RC Circuits

An RC circuit pairs a resistor and capacitor to create frequency-dependent behaviour. The combination shows up everywhere — filters, timing circuits, oscillators, coupling networks, debounce circuits. Understanding how RC circuits work is understanding a large chunk of practical analog electronics.

The Time Constant

τ = R × C    (in seconds, when R is ohms and C is farads)

τ (tau) is how long it takes the capacitor to charge to 63% of the supply voltage, or discharge to 37% of its initial voltage. After 5τ it's considered fully charged or discharged.

Example: R = 10kΩ, C = 100µF
τ = 10,000 × 0.0001 = 1 second
Fully charged in ~5 seconds

Charging and Discharging

Charging:    V(t) = V₀ × (1 - e^(-t/τ))
Discharging: V(t) = V₀ × e^(-t/τ)

The exponential shape is characteristic — fast at first, progressively slower as the capacitor approaches the final value. This makes RC circuits good for timing (the time to cross a threshold is predictable) but poor for constant-rate tasks.

As a Filter

The RC combination has a cutoff frequency where the output is 3dB below the input. Below the cutoff, signal passes. Above it, signal is attenuated:

f_c = 1 / (2π × R × C)

Low-Pass Filter

         R
Vin ──[   ]──┬── Vout
             |
            [C]
             |
            GND

Low frequencies see the capacitor as high impedance and pass through. High frequencies see it as low impedance — the signal takes the short path to ground instead. Result: low frequencies pass, high frequencies are blocked.

High-Pass Filter

        C
Vin ──||──┬── Vout
          |
         [R]
          |
         GND

Swap R and C. Now high frequencies pass (capacitor low impedance), low frequencies and DC are blocked (capacitor high impedance). Used for AC coupling — pass the audio signal, block the DC bias.

Cutoff Frequency Examples

Audio low-pass at 20kHz: R=1kΩ, C=8nF (use 8.2nF standard value)
f_c = 1 / (2π × 1000 × 0.000000008) ≈ 20kHz

Debounce filter at ~1kHz: R=1kΩ, C=100nF
f_c = 1 / (2π × 1000 × 0.0000001) ≈ 1.6kHz
(attenuates high-frequency contact bounce, passes slow button presses)

Timing Applications

The 555 timer in astable mode uses two resistors and a capacitor to set the frequency. The RC time constant determines how long it takes the capacitor to charge from 1/3 to 2/3 of the supply (high time) and discharge back (low time). This is one of the most common uses of RC circuits in hobby electronics.

Microcontroller debouncing also relies on RC: put a small RC filter on a button input and the mechanical bounce (millisecond-scale noise) is filtered before the signal reaches the digital input pin.

Limitations of Passive RC Filters

A single-stage RC filter has a -20dB/decade rolloff — fairly gentle. For steeper filtering (sharper transitions between passband and stopband) you need multiple stages or active filters using op-amps. The passband also loads the previous stage — a low-pass with R=10kΩ has 10kΩ output impedance, which affects the circuit driving it. Active filters using op-amps solve both problems.