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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

${\displaystyle C{\frac {dV}{dt}}+{\frac {V}{R}}=0\,,}$

where C is the capacitance of the capacitor.

Solving this equation for V yields the formula for exponential decay:

${\displaystyle V(t)=V_{0}e^{-{\frac {t}{RC}}}\,,}$

where V₀ is the capacitor voltage at time t = 0.

The time required for the voltage to fall to V₀/e is called the RC time constant and is given by,^[1]

${\displaystyle \tau =RC\,.}$

In this formula, τ is measured in seconds, R in ohms and C in farads.

Complex impedance

The complex impedance, Z_C (in ohms) of a capacitor with capacitance C (in farads) is

${\displaystyle Z_{C}={\frac {1}{sC}}}$

The complex frequency s is, in general, a complex number,

${\displaystyle s=\sigma +j\omega \,,}$

where

• j represents the imaginary unit: j² = −1,
• σ is the exponential decay constant (in nepers per second), and
• ω 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, ${\displaystyle \sigma =0}$ and the impedance becomes

${\displaystyle Z_{C}={\frac {1}{j\omega C}}=-{\frac {j}{\omega C}}\,.}$

Series circuit

By viewing the circuit as a voltage divider, the voltage across the capacitor is:

${\displaystyle V_{C}(s)={\frac {\frac {1}{Cs}}{R+{\frac {1}{Cs}}}}V_{\mathrm {in} }(s)={\frac {1}{1+RCs}}V_{\mathrm {in} }(s)}$

and the voltage across the resistor is:

${\displaystyle V_{R}(s)={\frac {R}{R+{\frac {1}{Cs}}}}V_{\mathrm {in} }(s)={\frac {RCs}{1+RCs}}V_{\mathrm {in} }(s)\,.}$

Transfer functions

The transfer function from the input voltage to the voltage across the capacitor is

${\displaystyle H_{C}(s)={\frac {V_{C}(s)}{V_{\mathrm {in} }(s)}}={\frac {1}{1+RCs}}\,.}$

Similarly, the transfer function from the input to the voltage across the resistor is

${\displaystyle H_{R}(s)={\frac {V_{R}(s)}{V_{\rm {in}}(s)}}={\frac {RCs}{1+RCs}}\,.}$

Poles and zeros

Both transfer functions have a single pole located at

${\displaystyle s=-{\frac {1}{RC}}\,.}$

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

${\displaystyle G_{C}={\big |}H_{C}(j\omega ){\big |}=\left|{\frac {V_{C}(j\omega )}{V_{\mathrm {in} }(j\omega )}}\right|={\frac {1}{\sqrt {1+\left(\omega RC\right)^{2}}}}}$

and

${\displaystyle G_R=|H_R(j\mathrm{\omega <!--\; \omega \; -->})}$Zdroj:https://en.wikipedia.org?pojem=RC_circuit
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