# RC circuit
An RC circuit is an electrical circuit built from a resistor (R) and a capacitor (C), usually driven by a voltage source. In a simple first-order RC circuit, the step response is exponential and is governed by the time constant τ = RC. The same two components serve as timers, filters, snubbers, and debouncers across analog and digital design.
| Key fact | Detail |
|---|---|
| Time constant | τ = R × C, in seconds (R in ohms, C in farads); at t = τ a charging capacitor reaches 1 − e⁻¹ ≈ 63.2% of its final voltage[1](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits) • [2](https://cdn.pasco.com/product_document/Resistor-Capacitor-Circuit-Manual-SE-9791.pdf) |
| Settling rule | After 3τ the response is about 95% complete and after 5τ more than 99%, so five time constants is the customary "fully settled" point[3](https://web.stanford.edu/class/archive/engr/engr40m.1178/slides/transient.pdf) • [4](http://kakukoto.free.fr/F/023/fundamentals-of-electrical-circuits/ch07/ch07.pdf) |
| Cutoff frequency | A first-order RC low-pass or high-pass filter has its −3.01 dB cutoff at fc = 1/(2πRC), where output is 70.7% of input and the phase-shift magnitude is 45°[5](https://www.electronics-tutorials.ws/filter/filter_2.html) • [6](https://electricalflux.com/learn-components/lowpass-rc-filter-cutoff-frequency-pwm-smoothing) |
| Roll-off | Attenuation above (or below) cutoff is −20 dB/decade (−6 dB/octave) for any single RC stage[5](https://www.electronics-tutorials.ws/filter/filter_2.html) |
| Real-capacitor limits | ESR typically 0.005–5 Ω and leakage resistance 10 MΩ–10 GΩ; leakage shortens RC timing intervals unpredictably[7](https://electricalflux.com/learn-components/resistance-of-capacitor-esr-leakage-guide) |
| Worked example | 1 kΩ with 1000 µF gives τ = 1 s[8](https://www.allaboutcircuits.com/textbook/experiments/chpt-3/capacitor-charging-and-discharging/) |
| Modern use | RC snubber design for cascode GaN HEMTs has been studied with a damping-ratio criterion of ζ ≥ 0.5[9](https://vbn.aau.dk/ws/portalfiles/portal/721051256/IET_final.pdf) |
## Time-domain behavior: charging, discharging, and the time constant
When a capacitor C charges from a source of emf ε through a resistor R, Kirchhoff's loop rule gives ε − R(dq/dt) − q/C = 0. Integrating yields the charge on the capacitor as a function of time:
\[ q(t) = C\varepsilon\left(1 - e^{-t/RC}\right) = Q\left(1 - e^{-t/\tau}\right) \]
where Q = Cε is the final charge. The curve is exponential because the resistor drops a voltage proportional to the current, and the current is proportional to how far the capacitor still has to charge; each increment of charge reduces the driving voltage, so the rate of change slows in proportion to what remains. The capacitor voltage follows the same shape, V = emf(1 − e^(−t/RC)), with e ≈ 2.718 the base of the natural logarithm[1](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits) • [10](https://openstax.org/books/college-physics-ap-courses/pages/21-6-dc-circuits-containing-resistors-and-capacitors).
**The time constant.** The product τ = RC has units of seconds (R in ohms, C in farads, not microfarads) and fixes the whole curve[1](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits) • [2](https://cdn.pasco.com/product_document/Resistor-Capacitor-Circuit-Manual-SE-9791.pdf). At t = τ the charge has reached 1 − e⁻¹ = 1 − 0.368 = 0.632 of its maximum[1](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits). Equivalently, τ is the time for a discharging response to decay to 1/e, or 36.8%, of its initial value[4](http://kakukoto.free.fr/F/023/fundamentals-of-electrical-circuits/ch07/ch07.pdf). A smaller resistance charges the capacitor faster because a larger current flows, and a smaller capacitance needs less charging time; both effects are captured in the product[10](https://openstax.org/books/college-physics-ap-courses/pages/21-6-dc-circuits-containing-resistors-and-capacitors).
