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Encyclopedia > RC circuit

resistor-capacitor circuit (RC circuit), or RC filter or RC network, is one of the simplest analogue electronic filters. It consists of a resistor and a capacitor, either in series or in parallel, driven by a voltage or current source. An analog filter handles analog stimuli (e. ... Television signal splitter consisting of a hi-pass and a low-pass filter. ... Resistor symbols (non-European) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... This article or section does not adequately cite its references or sources. ... This article or section does not adequately cite its references or sources. ... A voltage source is any device or system that produces an electromotive force between its terminals OR derives a secondary voltage from a primary source of the electromotive force. ... An ideal current source, I, driving a resistor, R, and creating a voltage V A current source is an electrical or electronic device that delivers or absorbs electric current. ...

# Introduction

This article relies on knowledge of the complex impedance representation of capacitors and on knowledge of the frequency domain representation of signals.

Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

The complex impedance ZC (in ohms) of a capacitor with capacitance C (in farads) is Electrical impedance or simply impedance is a measure of opposition to a sinusoidal electric current. ... The ohm (symbol: Î©) is the SI unit of electric resistance. ... The farad (symbol F) is the SI unit of capacitance (named after Michael Faraday). ...

$Z_C = frac{1}{sC}$

The angular frequency s is, in general, a complex number, It has been suggested that this article or section be merged into Angular velocity. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

$s = sigma + j omega$

where

j2 = − 1

Note: the identity j2 = − 1 is correctly written as such, and not as $j=sqrt{-1}$. The second expression is usually avoided, since the right-hand side has two possible values, one the negative of the other, and it is not desirable for the constant j to have two values at once. In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency Ï‰ (also called angular speed) is a scalar measure of rotation rate. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Sinusoidal steady state is a special case in which the input voltage consists of a pure sinusoid (with no exponential decay). As a result,

$sigma = 0$

and the evaluation of s becomes

$s = j omega$

## Series circuit

Series RC circuit

By viewing the circuit as a voltage divider, we see that the voltage across the capacitor is: Image File history File links Series-RC.svg Summary Series w:RC circuit. ... Image File history File links Series-RC.svg Summary Series w:RC circuit. ... This article or section does not adequately cite its references or sources. ... In electronics, a voltage divider or resistor divider or potential divider is a design technique used to create a voltage (Vout) which is proportional to another voltage (Vin). ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ...

$V_C(s) = frac{1/Cs}{R + 1/Cs}V_{in}(s) = frac{1}{1 + RCs}V_{in}(s)$

and the voltage across the resistor is:

$V_R(s) = frac{R}{R + 1/ Cs}V_{in}(s) = frac{ RCs}{1 + RCs}V_{in}(s)$.

### Transfer functions

The transfer function for the capacitor is A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...

$H_C(s) = { V_C(s) over V_{in}(s) } = { 1 over 1 + RCs } = G_C e^{j phi_C}$.

Similarly, the transfer function for the resistor is

$H_R(s) = { V_R(s) over V_{in}(s) } = { RCs over 1 + RCs } = G_R e^{j phi_R}$.

#### Poles and zeros

Both transfer functions have a single pole located at

$s = - {1 over RC }$ .

In addition, the transfer function for the resistor has a zero located at the origin. In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...

### Gain and phase angle

The gains across the two components are:

$G_C = | H_C(s) | = left|frac{V_C(s)}{V_{in}(s)}right| = frac{1}{sqrt{1 + left(omega RCright)^2}}$

and

$G_R = | H_R(s) | = left|frac{V_R(s)}{V_{in}(s)}right| = frac{omega RC}{sqrt{1 + left(omega RCright)^2}}$,

and the phase angles are:

$phi_C = angle H_C(s) = tan^{-1}left(-omega RCright)$

and

$phi_R = angle H_R(s) = tan^{-1}left(frac{1}{omega RC}right)$.

These expressions together may be substituted into the usual expression for the phasor representing the output: See wikibooks book on Phasors A phasor is a constant complex number representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. ...

$V_C = G_{C}V_{in} e^{jphi_C}$
$V_R = G_{R}V_{in} e^{jphi_R}$.

### Current

The current in the circuit is the same everywhere since the circuit is in series:

$I(s) = frac{V_{in}(s) }{R+1/ Cs} = { Cs over 1 + RCs } V_{in}(s)$

### 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 delta function. In the language of mathematics, the impulse response of a linear transformation is the image of Diracs delta function under the transformation. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function &#948;(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...

The impulse response for the capacitor voltage is

$h_C(t) = {1 over RC} e^{-t / RC} u(t) = { 1 over tau} e^{-t / tau} u(t)$

where u(t) is the Heaviside step function and The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

$tau = RC$

is the time constant. In physics and engineering, the time constant, usually denoted by the Greek letter Ï„ (tau), characterizes the frequency response of a first-order, linear time-invariant (LTI) system. ...

