Circuit Lab

Circuit Lab is a two-person event on electricity and electronics. Your team takes a written test and completes at least one hands-on task at a bench: wiring a breadboard, reading a multimeter, building a logic circuit or finding the fault in one. Under the 2027 rules the event has moved away from magnetism and capacitors and toward electronics. The written test now covers DC circuits in depth, diodes and LEDs, and combinational digital logic (gates, truth tables, Boolean algebra and adders) in both divisions. Division C adds flip-flops and timing, bipolar transistors and op-amps at State and National tournaments. Magnetism, electromagnets, transformers, motors, capacitance and Thevenin/Norton equivalents are no longer tested.

Event format

Class III is the most capable class allowed: any stand-alone calculator, including programmable and graphing models, but not calculator apps on phones, tablets, laptops or watches (see the calculator classes in the general rules). Printed books are now explicitly allowed in the notes collection, so a good electronics textbook with tabs can sit next to your binder.

What the rules cover

Both divisions' 2027 rules follow the same layout: a written test (Part I) with a list of content areas, a list of excluded topics, a list of allowed mathematics, then a hands-on part (Part II) and scoring. The written test must include at least one question from each listed area and from those areas only. The table summarizes the areas (Division B section 3.c, Division C section 3.c).

The 2027 rules also cap the mathematics a question may need:

The hands-on part is described by example. The listed tasks are: find a mystery resistor using only voltage readings; calculate the power supplied to a circuit; wire two LEDs so they glow equally brightly; read a USB charger label and work out its output power levels and a battery's charging time; build a logic circuit that matches a given truth table; troubleshoot a faulty circuit; wire a 3-way light switch with SPDT switches; and build a specified equivalent resistance from given resistors. The list is not exhaustive, so practice the underlying skills, not only these eight tasks.

History of electricity

Every test must ask at least one history question, and the 2027 rules name seven people: Ampère, Coulomb, Kirchhoff, Volta, Ohm, Hertz and Boole. Hertz and Boole are new for 2027; Faraday and Tesla are no longer on the list. History questions are cheap points if you know four things about each person: nationality and era, the discovery or invention, the publication or year tests quote, and the unit or law that carries the name. Many tests require the full first and last name, so learn the spelling.

The seven named scientists

More on Volta, because tests ask a lot about him. February 18 is observed in the U.S. as National Battery Day in his honor. His first published paper (1769) was a letter to Giambattista Beccaria on electrical attraction. The electrophorus makes static charge repeatedly by induction. He identified methane in marsh gas (1776–78). Against Galvani he argued that twitching frog legs only detected electricity made by the contact of two different metals, and he arranged metals in an early electromotive (contact) series: two different metals in contact develop a small contact potential, because their electrons are held with different energies (different work functions). He announced the pile in a letter to the Royal Society; his "crown of cups" used a row of cups of brine with metal strips. The pile's weakness was polarization: hydrogen bubbles collected on the copper plate and raised the internal resistance, so the current faded, a problem later cells such as Daniell's (1836) solved. Napoleon made him a count.

Other names that appear on tests

Older tests (and some 2027 writers) also ask about people outside the named seven. The 2027 history area lists only the seven above, so treat this table as background, but it explains terms such as galvanic, Leyden jar and the war of the currents.

Charges, fields and static electricity

Electric charge

Ordinary matter is made of protons (charge $+e$), electrons ($-e$) and neutrons (no charge), with $e = 1.602 \times 10^{-19}$ C. An object is neutral when it holds equal numbers of protons and electrons; it still contains huge amounts of both kinds of charge, which is why most objects show no static effects. An object becomes charged almost always by gaining or losing electrons: protons are locked in nuclei. So a positively charged object has a deficit of electrons, not "extra protons", and it still contains plenty of negative charge. A nucleus with atomic number $Z$ carries $Q = Ze$: a thorium nucleus ($Z = 90$) holds $90 \times 1.602 \times 10^{-19} = 1.44 \times 10^{-17}$ C, and that is the $Q$ to use in $E = kQ/r^2$ near it. An atom that has gained or lost electrons is an ion (negative ion: anion; positive ion: cation).

Two rules hold without exception. Conservation of charge: the total charge of an isolated system never changes, so rubbing a balloon on hair moves charge from one to the other but the pair's total stays zero. Quantization: every free charge is a whole-number multiple of $e$; no free particle with a fraction of $e$ has ever been observed (quarks carry thirds but are never found alone). Millikan's oil-drop experiment (1909) measured $e$ by balancing charged oil drops between plates.

Conductors have electrons free to move through the material (metals: each atom contributes outer electrons to a shared "sea"); insulators hold their electrons bound to atoms; semiconductors such as silicon sit in between and are covered with diodes below. Among metals, conductivity runs silver > copper > gold > aluminum > tungsten > iron. Good insulators include rubber, glass, plastics, porcelain and other ceramics, mica, dry wood, dry air and oil. The same free electrons carry heat, so good electrical conductors (metals) are also good thermal conductors, and most electrical insulators insulate heat too. Pure (distilled) water is a poor conductor; tap and salt water conduct because of dissolved ions, which is why water and electricity are a dangerous mix.

Coulomb's law

Charge comes in two signs. Like charges repel and unlike charges attract, and the force between two point charges grows with the product of the charges and falls with the square of the distance between them. Coulomb found this with his torsion balance: the twist of a fine wire measured the force.

The force obeys Newton's third law: the two charges push or pull on each other with equal and opposite forces even if one charge is much larger; the smaller or lighter object simply accelerates more. Coulomb's law is exact for charges at rest (electrostatics); moving charges also feel magnetic forces. It has the same inverse-square form as Newton's law of gravity, but it is enormously stronger: between an electron and a proton the electric force is about $10^{39}$ times their gravitational attraction.

Force is a vector, so a charge pushed by two others feels the vector sum. The rules let Division B use only 2D geometry (no trigonometry), so Division B problems are set up with forces along a line or at right angles, where the Pythagorean theorem gives the size. Division C may use trigonometry to split angled forces into components.

Where is the net force zero? For a third charge placed on the line through two charges:

• Like charges (same sign): the balance point lies between them, closer to the smaller charge, where $q_1/r_1^2 = q_2/r_2^2$, that is $r_1/r_2 = \sqrt{q_1/q_2}$.
• Unlike charges of different size: the balance point lies outside the pair, on the side of the smaller charge.
• Equal and opposite charges: there is no such point at any finite distance.

Electric field and potential

The electric field at a point is the force per unit charge a small positive test charge would feel there, $E = F/q$, in N/C (the same as V/m). A point charge makes $E = kQ/r^2$, so the field also falls as $1/r^2$: at $r$, $2r$ and $3r$ the fields are in the ratio $36 : 9 : 4$. A positive charge in a field is pushed along the field; a negative charge (an electron) is pushed against it. A proton and an electron in the same field feel forces of equal size in opposite directions, but the electron's acceleration is about 1836 times larger because its mass is that much smaller.

Field lines follow fixed rules, and tests ask for them directly:

• They start on positive charges (pointing radially outward) and end on negative charges (pointing radially inward).
• They never cross, because the field has one direction at each point.
• Their density shows the field strength: crowded lines mean a strong field.
• A charge's force is tangent to the line through it; the lines are not the paths charges follow.
• They point from high potential toward low potential and meet equipotential surfaces at right angles.
• They meet a conductor's surface at right angles, and none exist inside the metal of a conductor in equilibrium.

A dipole (equal and opposite charges close together) has a field that falls off faster than a single charge's, as $1/r^3$ far away, because the two charges' fields nearly cancel.

Electric potential is potential energy per unit charge, in volts (1 V = 1 J/C). A point charge makes $V = kQ/r$; potential is a scalar, so the potentials of several charges add as plain signed numbers, while fields add as vectors. Moving a charge $q$ through a potential difference $\Delta V$ changes its potential energy by $\Delta U = q\,\Delta V$, and the field does work $W = -q\,\Delta V$ on it (a battery that lifts the charge through a voltage $\Delta V$ does $W = q\,\Delta V$). The potential energy of two charges is $U = kq_1q_2/r$: positive for like charges, so it decreases as they fly apart (the energy becomes kinetic). Equipotential surfaces are sets of points at the same potential; moving along one takes no work, and they are always perpendicular to the field lines. The work done around any closed path in an electrostatic field is zero. Around a point charge the equipotentials are concentric spheres; surfaces drawn at equal steps of potential (say every 10 V) get farther apart with distance, because the field, and so the potential change per meter, weakens as $1/r^2$. In a uniform field between plates they are equally spaced parallel planes. An electric field also stores energy: the energy per unit volume is $u = \tfrac{1}{2}\varepsilon_0 E^2$ (J/m³).

The electronvolt is the energy one elementary charge gains crossing 1 V: 1 eV = $1.602 \times 10^{-19}$ J. A proton or electron accelerated through 9.0 V gains 9.0 eV; an alpha particle (charge $+2e$) gains 18 eV.

Charge can also be spread out: a charge density is charge per unit volume (C/m³), per unit area (C/m²) or per unit length (C/m). Gauss's law, which relates the field through a closed surface to the charge inside it, is equivalent to Coulomb's law for charges at rest. A wire carrying a steady current stays electrically neutral: as many electrons flow out of each section as flow in.

Scalars and vectors. Charge, potential (voltage), energy, work, power, resistance and circuit current are scalars (current has a direction along the wire, but currents add as signed numbers). Force, electric field, displacement, velocity and acceleration are vectors.

The medium matters too. Permittivity $\varepsilon$ measures how strongly a material polarizes in a field; $\varepsilon = \kappa\varepsilon_0$, in F/m, where $\kappa$ is the unitless relative permittivity (dielectric constant, 1 for vacuum, about 80 for water). Inside a material Coulomb's law becomes $F = q_1q_2/(4\pi\varepsilon r^2)$, weaker by the factor $\kappa$.

In a conductor at equilibrium the field inside the metal is zero, any excess charge sits on the outer surface (the charges repel each other as far apart as they can get), and the whole conductor, inside included, is at one potential. So for a charged hollow or solid metal sphere, the field is zero inside, greatest just outside the surface, and like a point charge $kQ/r^2$ beyond; the potential inside equals the surface value. Charge crowds onto sharp points, where the field is strongest; that is why lightning rods are pointed and why sparks and corona start at points. A closed conducting shell with no charge in its cavity has zero field in the cavity even in an external field: its surface charge rearranges until the inside field cancels. That is electrostatic shielding, the Faraday cage. Two conducting spheres joined by a wire sit at one potential, so their charges are proportional to their radii and their surface charge densities go as $1/r$: the smaller sphere has the denser charge and the stronger surface field. The "zero field inside" rule is for charges at rest. In a wire carrying current there is a field inside, $E = V/L$ along the wire; it is set up within nanoseconds by small surface charges that build up along the wire when the circuit is closed.

Static electricity: sources and hazards

Static electricity is an imbalance of charge that has built up on an object's surface and stays put. Objects become charged in three ways:

Polarization is induction without charge transfer: a nearby charge pulls the neutral object's charges slightly apart, its near side takes the opposite sign and feels a stronger pull than the repulsion on the far side, so the neutral object is attracted. This is why a charged balloon picks up paper scraps or sticks to a wall, and why a charged object attracts a neutral one whichever sign it carries. Induction and polarization do not change the object's net charge until a ground connection lets charge flow.

When identical conducting spheres touch, they end with equal charges, each holding half the total: +6.0 µC and +14.0 µC become +10.0 µC each; +2 C and −1 C become +0.5 C each. Conductors of different size end at the same potential, so the larger sphere takes the larger share: two spheres far apart and joined by a thin wire share charge in proportion to their radii.

The triboelectric series ranks materials by how readily they give up electrons. When two materials are rubbed, the one nearer the positive end becomes positive; the farther apart they are, the more charge moves. Exact orders differ between sources, but the classic pairs are consistent:

A longer version of the series, as commonly published (from the positive end): air, dry human skin, leather, rabbit fur, glass, human hair, nylon, wool, lead, cat fur, silk, aluminum, paper, cotton, steel, wood, amber, hard rubber, nickel and copper, brass and silver, gold and platinum, polyester, polystyrene (Styrofoam), plastic wrap, polyethylene, polypropylene, vinyl (PVC), silicone, Teflon (the negative end). Rankings of neighbors vary between lists, so test questions use widely separated pairs.

