A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.
RC circuits can be used to filter a signal by blocking certain frequencies and passing others. The two most common RC filters are the high-pass filters and low-pass filters; band-pass filters and band-stop filters usually require RLC filters, though crude ones can be made with RC filters.
Introduction
There are three basic, linear passive lumped analog circuit components: the resistor (R), the capacitor (C), and the inductor (L). These may be combined in the RC circuit, the RL circuit, the LC circuit, and the RLC circuit, with the acronyms indicating which components are used. These circuits, among them, exhibit a large number of important types of behaviour that are fundamental to much of analog electronics. In particular, they are able to act as passive filters. This article considers the RC circuit, in both series and parallel forms, as shown in the diagrams below.
Natural response
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. Once the circuit is closed, the capacitor begins to discharge its stored energy through the resistor. The voltage across the capacitor, which is time-dependent, can be found by using Kirchhoff's current law. 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. This results in the linear differential equation
where is the capacitance of the capacitor.
Solving this equation for yields the formula for exponential decay:
where is the capacitor voltage at time .
The time required for the voltage to fall to is called the RC time constant and is given by,
In this formula, is measured in seconds, in ohms and in farads.
Complex impedance
The complex impedance, (in ohms) of a capacitor with capacitance (in farads) is
The complex frequency is, in general, a complex number,
where
represents the imaginary unit: , is the exponential decay constant (in nepers per second), and is the sinusoidal angular frequency (in radians per second).
Sinusoidal steady state
Sinusoidal steady state is a special case in which the input voltage consists of a pure sinusoid (with no exponential decay). As a result, and the impedance becomes
Series circuit
By viewing the circuit as a voltage divider, the voltage across the capacitor is:
and the voltage across the resistor is:
Transfer functions
The transfer function from the input voltage to the voltage across the capacitor is
Similarly, the transfer function from the input to the voltage across the resistor is
Poles and zeros
Both transfer functions have a single pole located at
In addition, the transfer function for the voltage across the resistor has a zero located at the origin.
Gain and phase
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:
Current
The current in the circuit is the same everywhere since the circuit is in series:
Impulse response
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 is the Heaviside step function and is the time constant.
Similarly, the impulse response for the resistor voltage is
where is the Dirac delta function
Frequency-domain considerations
These are frequency domain expressions. Analysis of them will show which frequencies the circuits (or filters) pass and reject. This analysis rests on a consideration of what happens to these gains as the frequency becomes very large and very small.
As :
As :