Similarly, the impulse response for the resistor voltage is

$h_R(t) = delta (t) - {1 over RC} e^{-t / RC} u(t) = delta (t) - { 1 over tau} e^{-t / tau} u(t)$

where δ(t) is the Dirac delta function The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ...

### 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. Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

As $omega to infty$:

$G_C to 0$
$G_R to 1$.

As $omega to 0$:

$G_C to 1$
$G_R to 0$.

This shows that, if the output is taken across the capacitor, high frequencies are attenuated (rejected) and low frequencies are passed. Thus, the circuit behaves as a low-pass filter. If, though, the output is taken across the resistor, high frequencies are passed and low frequencies are rejected. In this configuration, the circuit behaves as a high-pass filter. A low-pass filter is a filter that passes low frequencies but attenuates (or reduces) frequencies higher than the cutoff frequency. ... A high-pass filter is a filter that passes high frequencies well, but attenuates (or reduces) frequencies lower than the cutoff frequency. ...

The range of frequencies that the filter passes is called its bandwidth. 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 This article or section does not cite its references or sources. ... The Butterworth filters frequency response, with cutoff frequency labeled. ...

$G_C = G_R = frac{1}{sqrt{2}}$.

Solving the above equation yields

$omega_{c} = frac{1}{RC} mathrm{rad/s}$

or

$f_c = frac{1}{2pi RC} mathrm{Hz}$

which is the frequency that the filter will attenuate to half its original power.

Clearly, the phases also depend on frequency, although this effect is less interesting generally than the gain variations.

As $omega to 0$:

$phi_C to 0$
$phi_R to 90^{circ} = pi/2^{c}$.

As $omega to infty$:

$phi_C to -90^{circ} = -pi/2^{c}$
$phi_R to 0$

So at DC (0 Hz), the capacitor voltage is in phase with the signal voltage while the resistor voltage leads it by 90°. As frequency increases, the capacitor voltage comes to have a 90° lag relative to the signal and the resistor voltage comes to be in-phase with the signal. Direct current (DC or continuous current) is the continuous flow of electricity through a conductor such as a wire from high to low potential. ... The hertz (symbol: Hz) is the SI unit of frequency. ...

### Time domain considerations

This section relies on knowledge of e, the natural logarithmic constant.

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 $jomega to s$. Assuming a step input (i.e. Vin = 0 before t = 0 and then Vin = V afterwards): The title given to this article is incorrect due to technical limitations. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

$V_{in}(s) = Vfrac{1}{s}$
$V_C(s) = Vfrac{1}{1 + sRC}frac{1}{s}$

and

$V_R(s) = Vfrac{sRC}{1 + sRC}frac{1}{s}$.
Capacitor voltage step-response.
Resistor voltage step-response.

Partial fractions expansions and the inverse Laplace transform yield: Image File history File links Series_RC_capacitor_voltage. ... Image File history File links Series_RC_capacitor_voltage. ... Image File history File links Series_RC_resistor_voltage. ... Image File history File links Series_RC_resistor_voltage. ... In algebra, the partial fraction decomposition of a rational function expresses the function as a sum of fractions, in each term of which, the denominator is an irreducible (i. ... In mathematics, the Laplace transform is a technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ...

$,!V_C(t) = Vleft(1 - e^{-t/RC}right)$
$,!V_R(t) = Ve^{-t/RC}$.

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. The equations can be modified for current and charge due to Ohm's Law and C=q/V, respectively.[verification needed] Look up charge in Wiktionary, the free dictionary. ... Look up Discharge in Wiktionary, the free dictionary. ... Look up current in Wiktionary, the free dictionary. ... Look up charge in Wiktionary, the free dictionary. ... For the phase law, see Ohms Phase Law. ...

Thus, the voltage across the capacitor tends towards V as time passes, while the voltage across the resistor tends towards 0, as shown in the figures. This is in keeping with the intuitive point that the capacitor will be charging from the supply voltage as time passes, and will eventually be fully charged and form an open circuit. Open circuit can mean:- In electronics, where there is nothing connected to a load and no current can flow. ...

These equations show that a series RC circuit has a time constant, usually denoted τ = RC being the time it takes the voltage across the component to either rise (across C) or fall (across R) to within 1 / e of its final value. That is, τ is the time it takes VC to reach V(1 − 1 / e) and VR to reach V(1 / e). The RC time constant, usually denoted by the Greek letter &#964; (tau), is a parameter that characterizes the frequency response of a resistance-capacitance (RC) circuit. ...