Glass rubbed with silk becomes positive (the glass loses electrons to the silk). A rubber balloon rubbed on hair, a rubber rod rubbed with fur and polyethylene rubbed with wool all become negative. Styrofoam packing peanuts cling to a cat because the fur becomes positive and the foam negative.

Devices and natural examples that tests name:

• An electroscope shows whether an object is charged (the leaves or needle spread apart); it does not by itself show the sign.
• An electrophorus (Volta) charges a metal plate by induction over and over from one charged insulating base.
• A Van de Graaff generator (invented by the American physicist Robert J. Van de Graaff around 1929–1931) uses a moving rubber belt: charge is produced by contact (triboelectric) charging at the lower roller and sprayed onto or taken off the belt by sharp metal combs (corona discharge); the upper comb transfers it to the dome, where it collects on the outer surface. Many textbooks summarize the combs' job as electrostatic induction plus the "action of points" (corona discharge), so a key may name induction. Hair stands up because each strand gets the same sign and they repel.
• Lightning is a giant static discharge, a one-way (DC) pulse of roughly tens of kiloamperes. Collisions between ice particles in a thunderstorm separate charge, usually leaving the cloud base negative. A lightning rod (Franklin) gives the strike a low-resistance path to ground through a heavy conductor, so the current bypasses the building.
• Corona discharge is a partial discharge around high-voltage conductors or sharp points that ionizes the air, giving a faint glow and a hiss; it wastes power on transmission lines.
• Piezoelectric crystals (quartz) make a voltage when squeezed and pyroelectric crystals when heated; both separate charge without friction. Light does not cause charge separation by rubbing; the photoelectric effect ejects electrons, a different process. Several of these effects run in reverse: a voltage applied to a piezoelectric crystal makes it flex (the inverse piezoelectric effect, used in buzzers and quartz oscillators), and an electric field applied to some materials changes their temperature (the electrocaloric effect, the reverse of pyroelectricity). Triboelectric charging has no such reverse: applying charge does not make surfaces rub. Corona and sparks also turn oxygen into ozone, which smells sharp and slowly cracks rubber insulation; radio-frequency RF burns come from high-frequency currents near transmitters, not from static.

Static hazards come from sudden discharge. A person walking on carpet can reach thousands of volts; you feel a spark at roughly 2–3 kV, but a much smaller discharge can still destroy a chip (electrostatic discharge, ESD). Electronics work uses grounded wrist straps, ESD mats, anti-static bags, grounded tools and controlled humidity. Sparks can ignite fuel vapor, solvent fumes and dust, which is why fuel trucks and aircraft are bonded and grounded before fueling (bonding joins conductive objects to each other so they sit at the same potential and cannot spark between them; grounding connects them to earth so the charge drains away), and why plastic dust-collection pipes in workshops are grounded. Flowing, splashing or free-falling flammable liquids also build up charge, so fuel is poured slowly through a bonded metal funnel or fill pipe that reaches the bottom of the container, never sprayed or dropped from a height, and portable gas cans are filled while standing on the ground rather than in a truck bed. Humid air and surface moisture bleed charge away, so static problems are worst on cold, dry winter days. Static builds up on insulators (plastics such as PVC, polyethylene and Teflon are the classic accumulators) because the charge cannot flow away; a conductor that is grounded or held in a bare hand drains its charge at once. In industry some of the most energetic discharges come from charged insulating sheets and linings (propagating brush discharges), which can ignite most flammable dusts.

Current, voltage, resistance, power and energy

Here $E$ means energy (some books write $W$ or $U$); in the electrostatics section $E$ means electric field. The units tell them apart: joules versus N/C.

Quantities and units

Every electrical quantity is a ratio of a few basic ones. Charge $Q$ is measured in coulombs; one electron carries $-e = -1.602 \times 10^{-19}$ C, so 1 C is $6.24 \times 10^{18}$ elementary charges. Current is the rate at which charge passes a point, $I = Q/t$. Voltage (potential difference, or emf for a source) is the energy given to or taken from each coulomb, $V = W/Q$; it is what pushes current, the electrical "pressure". Resistance is the opposition to current, $R = V/I$: how many volts it takes to push one ampere through. Power is the rate at which energy is transferred, $P = W/t$, and energy is power times time.

The table collects the units and their equivalences; any of the equivalent forms can be the answer to a "which unit equals…" question.

To build a base-unit expression, chain the definitions: J = N·m = kg·m²/s²; W = J/s = kg·m²/s³; V = W/A = kg·m²·s⁻³·A⁻¹; Ω = V/A = kg·m²·s⁻³·A⁻². Since 2019 the ampere has been defined by fixing the elementary charge at exactly $1.602176634 \times 10^{-19}$ C, so 1 A is $6.24 \times 10^{18}$ elementary charges per second. Older definitions used the force between two parallel wires; neither is likely to be asked in detail.

Other units that appear: horsepower (1 hp = 746 W, the electrical horsepower; a mechanical horsepower is 745.7 W), kilowatt-hour for electric bills, milliampere-hour for batteries and the electronvolt for the energy of single particles. Prefixes: m = 10⁻³, µ = 10⁻⁶, n = 10⁻⁹, k = 10³, M = 10⁶, G = 10⁹. Note that mΩ (milliohm) and MΩ (megohm) differ by a factor of 10⁹.

Conventional current is defined as the flow of positive charge, from the + terminal through the external circuit to the − terminal; the convention dates from Franklin (see History of electricity). In metal wires the moving carriers are free electrons only (the positive ions stay in the lattice), and they drift the opposite way, from − to +. In electrolytes and in the body both positive and negative ions move. Circuit analysis uses conventional current; a question about electron flow wants the reverse direction.

The electrons themselves move slowly. Their drift velocity is $v_d = I/(nqA)$, where $n$ is the number of free electrons per cubic meter ($8.5 \times 10^{28}$ m⁻³ for copper). For 1.0 A in a 1.0 mm² copper wire, $v_d = 1.0/(8.5 \times 10^{28} \times 1.60 \times 10^{-19} \times 1.0 \times 10^{-6}) = 7 \times 10^{-5}$ m/s, less than a tenth of a millimeter per second. The electrons' own random thermal motion is far faster: in a metal, conduction electrons move at the Fermi velocity, about $10^6$ m/s, but in random directions, so only the tiny drift adds up to a current. The average time between their collisions (the relaxation time, about $10^{-14}$ s in copper) sets the resistivity, $\rho = m/(ne^2\tau)$. A lamp still lights instantly because the electric field (the signal) travels along the wire at a large fraction of the speed of light and pushes all the electrons, already present throughout the wire, at once.

Ohm's law

Ohm's law says the current through a conductor is proportional to the voltage across it, so the ratio $R = V/I$ is constant: $V = IR$. At fixed voltage, more resistance means less current ($I = V/R$, inverse); at fixed resistance, current is proportional to voltage (double V, double I). Ohm found it empirically in 1827.

Ohmic and non-ohmic devices. A device is ohmic if its resistance stays constant as voltage and current change; its I–V graph is a straight line through the origin, and the slope of an I–V graph is $1/R$ (the slope of a V–I graph is $R$). Fixed resistors and metal wires at constant temperature are ohmic. Non-ohmic devices have curved graphs:

• An incandescent bulb's tungsten filament heats from room temperature to roughly 2000–3000 °C, and its resistance rises roughly tenfold, so the I–V curve bends over (less current than a straight line would predict at high voltage). A cold bulb's low resistance is why bulbs draw a surge and often fail at switch-on.
• A diode or LED conducts almost nothing until about 0.6–0.7 V (silicon) or its forward voltage, then current rises steeply; see Diodes, LEDs and the PN junction.
• Thermistors and light-dependent resistors (see Resistors, potentiometers and sensors) change resistance on purpose.
• A superconductor has zero resistance, so $V = IR$ gives zero voltage for any current; Ohm's law in the usual form does not describe it.

$R = V/I$ still defines the resistance of a non-ohmic device at one operating point; it just is not constant. So Ohm's law is not a universal law but a property of some materials under some conditions: it holds for metals and resistors at constant temperature, and fails for diodes, LEDs and transistors, gas discharges and vacuum tubes, electrolytes, filaments that heat up, superconductors, and at very high fields or frequencies.

Voltage drop means the decrease in potential across a component or along a conductor that carries current, $V = IR$. In wiring it is the loss along the supply wires: a long or thin extension cord drops part of the voltage, so the load receives less than the source supplies (and the cord heats). That is why long runs and high-current loads use thicker (lower-gauge) wire.

Power and Joule heating

Power is energy per time. A source does work $W = QV$ moving charge $Q$ through voltage $V$, so the rate is $P = (Q/t)V = IV$. Substituting Ohm's law gives the other two forms. In a resistor, the electrical energy becomes heat: the drifting electrons collide with the vibrating ions of the lattice and give up energy (Joule heating, Joule's first law: heat $\propto I^2R$). That is the working principle of toasters, kettles, hair dryers, incandescent bulbs, fuses and electric stoves, and the cause of power-line losses.

Choose the form by what stays fixed. Same current (series): $P = I^2R$, so the larger resistance dissipates more. Same voltage (parallel, or anything on a household outlet): $P = V^2/R$, so the smaller resistance dissipates more. That is why a 100 W, 120 V bulb has a smaller resistance than a 60 W, 120 V bulb.

Ratio and percent-change questions. Write the formula with the quantity that is fixed, substitute the changed values, and factor out the original.

Energy, kilowatt-hours and the electric bill

A utility charges for energy, measured in kilowatt-hours: one kWh is 1 kW used for 1 h, which is $1000\ \text{W} \times 3600\ \text{s} = 3.6 \times 10^6$ J. The US average residential price is roughly 15–20 cents per kWh. Bills use kWh rather than joules because the joule is tiny (a typical home uses a few billion joules a month) and because kW × hours matches appliance ratings and hours of use directly.

Efficiency

No device converts all its input energy into the form you want; the rest usually becomes heat. Efficiency is the fraction that becomes useful output:

Typical efficiencies: electric resistance heater nearly 100% (all the energy becomes heat, which is the goal); large electric motors 85–95%; LED lamp roughly 30–40% of input becomes light, an incandescent lamp only about 2–5%, the rest heat; solar panels 15–23%; power transmission and distribution lose about 5% overall.

Resistivity and wire resistance

Resistance belongs to an object; resistivity $\rho$ belongs to a material. Resistivity is the resistance of a 1 m cube of the material between opposite faces, in Ω·m, and it depends only on the material and its temperature, not on the shape. A wire's resistance grows with its length (electrons collide over a longer path) and shrinks with its cross-sectional area (more parallel paths side by side):

Metals conduct because each atom contributes one or more free (conduction) electrons that roam the whole crystal; resistance comes from those electrons scattering off lattice vibrations, impurities and defects. In insulators the electrons are tightly bound, and semiconductors lie in between.

The table gives resistivities at 20 °C (metals, alloys and semiconductors from OpenStax College Physics), grouped as pure metals, alloys, semiconductors, water and insulators, roughly from best conductor to best insulator.

Patterns to remember: conductivity Ag > Cu > Au > Al (resistivity runs the other way); alloys (nichrome, constantan) have much higher resistivity than pure metals; resistivities span more than 25 powers of ten from silver to quartz. Water's conductivity comes from dissolved ions, so seawater > tap water > distilled water. Copper's conductivity is $\sigma = 1/1.72 \times 10^{-8} = 5.8 \times 10^7$ S/m.

Scaling questions are the most common resistivity item. Write $R \propto L/A \propto L/d^2$ and substitute factors.

Stretching rule: a wire stretched to $n$ times its length (same volume) has $n^2$ times the resistance. Resistance per unit length: if a wire has 4 Ω per 1000 ft, 750 ft has 3.0 Ω.

