# Voltage divider
A voltage divider is a series circuit that divides the total voltage into fractional portions of constant ratio; for two resistors, the output taken across the lower resistor is the input multiplied by R2/(R1 + R2)[1](https://www.allaboutcircuits.com/textbook/direct-current/chpt-6/voltage-divider-circuits/). Dividers appear wherever a voltage must be scaled, referenced, or measured: microcontroller analog inputs, comparator thresholds[1](https://www.allaboutcircuits.com/textbook/direct-current/chpt-6/voltage-divider-circuits/), high-voltage instruments rated to hundreds of kilovolts[2](https://pubs.aip.org/aip/rsi/article/97/9/094709/3404792/Design-of-a-400-kV-resistive-voltage-divider-with), and laboratory ratio standards linear to a few parts per million[3](https://www.ietlabs.com/pdf/Datasheets/KVD-500.pdf).
| Key fact | Detail |
|---|---|
| Divider rule | The ratio of individual resistance to total resistance equals the ratio of individual voltage drop to total supply voltage; for two resistors, Vout = Vin·R2/(R1 + R2)[1](https://www.allaboutcircuits.com/textbook/direct-current/chpt-6/voltage-divider-circuits/) |
| Thévenin equivalent | A divider looks like a source of Vth = Vin·R2/(R1 + R2) in series with Rth = R1∥R2, so loading sags the output[4](https://circuits.mit.edu/_static/S23/handouts/lec03b/lecture03b.pdf) |
| Loading example | Two identical resistors on 24 V give 12 V unloaded, but 3.429 V into 200 Ω, 8.000 V into 1000 Ω, and 9.000 V into 1500 Ω[5](https://ibiblio.org/kuphaldt/socratic/model/mod_thevenin_norton.pdf) |
| Ratio accuracy | Division-ratio accuracy depends on the ratio between resistor values, not their absolute values[6](https://passive-components.eu/wp-content/uploads/2025/09/3_3_TT-Electronics-PCNS-2025-Paper.pdf) |
| Divider families | Resistive, capacitive, and inductive dividers; resistive types are chosen when operation from DC upward is required, capacitive types for fast pulses[7](http://www.hioki.com/euro-en/download/39431) • [8](https://www.osti.gov/servlets/purl/7189210) |
| Precision ratio device | The Kelvin-Varley divider reaches 0.1 ppm linearity with ten million dial settings in seven decades, using 11 precision resistors per decade[3](https://www.ietlabs.com/pdf/Datasheets/KVD-500.pdf) |
| High-voltage performance | A 400 kV compensated resistive divider achieved a response time below 4.7 ns, a division ratio of 1018.62, and about 2.1% expanded uncertainty (k = 2) over 30–180 kV[2](https://pubs.aip.org/aip/rsi/article/97/9/094709/3404792/Design-of-a-400-kV-resistive-voltage-divider-with) |
## The resistive divider and its derivation
In a series circuit the same current flows through every element, so each element's voltage drop is proportional to its resistance. Because a series circuit splits the supply voltage into fractional parts in a fixed proportion, it is known as a voltage divider: each resistance, compared with the whole resistance, bears the same relationship to one another as does that element's voltage drop compared with the entire supply voltage.[1](https://www.allaboutcircuits.com/textbook/direct-current/chpt-6/voltage-divider-circuits/) For two resistors R1 (top) and R2 (bottom) across an input V, the current is V/(R1 + R2) and the output across R2 is
\[ V_{\mathrm{out}} = V \cdot \frac{R_2}{R_1 + R_2} \]
MIT circuit notes derive the same relation vA = R2/(R1 + R2)·V from superposition and parallel-resistance analysis, showing the divider as a special case of Thévenin analysis[4](https://circuits.mit.edu/_static/S23/handouts/lec03b/lecture03b.pdf).
A practical consequence follows from the formula's structure: the accuracy of the division ratio depends on the ratio between resistor values rather than on their absolute values[6](https://passive-components.eu/wp-content/uploads/2025/09/3_3_TT-Electronics-PCNS-2025-Paper.pdf).