The rate of change is a fractional $left(1 - frac{1}{e}right)$ per τ. Thus, in going from t = Nτ to t = (N + 1)τ, the voltage will have moved about 63.2 % of the way from its level at t = Nτ toward its final value. So C will be charged to about 63.2 % after τ, and essentially fully charged (99.3 %) after about . When the voltage source is replaced with a short-circuit, with C fully charged, the voltage across C drops exponentially with t from V towards 0. C will be discharged to about 36.8 % after τ, and essentially fully discharged (0.7 %) after about . Note that the current, I, in the circuit behaves as the voltage across R does, via Ohm's Law. For the phase law, see Ohms Phase Law. ...

These results may also be derived by solving the differential equations describing the circuit: A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

$frac{V_{in} - V_C}{R} = Cfrac{dV_C}{dt}$

and

$,!V_R = V_{in} - V_C$.

The first equation is solved by using an integrating factor and the second follows easily; the solutions are exactly the same as those obtained via Laplace transforms. In mathematics, one solves certain ordinary differential equations by using an integrating factor. ...

#### Integrator

Consider the output across the capacitor at high frequency i.e.

$omega gg frac{1}{RC}$.

This means that the capacitor has insufficient time to charge up and so its voltage is very small. Thus the input voltage approximately equals the voltage across the resistor. To see this, consider the expression for I given above:

$I = frac{V_{in}}{R+1/jomega C}$

but note that the frequency condition described means that

$omega C gg frac{1}{R}$

so

$I approx frac{V_{in}}{R}$ which is just Ohm's Law.

Now, For the phase law, see Ohms Phase Law. ...

$V_C = frac{1}{C}int_{0}^{t}Idt$

so

$V_C approx frac{1}{RC}int_{0}^{t}V_{in}dt$,

which is an integrator across the capacitor. An integrator is a device to perform the mathematical operation known as integration, a fundamental operation in calculus. ...

#### Differentiator

Consider the output across the resistor at low frequency i.e.,

$omega ll frac{1}{RC}$.

This means that the capacitor has time to charge up until its voltage is almost equal to the source's voltage. Considering the expression for I again, when

$R ll frac{1}{omega C}$,

so

$I approx frac{V_{in}}{1/jomega C}$
$V_{in} approx frac{I}{jomega C} approx V_C$

Now,

$V_R = IR = Cfrac{dV_C}{dt}R$
$V_R approx RCfrac{dV_{in}}{dt}$

which is a differentiator across the resistor. In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ...

More accurate integration and differentiation can be achieved by placing resistors and capacitors as appropriate on the input and feedback loop of operational amplifiers. In calculus, the integral of a function is an extension of the concept of a sum. ... In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ... It has been suggested that this article or section be merged with Feedback loop. ... A 741 operational amplifier in a TO-5 metal can package An operational amplifier, usually referred to as an op-amp for brevity, is a DC-coupled high-gain electronic voltage amplifier with differential inputs and, usually, a single output. ...

## Parallel circuit

Parallel RC circuit

The parallel RC circuit is generally of less interest than the series circuit. This is largely because the output voltage Vout is equal to the input voltage Vin — as a result, this circuit does not act as a filter on the input signal unless fed by a current source. Image File history File links Parallel-RC.svg Summary Parallel w:RC circuit. ... Image File history File links Parallel-RC.svg Summary Parallel w:RC circuit. ... This article or section does not adequately cite its references or sources. ... An ideal current source, I, driving a resistor, R, and creating a voltage V A current source is an electrical or electronic device that delivers or absorbs electric current. ...

With complex impedances:

$I_R = frac{V_{in}}{R},$

and

$I_C = jomega C V_{in},$.

This shows that the capacitor current is 90° out of phase with the resistor (and source) current. Alternatively, the governing differential equations may be used:

$I_R = frac{V_{in}}{R}$

and

$I_C = Cfrac{dV_{in}}{dt}$.

For a step input (which is effectively a 0 Hz or DC signal), the derivative of the input is an impulse at t = 0. Thus, the capacitor reaches full charge very quickly and becomes an open circuit — the well-known DC behaviour of a capacitor. The hertz (symbol: Hz) is the SI unit of frequency. ... Direct current (DC or continuous current) is the continuous flow of electricity through a conductor such as a wire from high to low potential. ... The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function Î´(x) that has the value of infinity for x = 0, the value zero elsewhere. ... Open circuit can mean:- In electronics, where there is nothing connected to a load and no current can flow. ...

This article or section does not adequately cite its references or sources. ... This is a list of communications, computers, electronic circuits, fiberoptics, microelectronics, medical electronics, reliablity, and semiconductors. ...

Results from FactBites:

 RC circuit - Wikipedia, the free encyclopedia (1154 words) These may be combined in four important combinations: the RC circuit, the RL circuit, the LC circuit and the RLC circuit with the abbreviations indicating which components are used. This article considers the RC circuit, in both series and parallel as shown in the diagrams. It represents the response of the circuit to an input voltage consisting of an impulse or delta function.
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