Temperature, resistance and superconductors

In a metal, heating makes the ions vibrate harder, so the free electrons collide more often and resistance rises (positive temperature coefficient). In a semiconductor or carbon, heating frees more charge carriers, which outweighs the extra collisions, so resistance falls (negative coefficient). Over moderate ranges the change is close to linear:

An insulator behaves like a semiconductor with a very wide gap: heating frees a few more carriers, so its resistance falls as it warms, and hot or damp insulation leaks more current. Superconductors have exactly zero resistance below a critical temperature $T_c$, so a current in a superconducting loop flows indefinitely with no energy loss. Heike Kamerlingh Onnes discovered it in mercury at 4.2 K (1911). Most superconductors need cooling with liquid helium (4 K); "high-temperature" ceramic superconductors such as YBCO ($T_c \approx 92$ K) work in cheaper liquid nitrogen (77 K). The need for constant refrigeration is what makes superconducting devices expensive. Uses: MRI scanner magnets, particle accelerators, maglev trains, sensitive magnetic sensors and experimental power cables. To make a material superconduct, cool it.

Variable resistors and dimming

A potentiometer has three terminals: a resistive track between two ends and a sliding wiper. Used with all three terminals it is an adjustable voltage divider (volume knobs, joystick position sensors); used with only the wiper and one end it is a rheostat, an adjustable series resistance that controls current. A simple rheostat dimmer works but wastes power: the bulb is dimmer because part of the supply voltage is dropped across the rheostat, which turns that energy into heat. Modern solid-state dimmers instead switch the AC on for only part of each half cycle (phase control), so they waste very little. Their key parts are a triac (a semiconductor switch that conducts in both directions once triggered), a diac that triggers it, and a resistor-capacitor timing network set by the knob; older stage and lab dimmers used a variable autotransformer (Variac). A plain switch, fuse or diode bridge is not what does the dimming. Schematic symbols are in Schematics, open circuits and short circuits.

DC and AC

Characteristics and sources

Direct current (DC) is charge flow in one direction only; an ideal DC source holds a constant voltage, so a resistor across it carries a constant current. Alternating current (AC) periodically reverses direction: the voltage of an outlet swings smoothly positive, through zero, negative and back, tracing a sine wave. The rules test what each is, where it comes from, what it is used for and why it is hazardous, but not AC circuit theory: no impedance, reactance or phase. For a purely resistive load (heater, toaster, incandescent bulb), Ohm's law and the power formulas work with AC exactly as with DC as long as you use RMS values.

The table sums up the comparison tests ask about most.

DC came first: Volta's pile (1800) and Edison's first power station (Pearl Street, New York, 1882) were DC. AC distribution, championed by Tesla and Westinghouse, won the "war of the currents" in the 1880s–1890s because of the transformer (see History of electricity).

Measuring an AC wave: peak, RMS and frequency

A sine wave has several "sizes", and tests check that you know which is which.

• Peak (amplitude) $V_p$: the largest instantaneous value, measured from zero.
• Peak-to-peak $V_{pp} = 2V_p$: from the most negative to the most positive value.
• Average over a full cycle: zero, because the positive and negative halves cancel. (The average of one half cycle is $2V_p/\pi \approx 0.637\,V_p$; this matters only for rectified waves.)
• RMS (root mean square, the effective value): square the wave, average the square, take the square root. For a sine wave $V_{\text{RMS}} = V_p/\sqrt{2} = 0.707\,V_p$, and $V_p = \sqrt{2}\,V_{\text{RMS}} = 1.414\,V_{\text{RMS}}$. The same rule holds for current.
• Period $T$ is the time for one cycle; frequency $f = 1/T$ in hertz (cycles per second). At 60 Hz, $T = 16.7$ ms, and the current reverses direction 120 times a second. At 50 Hz, $T = 20$ ms.

Why RMS? Heating in a resistor goes as $V^2/R$, which is never negative. The RMS value is the DC voltage that delivers the same average power to a resistor. So "120 V AC" heats a toaster exactly as 120 V DC would, and meters, appliance labels and outlet ratings all quote RMS unless they say otherwise.

A signal can also be AC riding on DC: a steady DC level (the DC offset) with an AC variation added on top, $v(t) = V_{DC} + V_p\sin(2\pi f t)$. Its average is the DC offset, not zero. Examples: the bumpy output of a rectifier, an audio signal on a transistor's bias voltage, and the ripple on a cheap DC supply. A meter's DC setting reads the offset; the AC setting reads the varying part.

In a pure resistor on AC, voltage and current rise and fall in phase (together), because $v = iR$ holds at every instant.

Converting between AC and DC

Four device names come up constantly.

Most things that plug into the wall still run on DC inside. To classify a device on a test, ask what its working parts use:

A rating such as "12 V, 2 A" on a supply means it holds 12 V and can deliver up to 2 A; the load's resistance sets how much current is actually drawn ($I = V/R$). A charger label shows both: "Input 100–240 V ~ 50/60 Hz" (the ~ means AC) and "Output 5 V ⎓ 2 A" (the ⎓ means DC). See USB charger labels.

Processes that need DC: electrolysis and electroplating (the electrodes must stay positive and negative; with AC the reactions reverse every half cycle), charging a battery, and powering logic and LEDs. Things that need AC: transformers and the grid.

Power transmission: why the grid is AC

A power line has resistance $R_{\text{line}}$, and it wastes $P_{\text{loss}} = I^2 R_{\text{line}}$ as heat. A power station must deliver a power $P = VI$. For the same delivered power, raising the voltage lowers the current in proportion, and the loss falls with the square of the current: ten times the voltage means one tenth the current and one hundredth the loss. So electricity is generated at roughly 10–30 kV, stepped up by a transformer to 115–765 kV for transmission, then stepped down at substations to distribution voltages (about 4–35 kV) and finally by a pole-mounted or pad-mounted distribution transformer near your house to 120/240 V. Transformers keep power (nearly) constant, so when a step-down transformer lowers the voltage, the current on its low-voltage side goes up by the same factor. Transformers only work with AC; that is the main reason AC won.

Details tests ask about transmission:

• Conductors are mostly aluminum (usually aluminum strands around a steel core, "ACSR"), not copper. Aluminum conducts about 61% as well as copper per unit volume but is about one third as dense and much cheaper, so per kilogram it carries about twice the current; the steel core adds strength for long spans.
• Skin effect: alternating current crowds toward the outer surface of a conductor, more so at higher frequency, so the wire's effective resistance rises. At 60 Hz the effect is small for household wire but matters for thick transmission conductors; at radio frequencies conductors are made hollow or from many insulated strands (Litz wire). Steady DC uses the whole cross-section, one reason HVDC lines lose less.
• Long-distance lines are three-phase AC (three wires whose waves are offset by a third of a cycle). Three-phase analysis is excluded from the 2027 rules; just know the name.
• HVDC (high-voltage DC) is used where its lower line losses and lack of reactive effects beat the cost of converter stations at each end: very long overhead lines (hundreds of kilometers or more), undersea and underground cables (tens of kilometers or more, where AC cables waste power charging the cable's capacitance), and links between grids that are not synchronized (for example different frequencies, as between Japan's 50 Hz and 60 Hz halves). DC's main disadvantage is that its voltage cannot be stepped up or down with a simple transformer.
• Corona discharge (ionized air around very high-voltage conductors) wastes some power and makes a hiss or crackling buzz; see Static electricity: sources and hazards.
• Generation: most US electricity comes from rotating generators turned by steam, gas or water (natural gas is the largest single source, around 40%); solar panels are the main source that produces DC, which an inverter converts to grid AC.

Batteries and cells

A cell (electrochemical, voltaic or galvanic cell) turns chemical energy directly into electrical energy with a spontaneous redox reaction. A battery, strictly, is two or more cells connected together: an AA is a single cell, while a 9 V battery is six 1.5 V cells in series and a 12 V car battery is six 2 V lead-acid cells. A battery is a source of electrical energy made from stored chemical energy; it stores chemical energy, not a pile of separated charge as a capacitor does; its "capacity" in Ah is the total charge it can push around the circuit before the chemicals are used up. It is an active component (it supplies energy), unlike resistors, which are passive.

Every cell has three parts:

Memory aid: AN OX, RED CAT (anode oxidation, reduction cathode). Outside the cell, electrons flow from the negative anode through the load to the positive cathode; conventional current flows the other way, out of the + terminal. Inside the cell, the chemistry pushes positive charge from − to +, which is what the cell's emf does. A voltaic (galvanic) cell runs a spontaneous reaction and produces electricity; an electrolytic cell uses an external DC supply to drive a non-spontaneous reaction (electrolysis, electroplating, recharging). The definitions stay fixed (anode = oxidation, cathode = reduction), but the signs flip: in an electrolytic cell the anode is positive (connected to the supply's + terminal) and the cathode is negative.

Cells are classified two ways. Primary cells are used once and thrown away (the reaction is not practically reversible); secondary cells are rechargeable (an external DC source drives the reaction backward). Wet cells have a free liquid electrolyte (a flooded car battery, Volta's and Daniell's cells, a lemon cell); dry cells have the electrolyte as a paste or gel so they can be used in any orientation (zinc-carbon, alkaline, most modern rechargeables). The nominal voltage is set by the chemistry, not the size: AAA, AA, C and D alkaline cells are all 1.5 V; bigger cells hold more energy and deliver more current.

The table lists the chemistries and nominal voltages tests ask about.

A lemon or potato cell works because two different metals sit in an electrolyte (the juice): zinc is oxidized, and the copper surface is where reduction happens. Two identical metals give no voltage. Which metal becomes the anode follows the activity series (or the table of standard reduction potentials): the more reactive metal, the one more easily oxidized (zinc, magnesium, aluminum, iron), is the negative anode, and the less reactive one (copper, silver, gold) is the cathode. The farther apart they are in the series, the higher the cell voltage. The voltage is under 1 V and the internal resistance is very large (kilohms), so the current is under about a milliampere: enough for a low-power LCD clock, not for a bulb, and several cells in series are needed to light even a red LED. The potato or lemon supplies the electrolyte; the energy comes from the zinc being consumed. An ancient clay jar called the "Baghdad battery" is sometimes called the first battery, but whether it was one is disputed; Volta's pile (1800) is the first documented battery.

Battery capacity, run time and charging

Capacity is the total charge a battery can deliver, quoted in ampere-hours (Ah) or milliampere-hours (mAh). Despite the name, an ampere-hour is a unit of charge: 1 Ah = 3600 C and 1 mAh = 3.6 C. The energy stored is capacity times voltage, in watt-hours: $E = V \times Q$, so 1 Wh = 3600 J. Typical values: AA alkaline about 2000–3000 mAh (less at high drain), AA NiMH about 2000 mAh, phone 3000–5000 mAh at 3.85 V (12–19 Wh), laptop 40–100 Wh, car starter battery 40–100 Ah at 12 V.

Real capacity drops at high discharge currents and in the cold (a battery rated "700 Ah at 3.5 A" delivers less if you pull 100 A), but test problems use the simple $t = Q/I$ unless they say otherwise.

Cells in series and parallel. In series (+ of one to − of the next) the voltages add and the capacity in Ah stays that of one cell: four 1.5 V, 2500 mAh cells give 6.0 V and 2500 mAh. In parallel (all + together, all − together) the voltage stays that of one cell and the capacities add: four in parallel give 1.5 V and 10 000 mAh, so the pack runs longer and can supply more current. Either way the stored energy is $4 \times 1.5\ \text{V} \times 2.5\ \text{Ah} = 15$ Wh. Parallel cells must be identical and equally charged, or the stronger one drives current into the weaker. A cell connected backward in a series string subtracts its voltage: three cells with one reversed give $1.5 + 1.5 - 1.5 = 1.5$ V.

