RC circuit

Overview

**resistor–capacitor circuit**

**(RC circuit)**, or

**RC filter**or

**RC network**, is an electric circuit composed of resistors and capacitors driven by a voltage

Voltage source

In electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current...

or current source

Current source

A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...

. 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.

Unanswered Questions

Encyclopedia

A

or current source

. 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 four most common RC filters are the high-pass filter

, low-pass filter

, band-pass filter

, and band-stop filter

.

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 abbreviations 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.

.

Solving this equation for

where

The time required for the voltage to fall to is called the RC time constant

and is given by

s) of a capacitor with capacitance

The complex frequency

,

where

and the evaluation of

across the capacitor is:

and the voltage across the resistor is:.

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

.

In addition, the transfer function for the resistor has a zero

located at the origin

.

and,

and the phase angles are:

and.

These expressions together may be substituted into the usual expression for the phasor representing the output:.

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

and

is the time constant

.

Similarly, the impulse response for the resistor voltage is

where

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 :.

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

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.

Solving the above equation yields

or

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 :.

As :

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.

The most straightforward way to derive the time domain behaviour is to use the Laplace transforms of the expressions for and given above. This effectively transforms . Assuming a step input

(i.e. before and then afterwards):

and.

Partial fraction

s 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

).

Thus, the voltage across the capacitor tends towards

These equations show that a series RC circuit has a time constant

, usually denoted being the time it takes the voltage across the component to either rise (across C) or fall (across R) to within of its final value. That is, is the time it takes to reach and to reach .

The rate of change is a

.

These results may also be derived by solving the differential equation

s describing the circuit:

and.

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.

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 given above:

but note that the frequency condition described means that

so which is just Ohm's Law

.

Now,

so,

which is an integrator

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 again, when,

so

Now,

which is a differentiator

More accurate integration

and differentiation

can be achieved by placing resistors and capacitors as appropriate on the input and feedback

loop of operational amplifier

s (see

.

With complex impedances:

and.

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:

and.

When fed by a current source, the transfer function of a parallel RC circuit is

.

**resistor–capacitor circuit****(RC circuit)**, or**RC filter**or**RC network**, is an electric circuit composed of resistors and capacitors driven by a voltageVoltage source

In electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current...

or current source

Current source

A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...

. 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 four most common RC filters are the high-pass filter

High-pass filter

A high-pass filter is a device that passes high frequencies and attenuates frequencies lower than its cutoff frequency. A high-pass filter is usually modeled as a linear time-invariant system...

, low-pass filter

Low-pass filter

A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

, band-pass filter

Band-pass filter

A band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range.Optical band-pass filters are of common usage....

, and band-stop filter

Band-stop filter

In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter...

.

## Introduction

There are three basic, linear passive lumpedLumped element model

The lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions...

analog circuit components: the resistor

Resistor

A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...

(R), the capacitor

Capacitor

A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

(C), and the inductor

Inductor

An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...

(L). These may be combined in the RC circuit, the RL circuit

RL circuit

A resistor-inductor circuit ', or RL filter or RL network, is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series or in parallel, driven by a voltage source.-Introduction:The fundamental passive linear circuit...

, the LC circuit

LC circuit

An LC circuit, also called a resonant circuit or tuned circuit, consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C...

, and the RLC circuit

RLC circuit

An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively...

, with the abbreviations 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.

*This article relies on knowledge of the complex impedance*.Electrical impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

representation of capacitors and on knowledge of the frequency domainFrequency domainIn electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

representation of signals

## Natural response

The simplest RC circuit is a capacitor and a resistor in series. When a circuit consists of only a charged capacitor and a resistor, the capacitor will 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, where the current through the capacitor must equal the current through the resistor. This results in the linear differential equationLinear differential equation

Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...

.

Solving this equation for

*V*yields the formula for exponential decay:where

*V*is the capacitor voltage at time_{0}*t = 0.*The time required for the voltage to fall to is called the RC time constant

RC time constant

In an RC circuit, the value of the time constant is equal to the product of the circuit resistance and the circuit capacitance , i.e. \tau = R × C. It is the time required to charge the capacitor, through the resistor, to 63.2 percent of full charge; or to discharge it to 36.8 percent of its...

and is given by

## Complex impedance

The complex impedance,*Z*_{C}(in ohmOhm

The ohm is the SI unit of electrical resistance, named after German physicist Georg Simon Ohm.- Definition :The ohm is defined as a resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere,...

s) of a capacitor with capacitance

*C*(in farads) isThe complex frequency

*s*is, in general, a complex numberComplex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

,

where

*j*represents the imaginary unitImaginary unitIn mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

:- is the exponential decay constant (in radians per second), and
- is the sinusoidal angular frequencyAngular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

(also 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 evaluation of

*s*becomes## Series circuit

By viewing the circuit as a voltage divider, the voltageVoltage

Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

across the capacitor is:

and the voltage across the resistor is:.

