Resistors
Resistors are a passive electronic components that restrict the flow of current when a voltage is applied across them. They dissipate the energy lost as heat. Given their simplicity, usefulness and low price of manufacture, they are typically the most commonly used electronic component in any circuit design.
The mechanical equivalent of a resistor is friction. The larger the resistance, the larger the friction. This is when using the forcevoltage equivalence.
Resistors are used for a huge number of purposes, including:
 Resistor dividers: To produce an output voltage that is a fixed percentage of the input voltage.
 Voltage/current biasing: e.g. transistor amplifiers
 Current limiting: Power an LED from a fixed voltage rail.
 Opamp gain/feedback configuration: Resistors (along with the occasional capacitor) are used to set the gain/feedback of opamps.
 Impedance matching
 Current measuring: Resistors convert a current into a voltage, which is easily measurable by opamps/ADCs/e.t.c.
 GPIO/communication bus pullup/pulldowns: To drive nets into passive states, for example to keep the gate tied to source so it’s always OFF unless driven on. Also to drive communication bus lines into passive states, especially buses that use arbitration (e.g. I2C, CAN bus).
Schematic Symbols
The most commonlyused resistor schematic symbols are shown below. I prefer the Americanstyle resistor over the European purely because there are already many “boxlike” schematic symbols used for other components (e.g. fuses, ICs), and so the squiggle makes a resistor more distinguishable (given my distaste for the American imperial system, I never thought I would ever say that!). The American style is used throughout the rest of this website.
See the Potentiometers And Rheostats page for more schematic symbols related to resistors.
Resistors In Series And In Parallel
The behaviour of resistors when connected together in series and in parallel is exactly the same behaviour inductors exhibit, and exactly the opposite behaviour of what capacitors exhibit.
Resistors In Parallel
When two resistors are connected in parallel, the equivalent total resistance follows the inverse law:
$\begin{align} R_{total} = \frac{1}{\frac{1}{R1} + \frac{1}{R2}} \end{align}$It is usually easier to remember this equation as:
$\begin{align} \frac{1}{R_{total}} = \frac{1}{R1} + \frac{1}{R2} \end{align}$or as (and this is my favourite):
$\begin{align} R_{total} &= \frac{R1R2}{R1 + R2} \end{align}$This relationship is shown below:
Resistors In Series
When two resistors are connected in series, the total equivalent resistance is equal to the sum of individual resistances.
$\begin{align} R_{total} = R1 + R2 \end{align}$This is shown below:
Resistor Dividers
Resistor dividers are two or more resistors in a series configuration such that at some junction in the string, the voltage is a fixed proportion of the total voltage applied to the end’s of the string. In this way, they divide the input voltage (applied across the entire string) into a smaller output voltage (measured across only part of the string).
The simplest voltage divider consists of just two resistors in series, shown below:
The equation for $V_{OUT}$ is:
$\begin{align} V_{OUT} &= \frac{R2}{R1\ +\ R2} V_{IN} \\ \end{align}$The input voltage “divides” itself across the resistors proportionally based on relative resistances. The more resistance of any one resistor, the greater amount of voltage that will drop across it. You can easily reach the above equation by applying Ohm’s law to the circuit.
During circuit design, you will encounter times when you have three knowns from the equation above but have to solve for any one of the others. Thus it has be rearranged for every variable below for convenience (with $V_{IN}$ and $R1$ being able to be simplified slightly):
$\begin{align} V_{IN} &= \frac{R1 + R2}{R2} V_{OUT} \nonumber \\ &= \left( \frac{R1}{R2} + 1 \right) V_{OUT} \\ R1 &= \frac{V_{IN}  V_{OUT}}{V_{OUT}} R2 \nonumber \\ &= \left( \frac{V_{IN}}{V_{OUT}}  1 \right) R2 \\ R2 &= \frac{V_{OUT}}{V_{IN}  V_{OUT}} R1 \\ \end{align}$Thevenin Equivalent Circuit
The Thevenin equivalent circuit for a resistor divider is shown below. This is the equivalent circuit as looking from $V_{OUT}$ into the resistor divider.
Where the Thevenin voltage $V_{TH}$ is equal to the opencircuit output voltage:
$\begin{align} V_{TH} &= \frac{R2}{R1 + R2} V_{IN} \\ \end{align}$and the Thevenin resistance $R_{TH}$ is equal to the opencircuit output voltage divided by the shortcircuit current:
$\begin{align} R_{TH} &= R1  R2 \\ &= \frac{R1 \cdot R2}{R1 + R2} \\ \end{align}$The Thevenin resistance is sometimes called the output resistance or impedance $\b{Z_O}$. The Thevenin equivalent circuit is useful for calculating how long a capacitor attached to the output will take to charge to a certain level. It allows you to have a single resistance that you can then plug into the time constant equation $\tau = RC$. For more info on this, see the RC Charging Circuits page. It is also useful when looking at it’s frequency response. For more info on this, see the Analogue Filters page.