Battery care and temperature. Lithium-ion cells have no "memory effect": they do not need full discharges and last longest when kept between roughly 20% and 80% charge; deep discharge below their minimum voltage damages them, which is why packs contain protection circuits. Nickel-cadmium cells were the ones with a memory effect. Chargers fill a Li-ion cell in two stages, constant current (often 0.5–1 C) until the cell reaches 4.2 V (for common 3.6–3.7 V cells; LiFePO₄ stops at about 3.6 V), then constant voltage at 4.2 V while the current tapers off, then stop. A battery and a fixed resistor cannot do this safely: the current falls as the cell voltage rises, nothing stops it at exactly 4.2 V (overcharging is a fire risk), and a flat or shorted cell draws too much current. Cold slows a battery's chemistry and raises its internal resistance, so its voltage under load and its usable capacity drop (cars are hard to start in winter); heat lets it deliver current more easily but speeds up the side reactions that age it, shortening its life. Large home and grid energy-storage systems today use mostly lithium-ion batteries (often the LiFePO₄ chemistry), which have largely replaced lead-acid there.

Internal resistance and terminal voltage

A real cell behaves like an ideal voltage source, its emf $\mathcal{E}$, in series with a small internal resistance $r$ (from the electrolyte and electrodes). The emf is the energy the chemistry gives each coulomb, and it equals the terminal voltage only when no current flows (a voltmeter on an unloaded battery reads very close to $\mathcal{E}$). Under load, part of the emf is used up inside the battery:

Internal resistance explains several test favorites: headlights dim while a car's starter motor draws 100+ A; a dying battery shows nearly full voltage on a voltmeter but collapses under load (its $r$ has grown); and a wire straight across a battery (a short circuit) draws $I = \mathcal{E}/r$, a huge current that heats the battery, can melt the wire and can make a lithium cell catch fire. Other ways a battery is not an ideal source: its emf falls as it discharges, and its capacity falls with temperature and high current.

Other DC sources

Batteries are not the only sources tests name. Each converts some other form of energy into electrical energy.

Generators make AC. Any coil turning in a magnetic field produces a voltage that reverses every half turn, so power-station generators and car alternators produce AC; a car's alternator has built-in diodes (a rectifier) to charge the 12 V DC battery. Generators and batteries supply energy; they do not create charge.

Hazards and the human body

Electric shock is damage done by current passing through the body. Nerves and muscles are controlled by small electrical signals, so an external current can make muscles contract involuntarily, stop breathing, or throw the heart's rhythm into ventricular fibrillation (rapid, uncoordinated twitching that pumps no blood). Current also heats tissue (burns, as $I^2R$), most severely at the entry and exit points. Voltage matters only because it drives current through the body's resistance: a 12 V car battery can deliver hundreds of amperes to a short circuit but cannot push a dangerous current through dry skin, while a static spark of 10 kV carries too little charge to stop the heart.

The body's resistance comes mostly from the skin; the inside of the body (salty, water-rich tissue) conducts well. The table gives typical round numbers; real values vary widely.

Resistance falls further with higher voltage (the skin breaks down above a few hundred volts), larger contact area (gripping versus touching), and longer contact (the skin burns through). Standing barefoot on wet concrete, sweating, or being in a bathtub are the worst cases.

How much current does what

Effects of 60 Hz current, hand to foot, for about one second, as given in OSHA-based safety training:

When a test asks for "the lethal current" as a single number, the conventional answer is about 0.1–0.2 A (100–200 mA) through the chest, the range where fibrillation becomes likely; the OSHA table lists ventricular fibrillation with death likely in its 1–4.3 A (1,000–4,300 mA) row. Sources differ, so read the options for the one that matches this convention.

Charles Dalziel's experiments (1950s–60s) give the median thresholds below, which show how strongly the effect depends on frequency and on sex (body size).

At still larger currents (figures quoted range from a few hundred milliamperes to several amperes) the heart tends to clamp (stop) rather than fibrillate, which is how a defibrillator works: a large, brief current stops the fibrillating heart so its natural pacemaker can restart a normal rhythm. Shorter shocks need more current to cause fibrillation: Dalziel's estimate for short shocks is $I \approx 116/\sqrt{t}$ mA for a 50 kg person (157 for 70 kg; $t$ in seconds), which is why fast-acting protection (GFCIs trip in about 1/40 s) saves lives.

What makes a shock worse

• Current size (set by voltage and body resistance), the primary factor.
• Path: hand to hand or hand to foot crosses the chest and heart; a path through one hand's fingers is far less dangerous. This is why electricians work on live equipment with one hand (the other in a pocket) and test with the back of the hand (a shock makes the hand clench away, not grip).
• Duration: longer contact means more energy and a greater chance of hitting the vulnerable part of the heartbeat.
• Frequency: 50–60 Hz sits in the most dangerous range (roughly 15–100 Hz). Very high frequencies (above about 10 kHz) stimulate nerves less, so the let-go current is several times higher and keeps rising with frequency (a few tens of mA at 50 kHz is far less likely to freeze the hand than the same current at 60 Hz), but they still cause deep burns.
• Skin condition, contact area, body size and health.

AC versus DC on the body

Conclusion for tests: at household voltages and frequencies, AC is generally more dangerous than DC of the same voltage or current, mainly because of tetany and fibrillation. Both can burn and kill. Edison used AC's danger in his campaign against it (see History of electricity).

Burns, arcs and fires

• Electrical burns come from $I^2R$ heating of tissue, often worst inside the body along the current path and at the entry and exit wounds.
• An arc flash is the explosive release of energy when current jumps through air between conductors (a short in switchgear, a dropped tool). Arc temperatures can reach about 20 000 °C (35 000 °F), several times hotter than the Sun's surface; the arc blast throws molten metal and a pressure wave. Protection: de-energize first, keep distance, flame-resistant clothing and face shields for live work.
• Short circuits draw enormous currents that melt wires and start fires. Shorting a battery with a wire, a coin or keys in a pocket can make it red hot.
• Overloaded circuits start fires when too many loads share one branch circuit or extension cord: the wiring carries more current than it was sized for and heats up. Remedies: move some loads to a different circuit, have an electrician add a properly wired circuit of higher rating, and never "fix" a blowing fuse or tripping breaker by putting in a bigger one. Breaker sizing and trip calculations are in Relays, fuses, breakers and GFCIs.

Why some situations are safe and others are not

• Birds on a power line are safe because both feet touch the same wire at (nearly) the same potential; with no potential difference across the bird, no significant current flows through it. A bird or squirrel that touches two wires, or a wire and the grounded pole, completes a circuit and is electrocuted.
• Standing on rubber insulates you from ground, but touching two conductors at different potentials (hot and neutral) still completes a circuit through your body.
• Grounding the metal case of an appliance gives fault current a low-resistance path to earth, so the breaker trips instead of the case becoming live; a GFCI shuts off power when a few milliamperes leak away (see Relays, fuses, breakers and GFCIs).
• Series versus parallel with a load: if you touch the two ends of a toaster element, you are in parallel with it and get the full voltage across you. In series with the element, its resistance would limit your current, which is safer.
• Downed power lines: stay at least 10 m (about 35 ft) away; voltage spreads through the ground, so a long stride can put a voltage between your feet ("step potential"). If a line falls on your car, stay inside unless there is a fire.

Voltage guidance and safe practice

There is no single "safe voltage", because current depends on resistance, but standards treat about 50 V as the threshold for a shock hazard: OSHA requires live parts at 50 V or more to be guarded, and the international "extra-low voltage" limit is 50 V AC (120 V ripple-free DC). In wet conditions much lower voltages can be dangerous. The 9 V and 1.5 V sources used in Circuit Lab hands-on tasks cannot push a harmful current through skin, but a shorted battery can still burn you.

Safe practice tests ask about: unplug or switch off and lock out before working on anything; use one hand on live circuits; keep water away; use insulated tools and fiberglass or wooden (not aluminum) ladders near wires; never remove the grounding pin of a plug; and replace damaged cords.

DC circuits

Schematics, open circuits and short circuits

An electric circuit is a complete, closed path from a source through components and back. A schematic shows how components connect, not where they sit. Lines are wires; two lines crossing with a dot are connected, and lines crossing without a dot are not.

A flashlight is the simplest real circuit: batteries in series (the source), a switch, and a bulb or LED (the load), joined by metal strips that act as wires.

An open circuit is a break in the path: its resistance is effectively infinite, no current flows anywhere in that series path, and the full source voltage appears across the gap. A short circuit is a near-zero-resistance path around a component or straight across the source: the shorted part gets no voltage and no current, while the source pushes a very large current through the short. That current heats the wiring ($P = I^2R$ in the wires and the source's internal resistance) and can melt insulation or start a fire, which is what fuses and breakers exist to stop. Replacing a lamp with a plain wire creates exactly this hazard. On a bench, an open shows up as zero current with the full voltage across the break; a short shows up as zero volts across the bypassed part.

Resistors, potentiometers and sensors

A resistor limits current or sets a voltage by turning electrical energy into heat; its power rating (commonly 1/4 W for small through-hole parts) is the most heat it can shed safely. A resistor has two terminals, a one-port component.

• Potentiometers (three terminals, an adjustable voltage divider) and rheostats (two terminals, an adjustable series resistance) are covered with a worked example in Variable resistors and dimming.
• A thermistor is a temperature-sensitive resistor. NTC thermistors lose resistance as they warm (temperature sensors, inrush limiters); PTC thermistors gain resistance (resettable overcurrent protection).
• A light-dependent resistor (LDR, photoresistor) loses resistance in bright light: night-lights and light meters.
• A varistor (metal-oxide varistor, MOV) has very high resistance at normal voltage and drops sharply above a threshold, so it clamps voltage spikes: the heart of a plug-in surge protector.
• A flex sensor is a strip whose resistance rises as it is bent (glove controllers, robot joints).
• A strain gauge changes resistance slightly when stretched; it is read with a Wheatstone bridge (below).

Most of these are transducers, devices that convert energy or a signal from one form to another. Input transducers (sensors) turn a physical quantity into an electrical signal: microphone (sound), thermocouple and thermistor (temperature), photodiode and LDR (light), strain gauge and piezoelectric crystal (force or pressure). Output transducers (actuators) turn an electrical signal into something physical: loudspeaker and buzzer (sound), LED and lamp (light), motor (motion), heater (heat). An antenna converts between electrical signals and radio waves. A sensor's sensitivity is the change in output per change in the measured quantity, the slope of its calibration curve (a type K thermocouple: about 41 µV/°C); its range, resolution (smallest detectable change), accuracy and response time complete the description.

Resistor color code

Small resistors are marked with colored bands. Hold the resistor with the tolerance band (gold or silver, often set apart from the others) on the right and read from the left. On a 4-band resistor the first two bands are digits, the third is the multiplier (the power of ten, "number of zeros"), and the fourth is the tolerance. A 5-band precision resistor has three digit bands, then the multiplier, then the tolerance. A 3-band resistor has no tolerance band and is ±20%.

A common mnemonic for the digit order is "Bad Beer Rots Our Young Guts But Vodka Goes Well" (black, brown, red, orange, yellow, green, blue, violet, gray, white). Silver (±10%) is the least precise common band.

Resistors are sold only in standard (preferred) values. The ±10% E12 series is 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 times a power of ten; the ±5% E24 series adds 11, 13, 16, 20, 24, 30, 36, 43, 51, 62, 75, 91. When a calculation gives an in-between value for a current-limiting resistor, pick the next higher standard value so the current stays at or below the target (267 Ω becomes 270 Ω).

Series and parallel

In series, every element carries the same current and the voltages add to the source voltage. In parallel, every branch has the same voltage and the branch currents add to the total.

Symmetry solves networks that are neither series nor parallel: points that must be at the same potential can be joined (or a resistor between them removed, since it carries no current). For a tetrahedron of six equal resistors $R$, the two corners not on the measured edge sit at the same potential, so the resistor between them carries nothing; what remains is $R$ in parallel with two paths of $2R$, giving $R_{eq} = R/2$. The classic cube of twelve equal resistors $R$ gives $\tfrac{5}{6}R$ between the two ends of a body diagonal (split the current 3 ways, then 6, then 3), $\tfrac{3}{4}R$ across a face diagonal and $\tfrac{7}{12}R$ across one edge.

Reduce a mixed network one block at a time: combine the innermost group, redraw, repeat. Then work back out: find the total current, then the voltage across each block, then each branch current.

A ring of resistance wire with total resistance $R$, tapped at two points, is two arcs in parallel. The equivalent is largest when the arcs are equal: $(R/2)(R/2)/R = R/4$. Any unequal split gives less.