**Rules of thumb.** After one time constant the circuit has moved 1 − e⁻¹ ≈ 63% of the way to its new steady state, after 3τ about 95%, and after 5τ more than 99%[3](https://web.stanford.edu/class/archive/engr/engr40m.1178/slides/transient.pdf). Textbooks accordingly treat the capacitor as fully charged or discharged after five time constants, by which point the remaining voltage is under 1% of the initial value[4](http://kakukoto.free.fr/F/023/fundamentals-of-electrical-circuits/ch07/ch07.pdf). For a charging circuit driven by Vs, vC(τ) = 0.63Vs; for a discharge from V0, vC(τ) = 0.37V0[11](https://www.cs.cmu.edu/~tdear/ee/rc_rl.pdf).
Every first-order step response, charging or discharging, fits one formula:
\[ x(t) = x(\infty) + \left[x(0) - x(\infty)\right]e^{-t/\tau} \]
so knowing the initial value, the final value, and τ is enough to solve the circuit[4](http://kakukoto.free.fr/F/023/fundamentals-of-electrical-circuits/ch07/ch07.pdf).
## Frequency-domain behavior: impedance, phase, and RC filters
The same series RC arrangement acts as a filter depending on where the output is taken. Output across the capacitor gives a low-pass filter, and the response falls at −20 dB/decade (−6 dB/octave), a slope identical for any RC combination; cascading n stages gives n × −20 dB/decade[5](https://www.electronics-tutorials.ws/filter/filter_2.html).
**The −3 dB point.** The cutoff frequency is where the capacitive reactance equals the resistance, R = Xc, which gives
\[ f_c = \frac{1}{2\pi RC} \]
with R in ohms and C in farads. At this frequency the output is 0.707 of the input, an attenuation of exactly −3.01 dB, not half the voltage, because the input voltage is the phasor sum of the resistive and capacitive voltage drops[5](https://www.electronics-tutorials.ws/filter/filter_2.html) • [6](https://electricalflux.com/learn-components/lowpass-rc-filter-cutoff-frequency-pwm-smoothing). The output phase lags the input by −45° at cutoff[5](https://www.electronics-tutorials.ws/filter/filter_2.html).
**Cascading and loading.** Connecting filter stages directly makes each stage load the previous one and shifts the actual cutoff. The standard passive fix is to make each following stage's impedance 10× the previous one, R2 = 10 × R1 and C2 = 1/10 C1, keeping the same cutoff frequency[5](https://www.electronics-tutorials.ws/filter/filter_2.html).
## Real components and non-ideal behavior
A real capacitor is not the pure C of the textbook model. It has an equivalent series resistance (ESR) in series with the ideal capacitance, typically 0.005 Ω to 5 Ω, responsible for [AC power](https://www.edgepedia.org/ac-power) loss and heating, and an insulation (leakage) resistance in parallel, typically 10 MΩ to 10 GΩ[7](https://electricalflux.com/learn-components/resistance-of-capacitor-esr-leakage-guide). Typical ESR by dielectric: aluminum electrolytic 0.05–2.0 Ω (±20% tolerance), solid tantalum 0.05–0.5 Ω, X7R/X5R MLCCs 0.005–0.05 Ω, C0G/NP0 MLCCs below 0.01 Ω, and polypropylene film 0.005–0.05 Ω[7](https://electricalflux.com/learn-components/resistance-of-capacitor-esr-leakage-guide).
**Leakage ruins timing.** In RC timing networks, a high-leakage capacitor shortens the timing interval unpredictably because the capacitor cannot hold its charge against the leakage path; timing applications therefore need low-leakage dielectrics[7](https://electricalflux.com/learn-components/resistance-of-capacitor-esr-leakage-guide).
**ESR and heat.** Heat generation in a capacitor follows P = I² × ESR, so lower ESR reduces heating, improves charge/discharge current capability, and lets ripple-filter capacitors work at higher AC noise frequencies in switch-mode power supply output filters[12](https://www.vishay.com/docs/40256/lowesrtantcapsincircdes.pdf).
