RC circuit

The simplest RC circuit consists of a resistor and a charged capacitor connected to one another in a single loop, without an external voltage source.

The current through the resistor must be equal in magnitude (but opposite in sign) to the time derivative of the accumulated charge on the capacitor.

Solving this equation for V yields the formula for exponential decay: where V0 is the capacitor voltage at time t = 0.

The magnitude of the gains across the two components are and and the phase angles are and These expressions together may be substituted into the usual expression for the phasor representing the output: The current in the circuit is the same everywhere since the circuit is in series: The impulse response for each voltage is the inverse Laplace transform of the corresponding transfer function.

It represents the response of the circuit to an input voltage consisting of an impulse or Dirac delta function.

The impulse response for the capacitor voltage is where u(t) is the Heaviside step function and τ = RC is the time constant.

Similarly, the impulse response for the resistor voltage is where δ(t) is the Dirac delta function These are frequency domain expressions.

The point at which the filter attenuates the signal to half its unfiltered power is termed its cutoff frequency.

This requires that the gain of the circuit be reduced to Solving the above equation yields which is the frequency that the filter will attenuate to half its original power.

The most straightforward way to derive the time domain behaviour is to use the Laplace transforms of the expressions for VC and VR given above.

This effectively transforms jω → s. Assuming a step input (i.e. Vin = 0 before t = 0 and then Vin = V afterwards): Partial fractions expansions and the inverse Laplace transform yield: These equations are for calculating the voltage across the capacitor and resistor respectively while the capacitor is charging; for discharging, the equations are vice versa.

These equations can be rewritten in terms of charge and current using the relationships C = ⁠Q/V⁠ and V = IR (see Ohm's law).

Further, the critical frequency nearest the origin must be a pole, assuming the rational function represents an impedance rather than an admittance.