Two series resistors split the source voltage in proportion to their resistances, the voltage divider: $V_2 = V_s R_2/(R_1 + R_2)$. Two parallel resistors split the current inversely, the current divider: $I_1 = I_{\text{total}} R_2/(R_1 + R_2)$, so the smaller resistor takes the larger share.

Adding a resistor in parallel always lowers the total resistance and raises the current drawn from the source, so the source delivers more power; adding one in series raises the total resistance and lowers the current and power. Houses are wired in parallel so every appliance gets the full 120 V and works independently, and each extra appliance switched on draws more total current from the circuit.

Bulb questions are series-parallel questions in disguise, since brightness follows power. Identical bulbs in series share the voltage and are dim, and removing one breaks the loop so all go out. Identical bulbs in parallel each get the full voltage, and removing one leaves the others unchanged (but the battery drains faster, since it supplies more total current). If bulbs differ, use $P = I^2R$ in series (larger resistance is brighter) and $P = V^2/R$ in parallel (smaller resistance, such as a thicker filament, is brighter).

Old series strings of holiday lights went dark when one filament broke. Newer strings put a shunt wire in each bulb's base: when the filament fails, the full string voltage appears across that bulb, breaks down the shunt's insulating coating, and the shunt then carries the current around the dead filament, so the rest stay lit. Each remaining bulb then gets a slightly larger share of the voltage. With 50 identical bulbs in series on 120 V, each gets $120/50 = 2.4$ V.

Building a target resistance

One listed hands-on task asks you to make a specified equivalent resistance from a handful of resistors. Work backward from the target:

1. If the target is larger than any single resistor, you need series pieces; subtract the largest resistor that fits and solve for the rest.
2. If the target is smaller than the smallest resistor, you need parallel pieces; two equal resistors in parallel halve the value, three give a third.
3. Fractions such as 1.5 or 2/3 of a value come from combining both: $R + R/2 = 1.5R$, and $R \parallel 2R = 2R/3$.

The Wheatstone bridge

A Wheatstone bridge (popularized by Charles Wheatstone in 1843; first described by Samuel Hunter Christie in 1833) measures an unknown resistance precisely by comparing it with known ones. It is two voltage dividers side by side, fed by the same source, with a sensitive meter (a galvanometer, the null detector) between their midpoints. A bridge has four resistors: two known fixed, one known and adjustable, and the unknown.

Procedure: connect the source, then adjust the variable resistor until the meter reads zero (no current through it, no voltage across it). The bridge is then balanced: the top-to-bottom resistance ratio is the same in both arms, so the unknown follows from the three known values. Because the meter only has to detect zero, the result does not depend on the meter's calibration or on the exact source voltage.

A slide-wire (meter bridge) version replaces two arms with a uniform wire and a sliding contact, so the ratio of wire lengths gives the resistance ratio. A sensor in one arm turns the bridge into a measuring instrument: a strain gauge (force, weight, pressure in load cells), a thermistor or platinum resistance thermometer (temperature), or a light-dependent resistor (light). It measures quantities that change a resistance, not things like time.

Kirchhoff's laws

Kirchhoff's two laws govern every network. The current law (KCL, the junction rule) says the current into any node equals the current out: the algebraic sum of currents at a node is zero, because charge does not pile up at a junction (conservation of charge). The voltage law (KVL, the loop rule) says the voltage rises and drops around any closed loop sum to zero, because a charge that goes around a loop returns to the same potential (conservation of energy).

Two systematic methods do most of the work. Node-voltage analysis uses KCL: pick a reference node (ground), name the other node voltages, and write one current equation per unknown node. Mesh analysis uses KVL: give each window of the circuit a loop current and write one voltage equation per loop. Choose whichever gives fewer unknowns. Superposition (solve with one source at a time, then add) is a separate technique based on linearity, not one of Kirchhoff's rules. To switch a source off for superposition, replace an ideal voltage source with a short (0 V) and an ideal current source with an open (0 A), keeping any internal resistances in place. Dependent (controlled) sources, whose value is set by a voltage or current elsewhere in the circuit (a transistor model, for example), are never switched off.

The same circuit by mesh analysis takes two equations, which is the most a Division B question may need. With clockwise loop currents $I_a$ (left window) and $I_b$ (right window), the shared 6.0 Ω carries $I_a - I_b$: walk each loop clockwise. Left loop, starting at ground: $+12 - 4.0I_a - 6.0(I_a - I_b) = 0$. Right loop: $-6.0(I_b - I_a) - 2.0I_b - 6.0 = 0$ (the last term is crossing the 6.0 V battery from + to −). Rearranged: left loop $10I_a - 6I_b = 12$, right loop $6I_a - 8I_b = 6$. Solving gives $I_a = 1.36$ A and $I_b = 0.27$ A, and the shared resistor carries $1.36 - 0.27 = 1.09$ A, matching the node method. Solve two equations by substitution or elimination; matrix methods are never required.

Control devices

Switches

A switch opens or closes a circuit by hand. Switches are named by poles (how many separate circuits one lever controls) and throws (how many positions each pole can connect to).

Switches can also be worked by something other than a hand: a float switch is moved by a rising or falling liquid level (sump pumps, tanks), a flow switch by liquid or air moving through a pipe or duct, a pressure switch by pressure, a thermostat by temperature (often a bimetal strip), a limit switch by a moving part reaching the end of its travel, and a reed switch by a nearby magnet.

A momentary (push-button) switch only changes state while pressed and springs back. Normally open (NO) means the contacts are open at rest and close when pressed; normally closed (NC) means the reverse. Pressing a momentary NC DPST button therefore breaks two connections at once, one per pole. Other forms: toggle, slide, rocker, rotary (multi-position) switches that select one of several circuits (a multimeter's range dial, a fan's speed selector), microswitches (small snap-action switches that change state at a precise small movement, used as limit sensors and in mouse buttons) and keypads.

Switch arrangements are logic gates. Two switches in series with a lamp light it only if both are closed: AND. Two switches in parallel light it if either is closed: OR. So if either of two people must be able to fire a circuit, put their switches in parallel with each other and the pair in series with the load. A switch belongs in the hot wire, so that the device is dead when switched off.

The 3-way switch

A 3-way switch lets two switches, for example at the top and bottom of a stair, each control the same light. Each switch is an SPDT. The source feeds the common terminal of switch 1; its two throw terminals connect by two wires (the travelers) to the two throw terminals of switch 2; the common terminal of switch 2 goes to the lamp, and the lamp returns to the source. The lamp is on when both switches select the same traveler and off when they select different ones, so flipping either switch always changes the state. In logic terms the lamp is the XNOR (or, depending on labeling, XOR) of the two switch positions. A circuit with a broken traveler wire behaves like a single switch that only works when the other switch is in one particular position.

To control a light from three or more places, insert a 4-way switch (a DPDT wired as a crossover) in the travelers between the two 3-way switches for each extra location: three locations take two 3-way switches and one 4-way switch, three switches in all. Outside North America the 3-way switch is called a two-way switch and the 4-way switch an intermediate switch (so the three-location setup is "intermediate switching").

At the bench, identify each SPDT's common terminal first (use the continuity beeper: the common connects to one throw in each lever position). Then wire battery + to common 1, throws to throws, common 2 through a resistor and LED to battery −, and test all four switch combinations.

Relays, fuses, breakers and GFCIs

A relay is an electrically operated switch. Its parts are a coil wound on an iron core, a movable iron armature, a spring that holds the armature at rest, and one or more sets of contacts. A small current through the coil magnetizes the core, which pulls the armature and moves the contacts; when the current stops, the spring returns them. When the coil is energized, normally open contacts close and normally closed contacts open. Because the coil circuit and the contact circuit are separate, a low-power control circuit (a button, a microcontroller, a remote receiver) can safely switch a separate high-current or high-voltage load while staying electrically isolated from it: car starter and headlight circuits, garage-door openers, heating systems. Relay contacts are described with the same pole/throw names as switches (an SPDT relay has a common, an NO and an NC contact), and coils and contacts have separate voltage and current ratings. A solid-state relay does the same job with semiconductors and no moving parts. A diode across the coil absorbs the voltage spike when the coil switches off. You need to know what a relay does; the magnetism inside the coil is no longer a tested area.

Protective devices open a circuit automatically. They go in the hot wire.

Fuses come in fast-acting types for sensitive electronics and slow-blow (time-delay) types that ride through brief inrush current, such as the surge when a motor starts or a cold filament is first switched on. A fuse rated only slightly above a motor's running current can fail repeatedly from its starting surges. Compared with breakers, fuses are cheaper, simpler and react very fast to a short, but must be replaced each time; breakers cost more but can be reset and switched off by hand.

A fuse or breaker protects the wiring from overheating; it trips at amperes, far above the current that can kill. A GFCI protects people: it trips on a few milliamperes of imbalance within a fraction of a second even when the total current is small, which is why codes require GFCIs in bathrooms, kitchens, garages and outdoors. A GFCI does not respond to a balanced overload. GFCIs come as receptacles (outlets with TEST and RESET buttons), as breakers in the panel, and as portable plug-in or cord units; press TEST monthly to check one. A GFCI needs no ground wire to work, because it only compares hot and neutral, so a GFCI receptacle is the code-approved way to add shock protection to an old two-wire ungrounded outlet (it is then labeled "No equipment ground"). GFCIs are made for AC household circuits. Equipment with large inrush or small normal leakage (such as an old refrigerator) can cause nuisance tripping. A switch controls a circuit but protects nothing.

Remedies for an overloaded circuit are in Burns, arcs and fires.

Household wiring

U.S. homes receive split-phase power: two 120 V hot lines and a neutral, giving 120 V for ordinary outlets and 240 V across the two hots for ovens, dryers and air conditioners. Each branch circuit runs from a breaker in the panel, and every outlet and fixture on it is in parallel. The rules for which wire is which are fixed by the National Electrical Code (NEC), which also sets the current each wire size may carry.

In low-voltage DC and electronics wiring the convention is different: red is positive and black is negative or ground, as on meter leads and battery clips. Do not carry the house-wiring rule (black = hot) over to a breadboard.

A polarized plug has one wider blade so it fits only one way, keeping a lamp's switch and shell on the correct wires. The third, round prong is the ground connection.

Diodes, LEDs and the PN junction

Why silicon: crystals, bands and carriers

A silicon atom (atomic number 14, group 14, a metalloid) has four valence electrons; germanium (32) sits just below it. Boron, the usual p-type dopant, is also a metalloid and a poor conductor on its own, while metals such as nickel, copper and aluminum conduct well. In a silicon crystal each atom shares one electron pair with each of four neighbors, so every valence electron is locked in a covalent bond and none is free to carry current. At room temperature a few bonds are shaken loose by heat. Each broken bond releases a free electron and leaves behind a hole, a missing electron that a neighbor's electron can hop into, so the hole moves through the crystal like a positive charge. Pure (intrinsic) silicon at 300 K has only about $10^{10}$ free electrons per cm³ (and as many holes), against about $5 \times 10^{22}$ atoms per cm³, which is why it conducts poorly.

The band picture says the same thing. Electrons in a solid can only take energies inside allowed bands. In a metal the highest occupied band is only partly full, so electrons move freely. In an insulator a wide forbidden band gap separates the full valence band from the empty conduction band. A semiconductor has a small gap, so heat or light can lift a few electrons across it. Rough rule: metals have no gap (0 eV), semiconductors have gaps up to about 3–4 eV, and insulators above about 4 eV. The gap widens slightly as a semiconductor cools, so an LED dipped in liquid nitrogen shifts toward shorter (bluer) wavelengths and needs a higher forward voltage, while a hot LED shifts redder and dims.

Because more heat frees more carriers, a semiconductor's resistance decreases as temperature rises (a negative temperature coefficient), the opposite of a metal. Silicon is preferred over germanium because its larger band gap gives far less leakage current, higher breakdown voltage and operation at higher temperatures, and because silicon is cheap and forms a stable insulating oxide (SiO₂) for making chips.

Doping adds a tiny amount of another element to control the carriers. Typical doping is around one impurity atom per million silicon atoms or less.