## Loading and output impedance
A divider drives the next stage through its own output resistance, and the load forms a second divider with it. The loaded-divider relation v = R2/(R1 + R2)·V + i·R1R2/(R1 + R2) shows that the Thévenin equivalent of a divider is a source of Vth = R2/(R1 + R2)·V in series with Rth = R1∥R2[4](https://circuits.mit.edu/_static/S23/handouts/lec03b/lecture03b.pdf). Voltage signals should come from a source with an output impedance much lower than the input impedance of the next stage; the lower the output impedance, the larger the fraction of the divided voltage appears across the load[4](https://circuits.mit.edu/_static/S23/handouts/lec03b/lecture03b.pdf).
The sag is quantified in a worked example from Kuphaldt's lesson series. Two identical resistors on a 24 V source output 12 V unloaded, but the Thévenin equivalent is a 12 V source in series with 500 Ω, and the load voltage falls to 3.429 V at 200 Ω, 8.000 V at 1000 Ω, and 9.000 V at 1500 Ω[5](https://ibiblio.org/kuphaldt/socratic/model/mod_thevenin_norton.pdf). A load of three times Rth (1500 Ω) loses exactly a quarter of the output (9.000 V versus 12 V)[5](https://ibiblio.org/kuphaldt/socratic/model/mod_thevenin_norton.pdf). Thévenizing the loaded divider reduces what would otherwise be a fresh series-parallel analysis for each load value to a single application of the divider formula[5](https://ibiblio.org/kuphaldt/socratic/model/mod_thevenin_norton.pdf).
## Divider types: resistive, capacitive, inductive
Three impedance families divide voltage. Hioki's account of a standard high-voltage divider lists resistive, capacitive, and inductive types, and chose a resistive divider because the instrument had to operate from DC to 10 kHz[7](http://www.hioki.com/euro-en/download/39431).
**Resistive dividers** work at DC but run into two limits at high voltage and high speed. For pulses of several hundred thousand volts lasting up to about 20 microseconds, a resistive divider is unacceptable because of circuit loading from distributed capacitance, the high resistance values and frequency compensation needed, and the excessive physical size of reliable constant-use dividers[8](https://www.osti.gov/servlets/purl/7189210). At several hundred kilovolts, stray capacitance between the high-voltage arm and ground distorts fast transient waveforms and increases response time[2](https://pubs.aip.org/aip/rsi/article/97/9/094709/3404792/Design-of-a-400-kV-resistive-voltage-divider-with). Such stray capacitances arise between conductors, between resistor bodies, and between conductors and resistor bodies, and fully calculating them in simulation is not a common approach[9](https://iris.inrim.it/retrieve/handle/11696/52432/dd2573c2-cee2-e71c-e053-d805fe0ad5dc/TPWRD-01071-2015%20-%20%20Design%20of%20MV-HV%20Dividers%20_%20rev_final.pdf). Early NIST work on dividers with resistance and self-capacitance showed that only under special conditions is oscillograph deflection proportional to the applied voltage, confirmed by oscillograms at 20 to 1000 kc[10](https://nvlpubs.nist.gov/nistpubs/jres/9/jresv9n1p81_A2b.pdf).
**Capacitive dividers** suit fast, high-voltage pulses. They can be designed with very little circuit loading, on the order of 10 picofarads, and work well with rapidly changing voltages; their disadvantages are poor low-frequency response, electrostatic shielding effects from nearby objects, internal corona, and changes in the dielectric nearby and inside the divider[8](https://www.osti.gov/servlets/purl/7189210).
**Compound RC dividers** combine the two. For DC high-voltage measurement, the resistance-capacitance type is usually preferred[11](https://www.mdpi.com/1996-1073/14/22/7744), and an HVDC divider based on RC compound connection developed in Sweden reaches a DC measurement uncertainty of 16 × 10⁻⁶, while a resistive divider developed by T. Harada has a step response time of about 10 ns[12](https://www.mdpi.com/1996-1073/15/2/451). Compensation can also be deliberate: a filtering or compensation capacitor in parallel with the low-side resistor permits gain-shaping, though an excessively large value slows the step response[13](https://www.ti.com/lit/ds/symlink/res60a-q1.pdf).
