Circuits and Measurement

Chapter 1 gave you Ohm's law and the series and parallel rules for a single loop. Real circuits are networks: several loops sharing parts, with voltages and currents you cannot see until you measure them. This chapter gives you three tools for that: the conventions of the schematic (how to read and draw a circuit), Kirchhoff's two laws (the bookkeeping rules that every circuit obeys), and the multimeter (how to check what is really happening, and how the meter itself can mislead you). It also explains the breadboard layout properly, so the circuits in later chapters go together on the first attempt.

1. Reading a schematic

A schematic is a map of connections, not a picture of the board. Part positions on the page mean nothing; only which terminals are joined matters. By convention, the highest voltage (the + supply) is drawn at the top, ground at the bottom, and signals flow left to right.

Each part also has a reference designator: a letter plus number. R is a resistor, C a capacitor, D a diode or LED, Q a transistor, U an integrated circuit, S or SW a switch, B or BT a battery. R1, R2 and R3 are three different resistors; the values are written next to them (1k, 2.2k, 470). A value such as 4k7 means 4.7 kΩ: the letter replaces the decimal point, so it cannot be lost when a drawing is photocopied.

Nodes

A node is any set of points joined by wire. Everything on a node is at the same voltage, because wire has (almost) no resistance. This is the key reading skill: trace each wire until it reaches a part, and mark every stretch that is one node. In the circuit below there are only three nodes: the + rail, node X and ground, even though the drawing has many wire segments.

2. Kirchhoff's laws

Ohm's law describes one part. Kirchhoff's laws (Gustav Kirchhoff, 1845) describe how parts connect, and together with Ohm's law they solve every resistor network. Both are simple statements of conservation.

Kirchhoff's current law (KCL): what flows in must flow out

Charge is not created or stored at a junction (a wire holds no charge, to the precision we care about), so the total current into any node equals the total current out of it.

Kirchhoff's voltage law (KVL): round a loop, the voltages sum to zero

Voltage is energy per coulomb. A charge that goes round a closed loop returns to where it started, so the energy it gained from the sources equals the energy it gave up in the parts. In practice: the rises equal the drops.

Worked example 1: a series-parallel network (the chapter's circuit)

Nine volts feeds R1 = 1 kΩ in series with a parallel pair, R2 = 2.2 kΩ and R3 = 3.3 kΩ. We want every current and voltage.

Worked example 2: two sources, one unknown (node method)

Series and parallel reduction fails when two sources feed the same network. Kirchhoff handles it by choosing one unknown, the voltage V_A at the shared node, and applying KCL there.

Why the negative sign matters: the larger source (9 V) overpowers the smaller one and pushes current backwards through it. A 3 V source being pushed backwards is, in effect, being charged. Keep that in mind whenever two supplies meet at a node, a situation you will meet again with battery packs and regulators in Basic Electronics 9.

3. The multimeter

A digital multimeter (DMM) has a dial, a display and three sockets. It measures voltage, current and resistance, usually also continuity and diode forward voltage. A basic model is under £15 and is all this course needs.

How to measure each quantity

1. Voltage: set the dial to DC V at a range above the expected voltage (or auto-range). Touch the black probe to the lower-voltage side (usually ground) and the red probe to the point of interest. The meter reads the potential difference between the two probe tips. To measure the drop across R1 in the chapter circuit, put one probe on each leg of R1.
2. Current: turn the power off, break the loop at the point you want, set the dial to DC mA, move the red lead to the mA socket, and complete the loop through the meter. The same current now flows through the meter. Afterwards move the red lead back to VΩ: leaving it in the current socket and then measuring a voltage is the classic way to blow the meter's fuse and short the circuit.
3. Resistance: power off, and ideally take one leg of the part out of the circuit. If you measure in place, every other path between the two points is in parallel with the part you wanted, and the reading is lower than the part's value.
4. Continuity: test a jumper wire (should beep), the two ends of a breadboard rail (does the rail run the full length?), or two points that should not be connected (must not beep). It is the fastest fault-finding tool you own.

The meter is part of the circuit: loading

Measuring changes the thing you measure. A voltmeter has a high input resistance (typically 10 MΩ, as little as 1 MΩ on a cheap or analogue meter) and draws a little current through it. An ammeter has a small resistance (its burden, a few ohms on the mA range) added into the loop. Both change the circuit slightly; the effect matters when the circuit resistance is comparable with the meter's.