Note that the output impedance of a resistor divider is normally quite high, compared to other “standard” voltage sources. For this reason, you cannot normally use a resistor divider to drop the voltage and provide power to a device. This is a common mistake that people learning electronics do, when in reality you should either be using a linear regulator, a SMPS, or a transformer. Voltage dividers should normally only be used to provide a voltage to a highimpedance input (e.g. opamp input, comparator input, microcontroller ADC input, or voltagelevel translation for comms signals).
The exception to the above rule is when the two following conditions are met:
 The device will draw a small enough current that the voltage sag due to the extra current through $R1$ is acceptable.
 Voltage fluctuations when the load changes are acceptable (i.e. this is not active regulation…there is no feedback).
 The current going through $R1$ will not cause it to overheat.
Applications
Resistor dividers are used everywhere in circuit design! They are commonly used for:
 Scaling down a higher voltage (e.g. a 012V voltage rail) down to something that can be measured by an ADC (e.g. 02.5V).
 Biasing transistors.
 Providing the correct voltages to the inputs of opamps.
An interesting example I have seen of a resistor divider powering a circuit was a lowpower microcontroller being powered directly from a resistordivider, diode and capacitor from mains supply (240VAC). The microcontroller only drew $uA$ so met the above criteria for using a resistor divider as a power supply.
Online Calculators
NinjaCalc has a calculator that can work out voltages, resistances and currents of a resistor divider.
Tolerances
The tolerance of a resistor defines how accurately the resistor the actual resistors value is to the specified value, usually in terms of a percentage. 5% and 1% resistors are the most common. Typically the cost of a resistor goes up as the tolerance reduces, as it requires increased manufacturing precision.
5% resistors are normally fine for the most resistor jobs such as:
 Currentlimiting
 ESD protection
 Pullups/pulldowns
 Termination resistors
1% or lower precision is usually required for:
 Resistor dividers for ADC inputs
 Opamp gain resistors
 Current measuring resistors (0.1% precision may be needed here, and they start costing a bit!)
With the advent of SMD resistors, the difference in price between 1% and 5% resistors is so insignificant that for most purposes you can get away with using 1% tolerance resistors for everything in your circuit design.
Can I Put Resistors In Series Or Parallel For Better Tolerances?
The short answer. No. 2x $1k\Omega$ 1% resistors in series is the equivalent to 1x $2k\Omega$ 1% resistor.
The long answer. You will never get a worse tolerance by putting two resistors in series or parallel. BUT, you may get a better distribution of values, depending on the distribution of the original resistors. If you assume (and this is a bad assumption) that the resistor values followed a Gaussian distribution, then the resulting distribution is a better Gaussian distribution (skinnier/smaller deviation). If the original resistors had a flat distribution, the resulting distribution is a triangle shape.
However, the distribution of resistor values could be any number of shapes. For example, the manufacturer might make heaps of 5% $1k\Omega$ resistors, which are then measured. If the resistance falls within 1% of $1k\Omega$, then they are made into 1% resistors. This would leave the 5% resistor bin with a double peak, with a valley right in the middle of the distribution.
Also, correlation between resistors from the same manufacturing batch run may mean that you do not get any standard deviation improvements.
Manually Tweaking Resistance
For nonrepetitive, high precision values, you can actually tweak a resistors value by grinding some of the resistor away with a metal file. This only works for the metal film style resistors. See this video as an example.
The E Series
Practically all resistors follow an E series, a scale of predefined resistances that have standardised by IEC 60063. This type of sequence is called preferred numbers. Common E series are the E12, E24, E48, E96 and E192 series. The series divides the numbers between 1 and 10 into 12, 24, 48, e.t.c steps. The steps are chosen so that maximum relative error between any resistance you want and the closest resistance in the series is fixed (i.e. constant).
Simply, this means that each series guarantees you will be able to find a resistor that equals the resistance you need within a fixed maximum percentage error.
Series  Maximum Percentage Error 