**DC bias in MLCCs.** Class II ceramic capacitors lose capacitance with temperature, applied voltage, and DC-bias aging; a rule of thumb is 15–20% capacitance loss for X7R at rated voltage and 75–80% for Y5V, regardless of rated voltage[13](https://www.kyocera-avx.com/docs/techinfo/CeramicCapacitors/mlc-tant.pdf) • [14](https://passive-components.eu/snubber-capacitors-functionality-and-selection-guide/). C0G dielectrics hold their value but are available only in small capacitances at significantly higher cost[14](https://passive-components.eu/snubber-capacitors-functionality-and-selection-guide/).
## Applications in practice
**Timers.** RC circuits serve as timers in intermittent windshield wipers, pacemakers, and strobe lights, and the 555 timer's pulse interval is set by an RC network; increasing the resistance lengthens the interval between wiper sweeps[1](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits). RC and RL circuits also appear in filtering in DC power supplies and smoothing circuits in digital communications[4](http://kakukoto.free.fr/F/023/fundamentals-of-electrical-circuits/ch07/ch07.pdf).
**Debouncing.** A simple RC filter is one of the cheapest and simplest switch-debounce circuits: when the switch is open the capacitor charges through R1 + R2, and when closed it discharges through R2 at a controlled rate, with a charging time constant τ_charge = (R1 + R2)·C1 and a discharging time constant τ_discharge = R2·C1[15](https://www.we-online.com/en/components/media/o185480v410%20SN015_Contact%20debounce%20circuit%20for%20switches.pdf). The mechanism is a low-pass filter: fast contact spikes do not have time to charge or discharge the capacitor through the resistor, while a deliberate press follows the exponential curve[16](https://docs.openmv.io/dev/openmvcam/tutorial/hardware/gpio-input/debouncing.html). Worked values: for a 10 ms bounce specification with R1 = 1 kΩ on a 5 V supply, Würth proposes R2 = 10 kΩ with C1 = 1 µF, or R2 = 47 kΩ with C1 = 220 nF; the output reaches 63% (3.15 V) of its final 5.0 V after approximately one time constant (about 11 ms for the 1 µF example)[15](https://www.we-online.com/en/components/media/o185480v410%20SN015_Contact%20debounce%20circuit%20for%20switches.pdf). A common microcontroller pairing is R1 = 10 kΩ pull-up, R2 = 10 kΩ series, C = 100 nF, giving τ_charge = 2 ms and τ_discharge = 1 ms[16](https://docs.openmv.io/dev/openmvcam/tutorial/hardware/gpio-input/debouncing.html). Adding a diode across R2 lets charge and discharge times be controlled separately[15](https://www.we-online.com/en/components/media/o185480v410%20SN015_Contact%20debounce%20circuit%20for%20switches.pdf).
**Touch sensing.** [STMicroelectronics](https://www.edgepedia.org/stmicroelectronics)' RC acquisition method charges a sense electrode's capacitance, set by electrode area, the insulator's relative dielectric constant, the permittivity of air, and electrode spacing, through a fixed resistor; a finger changes C and therefore the measured charge time[17](https://www.manuallib.com/download/ST-AN2927-APPLICATION-NOTE.PDF).
## RC snubbers and wide-bandgap power electronics
An RC snubber placed across a power switch damps the voltage ringing caused by parasitic inductance and device capacitance. The capacitor blocks DC but looks like a short circuit during the fast switching edge, letting the resistor shunt current out of the switch node; the tradeoff is extra turn-on current the transistor must carry[14](https://passive-components.eu/snubber-capacitors-functionality-and-selection-guide/).