Boron is by far the most common p-type dopant for silicon, and phosphorus (with arsenic and antimony) the common n-type dopants.

In band terms, n-type doping raises the Fermi level (the energy where electron states are half likely to be filled) toward the conduction band, and p-type doping lowers it toward the valence band; in a junction the Fermi levels line up, which is what creates the built-in potential.

A doped piece of silicon is still electrically neutral: every free electron from a donor is matched by the donor's fixed positive ion. "n" and "p" name the sign of the majority carrier, not the net charge. In equilibrium the electron and hole densities obey the mass-action law $np = n_i^2$, so adding donors to $n = 10^{16}$ cm⁻³ pushes the holes down to $p = (10^{10})^2/10^{16} = 10^4$ cm⁻³. In compound semiconductors such as GaAs the same logic applies by site: a group 12 atom (zinc) replacing gallium makes p-type, and a group 16 atom (selenium, tellurium) replacing arsenic makes n-type.

How a PN junction works

Where p-type meets n-type silicon, free electrons diffuse from the n side into the p side and holes diffuse the other way, and they recombine near the boundary. That leaves a thin depletion region (space-charge region) with no free carriers, only the fixed ions the carriers left behind: negative acceptor ions on the p side and positive donor ions on the n side. Their charge creates a built-in electric field pointing from n to p and a built-in potential of roughly 0.6–0.8 V in silicon (about 0.3 V in germanium). At zero bias the field's drift exactly balances diffusion, so no net current flows and the depletion region acts as an insulator.

• Forward bias (p side, the anode, more positive than the n side, the cathode): the applied voltage opposes the built-in barrier and the depletion region narrows. Once the voltage exceeds about 0.6–0.7 V in silicon, carriers diffuse across in large numbers (diffusion current dominates), current rises steeply, and the diode's voltage stays close to that value. Each side injects minority carriers into the other (electrons into the p side, holes into the n side), and they recombine with majority carriers near the junction; in an LED that recombination makes the light.
• Reverse bias (n side more positive): the depletion region widens and its field grows stronger. Only a tiny leakage (saturation) current of minority carriers flows, nanoamperes in silicon, until breakdown at a large reverse voltage.
• Zero bias: no external voltage, no net current. These are the three bias conditions tests list.

Breakdown comes in two kinds. Zener breakdown happens in heavily doped junctions with a very thin depletion region, at low voltages (below about 5 V): the strong field pulls electrons straight through the barrier by quantum tunneling. Avalanche breakdown happens at higher voltages in lightly doped junctions: carriers accelerated by the field knock more electrons loose, which knock out more. Both are reversible if the current is limited, and "Zener diodes" sold above about 5–6 V actually use avalanche breakdown. Breakdown is not damage; overheating from too much current is.

A diode is therefore a one-way valve with a nearly fixed forward drop: low resistance one way, nearly infinite the other. The ideal diode (Shockley) equation describes the curve: $I = I_S\left(e^{V/(nV_T)} - 1\right)$, where $I_S$ is the tiny saturation current, $n$ an ideality factor between 1 and 2, and $V_T = kT/q \approx 25.9$ mV at 300 K (the thermal voltage). Forward current grows exponentially with voltage, which is why the drop barely changes once the diode conducts. Heating a silicon diode lowers its forward voltage by about 2 mV per °C and raises its leakage current. Its I–V curve is flat near zero in reverse, has a knee near 0.7 V forward where current shoots up, and drops sharply at the reverse breakdown voltage. "Which side is negative?" has two answers: the n side (cathode) holds the negative mobile carriers and is connected to the negative terminal in forward bias, while inside the depletion region the p side holds the negative fixed ions. A key that says "cathode" means the n side. The symbol's triangle points in the direction of conventional current, from anode to cathode; the bar marks the cathode (n side), which is the banded end of a real diode. Conventional current enters at the anode.

Uses of single-junction devices

Several diodes in one circuit: guess which diodes conduct, replace each with a 0.7 V drop (on) or an open (off), solve, then check that every "on" diode carries forward current and every "off" diode has less than 0.7 V forward across it; if not, change the guess. Diode logic: two diodes with anodes as inputs and cathodes joined to a resistor to ground make OR; two diodes with cathodes as inputs and anodes joined to a pull-up resistor make AND (each output sits about 0.7 V from the ideal level).

Active and passive components. An active component can supply energy to a circuit or control a larger current with a smaller signal: batteries and other sources, transistors, op-amps and other ICs. A passive component cannot add energy or amplify: resistors, capacitors, inductors, switches, fuses. Diodes are usually listed as passive (they cannot amplify), though some texts group all semiconductors as active; read the question's choices.

Rectifiers

Rectifiers (defined in Converting between AC and DC) are built from diodes.

• Half-wave (one diode): passes one half of each cycle and blocks the other, so the output is pulsing DC that is off half the time, at the line frequency (60 Hz).
• Full-wave (a four-diode bridge, or two diodes with a center-tapped transformer): flips the negative half cycles so both halves come out positive. The output pulses at twice the line frequency (120 Hz) and delivers more average power. A bridge loses two diode drops (about 1.4 V) because current passes through two diodes on each half cycle.

Real supplies follow the rectifier with a filter to smooth the ripple (that filter uses a capacitor, which is outside the 2027 topics) and often a regulator.

LEDs

In a light-emitting diode electrons and holes recombine at the forward-biased junction and release their energy as photons. The semiconductor compound sets the band gap, and the band gap sets the photon energy, so it sets the color (wavelength) and roughly the forward voltage. The color does not depend on the plastic lens tint or on the current. Silicon cannot make an efficient LED: it has an indirect band gap, so recombination mostly releases heat. LEDs use direct-gap III–V compounds.

The first visible (red) LED was made by Nick Holonyak in 1962; bright blue LEDs, the last primary color and the key to white LED lighting, came in the early 1990s (Nobel Prize 2014 to Akasaki, Amano and Nakamura). Red has the lowest forward voltage of the visible colors; blue and violet have the highest. The order of development: infrared and red (early 1960s), yellow and yellow-green (late 1960s–1970s), bright blue last (1993), and white LEDs (1996) only after blue, since a white LED is a blue chip plus phosphor. LED efficiency is measured three ways: external quantum efficiency is photons emitted per second divided by electrons passing per second, $(P_{\text{light}}\lambda/hc)/(I/e)$; wall-plug efficiency is optical power out over electrical power in; luminous efficacy is lumens per watt.

LEDs versus incandescent bulbs. An incandescent bulb heats a tungsten filament until it glows and turns only a few percent of its power into visible light; the rest is heat. LEDs are several times more efficient, last tens of thousands of hours, switch instantly and can be dimmed by rapid on-off switching (pulse-width modulation). They still waste a large share of their input as heat, which is why an LED bulb contains a heat sink along with the LED chips and a driver circuit that turns 120 V AC into a controlled DC current. LEDs work best on DC: on AC they light only on one half cycle and flicker, and their reverse breakdown voltage is low (often about 5 V).

Because an LED's voltage barely changes once it conducts, connecting it straight across a battery bigger than $V_F$ lets the current climb until the LED overheats (warmer LEDs conduct even more: thermal runaway). Always use a series current-limiting resistor. A standard 5 mm indicator LED runs at about 20 mA, with an absolute maximum around 30 mA.

LEDs in series carry the same current and their forward voltages add, so a supply can light only so many in a chain: the resistor needs $V_s - \sum V_F$. The long leg of an LED is the anode (+, toward the positive terminal); the short leg and the flat edge of the rim mark the cathode. A backwards LED simply stays dark (it does not change color), so if an LED will not light, flip it before anything else.

Digital logic

Digital circuits treat a voltage as one of two values: high (1, true) or low (0, false). They run on steady DC because each logic level must sit in a defined voltage band (for TTL, high above 2.0 V and low below 0.8 V); an AC supply would sweep every signal through the undefined region many times a second. Everything in this section is in both divisions at every level.

Binary numbers

Each binary digit (bit) is a power of two. Reading right to left the place values are 1, 2, 4, 8, 16, 32, 64, 128. The rightmost bit is the least significant bit (LSB) and the leftmost the most significant bit (MSB). Four bits make a nibble and eight bits a byte. With $n$ bits you can represent $2^n$ values, 0 to $2^n - 1$: four bits give 0–15 and eight bits 0–255.

• Binary to decimal: add the place values of the 1s. $110101_2 = 32 + 16 + 4 + 1 = 53$.
• Decimal to binary: subtract the largest power of two that fits and repeat, or divide by 2 repeatedly and read the remainders from the bottom up. $45 = 32 + 8 + 4 + 1 = 101101_2$.

Binary addition works like decimal addition with a carry whenever a column reaches 2: $0+0 = 0$, $0+1 = 1$, $1+1 = 10$ (write 0, carry 1), $1+1+1 = 11$ (write 1, carry 1).

The rules limit binary to whole numbers and addition, so negative numbers (two's complement), subtraction, multiplication, floating point and other bases are not required. You may still see hexadecimal on a test: each hex digit stands for four bits (0–9, then A = 10 through F = 15), so $1011\,0110_2 = \text{B6}_{16}$. Octal (base 8) groups bits in threes from the right: $101\,101_2 = 55_8 = 5 \times 8 + 5 = 45$.

Gates and truth tables

A truth table lists the output for every combination of inputs; $n$ inputs give $2^n$ rows, written in binary counting order (000, 001, 010, ...).

NOT (the inverter) has one input and outputs its opposite, $\overline{A}$ (also written $A'$ or ¬A). A buffer outputs its input unchanged (it restores signal strength). Of the standard gates only NOT (and the buffer) is reversible, meaning the input can always be recovered from the output; AND, OR, XOR and the rest map two input bits to one output bit, so information is lost. Reversible logic uses special gates with as many outputs as inputs, such as the controlled-NOT (Feynman), Toffoli and Fredkin gates. A small circle (bubble) on a gate symbol means inversion: NAND is an AND symbol with a bubble on the output.

Ways to remember the gates, including versions with more than two inputs:

• AND: output 1 only when all inputs are 1. NAND: output 0 only when all inputs are 1.
• OR: output 0 only when all inputs are 0. NOR: output 1 only when all inputs are 0.
• XOR ("exclusive OR", symbol ⊕): output 1 when the inputs differ; for more inputs, when an odd number are 1 (a parity checker). $A \oplus B = A\overline{B} + \overline{A}B$.
• XNOR (equivalence): output 1 when the inputs are equal; for more inputs, when an even number are 1.

To evaluate a nested expression, work from the innermost parentheses outward; without parentheses, NOT binds first, then AND, then OR.

Logic families and chips. Bench tasks use 14-pin chips from the 7400 family. In the common pinout, pin 14 is $V_{CC}$ (+5 V for TTL) and pin 7 is ground. Know the part numbers: 7400 quad 2-input NAND, 7402 quad NOR, 7404 hex inverter, 7408 quad AND, 7432 quad OR, 7486 quad XOR. Letters in the middle name the family (74LS00 is low-power Schottky TTL; 74HC00 is high-speed CMOS), not the function.

A gate reads any input voltage below its $V_{IL}$ (maximum input-low voltage) as 0 and any above its $V_{IH}$ (minimum input-high voltage) as 1; voltages between are undefined. Its outputs are guaranteed to stay below $V_{OL}$ when low and above $V_{OH}$ when high. The noise margins are the slack between them: $NM_H = V_{OH} - V_{IH}$ and $NM_L = V_{IL} - V_{OL}$. For standard TTL ($V_{OH} = 2.4$ V, $V_{IH} = 2.0$ V, $V_{IL} = 0.8$ V, $V_{OL} = 0.4$ V) both margins are 0.4 V.

Bitwise operations apply a gate to each pair of bits in two whole numbers, column by column, with no carries. $1100_2$ AND $1010_2 = 1000_2$; OR gives $1110_2$; XOR gives $0110_2$; NOT $1100_2$ (in 4 bits) is $0011_2$. Programming languages write these as & (AND), | (OR), ^ (XOR) and ~ (NOT).

