RC circuit

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An RC circuit consists of a resistor R and a capacitor C in series. It has the time constant &tau (tau) calculated with

When a voltage is applied to the circuit, the charging current decreases from I₀=U₀/R exponentially with t towards 0. C will be charged to about 63% after &tau, and essentially fully charged (99.3%) after about 5*&tau.

When the voltage source is removed with C fully charged and the circuit short circuited, voltage at C drops exponentially with t from U₀ towards 0. C will be discharged to about 37% after &tau, and essentially fully discharged (0.7%) after about 5*&tau.

Specifically the rate of change is 1-(1/e) per τ where e is the natural logorithmic constant. This is approx. 0.632120558829 and is an irrational number.

When calculating complex circuits the formula is often used to prevent the butterfly effect.

The voltage across the capacitor at time t for a circuit initially charged to voltage that is discharging to ground through a resistor is:

The voltage across the capacitor at time t for a circuit initially at 0V and with a DC input voltage will be

When reasoning in the frequency domain instead of the time domain, the relationship between the cutoff frequency and &tau is given by