**Design has become quantitative.** Recent work on cascode GaN HEMTs uses a damping-ratio criterion: switching oscillation is satisfactorily damped when the snubber design achieves ζ > 0.4, and with a 0.1 safety margin a criterion of ζ ≥ 0.5 is applied[9](https://vbn.aau.dk/ws/portalfiles/portal/721051256/IET_final.pdf). The snubber capacitor is the key design element, chosen as a tradeoff between maximum damping and snubber losses, and its choice predetermines the optimal resistor value[18](https://publications.rwth-aachen.de/record/774287/files/774287.pdf). During the switching transient the snubber capacitor slows the switching speed and adds power losses, so it should be as small as possible; achieving ζ = 1 would require a large capacitor with high losses, so designs trade minimum capacitance with effective damping against maximum damping at higher loss[9](https://vbn.aau.dk/ws/portalfiles/portal/721051256/IET_final.pdf).
**Why it matters now.** Without a snubber, a fast SiC cascode JFET such as the UF3C120040K3S must be substantially slowed down to prevent VDS spikes and ringing at turn-off; with a snubber it can switch fast without that penalty[19](https://www.onsemi.com/download/application-notes/pdf/and90319-d.pdf). Integration is following: the Melexis MLX91299 puts a series resistor and capacitor on a 1.5 × 5 mm silicon die for SiC power modules, rated 1,200 V with 0.2 to 5.8 nF capacitance and 1 to 35 Ω internal resistance; the capacitor charges at turn-off, absorbing energy that would otherwise cause ringing, and the resistor dissipates it as heat[20](https://www.electronicdesign.com/technologies/power/product/55379074/electronic-design-melexis-all-silicon-rc-snubber-removes-ringing-from-power-mosfets).
## Choosing R and C, by the numbers
Component values come from the IEC 60063 series of preferred values for resistance and capacitance, which underpins component marking and coding[21](https://webstore.iec.ch/en/publication/22011). The design arithmetic is direct:
- 1 kΩ with 1000 µF gives τ = 1 s[8](https://www.allaboutcircuits.com/textbook/experiments/chpt-3/capacitor-charging-and-discharging/).
- A 10 kΩ resistor with 100 nF gives τ = 1 ms, and (10 kΩ + 10 kΩ) with 100 nF gives 2 ms[16](https://docs.openmv.io/dev/openmvcam/tutorial/hardware/gpio-input/debouncing.html).
- Würth's 10 ms debounce solutions pair 11 kΩ total with 1 µF, or 48 kΩ with 220 nF[15](https://www.we-online.com/en/components/media/o185480v410%20SN015_Contact%20debounce%20circuit%20for%20switches.pdf).
The tradeoffs run in opposite directions. Higher capacitance in a snubber gives a greater safety margin but increases resistor losses, while a lower value can result in higher voltage peaks; estimated peak reverse voltage should typically not exceed 80% of the device's maximum voltage blocking[22](https://www.dynexsemi.com/Portals/0/PDF/DNX_AN6481.pdf). In timing networks, a high-leakage capacitor shortens the timing interval unpredictably[7](https://electricalflux.com/learn-components/resistance-of-capacitor-esr-leakage-guide).
## Where the ideal model breaks down
The textbook RC model assumes a constant C, a constant R, and no other paths. Three departures matter most in practice. First, capacitance itself moves: Class II MLCCs lose 15–20% (X7R) to 75–80% (Y5V) of their capacitance at rated DC bias, so a filter or timer built around them runs faster than the nominal calculation suggests[13](https://www.kyocera-avx.com/docs/techinfo/CeramicCapacitors/mlc-tant.pdf). Second, ESR adds real heating, which is why low-ESR parts are specified for SMPS output filters[12](https://www.vishay.com/docs/40256/lowesrtantcapsincircdes.pdf). Third, snubber design has no single closed-form answer: the damping criterion ζ ≥ 0.5 is a chosen safety margin over a lower ζ > 0.4 used elsewhere, and the capacitor value is explicitly a loss-versus-damping tradeoff rather than a computed optimum[9](https://vbn.aau.dk/ws/portalfiles/portal/721051256/IET_final.pdf).