Boolean algebra and De Morgan's law

Boolean expressions use $\cdot$ (AND, often just written $AB$), $+$ (OR) and an overbar (NOT). The rules look like ordinary algebra with a few twists:

Universal gates

A gate set is universal (functionally complete) if any logic function can be built from it. NAND alone is universal, and so is NOR alone; that is why chips full of only NAND gates were so common. {AND, OR, NOT} is also complete, but AND alone or OR alone is not, because neither can invert.

XOR from four NANDs: gate 1 makes $N = \overline{AB}$; gate 2 takes A and N; gate 3 takes B and N; gate 4 NANDs the outputs of gates 2 and 3 to give $A \oplus B$. From NORs: NOT is a NOR with tied inputs; OR is a NOR followed by a NOR inverter; AND is NOR fed by $\overline{A}$ and $\overline{B}$ (two NOR inverters first).

De Morgan explains the 3-gate rows: $\overline{\overline{A} \cdot \overline{B}} = A + B$, so a NAND fed inverted inputs is an OR.

From a truth table to a circuit

A listed hands-on task asks you to build a circuit that reproduces a given truth table, and written questions ask the same on paper. The reliable method is sum of products (SOP):

1. Find every row where the output is 1.
2. For each such row write an AND term (a minterm) that is 1 only on that row: use the variable where the input is 1 and its complement where it is 0.
3. OR the terms together.
4. Simplify with Boolean algebra or a Karnaugh map, then build it.

With $n$ inputs a truth table has $2^n$ rows and each row's output can be 0 or 1, so there are $2^{2^n}$ possible Boolean functions: 16 for two inputs (AND, OR, XOR, NAND, NOR, XNOR, the two constants, implication and the rest).

The mirror image, product of sums (POS), ORs together the complements on each 0 row and ANDs those terms; it is shorter when the table has few 0s. For each 0 row write a maxterm, a sum that is 0 only on that row: use the variable where the input is 0 and its complement where it is 1 (row 101 gives $(\overline{A} + B + \overline{C})$). AND the maxterms, or group the 0s on a K-map and write one sum term per group. Example: 0s at rows 001, 011 and 100 give $F = (A + B + \overline{C})(A + \overline{B} + \overline{C})(\overline{A} + B + C)$; the first two combine to $(A + \overline{C})$, so $F = (A + \overline{C})(\overline{A} + B + C)$.

A Karnaugh map (K-map) lays the truth table out as a grid whose rows and columns are labeled in Gray-code order (00, 01, 11, 10), so neighboring cells differ in one variable. Circle groups of adjacent 1s whose sizes are powers of two (1, 2, 4, 8), as large as possible; groups may wrap around the edges and overlap. Each group becomes one product term containing only the variables that stay constant across it.

A pull-up resistor (typically 1–10 kΩ) ties an input to $V_{CC}$ so it reads 1 until a switch pulls it to ground; a pull-down ties it to ground so it reads 0 until a switch connects $V_{CC}$. Never leave an input floating: an open TTL input usually reads as 1, and an open CMOS input drifts unpredictably. On the common 14-pin gate chips (7400, 7402, 7404, 7408, 7432, 7486) pin 14 is $V_{CC}$ and pin 7 is ground, but gate pin assignments differ: the 7402 NOR puts each gate's output first (pin 1 = output, pins 2–3 = inputs), the reverse of the 7400. Always check the supplied pinout.

On the bench, logic chips need power and ground pins connected before anything works (check the pinout the supervisor gives you), unused inputs should be tied high or low rather than left floating, and an LED with a resistor on the output shows the state.

Combinational circuits: adders and multiplexers

A combinational circuit's output depends only on its present inputs (no memory). The two examples named in the rules are adders and multiplexers.

A half adder adds two bits. The sum bit is 1 when exactly one input is 1 (XOR), and the carry is 1 only when both are (AND): $S = A \oplus B$, $C = A \cdot B$.

A full adder also accepts a carry-in, so adders can be chained: $S = A \oplus B \oplus C_{\text{in}}$ and $C_{\text{out}} = AB + C_{\text{in}}(A \oplus B)$. Its truth table is just binary counting: the sum is 1 when an odd number of the three inputs are 1, and the carry-out is 1 when at least two are (the majority function above).

A 4-bit ripple-carry adder is four full adders with each carry-out wired to the next carry-in (the first carry-in is 0, or the first stage is a half adder); the worked binary addition above is exactly what it computes. Its weakness is speed: the last sum is not valid until the carry has rippled through every stage.

A multiplexer (MUX) is a data selector: select lines choose which of several inputs passes to the single output. A 2-to-1 MUX is $Y = \overline{S}D_0 + SD_1$. A 4-to-1 MUX uses two select bits $S_1 S_0$ to pick input 0, 1, 2 or 3; in general $2^n$ inputs need $n$ select lines. A MUX can implement any function: wire the truth table's output column to the data inputs and the function's variables to the select lines.

Other combinational blocks that appear on tests:

Sequential logic and timing (Division C)

Sequential circuits have memory: the output depends on the inputs and on the stored state. Their timing is set by a clock, a square wave alternating between 0 and 1. Its period $T$ is the time for one cycle, its frequency is $f = 1/T$, and its duty cycle is the fraction of the period spent high (a clock high for 2 µs of every 8 µs has a 25% duty cycle and a frequency of 125 kHz). A rising (positive) edge is the 0→1 transition and a falling (negative) edge is 1→0.

Latches and flip-flops

The simplest memory is the SR latch: two cross-coupled NOR gates (or NAND gates with active-low inputs). In the NOR version, S = 1 sets Q to 1, R = 1 resets Q to 0, and S = R = 0 holds the last value. S = R = 1 is forbidden: it forces both Q and $\overline{Q}$ to 0, and if both inputs then drop to 0 together the final state depends on which gate is faster, a race condition.

A latch is level-sensitive (it follows its input while enabled); a flip-flop is edge-triggered (it samples its input only at a clock edge). On a symbol, a small triangle at the clock input means edge-triggered, and a bubble before it means the falling edge.

Many flip-flops also have asynchronous preset and clear inputs that force Q to 1 or 0 immediately, without waiting for the clock.

Applications follow from these behaviors. A T flip-flop with T = 1 (or a JK with J = K = 1, or a D flip-flop with $\overline{Q}$ fed back to D) toggles once per clock cycle, so its output is a square wave at half the clock frequency: chaining $n$ of them divides the frequency by $2^n$ and the outputs count in binary (a ripple counter, which counts from 0 to $2^n - 1$). A row of D flip-flops sharing one clock is a register that stores a binary word; feeding each output into the next D input makes a shift register.

Timing diagrams and delays

A timing diagram draws each signal against a shared time axis so you can read what happens at each clock edge. To trace a D flip-flop, look at D just before each rising edge and copy that value to Q until the next edge; changes in D between edges do not matter.

Real gates are not instantaneous:

• Propagation delay ($t_{pd}$) is the time from an input change to the resulting output change, usually measured between the 50% points of the two transitions. Low-to-high and high-to-low delays can differ ($t_{PLH}$, $t_{PHL}$).
• Rise time is how long the output takes to go from 10% to 90% of its final value; fall time is the 90% to 10% transition.
• Setup time is how long a flip-flop's input must be steady before the clock edge, and hold time how long it must stay steady after. Violating them can leave the flip-flop briefly undecided (metastable).
• Delays add along a path, so the slowest path (the critical path) limits how fast the clock can run.

A race condition is any situation where the result depends on which of two signals arrives first. In combinational logic, unequal path delays can produce a brief wrong output pulse, a glitch (for example $A \cdot \overline{A}$ should always be 0, but if the inverter is slow both inputs of the AND are briefly 1). Synchronous design avoids races by letting every flip-flop sample on the same clock edge after the signals have settled.

Bipolar transistors (Division C)

A bipolar junction transistor (BJT) has three regions, either N-P-N or P-N-P, making two back-to-back junctions. The terminals are the emitter (heavily doped), the base (the thin, lightly doped middle layer) and the collector. A small base current controls a much larger collector current, so the transistor can act as an amplifier or a switch. "Bipolar" means both electrons and holes carry current. The arrow in the symbol sits on the emitter and points the way conventional current flows: out of the emitter for NPN ("Not Pointing iN"), into it for PNP. In an NPN, conventional current flows from collector to emitter; in a PNP, from emitter to collector.

The point-contact transistor was invented at Bell Labs in 1947 by John Bardeen and Walter Brattain, and William Shockley developed the junction transistor in 1948; the three shared the 1956 Nobel Prize. Transistors replaced vacuum tubes, and today's chips hold billions of them (Moore's law: the transistor count on a chip doubles about every two years). NPN transistors are more common than PNP because electrons are more mobile than holes, which makes NPN devices faster for the same size.

How transistor action works (NPN): forward-biasing the base-emitter junction makes the heavily doped emitter inject electrons into the base. The base is thin and lightly doped, so only a few of them recombine there (that small flow is the base current); most diffuse across into the reverse-biased base-collector junction, whose field sweeps them into the collector. So $I_C$ is almost all of $I_E$ and many times $I_B$. Two separate diodes cannot do this, because there is no thin shared base.

Operating regions and gain

For a PNP every polarity flips: it turns on when the base is pulled about 0.7 V below the emitter, and the emitter usually sits at the positive supply.

Biasing means choosing resistors so the transistor sits in the right region with no signal applied (its Q-point, or quiescent operating point). For an amplifier you want the active region with $V_{CE}$ comfortably between 0 and the supply, so the signal can swing both ways.

In an amplifier it is really $V_{BE}$ that controls the collector current: $I_C$ grows exponentially with $V_{BE}$ (about tenfold for every 60 mV), so the small input signal, added on top of the bias, wiggles $V_{BE}$ by millivolts and makes $I_C$ swing much more. Describing the same thing as $I_C = \beta I_B$ is equivalent and easier for DC bias problems.

As a switch, a transistor is driven hard into saturation. To turn on a load drawing 20 mA with $\beta = 100$, the base needs at least 0.2 mA; designers give two to ten times that so the transistor saturates reliably. From a 5.0 V logic output, $R_B = (5.0 - 0.7)/1.0\ \text{mA} = 4.3$ kΩ gives 1.0 mA of base current. An NPN switches the low side of a load (between the load and ground); a PNP, with its emitter at the supply, switches the high side and turns on when its base is pulled low.

Transistor-level logic gates

Gates were first built from transistors and resistors (resistor-transistor logic, RTL). With NPN transistors whose emitters go to ground and a shared collector resistor to $V_{CC}$, the output is the collector node:

Parallel transistors give OR-type behavior and series transistors give AND-type behavior, each with an inverted output, because the transistor pulls the output down.

Operational amplifiers (Division C)

Amplifiers are named by what goes in and out: a voltage amplifier (V in, V out), a current amplifier (I in, I out), a transresistance (transimpedance) amplifier (current in, voltage out; an op-amp with a feedback resistor turns a photodiode's current into $V_{\text{out}} = -IR_f$) and a transconductance amplifier (voltage in, current out).

An op-amp is a high-gain differential amplifier: it amplifies the difference between its non-inverting (+) input and its inverting (−) input, $V_{\text{out}} = A(V_+ - V_-)$, with an enormous open-loop gain $A$. A basic op-amp has five terminals: the two inputs, the output and the two supply pins ($+V_S$ and $-V_S$). Its output cannot go beyond its supply voltages: when the math asks for more, it saturates ("clips") near the rails.

The open-loop gain (written $A_{OL}$) is the op-amp's own gain with no feedback; the closed-loop gain ($A_{CL}$) is the gain of the whole circuit once feedback resistors are added, set almost entirely by those resistors. A non-inverting amplifier is the usual way to boost a small sensor or microphone signal without loading it. An ideal op-amp is analyzed with two golden rules:

1. No current flows into either input (infinite input resistance).
2. With negative feedback (output connected back to the inverting input), the output adjusts until $V_+ = V_-$, so the voltage difference between the inputs is 0.