Worked example: voltmeter loading. Two 1 MΩ resistors divide 10 V. The midpoint should be 5 V. A meter with 10 MΩ input is placed across the lower resistor, so the meter's resistance is in parallel with it:

Worked example: ammeter burden. A 5 V supply across a 100 Ω resistor should carry I = 5 / 100 = 50 mA. If the meter's mA range has 10 Ω of internal resistance, the loop becomes 110 Ω and I = 5 / 110 = 45.5 mA, a 9 % error. In the 9 V, 2.32 kΩ circuit of section 2 the same meter changes the current by only 9 / 2330 versus 9 / 2320, a 0.4 % effect.

4. The breadboard

A solderless breadboard is a plastic block with spring-clip contacts hidden under a grid of holes. You push component legs (up to about 0.8 mm thick) and solid jumper wires into the holes. Nothing is soldered, so circuits can be changed in seconds, but only if you know which holes are connected underneath.

• Each column is two groups of five: a to e together, and f to j together, with no connection across the central gap. The gap is sized to straddle a standard 0.3-inch (7.62 mm) DIP integrated circuit, so each pin gets its own isolated column of four spare holes.
• The power rails (the long strips at the edges) run along the length of the board. On a full-size board some rails are split in the middle, so check with the multimeter's continuity setting before assuming a rail is live at both ends.
• Red conventionally marks +, blue or black marks −. Use one rail for + and one for ground and keep jumper wires short and flat, which makes the layout readable and easy to debug.
• Breadboard contacts add a little resistance and are not for high current. A rule of thumb is to keep to under about 1 A in total, which is plenty for sensors, LEDs and logic, and not for motors (Basic Electronics 9).

5. Build it: the chapter circuit

Build the series-parallel network from section 2 (Worked example 1) and check your calculations against meter readings.

Parts list

Real breadboard layout

Each connection below is one column. Numbers refer to the column; the letters are the rows used.

1. Wire everything with the battery disconnected, as listed above.
2. Before connecting power, switch the multimeter to resistance (Ω) and put the probes on the + and − rails. You should read about 2.32 kΩ. A very low reading means a short somewhere (do not connect the battery), and an open-circuit reading means a missing connection.
3. Connect the battery (red to +, black to −) and switch to DC volts.
4. Measure the battery voltage first, then the voltage across R1, then between node X (column 10) and the − rail.
5. Break the circuit at R1 (lift one leg out of its hole) and put the meter, on DC mA, between the loose leg and the hole: that is the total current. Repeat by lifting the R2 leg at column 20 for R2's branch current.

Simulator version

Falstad Circuit Simulator (falstad.com/circuit): add a DC voltage source set to 9 V and three resistors set to 1 k, 2.2 k and 3.3 k (double-click a part to change its value), wired as in the schematic, with a ground symbol on the lower rail. Hover over any component to see its voltage and current; use the right-click menu (Add Ammeter, or the Scope option) for a persistent display.

Tinkercad Circuits (tinkercad.com): place a 9 V battery and three resistors on a breadboard, wire per the hole list above, set each resistor's value in its panel, then add a multimeter. Click it to choose the mode: connect across a part for volts, in series for amps, and across the rails with the power disconnected for ohms. Click Start Simulation to take readings.

Expected readings

The real readings scale with the real battery voltage: a fresh 9 V battery that measures 9.4 V gives 9.4 / 2320 = 4.05 mA, for instance. Resistor tolerance (±5 % is typical for carbon-film resistors) adds a few percent. The checks that always hold, to within meter accuracy: the two voltages add to the battery voltage (KVL), and the two branch currents add to the total current (KCL).

Build: the two-source circuit

For Worked example 2, wire a 9 V battery, a pack of two AA cells (3 V), and three 1 kΩ resistors with all the negative terminals joined to the same rail: R1 from the 9 V + terminal to node A, R2 from the 3 V + terminal to node A, R3 from node A to the rail. Expected (calculated, then real within a few percent): V_A = 4.0 V; voltage across R1 = 5.0 V; across R3 = 4.0 V; R1 current 5.0 mA, R3 current 4.0 mA, and the R2 current reads −1.0 mA (the meter shows a minus sign when current flows against the direction its leads define, here out of node A towards the AA pack).

6. Challenges

Next, Basic Electronics 3 looks at resistors in depth (colour codes, power ratings), then uses the series rule to build voltage dividers, pull-up and pull-down resistors and a light-sensor circuit.