Resistors
Resistors
A resistor is a passive component that opposes the flow of current in a circuit. It converts some of the electrical energy passing through it into heat, producing a controlled voltage drop across itself.
Resistance is measured in ohms (Ω). Resistors are used to limit current, to divide voltage, to set the operating point of a transistor or sensor, and to safely discharge stored charge.
The ohm is named after Georg Ohm, whose 1827 book describing the relationship between voltage, current and resistance was so poorly received that he resigned his teaching post in disgust. It took another decade for the work to be recognised.
Types of Resistor
Fixed Resistor
A single, unchangeable value. The everyday component.
- Carbon film – cheap, general purpose, ±5% typical.
- Metal film – tighter tolerance (±1%), lower noise, more stable with temperature.
- Wirewound – a resistance wire wound on a former; used for high power ratings, but the coil makes it slightly inductive.
Variable Resistors
- Potentiometer – a resistive track with a moving wiper, giving three terminals. Wired across a supply it acts as an adjustable voltage divider. Used for volume knobs, joysticks, and dimmers.
- Rheostat – the same device wired with only two terminals (one end and the wiper), acting as a variable resistance in series with the load, controlling current directly.
- Trimpot – a small potentiometer adjusted with a screwdriver, set once during calibration and then left alone.
A potentiometer and a rheostat are often the same physical component wired differently. If three terminals are used, it acts as a voltage divider (potentiometer). When two terminals used, it provides variable resistance (rheostat).
Sensing Resistors
These are transducers: their resistance changes in response to a physical quantity.
- LDR (light dependent resistor) – resistance falls as light increases. Dark: hundreds of kΩ. Bright: a few hundred Ω.
- Thermistor – resistance changes with temperature. An NTC (negative temperature coefficient) thermistor's resistance falls as it gets hotter; a PTC thermistor's resistance rises as it gets hotter.
Each is almost always used as one half of a voltage divider, so that a change in resistance becomes a change in voltage that a circuit can actually read.
Pull-Up and Pull-Down Resistors
A microcontroller input pin left unconnected is floating — it will drift and read random values. A resistor (typically 10 kΩ) ties it to a known state:
- Pull-up – resistor from the pin to the positive supply. The pin reads HIGH until a switch pulls it LOW.
- Pull-down – resistor from the pin to ground. The pin reads LOW until a switch pulls it HIGH.
Power and Heat
Every resistor turns electrical energy into heat. The rate at which it does this is the power dissipation, measured in watts (W):
$$ P = V \times I $$
Every resistor has a power rating — the maximum it can dissipate before it overheats, discolours, drifts in value, or burns out. Common ratings are ⅛ W, ¼ W, ½ W, 1 W and upward. Always choose a resistor rated comfortably above the calculated dissipation.
The Resistor Colour Code
Most small resistors are too small to print a number on, so the value is encoded in coloured bands. Reading bands will not appear in the exam, but it is a practical skill to have in this class.
| Colour | Digit | Multiplier | Tolerance |
|---|---|---|---|
| Black | 0 | ×1 | – |
| Brown | 1 | $\times 10^1$ | ±1% |
| Red | 2 | $\times 10^2$ | ±2% |
| Orange | 3 | $\times 10^3$ | – |
| Yellow | 4 | $\times 10^4$ | – |
| Green | 5 | $\times 10^5$ | ±0.5% |
| Blue | 6 | $\times 10^6$ | – |
| Violet | 7 | – | – |
| Grey | 8 | – | – |
| White | 9 | – | – |
| Gold | – | ×0.1 | ±5% |
| Silver | – | ×0.01 | ±10% |
Four-band resistor: digit, digit, multiplier, tolerance. Five-band resistor: digit, digit, digit, multiplier, tolerance (used for precision resistors).
The tolerance band (usually gold or silver) is separated from the others by a wider gap and sits at the right. Read the bands with that band last. If you read from the wrong end you will get a wildly wrong value — a resistor read backwards is almost always an implausible number, which is your clue to flip it.
Tolerance
No resistor is exactly its stated value. Tolerance states how far the real value may stray from the nominal (marked) value.
$$ \text{range} = \text{nominal} \pm (\text{nominal} \times \text{tolerance}) $$
A resistor has the bands brown, red, yellow, gold. What is its value and its acceptable range?
Step 1 — Digits. Brown = 1, Red = 2 → the significant digits are 12.
Step 2 — Multiplier. Yellow = ×10 000.
$$ R = 12 \times 10\,000 = 120\,000\ \Omega = 120\text{ k}\Omega $$
Step 3 — Tolerance. Gold = ±5%.
$$ 120\,000 \times 0.05 = 6\,000\ \Omega $$
Step 4 — Range.
$$ 120\,000 \pm 6\,000 \;\Rightarrow\; 114\text{ k}\Omega \text{ to } 126\text{ k}\Omega $$
A resistor measuring anywhere in that band is working correctly and within specification.
Resistors are not made in every conceivable value. They come in standard series — the E12 series (10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 and their decade multiples) covers ±10% tolerance parts, spaced so that the tolerance bands of adjacent values just touch. This is why 4.7 kΩ and 2.2 kΩ are everywhere and 5 kΩ is not.
Resistors in Series
Resistors connected end to end, in a single path, so the same current flows through all of them. Their resistances simply add:
$$ R_{t} = R_{1} + R_{2} + R_{3} + \dots $$
The total is always larger than the largest individual resistor.
Key behaviours of a series circuit: - Current is the same at every point. - Voltage divides across the resistors, in proportion to their resistance. - The individual voltage drops sum to the supply voltage.
A 12 V supply is connected to a 220 Ω resistor in series with a 380 Ω resistor. Find the total resistance, the current, and the voltage across each resistor.
