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τ = R × C
Time constant
fc = 1 / (2πRC)
Cutoff frequency
V(t) = V₀(1 - e-t/τ)
Charging
V(t) = V₀ × e-t/τ
Discharging
| Time | Charging % | Discharging % |
|---|---|---|
| 1τ | 63.2% | 36.8% |
| 2τ | 86.5% | 13.5% |
| 3τ | 95% | 5% |
| 4τ | 98.2% | 1.8% |
| 5τ | 99.3% | 0.7% |
An RC circuit (resistor-capacitor circuit) is one of the most fundamental analog circuits in electronics, consisting of a resistor and capacitor connected in series or parallel. The defining characteristic of an RC circuit is its time constant, expressed as τ = R × C, which determines how quickly the capacitor charges or discharges through the resistor. Measured in seconds, the time constant represents the time required for the voltage across the capacitor to reach approximately 63.2% of its final value during charging, or to decay to 36.8% during discharging. RC circuits serve as the building blocks for filters, timing circuits, oscillators, and signal coupling/decoupling stages found in virtually every electronic device.
Determine the resistance in ohms (Ω). If multiple resistors are present, calculate the equivalent resistance seen by the capacitor. For series resistors, add them directly; for parallel, use the reciprocal formula.
Find the capacitance in farads (F). Common units include microfarads (µF = 10⁻⁶ F), nanofarads (nF = 10⁻⁹ F), and picofarads (pF = 10⁻¹² F). Convert to farads before calculating.
Calculate τ = R × C. For example, a 10 kΩ resistor with a 100 µF capacitor gives τ = 10,000 × 0.0001 = 1 second. The result is always in seconds.
After 1τ the capacitor is 63.2% charged; after 5τ it is 99.3% charged and considered fully charged. For cutoff frequency, use fc = 1 / (2πτ). This determines the -3 dB point for filter design.
RC circuits form the basis of low-pass and high-pass filters used in audio processing, power supply smoothing, and signal conditioning. Knowing the time constant lets you precisely set the cutoff frequency to pass desired signals while attenuating noise and unwanted frequencies.
From debouncing switches to creating delays in microcontroller circuits, RC timing is essential. Timer ICs like the 555 use RC networks internally to generate precise time delays and oscillation frequencies, making accurate time constant calculation critical for reliable circuit behavior.
Every digital IC requires bypass capacitors that form RC circuits with trace resistance. Proper selection ensures transient currents are supplied locally rather than through long power traces, preventing voltage droops and high-frequency noise that cause logic errors and EMI issues.
| R Value | C Value | Time Constant (τ) | Cutoff Freq (fc) | Typical Application |
|---|---|---|---|---|
| 1 kΩ | 1 µF | 1 ms | 159 Hz | Audio low-pass filter |
| 10 kΩ | 100 nF | 1 ms | 159 Hz | Signal conditioning |
| 10 kΩ | 10 µF | 100 ms | 1.59 Hz | Switch debouncing |
| 1 MΩ | 1 µF | 1 s | 0.16 Hz | Long delay timer |
| 100 Ω | 100 nF | 10 µs | 15.9 kHz | Power supply decoupling |
| 4.7 kΩ | 470 pF | 2.2 µs | 72 kHz | RF filtering |
| 47 kΩ | 22 µF | 1.03 s | 0.15 Hz | 555 timer delay |
| 100 kΩ | 10 nF | 1 ms | 159 Hz | Sensor signal smoothing |
After 5τ, the capacitor has reached 99.3% of its final voltage and is considered fully charged (or discharged) for practical purposes. Engineers use the 5τ rule as the standard settling time for RC circuits in both analog and digital applications.
The time constant directly determines the cutoff frequency: fc = 1/(2πτ). A larger time constant means a lower cutoff frequency, passing only slow-changing signals. A smaller time constant yields a higher cutoff, allowing faster signals through while blocking low-frequency drift.
Yes, RC circuits are commonly used for timing. By monitoring when the capacitor voltage crosses a threshold (often 63.2% or 50% of supply voltage), you can create predictable time delays. The 555 timer IC uses this principle with internal comparators set at 1/3 and 2/3 of VCC.
In a series RC circuit, the same current flows through both components, and it acts as a voltage divider that varies with frequency -- forming a low-pass or high-pass filter. In a parallel RC circuit, both components share the same voltage, and the circuit acts as a current divider. Series is more common in filter and timing applications.
The time constant accuracy depends on both resistor and capacitor tolerances. A 5% resistor with a 20% capacitor could produce a time constant varying by up to 25%. For precision timing, use 1% resistors and C0G/NP0 capacitors (which have tight tolerances and minimal drift with temperature).
Calculate total impedance of series and parallel RLC circuits including phase angle and resonant frequency.
Compute capacitive and inductive reactance at any frequency for filter design and impedance matching.
Find equivalent capacitance for series and parallel capacitor combinations and stored energy calculations.