)×10mult ±tol | Tolerance & power rating |
| Resistor power | P = V²/R; Vmax=√(P·R) | Derate 2–4× |
| Resistors series/parallel | Rs=ΣR; 1/Rp=Σ1/R | — |
| Capacitor | Q=CV; Cpar=ΣC; 1/Cser=Σ1/C | ESR / ESL |
| Cap filter | fc=1/(2πRC); Vrip≈I/(fC) | Ripple, ESR heating |
| Inductor | V=L·di/dt; Ls=ΣL; 1/Lp=Σ1/L | Saturation, kickback |
| Transformer | Vs=Vp(Ns/Np); Np/Ns=√(Zp/Zs) | 65–99% efficient |
| Fuse / breaker | Ifuse≈1.5×Inormal | Inrush → use slow-blow |
You can now decode a resistor, predict a battery's life, size a fuse, and read a transformer. Next, we make these parts do something active.