The transition between resistive and capacitive behavior is visible in an integrated part. The RES60A-Q1, rated 1400 VDC, acts as a resistive divider at low and near-DC frequencies with a −3 dB bandwidth around 70 kHz (ratio dependent), and transitions to capacitive division at roughly 100–200 kHz as parasitic capacitances dominate[13](https://www.ti.com/lit/ds/symlink/res60a-q1.pdf). A 400 kV research divider with explicit stray-capacitance compensation achieved a response time below 4.7 ns, withstood impulse voltages exceeding 400 kV, and calibrated to a division ratio of 1018.62 with about 2.1% expanded relative uncertainty over the directly calibrated 30–180 kV range[2](https://pubs.aip.org/aip/rsi/article/97/9/094709/3404792/Design-of-a-400-kV-resistive-voltage-divider-with).
## Adjustable dividers: potentiometers and precision ratio devices
A [potentiometer](https://www.edgepedia.org/potentiometer) is a variable voltage divider whose division ratio is set by wiper position; at the 50 percent (center) position, with no load on the wiper, the wiper-to-terminal voltage is half the applied voltage, regardless of the applied voltage magnitude or the end-to-end resistance[1](https://www.allaboutcircuits.com/textbook/direct-current/chpt-6/voltage-divider-circuits/).
For laboratory precision, the Kelvin-Varley divider (KVD) is the classic ratio device. In its most developed form it is linear to 0.1 ppm and features ten million individual dial settings arranged in seven decades[3](https://www.ietlabs.com/pdf/Datasheets/KVD-500.pdf). Its economy comes from iteration: a plain tapped string needs 1000 precision resistors for 1-part-in-1000 resolution, while each Kelvin-Varley stage provides a decade of resolution with only 11 precision resistors, and cascading three stages permits any division ratio from 0 to 1 in increments of 0.001[3](https://www.ietlabs.com/pdf/Datasheets/KVD-500.pdf). The device can be viewed as a three-terminal potentiometer with fixed end-to-end resistance and a seven-decade switched wiper position, and it has been used to verify the roughly 2 ppm linearity of precision ADCs[3](https://www.ietlabs.com/pdf/Datasheets/KVD-500.pdf). Design procedures for KVD-based standards, including one with a span of 2111.111 units for use with multimeters and Universal Ratio Sets, treat it as the basis of precision voltage-ratio metrology[14](https://www.osti.gov/servlets/purl/6861036).
The same principle underlies modern cascaded balanced dividers, which use cascaded stages of N identical sub-elements so the overall ratio approximates the product of the individual stage ratios, with precision maintained by in-stage matching rather than absolute values[6](https://passive-components.eu/wp-content/uploads/2025/09/3_3_TT-Electronics-PCNS-2025-Paper.pdf).
## Applications
**ADC input scaling.** A divider into a high-impedance input is commonly used to bring a voltage within an ADC's measurable range, for example a 4:1 divider to measure a 3.6 V battery with a 1 V-reference ADC[15](https://electronics.stackexchange.com/questions/106718/when-would-i-use-a-voltage-regulator-vs-voltage-divider). Integrated parts push this to high voltage: TI's RES60 is a 1.4 kV precision divider for high-voltage battery monitoring, and its evaluation module uses a 500:1 resistor divider followed by an operational-amplifier buffer on a split supply of ±0.9 to ±2.75 V[16](https://www.electronicdesign.com/technologies/components/passives/article/55293573/electronic-design-tis-integrated-14-kv-precision-resistor-dividers-extend-hv-battery-life).