### Transfer functions

The transfer functionTransfer function

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

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 resistor has a zero

Zero (complex analysis)

In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...

located at the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

.

### Gain and phase angle

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 responseImpulse response

In signal processing, the impulse response, or impulse response function , of a dynamic system is its output when presented with a brief input signal, called an impulse. More generally, an impulse response refers to the reaction of any dynamic system in response to some external change...

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

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

.

The impulse response for the capacitor voltage is

where

*u*(*t*) is the Heaviside step functionHeaviside step function

The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

and

is the time constant

Time constant

In physics and engineering, the time constant, usually denoted by the Greek letter \tau , is the risetime characterizing the response to a time-varying input of a first-order, linear time-invariant system.Concretely, a first-order LTI system is a system that can be modeled by a single first order...

.

Similarly, the impulse response for the resistor voltage is

where

*δ*(*t*) is the Dirac delta functionDirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

### Frequency-domain considerations

These are frequency domainFrequency domain

In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

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 :.

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 aLow-pass filter

A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

*high-pass filter*

.High-pass filter

A high-pass filter is a device that passes high frequencies and attenuates frequencies lower than its cutoff frequency. A high-pass filter is usually modeled as a linear time-invariant system...

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

Cutoff frequency

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.Typically in electronic systems such as filters and...

. This requires that the gain of the circuit be reduced to.

Solving the above equation yields

or

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 :.

As :

So at DC

Direct current

Direct current is the unidirectional flow of electric charge. Direct current is produced by such sources as batteries, thermocouples, solar cells, and commutator-type electric machines of the dynamo type. Direct current may flow in a conductor such as a wire, but can also flow through...

(0 Hz

Hertz

The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

), 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.

### 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 and given above. This effectively transforms . Assuming a step input

Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....

(i.e. before and then afterwards):

and.

Partial fraction

Partial fraction

In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function ....

s 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

Electric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

; 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

Ohm's law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

).

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.These equations show that a series RC circuit has a time constant

RC time constant

In an RC circuit, the value of the time constant is equal to the product of the circuit resistance and the circuit capacitance , i.e. \tau = R × C. It is the time required to charge the capacitor, through the resistor, to 63.2 percent of full charge; or to discharge it to 36.8 percent of its...

, usually denoted being the time it takes the voltage across the component to either rise (across C) or fall (across R) to within of its final value. That is, is the time it takes to reach and to reach .

The rate of change is a

*fractional*per . Thus, in going from to , the voltage will have moved about 63.2 % of the way from its level at toward its final value. So C will be charged to about 63.2 % after , 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 towards 0. C will be discharged to about 36.8 % after , and essentially fully discharged (0.7 %) after about . Note that the current, , in the circuit behaves as the voltage across R does, via Ohm's LawOhm's law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

.

These results may also be derived by solving the differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s describing the circuit:

and.

The first equation is solved by using an integrating factor

Integrating factor

In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an...

and the second follows easily; the solutions are exactly the same as those obtained via Laplace transforms.

#### Integrator

Consider the output across the capacitor at*high*frequency i.e..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 given above:

but note that the frequency condition described means that

so which is just Ohm's Law

Ohm's law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

.

Now,

so,

which is an integrator

Integrator

An integrator is a device to perform the mathematical operation known as integration, a fundamental operation in calculus.The integration function is often part of engineering, physics, mechanical, chemical and scientific calculations....

*across the capacitor*.#### Differentiator

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

so

Now,

which is a differentiator

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

*across the resistor*.More accurate integration

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

and differentiation

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

can be achieved by placing resistors and capacitors as appropriate on the input and feedback

Feedback

Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

loop of operational amplifier

Operational amplifier

An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output...

s (see

*operational amplifier integrator*and*operational amplifier differentiator*).## Parallel circuit

The parallel RC circuit is generally of less interest than the series circuit. This is largely because the output voltage is equal to the input voltage — as a result, this circuit does not act as a filter on the input signal unless fed by a current sourceCurrent source

A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...

.

With complex impedances:

and.

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:

and.

When fed by a current source, the transfer function of a parallel RC circuit is

.

## See also

- RL circuitRL circuitA resistor-inductor circuit ', or RL filter or RL network, is one of the simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series or in parallel, driven by a voltage source.-Introduction:The fundamental passive linear circuit...
- LC circuitLC circuitAn LC circuit, also called a resonant circuit or tuned circuit, consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C...
- RLC circuitRLC circuitAn RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively...
- Electrical networkElectrical networkAn electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...
- List of electronics topics
- Step responseStep responseThe step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from...
- RC Circuit and continuous-repayment mortgage