# RC circuit

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Series RC circuit

A RC circuit, or RC filter or RC network, is the simplest analog infinite impulse response electronic filter. It consists of a resistor and a capacitor, either in series or in parallel.

## Series RC circuit

A series RC circuit has a time constant, usually denoted [itex]\tau[itex], the time it takes the current in the circuit to decrease to [itex]1/e[itex] (about 37%) of its initial value, calculated with

[itex]\tau = RC[itex]

When a voltage is applied to the circuit, the charging current decreases from [itex]I_0[itex] = [itex] \begin{matrix} \frac{V_0}{R} \end{matrix} [itex] exponentially with t towards 0. C will be charged to about 63% after [itex]\tau[itex], and essentially fully charged (99.3%) after about [itex]5\tau[itex].

When the voltage source is replaced with a short-circuit, with C fully charged, the voltage at C drops exponentially with t from [itex]V_0[itex] towards 0. C will be discharged to about 37% after [itex]\tau[itex], and essentially fully discharged (0.7%) after about [itex]5\tau[itex].

Specifically the rate of change is 1 − [itex] \begin{matrix} \frac{1}{e} \end{matrix} [itex] per [itex]\tau[itex]; where e is the natural logarithmic 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 [itex]V_0[itex] that is discharging to ground through a resistor is:

[itex] V_0 \over e^{t \over RC}[itex]

The voltage across the capacitor at time t for a circuit initially at 0V and with a DC input voltage [itex]V_{in}[itex] will be

[itex]V_{in} - \frac{V_{in}}{e^{t \over RC}}[itex]

## RC filter

An electrical network that is constructed using a resistor and capacitor in series acts as an effective high-pass filter. When reasoning in the frequency domain instead of the time domain, the relationship between the cutoff frequency and RC is given by

[itex]f_c = {1 \over {2\pi RC}}[itex]

where

• C is the capacitor value, and
• R is the source impedance.

One application of the RC circuit is the pacemaker.

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