**Comparator thresholds.** When a divider sets a comparator threshold, the pin's input leakage current shifts the actual sensed voltage away from the desired value; TI's design equations select R2 first, then solve for R1, sizing the divider to meet an accuracy requirement while trading accuracy against current consumption[17](https://www.ti.com/lit/an/slva450b/slva450b.pdf). If the leakage current is negative, flowing out of the pin, the threshold voltage is lower than expected[17](https://www.ti.com/lit/an/slva450b/slva450b.pdf).
**Reference and bias generation.** A divider can provide a secondary reference voltage, such as halving a 3.6 V supply to obtain a 1.8 V reference for biasing an AC signal with a DC offset, feeding an op-amp buffer instead of an expensive voltage-reference IC[15](https://electronics.stackexchange.com/questions/106718/when-would-i-use-a-voltage-regulator-vs-voltage-divider). Precision dividers built from carefully matched, low-tolerance resistors are used for stable voltage scaling in signal conditioning, reference-voltage generation, and measurement inputs[18](https://www.edn.com/splitting-voltage-with-purpose-a-guide-to-precision-voltage-dividers/).
## By the numbers
- **Resistor selection.** E24-series resistors are now available with 1% tolerance, giving 18 values not available in the E96 series, which matters for divider output-voltage accuracy in converters[19](https://www.allegromicro.com/-/media/files/application-notes/an296255-resistive-voltage-dividers.pdf?sc_lang=en).
- **Technology limits.** In thick-film resistor technology the practical accuracy limit is around 0.1% tolerance and 25 ppm/K temperature coefficient of resistance[6](https://passive-components.eu/wp-content/uploads/2025/09/3_3_TT-Electronics-PCNS-2025-Paper.pdf).
- **Commercial precision parts.** Type HVD ultra-precision dividers are rated up to 5000 V DC with a standard ratio tolerance of 0.05% and ratio temperature-coefficient options of 5 ppm/°C and 10 ppm/°C[20](https://datasheet.lcsc.com/datasheet/pdf/2d3b03ffd0e7404853d7089f3d969122.pdf?productCode=C6274067).
- **Power budget.** A 1 kV multirange resistive divider design uses series groups of five parallel 10 kΩ resistors, 2 kΩ per group, giving 100 mW of power dissipation per resistor, with parallel 8.2 pF capacitors per group for phase compensation; ranges of 240, 600, and 1000 V have nominal ratios of 401, 1001, and 1667[21](https://www.imeko.org/publications/tc4-2022/IMEKO-TC4-2022-11.pdf).
- **Calibration uncertainty.** NIST offers 1 kV to 10 kV divider calibration with estimated uncertainty of the order of ±10 × 10⁻⁶[22](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir5566.pdf).
- **AC bandwidth.** Precision AC resistance dividers serve harmonic measurements from power frequency to the 60th harmonic (3 kHz) and audio power measurements up to 100 kHz; a calibration method verified on a 480 V divider corrects ratio-difference phase from 50 Hz to 100 kHz[23](https://pmc.ncbi.nlm.nih.gov/articles/PMC10458715/).
- **Sensor classes.** A prototype wideband resistive-divider voltage sensor achieved accuracy class 0.2 for 20 Hz to 5 kHz and class 0.5 for 1 Hz to 20 Hz[24](https://pmc.ncbi.nlm.nih.gov/articles/PMC5712857/).
## How it compares with regulators and active references
A divider scales or references a voltage but cannot power circuitry: any current drawn from the tap flows through the divider resistors and shifts the output, as the loading analysis shows. A voltage regulator, by contrast, is designed to provide a fixed voltage and is commonly used to provide power to other circuitry; a linear regulator lowering 10 V to 5 V wastes the difference as heat, ideally running at 50% efficiency[15](https://electronics.stackexchange.com/questions/106718/when-would-i-use-a-voltage-regulator-vs-voltage-divider). The divider's role is therefore signals into high-impedance inputs and cheap secondary references, where a buffered divider can stand in for a dedicated reference IC[15](https://electronics.stackexchange.com/questions/106718/when-would-i-use-a-voltage-regulator-vs-voltage-divider). The Thévenin view makes the boundary explicit: the divider is a source with output impedance R1∥R2, and it holds its set voltage only while the load impedance stays much larger than that[4](https://circuits.mit.edu/_static/S23/handouts/lec03b/lecture03b.pdf).