E6  20% 
E12  10% 
E24  5% 
E48  2% 
E96  1% 
E192  0.5% 
The E192 series is also used for 0.25% and 0.1% error resistors.
Below are all the actual values for the common E series. They are written as initialised arrays in the C programming language, so that you can copy them and use them in code easily (some modifications may be required for other programming languages).
Note how there are two digits of precision for E6, E12, and E24 values, while 3 digits of precision for E48, E96 and E192 values. These two sets of three series share special properties with one another. E6 is every second value from the E12 series, and E12 is every second value from the E24 series. Similarly, E48 is every second value from the E96 series, and E96 is every second value from the E192 series.
The values come from the exponential number series, using the equation:
$\begin{align} v(i, n) = 10^{i/n} \end{align}$where:
$i$ = the i’th element in the series
$n$ = the total number of elements in the series
For any Eseries range, this pattern is applied for every decade of resistance, e.g. between $110\Omega$, $10100\Omega$, $1001k\Omega$ and so on. standard families of resistors will start at about $1\Omega$ and continue up to $1020M\Omega$. For values outside of this range you generally have to find specialist products (e.g. precision current measuring resistors) and pay a little more for them.
The NinjaCalc Standard Resistance Finder calculator, can easily find the closest Eseries resistance to your desired resistance.
See Wikipedia  Preferred Number for more information on these series.
Resistor Manufacturing Processes
Wire Wound
Wirewound resistors are the oldest type of resistor, and are formed by coiling up a piece of wire to get a desired resistance. They are only typically used in modern times in high power applications and for things like fuses, with ratings up into the 100’s of Watts.
Given they are normally a coil of wire, they can have a significant parasitic inductance and be give off/be susceptible to magnetic fields.
Metal Film
Metal film resistors are the most popular resistor in today’s market. They have replaced [^_carbon_film, carbon film resistors] in most applications due to their cheaper cost, lower noise and smaller size. Metal film resistors can be split into two subcategories, thick metal film and thin metal film.
Thick Metal Film
Thick film is the most common type of metal film resistor. Most 1% and 5% SMD chip package resistors (0402, 0603, e.t.c) use thick film technology.
Thin Metal Film
Most 0.1% SMD chip package resistors (0402, 0603, e.t.c) use thin film technology. Thin film resistors can be split into two subcategories, commercial thinfilm and precision thinfilm.
Metal Foil
Even rarer than thick and thin film resistors, metal foil resistor technology allows the most precise resistors to be made.
Carbon Film
Carbon film resistors are formed by forming a conductive carbon film on a ceramic substrate. Carbon film resistors have been replaced in most applications by metal film resistors.
Power Resistors
Power resistors is a term used with resistors which are usually rated to dissipate 1W or more of power (without heatsinking).
Power resistors can be used to intentionally heat things. The below image shows a common 5W resistor being used to heat a small container of oil, with a copper thermostat from a hot water cylinder being used to control the temperature.
CurrentSense Resistors
Currentsense resistors are a label given to lowvalued, high precision (1% or better), and high power resistors that are good for using in currentsense circuits. Sometimes there is nothing special about these resistors (it’s purely a marketing term), other times they may have two additional terminals for Kelvin sensing. A four terminal resistor is also called an ammeter shunt. Two of the terminals are used to pass the high current, the other two are used to measure to voltage drop across the resistor. This gets rid of measurement errors due to voltage drop in the wires going to the resistor (when the sense line and highcurrent path are the same thing).
More information and schematics on how to make currentsense circuits can be found on the CurrentSensing page.
Jumpers
Most resistor series also include a 0Ω jumper resistor. Jumper resistors are great for jumping tracks when doing PCB design, as well as providing a convenient and cheap way of connecting/disconnecting certain tracks in different PCB variants.
Note that sometimes these jumper resistors can handle much less current than you expect! For example, most 0603/0805 sized SMD jumper resistors are only rated to 12A, even though at this current the I*R power dissipation is well below the continuous maximum (0.10.5W). However, some can handle some decent current, for example, the Susumu YJP1608R001 0603 jumper resistor, which can handle 10A continuous.
Jumper resistors are not specified with a percentage tolerance as most other resistors, as this makes no sense (what is 5% of 0Ω?). Instead, they are printed as 0Ω, and a maximum resistance is given in the datasheet, which is usually in the order of 150mΩ.
Volume Resistance (Bulk Resistance)
Volume resistance (also known as just resistivity, electrical resistivity, or bulk resistance) has the SI units $\Omega m$. It is a measure of how well a particular material conducts electricity, and is an intrinsic property of that material (it does not depend on how much of the material or what shape it is in). If the resistance between two conducting plates on opposite faces of a $1 \times 1 \times 1m$ cube of material is measured to be $1\Omega$, then the material has a volume resistivity of $1\Omega m$.
Parasitic Elements
For most daytoday applications, resistors can just be treated as if they have a resistance. However, in high frequency circuits, there are other parasitic elements to a resistor that you must consider. Typically, the main parasitics are modelled as a extra inductor and capacitor, although the is no standard way of wiring them (depends on what literature you are reading). One popular configuration is shown below:
The below image shows another model which is popular as it models the resistance in parallel with the end cap capacitance and this in series with the lead inductance^{4}.
The below table shows resistor types and the ranges of their parasitic inductance^{5}:
Resistor Type  Capacitance  Inductance 