The non-inverting amplifier's output is in phase with its input; the inverting amplifier's is upside down (180° out of phase). Without feedback the op-amp is a comparator: the output swings to the positive rail when $V_+ > V_-$ and to the negative rail otherwise. That turns an analog voltage (for example from a divider with a light- or temperature-sensitive resistor) into a logic-level signal. The follower is used as a buffer: it draws no current from the source but can drive a load, so it stops a meter or load from disturbing a voltage divider. Other applications named on tests: active rectifiers, filters, integrators and differentiators (these last three use capacitors, outside the 2027 topics). Op-amps are DC-coupled (they amplify DC as well as AC) and never act as transformers.

Hands-on tasks

The bench part is worth 25–50% of the score. The 2027 rules guarantee you will not solder, touch high-power circuits or work with AC, so every task runs on batteries or low-voltage DC supplies.

Multimeters

A multimeter measures voltage, current and resistance (most also test continuity and diodes). Each function must be connected in its own way:

Two wiring mistakes cost points. An ammeter placed across a source or a part is a near short circuit: a huge current flows and usually blows the meter's internal fuse. A voltmeter placed in series nearly opens the circuit: the current drops to almost zero and the meter reads close to the full source voltage. Move the red lead to the current jack only when measuring current, and move it back afterward. Start on the highest range and work down. A digital meter displays the potential of the red lead minus that of the black (COM) lead, so a negative reading simply means the leads are reversed. On the DC setting a meter averages the input, so a symmetric AC voltage reads close to zero; use the AC setting, which reports an RMS value (accurate for non-sine waveforms only on a "true RMS" meter).

Digital or analog? A digital meter reads more precisely, has a high (about 10 MΩ) input resistance on its voltage ranges and auto-polarity; an analog meter shows trends, fluctuations and the approach to a null (as in a Wheatstone bridge) more clearly. Neither is better for every job.

Analog meters are built around a galvanometer, a sensitive needle movement with a small full-scale current. A small shunt resistor in parallel with it turns it into an ammeter for larger currents; a large multiplier resistor in series turns it into a voltmeter.

An analog ohmmeter is a galvanometer in series with an internal battery and an adjustable resistor. Shorting the leads and turning the zero-adjust knob sets full-scale deflection as 0 Ω; an open circuit gives no deflection, ∞ Ω. So the ohms scale runs backward (zero at the right) and is crowded at the high end. On any analog meter, read the scale that matches the function and multiply by the range setting: on the ×100 ohms range, a needle at 47 means 4.7 kΩ; on a 10 V range a needle at 6.2 on the 0–10 scale means 6.2 V.

A real voltmeter loads the circuit: it is a resistor in parallel with whatever it measures. The effect matters when the circuit's resistances approach the meter's.

Breadboards

A breadboard lets you prototype circuits without soldering. In a typical board each short row of five holes is connected inside by a metal clip, the central channel separates the two halves (so a chip straddles it, with each pin on its own row), and the long rails along the edges carry power and ground. Some boards split the rails halfway along, which is why the rules let you ask the supervisor how the board is wired. Two holes in the same strip are one node: putting both legs of a resistor in the same strip shorts it out.

A push button that seems to do nothing, or an LED that stays on whether or not the button is pressed, usually means the circuit uses two of the button's internally always-connected pins, or two pins that should be switched sit in the same strip; rotate the button 90° (four-pin buttons normally straddle the central channel).

Mystery resistor from voltages only

Place the unknown $R_x$ in series with a known $R_k$. The same current flows through both, so each voltage is proportional to its resistance, and you never need the source voltage or the current.

If the mystery resistor is buried in a larger circuit, find the voltage across it and the voltage across a known resistor that carries the same current, then use the same ratio.

Power supplied to a circuit

Measure the voltage across the whole circuit and the current from the source (ammeter in series), then $P = VI$. With only voltages and known resistors, use $P = V^2/R$ for each resistor and add. A station that measures 6.0 V and 0.25 A has $P = 1.5$ W; over 2.0 minutes that is $1.5 \times 120 = 180$ J.

Two LEDs, equal brightness

Equal brightness means equal current. Two matched LEDs in series with one resistor carry the same current automatically, but the supply must exceed both forward voltages added together. If the LEDs differ (say red at 2.0 V and blue at 3.0 V), put them in parallel branches, each with its own resistor sized by the LED formula for the same current. One shared resistor feeding two parallel LEDs does not work: the LED with the lower $V_F$ hogs the current and the other stays dim or dark.

USB charger labels

A charger label lists output modes such as "5 V ⎓ 3 A, 9 V ⎓ 2 A, 12 V ⎓ 1.5 A" (⎓ is the DC symbol; the input side shows "100–240 V ~ 50/60 Hz" for AC). Each mode's maximum power is $P = VI$: 15 W, 18 W and 18 W here.

Troubleshooting a faulty circuit

Work from the schematic and measure rather than guess:

1. Check the obvious first: is the supply on and at the right voltage, are the meter leads in the right jacks, is every LED and diode the right way round, is every chip powered?
2. With power on, measure the voltage at each node relative to ground and compare with what the schematic predicts.
3. Read the symptoms. A part with the full supply voltage across it and no current is an open (broken lead, wire in the wrong strip, part not seated). A part with 0 V across it while current flows elsewhere is shorted (both legs in one strip, a stray wire). A node that should be high but reads low points to a short to ground or a missing pull-up.
4. With power off, use continuity to confirm the suspect connection, fix it, and re-measure.

Record what you measured and why it showed the fault; points are given for data analysis, not only for the fix.

Common pitfalls

How tournaments test this

The figures below come from Scio.ly's Circuit Lab question bank: 9,529 distinct questions from past invitationals, regionals, states and tryouts. Under the 2027 rules about 6,520 of them (68%) are still in scope; the other 3,000 are mostly magnetism and electromagnetic induction, capacitors and RC circuits, and antenna and radio questions from earlier rotations. Of the in-scope questions, 57% are multiple choice (including true/false and matching) and 43% free response; about 70% come from Division C tests.

Digital logic, flip-flops, transistors and op-amps are underrepresented because they were only partly tested before 2027; every 2027 test must include at least one question from each listed area, and State and National tests must cover the level-restricted ones, so expect their share to grow. Practice them from the worked examples here and from a digital-logic text.

Question shapes repeat from test to test:

• Definitions and identification (about a third of the bank): name the law, unit, device or scientist; true/false statements about field lines, conductors, GFCIs, diodes; "which of these is not ...".
• One-step calculations: Ohm's law, $P = IV$ and its variants, kWh and cost, Coulomb's law with a ratio change, number of electrons from a charge, RMS from peak, color code to ohms, LED resistor.
• Multi-step calculations: series-parallel reduction, voltage across one resistor, Kirchhoff with two loops, internal resistance, battery run time or charging time, heater efficiency, Wheatstone balance, transistor bias, op-amp gain.
• Explain why: AC for transmission, why wet skin is dangerous, why a GFCI trips, why an LED needs a resistor, why silicon is used, why charge sits on a conductor's surface.
• Logic: complete a truth table, evaluate a nested expression, simplify with De Morgan, build a function from NAND gates, read an adder.

Across the bank the most common words are "current", "circuit", "voltage", "resistance" and "resistor", and the commonest phrases are "equivalent resistance", "voltage across", "potential difference", "voltage drop" and "internal resistance". Free-response items reward showing the law, the substitution and the result with its unit, since the rules award points for calculations and data analysis as well as final answers. Multiple-choice distractors usually encode a standard mistake: series and parallel rules swapped, a square dropped, RMS and peak confused, or electron flow confused with conventional current.

Strategy

Split by topic, then swap. One partner leads circuit calculations (Ohm's law, series-parallel, Kirchhoff, and for Division C transistors and op-amps); the other takes history, electrostatics, DC/AC and safety, and digital logic. Swap at the halfway mark to check each other's work. At the bench, one partner wires while the other reads the schematic, takes readings and records them.

Build the binder around methods. Tabs for: formulas with units; series-parallel and Kirchhoff steps with one solved two-equation example; gate symbols, truth tables, Boolean laws and De Morgan; NAND/NOR conversions; sum-of-products steps; adder and MUX diagrams; history table; safety and control devices; meter and breadboard rules; and for Division C, flip-flop tables, a timing-diagram example, BJT bias examples and the op-amp formula sheet. Because books are allowed, add a tabbed introductory electronics text.

Prepare the bench skills. Buy a cheap breadboard, multimeter, resistor kit, LEDs, two SPDT switches and a few logic chips (a quad NAND is enough to build every gate). Practice each listed task against a timer until you can wire, measure and record without looking anything up.

Manage the clock. With about 50 minutes for both parts, agree on when the hands-on work starts and protect that time; it is worth up to half the score. On the written test take the quick points first (history, definitions, truth tables, safety), then the multi-step calculations, writing each step with units for partial credit.

Check the calculators. Both partners should know how to enter scientific notation and fractions on their Class III calculators. Division C students should check the angle mode before each trig step: degrees for force-vector angles, radians for $v(t) = V_p\sin(\omega t)$.

Test yourself

The first two blocks use official-key archive questions; the last three use tournament questions from Scio.ly's bank whose keys were re-checked by hand. All are inside the 2027 rules, and each answer comes with a one-line explanation.

Test yourselfTest yourself: DC circuits, power, Kirchhoff and measurement

0 / 0 correct

1. 1

   If a 12 V battery is connected to a resistor and draws a current of 3 A, what is the resistance of the resistor?

   Question 1 options

   A36 ΩB15 ΩC9 ΩD4 Ω

   From: Cypress Falls - TX, 2025-26, Division C

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Test yourselfTest yourself: LEDs, electrostatics, safety and history

0 / 0 correct

1. 1

   An LED has the following specs V_F = 2.0V, I_F = 20mA. If the circuit is powered by 5V, what size current limiting resistor should be used?

   Question 1 options

   A1.5kΩB150ΩC330ΩD200ΩE100Ω

   From: Northview Invitational, 2025-26, Division C

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Test yourselfTest yourself: digital logic, diodes and LEDs

0 / 0 correct

1. 1

   What is the base of much of the data storage in computers?

   Question 1 options

   ABase 16 (Hexadecimal)BBase 3CBase 4DBase 2 (Binary)

   From: Centerville Invitational 2020, Division B

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Test yourselfTest yourself: op-amps, AC and DC, power and safety

0 / 0 correct

1. 1

   An operational amplifier amplifies what?

   Question 1 options

   AVoltageBCurrentCResistanceDTemperature

   From: LISO Invitational 2019-2020, Division C

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Test yourselfTest yourself: control devices, circuits, static and history

0 / 0 correct

1. 1

   When different resistors are connected in parallel across an ideal battery, we can be certain that

   Question 1 options

   Athe same current flows in each one.Bthe potential difference across each is the same.Cthe power dissipated in each is the same.Dtheir equivalent resistance is greater than the resistance of any one of the individual resistances.

   From: CSRA Invitational 2020, Division C

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Condensed notesheet

A print-ready summary of the formulas, constants and lookup facts above. It is a reminder sheet, not a substitute for the worked methods.

References and further reading

• Science Olympiad, 2027 Circuit Lab rules for Divisions B and C, in the 2027 rules manual from soinc.org. The general rules there define calculator classes and eye protection.
• OpenStax, University Physics Volume 2, free at openstax.org: chapters on electric charges and fields, current and resistance, and DC circuits (Kirchhoff's rules).
• OpenStax, College Physics 2e, free at openstax.org: the electric hazards and the human body section.
• OSHA, Controlling Electrical Hazards and the NIOSH publication Electrical Safety: Safety and Health for Electrical Trades, for current-effect thresholds: osha.gov and cdc.gov/niosh.
• All About Circuits, free online textbook, volumes on DC, semiconductors and digital circuits (logic gates, Boolean algebra, Karnaugh maps, flip-flops, BJTs, op-amps): allaboutcircuits.com/textbook.
• Horowitz and Hill, The Art of Electronics (3rd ed., Cambridge University Press), for transistor biasing and op-amp circuits.
• NIST, Fundamental Physical Constants: physics.nist.gov/cuu/Constants.
• HyperPhysics (Georgia State University), concept pages on electricity, semiconductors and electronics: hyperphysics.phy-astr.gsu.edu.
• SciOly Wiki, Circuit Lab page, for the community's view of past tests: scioly.org/wiki. This guide does not reproduce that page.