## History and metrology
The analytical foundation predates its common name. The voltage-source equivalent underlying divider and Thévenin analysis was published in 1853 by [Hermann von Helmholtz](https://www.edgepedia.org/hermann-von-helmholtz) in Poggendorff's *Annalen der Physik und Chemie* (page 211), four years before Thévenin was born[25](https://jontallen.ece.illinois.edu/uploads/537.F18/Papers/DonJohnson1-03.pdf). The potentiometric method, in which a potentiometer or precision voltage divider with a separate source is adjusted until a galvanometer reads null against the voltage under test, made the divider a primary measuring instrument[26](https://www.ietlabs.com/pdf/GenRad_History/A_History_of_Z_Measurement.pdf).
National metrology institutes built on the Kelvin-Varley principle. NIST, then the National Bureau of Standards, constructed a 10 kV DC resistive ratio standard in 1975 using three tapped resistor decades and a Kelvin-Varley circuit for the lowest two decades, with guard resistors and a temperature-controlled oil bath[22](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir5566.pdf). Its lowest decade uses 2 kΩ individual resistors with a shunt making the input resistance a nominal 20 kΩ, and imperfect linearity added only 1.5 ppm of uncertainty across the four stages[22](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir5566.pdf). Calibration uses a high-precision digital voltmeter at the 10 V level to establish a 10:1 ratio per decade, combined to an overall 10,000:1 ratio, with linearity checked against a Josephson-array voltage standard[22](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir5566.pdf).
Modern practice is codified in standards and continues to develop. IEC 60060-2:2025 specifies tests for approved high-voltage measuring systems, including determination of the scale factor and calibration by comparison with a reference measuring system[27](https://cdn.standards.iteh.ai/samples/iec/iec-60060-2-2025/4c4d040691194886b57627976ac2af62/redline-iec-60060-2-2025-rlv.pdf). A 2024 ultra-stable 3D-printed precision voltage divider was developed for calibrations and experiments in nuclear and atomic physics facilities, integrated into setups such as COALA (TU [Darmstadt](https://www.edgepedia.org/darmstadt)), BECOLA ([Michigan State University](https://www.edgepedia.org/michigan-state-university)), and COLLAPS[28](https://arxiv.science/abs/2407.06700).
## References
1. [Voltage Divider Circuits, All About Circuits textbook](https://www.allaboutcircuits.com/textbook/direct-current/chpt-6/voltage-divider-circuits/)
2. [Design of a 400 kV resistive voltage divider with stray capacitance compensation, Review of Scientific Instruments](https://pubs.aip.org/aip/rsi/article/97/9/094709/3404792/Design-of-a-400-kV-resistive-voltage-divider-with)
3. [Kelvin-Varley Voltage Divider, IET Labs KVD-500 application note](https://www.ietlabs.com/pdf/Datasheets/KVD-500.pdf)
4. [MIT 6.200 Notes: Thévenin-Helmholtz and Mayer-Norton Theorems](https://circuits.mit.edu/_static/S23/handouts/lec03b/lecture03b.pdf)
5. [Thévenin's and Norton's theorems applied to loaded voltage dividers, ModEL project (Tony Kuphaldt)](https://ibiblio.org/kuphaldt/socratic/model/mod_thevenin_norton.pdf)
6. [A New Approach to the Design of High Precision Integrated Resistive Voltage Dividers, TT Electronics, PCNS 2025](https://passive-components.eu/wp-content/uploads/2025/09/3_3_TT-Electronics-PCNS-2025-Paper.pdf)
7. [Design of the standard high-voltage divider, Hioki](http://www.hioki.com/euro-en/download/39431)
8. [An Improved System for the Measurement of High-Voltage Pulses, OSTI](https://www.osti.gov/servlets/purl/7189210)