Step 1 — Total resistance.
$$ R_{t} = 220 + 380 = 600\ \Omega $$
Step 2 — Current (same everywhere in a series loop).
$$ I = \frac{V}{R_{t}} = \frac{12}{600} = 0.02\text{ A} = 20\text{ mA} $$
Step 3 — Voltage across each resistor.
$$ V_{1} = I \times R_{1} = 0.02 \times 220 = 4.4\text{ V} $$
$$ V_{2} = I \times R_{2} = 0.02 \times 380 = 7.6\text{ V} $$
Check: $4.4 + 7.6 = 12\text{ V}$ ✓ — the drops must add back up to the supply.
Resistors in Parallel
Resistors connected across the same two points, so the same voltage appears across all of them and the current splits between them.
$$ \frac{1}{R_{t}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \dots $$
The total is always smaller than the smallest individual resistor — adding a parallel path always gives current somewhere else to go.
This formula gives you $\frac{1}{R_t}$, **not** $R_t$. Forgetting the final step and writing down the reciprocal as the answer is the classic parallel-resistor mistake. If your total is bigger than the smallest resistor, you have gone wrong.
Find the total resistance of 200 Ω, 400 Ω and 400 Ω in parallel.
$$ \frac{1}{R_{t}} = \frac{1}{200} + \frac{1}{400} + \frac{1}{400} $$
$$ \frac{1}{R_{t}} = 0.005 + 0.0025 + 0.0025 = 0.01 $$
$$ R_{t} = \frac{1}{0.01} = 100\ \Omega $$
Combination Circuits
Real circuits mix the two. The method never changes:
- Find any group of resistors that is purely series or purely parallel.
- Reduce that group to a single equivalent resistance.
- Repeat until one resistance remains.
A 50 Ω resistor is in parallel with a series pair of 2.5 Ω and 10 Ω, across a 10 V supply. Find the total resistance, and the current drawn.
Step 1 — Reduce the series pair.
$$ R_{s} = 2.5 + 10 $$
Step 2 — Calculate $R_{s} and the 50 $\Omega$ resistor.
$$ \frac{1}{R_{t}} = \frac{1}{12.5} + \frac{1}{50} = \frac{1}{0.08 + 0.02} = 10 \Omega $$
Step 3 — Total current.
$$ I = \frac{V}{R} = \frac{10}{10} = 1\text{ A} $$
Two identical resistors in parallel always give half the value. Three identical give a third. In general, $R_t = R \div n$ for $n$ identical resistors. This shortcut saves a lot of arithmetic.
The Voltage Divider
Two resistors in series across a supply produce an intermediate voltage at their junction. This is a very common resistor circuit in electronics.
$$ V_{x} = \frac{R_{2}}{R_{total}} \times V_{supply} $$
The output is a fraction of the supply, set purely by the ratio of the two resistors.
A 9 V supply feeds a 1 kΩ resistor in series with a 2 kΩ resistor. What is the output voltage?
$$ V_{x} = \frac{R_{2}}{R_{1} + R_{2}} \times V_{supply} = \frac{2000}{1000 + 2000} \times 9 $$
$$ V_{x} = \frac{2}{3} \times 9 = 6\text{ V} $$
The 2 kΩ resistor takes two-thirds of the resistance, so it drops two-thirds of the voltage.
Replace one of the two resistors (usually R2) with a sensor whose resistance changes — an LDR, a thermistor, a force-sensing resistor — and the output voltage now varies with light, heat, or pressure. That varying voltage can be fed straight into a microcontroller's analog input. This is how the majority of simple sensor circuits work.
Current-Limiting Resistors
An LED is not a resistor and does not obey Ohm's law. Connected directly to a supply it will draw whatever current the supply can deliver and destroy itself. A series resistor sets the current.
The LED drops a roughly fixed forward voltage ($V_f$, typically about 2 V for a red LED). The resistor must drop everything that is left over:
$$ R = \frac{V_{supply} - V_{f}}{I_{LED}} $$
A red LED ($V_f = 2\text{ V}$, desired current 20 mA) is to run from a 9 V supply. What series resistor is needed?
Step 1 — Voltage the resistor must drop.
$$ V_{R} = 9 - 2 = 7\text{ V} $$
Step 2 — Resistance for the desired current. Remember to work in amps.
$$ R = \frac{V_{R}}{I} = \frac{7}{0.02} = 350\ \Omega $$
Step 3 — Choose a real resistor. 350 Ω is not a preferred value. Round up to the next standard value, 390 Ω, which gives slightly less current and keeps the LED safe.
Step 4 — Check the power rating.
$$ P = I^{2}R = (0.02)^{2} \times 390 = 0.156\text{ W} $$
A ¼ W resistor is fine.
Summary of Formulas
| Quantity | Formula |
|---|---|
| Ohm's law | $V = IR$ |
| Power | $P = VI |
| Tolerance range | $\text{nominal} \pm (\text{nominal} \times \text{tolerance})$ |
| Series total | $R_{t} = R_{1} + R_{2} + R_{3} + \dots$ |
| Parallel total | $\dfrac{1}{R_{t}} = \dfrac{1}{R_{1}} + \dfrac{1}{R_{2}} + \dots$ |
| Voltage divider | $V_{x} = \dfrac{R_{x}}{R_{total}} \times V_{supply}$ |
Series vs Parallel at a Glance
| Series | Parallel | |
|---|---|---|
| Current | Same through every resistor | Splits between branches |
| Voltage | Divides across the resistors | Same across every resistor |
| Total resistance | Larger than the largest | Smaller than the smallest |
| If one fails open | The whole circuit stops | The other branches keep working |