## References
1. [10.5 RC Circuits, OpenStax University Physics Volume 2](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits)
2. [Resistor-Capacitor Circuit Manual SE-9791, PASCO](https://cdn.pasco.com/product_document/Resistor-Capacitor-Circuit-Manual-SE-9791.pdf)
3. [Transient response of RC and RL circuits, Stanford ENGR 40M](https://web.stanford.edu/class/archive/engr/engr40m.1178/slides/transient.pdf)
4. [Fundamentals of Electric Circuits (Alexander & Sadiku), Chapter 7: First-Order Circuits](http://kakukoto.free.fr/F/023/fundamentals-of-electrical-circuits/ch07/ch07.pdf)
5. [Passive Low Pass Filter Circuit – A Passive RC Filter Tutorial, Electronics Tutorials](https://www.electronics-tutorials.ws/filter/filter_2.html)
6. [Designing a Lowpass RC Filter: Cutoff Frequency Tables and PWM Smoothing, ElectricalFlux](https://electricalflux.com/learn-components/lowpass-rc-filter-cutoff-frequency-pwm-smoothing)
7. [Understanding the Resistance of a Capacitor: ESR, Leakage, and Part Selection, ElectricalFlux](https://electricalflux.com/learn-components/resistance-of-capacitor-esr-leakage-guide)
8. [DC Lab – Capacitor Charging and Discharging, Electronics Textbook (All About Circuits)](https://www.allaboutcircuits.com/textbook/experiments/chpt-3/capacitor-charging-and-discharging/)
9. [An RC snubber design method to achieve optimized switching noise-loss trade-off of cascode GaN HEMTs, IET (Aalborg University)](https://vbn.aau.dk/ws/portalfiles/portal/721051256/IET_final.pdf)
10. [21.6 DC Circuits Containing Resistors and Capacitors, OpenStax College Physics for AP Courses](https://openstax.org/books/college-physics-ap-courses/pages/21-6-dc-circuits-containing-resistors-and-capacitors)
11. [The Canonical Charging and Discharging RC Circuits, Carnegie Mellon University](https://www.cs.cmu.edu/~tdear/ee/rc_rl.pdf)
12. [Low-ESR Tantalum Capacitors in Circuit Design, Vishay application note 40256](https://www.vishay.com/docs/40256/lowesrtantcapsincircdes.pdf)
13. [Comparison of MLCC and Tantalum Technologies, Kyocera AVX](https://www.kyocera-avx.com/docs/techinfo/CeramicCapacitors/mlc-tant.pdf)
14. [Snubber Capacitors Functionality and Selection Guide, passive-components.eu](https://passive-components.eu/snubber-capacitors-functionality-and-selection-guide/)
15. [Contact debounce circuit for switches, Würth Elektronik application note SN015](https://www.we-online.com/en/components/media/o185480v410%20SN015_Contact%20debounce%20circuit%20for%20switches.pdf)
16. [Debouncing, OpenMV MicroPython documentation](https://docs.openmv.io/dev/openmvcam/tutorial/hardware/gpio-input/debouncing.html)
17. [RC acquisition principle for touch sensing applications, STMicroelectronics AN2927](https://www.manuallib.com/download/ST-AN2927-APPLICATION-NOTE.PDF)
18. [RC Snubber Design Procedure for Enhanced Oscillation Damping in Wide-Bandgap Switching Cells, RWTH Aachen](https://publications.rwth-aachen.de/record/774287/files/774287.pdf)
19. [AND90319 – Switching Fast SiC Cascode JFETs with a Snubber, onsemi](https://www.onsemi.com/download/application-notes/pdf/and90319-d.pdf)
20. [Melexis All-Silicon RC Snubber Removes Ringing from Power MOSFETs, Electronic Design](https://www.electronicdesign.com/technologies/power/product/55379074/electronic-design-melexis-all-silicon-rc-snubber-removes-ringing-from-power-mosfets)
21. [IEC 60063:2015, IEC Webstore](https://webstore.iec.ch/en/publication/22011)
22. [AN6481 – Snubber design application note, Dynex Semiconductor](https://www.dynexsemi.com/Portals/0/PDF/DNX_AN6481.pdf)
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*Topic: Encyclopedia › Technology and the built world › Engineering and manufacturing › Electrical and electronics engineering › Circuits and signal processing*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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