9. [Design of MV-HV Dividers, INRIM repository](https://iris.inrim.it/retrieve/handle/11696/52432/dd2573c2-cee2-e71c-e053-d805fe0ad5dc/TPWRD-01071-2015%20-%20%20Design%20of%20MV-HV%20Dividers%20_%20rev_final.pdf)
10. [Theory of voltage dividers and their use with cathode ray oscillographs, NBS Journal of Research](https://nvlpubs.nist.gov/nistpubs/jres/9/jresv9n1p81_A2b.pdf)
11. [Method to Evaluate the Resistance–Capacitance Voltage Divider and Uncertainty Analysis, Energies](https://www.mdpi.com/1996-1073/14/22/7744)
12. [Development of Broadband Resistive–Capacitive Parallel-Connection Voltage Divider, Energies](https://www.mdpi.com/1996-1073/15/2/451)
13. [RES60A-Q1 Automotive, 1400 VDC, Precision Resistive Divider datasheet, Texas Instruments](https://www.ti.com/lit/ds/symlink/res60a-q1.pdf)
14. [Development of a procedure for precision voltage dividers, OSTI](https://www.osti.gov/servlets/purl/6861036)
15. [When would I use a voltage regulator vs a voltage divider?, Electronics Stack Exchange](https://electronics.stackexchange.com/questions/106718/when-would-i-use-a-voltage-regulator-vs-voltage-divider)
16. [TI's Integrated 1.4-kV Precision Resistor Dividers Extend HV Battery Life, Electronic Design](https://www.electronicdesign.com/technologies/components/passives/article/55293573/electronic-design-tis-integrated-14-kv-precision-resistor-dividers-extend-hv-battery-life)
17. [Optimizing Resistor Dividers at a Comparator (Rev. B), Texas Instruments application note](https://www.ti.com/lit/an/slva450b/slva450b.pdf)
18. [Splitting voltage with purpose: A guide to precision voltage dividers, EDN](https://www.edn.com/splitting-voltage-with-purpose-a-guide-to-precision-voltage-dividers/)
19. [Resistor Divider Spreadsheet Provides Standard Resistor Values, Allegro MicroSystems application note](https://www.allegromicro.com/-/media/files/application-notes/an296255-resistive-voltage-dividers.pdf?sc_lang=en)
20. [Type HVD Ultra-Precision Voltage Dividers datasheet](https://datasheet.lcsc.com/datasheet/pdf/2d3b03ffd0e7404853d7089f3d969122.pdf?productCode=C6274067)
21. [Design and Characterisation of 1 kV Multirange Resistive Voltage Divider, IMEKO TC4 2022](https://www.imeko.org/publications/tc4-2022/IMEKO-TC4-2022-11.pdf)
22. [10 kV DC resistive divider calibration, NISTIR 5566](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir5566.pdf)
23. [Calibration Method of a Wideband AC Resistance Voltage Divider Based on an Equivalent Model](https://pmc.ncbi.nlm.nih.gov/articles/PMC10458715/)
24. [The Design and Characterization of a Prototype Wideband Voltage Sensor Based on a Resistive Divider, Sensors](https://pmc.ncbi.nlm.nih.gov/articles/PMC5712857/)
25. [Origins of the equivalent circuit concept: the voltage-source equivalent, Proceedings of the IEEE](https://jontallen.ece.illinois.edu/uploads/537.F18/Papers/DonJohnson1-03.pdf)
26. [A History of Impedance Measurements, GenRad/IET Labs](https://www.ietlabs.com/pdf/GenRad_History/A_History_of_Z_Measurement.pdf)
27. [IEC 60060-2:2025 redline preview](https://cdn.standards.iteh.ai/samples/iec/iec-60060-2-2025/4c4d040691194886b57627976ac2af62/redline-iec-60060-2-2025-rlv.pdf)
28. [Ultra-stable 3D-printed precision voltage divider for calibrations and experiments, arXiv](https://arxiv.science/abs/2407.06700)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electromagnetic quantities and history › Electromagnetic quantities › Electric potential and voltage quantities*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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