Wirewound  0.0356uH  
Metal oxide  3200nH  
Metal foil  0.05pF  <80nH 
Metal film  <2nH 
Helical wirewound resistors have an especially large parasitic inductance because they are wound in a coil. They are also especially susceptible to magnetic pickup (inducing electrical noise into the circuit due to nearby magnetic fields). There are also special noninductive wirewound resistors in where the wire is wound back and forth rather than in a coil to drastically reduce the inductance (they use the Ayrton–Perry winding technique).
Resistor Noise
Resistors add thermal (JohnsonNyquist) noise into circuits. See the Electrical Noise page for more info.
Packages
Resistor come in many packages, from large, wirewound power resistors that come in coils and blocks, to smaller throughhole resistors, to even smaller still SMD resistors. You can find more about resistor packages on the Component Package Database page.
Throughhole resistors use the older color code scheme (the current international standard as of 2013 is IEC 60062). Newer surfacemount resistors usually have the value printed directly on them (a threedigit number is most common, with the third digit being the multiplier).
Once taken out of the tape, they don’t look like much!
The below image was me trying to be artyfarty with the leftovers from putting about 30,000 reeled 0603 resistors into containers for prototyping with.
Resistor Optimizer Tool
There is a great, free tool by Janne Ahonen called Resistor Optimizer (download it here) which finds optimal values for resistor dividers and optimal values for series/parallel resistor combinations to achieve the desired total resistance. It runs on Windows (Win32 application)^{6}.
Further Reading
 For information on positive temperature coefficient resistors used as “fuses” in circuit protection applications, see the PTC Resettable Fuses page.
 For info on mechanically variable resistors, see the Potentiometers and Rheostats page.
 For info on digital controlled variable resistors, see the Digital Potentiometers (DPOTs) page.
Footnotes

Geoffrey Hunter (2024, Feb 16). RC Charging Circuits. mbedded.ninja. Retrieved 20240216, from https://blog.mbedded.ninja/electronics/circuitdesign/rcchargingcircuits/. ↩

Yageo. Thick Film General Purpose (product page). Retrieved 20221005, from https://www.yageo.com/en/Product/Index/rchip/thick_film_general_purpose. ↩

TT Electronics (2020, Jun). Vitreous Enamelled Wirewound Resistors: W20 Series (datasheet). Retrieved 20220421, from https://www.mouser.com/datasheet/2/414/TTRB_S_A0010754439_12565592.pdf. ↩

Wyatt, Kenneth (20131029). Resistors aren’t resistors. EDN. Retrieved 20210815, from https://www.edn.com/resistorsarentresistors/. ↩

EE Power. Resistor Inductance. Retrieved 20210813, from https://eepower.com/resistorguide/resistorfundamentals/resistorinductance/#. ↩

Janne Ahonen (2018, Jun 17). Resistor optimizer (product page). Retrieved 20220826, from http://jahonen.kapsi.fi/Electronics/ResOptimizer/. ↩ ↩^{2}