A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage 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.

There are three basic, linear passive lumped analog circuit components: the resistor (R), capacitor (C) and 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, between 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 as shown in the diagrams.

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 equation

Series circuitSeries RC 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 for the capacitor is

Similarly, the transfer function for 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 located at the origin.

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 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

is the time constant.

Similarly, the impulse response for the resistor voltage is

where δ(t) 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.

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.

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.

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.

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

. Assuming a step input (i.e. Vin = 0 before t = 0 and then Vin = V afterwards):

and

.

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).

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.

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 rate of change is a fractional 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 5τ. 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 5τ. Note that the current, I, in the circuit behaves as the voltage across R does, via Ohm's Law.

These results may also be derived by solving the differential equations 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.

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 I given above:

but note that the frequency condition described means that

so

which is just Ohm's Law.

Now,

so

,

which is an integrator 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 I again, when

,

so

Now,

which is a differentiator across the resistor.

More accurate integration and differentiation can be achieved by placing resistors and capacitors as appropriate on the input and feedback loop of operational amplifiers (see operational amplifier integrator and operational amplifier differentiator).

Parallel circuitThe 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.

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

.

Low-pass filterA low-pass filter is a filter that passes low-frequency signals but attenuates (reduces the amplitude of) 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 when used in audio applications. A low-pass filter is the opposite of a high-pass filter, and a band-pass filter is a combination of a low-pass and a high-pass.

Low-pass filters exist in many different forms, including electronic circuits (such as a hiss filter used in audio), digital filters for smoothing sets of data, acoustic barriers, blurring of images, and so on. The moving average operation used in fields such as finance is a particular kind of low-pass filter, and can be analyzed with the same signal processing techniques as are used for other low-pass filters. Low-pass filters provide a smoother form of a signal, removing the short-term fluctuations, and leaving the longer-term trend.

Examples of low-pass filters

Acoustic

A stiff physical barrier tends to reflect higher sound frequencies, and so acts as a low-pass filter for transmitting sound. When music is playing in another room, the low notes are easily heard, while the high notes are attenuated.

Electronic

In an electronic low-pass RC filter for voltage signals, high frequencies contained in the input signal are attenuated but the filter has little attenuation below its cutoff frequency which is determined by its RC time constant.

For current signals, a similar circuit using a resistor and capacitor in parallel works in a similar manner. See current divider discussed in more detail below.

Electronic low-pass filters are used to drive subwoofers and other types of loudspeakers, to block high pitches that they can't efficiently broadcast.

Radio transmitters use low-pass filters to block harmonic emissions which might cause interference with other communications.

The tone knob found on many electric guitars is a low-pass filter used to reduce the amount of treble in the sound.

An integrator is another example of a single time constant low-pass filter.[1]

Telephone lines fitted with DSL splitters use low-pass and high-pass filters to separate DSL and POTS signals sharing the same pair of wires.

Low-pass filters also play a significant role in the sculpting of sound for electronic music as created by analogue synthesisers. See subtractive synthesis.

Ideal and real filters

The sinc function, the impulse response of an ideal low-pass filter.

An ideal low-pass filter completely eliminates all frequencies above the cutoff frequency while passing those below unchanged: its frequency response is a rectangular function, and is a brick-wall filter. The transition region present in practical filters does not exist in an ideal filter. An ideal low-pass filter can be realized mathematically (theoretically) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with its impulse response, a sinc function, in the time domain.

However, the ideal filter is impossible to realize without also having signals of infinite extent in time, and so generally needs to be approximated for real ongoing signals, because the sinc function's support region extends to all past and future times. The filter would therefore need to have infinite delay, or knowledge of the infinite future and past, in order to perform the convolution. It is effectively realizable for pre-recorded digital signals by assuming extensions of zero into the past and future, or more typically by making the signal repetitive and using Fourier analysis.

Real filters for real-time applications approximate the ideal filter by truncating and windowing the infinite impulse response to make a finite impulse response; applying that filter requires delaying the signal for a moderate period of time, allowing the computation to "see" a little bit into the future. This delay is manifested as phase shift. Greater accuracy in approximation requires a longer delay.

An ideal low-pass filter results in ringing artifacts via the Gibbs phenomenon. These can be reduced or worsened by choice of windowing function, and the design and choice of real

filters involves understanding and minimizing these artifacts. For example, "simple truncation [of sinc] causes severe ringing artifacts," in signal reconstruction, and to reduce these artifacts one uses window functions "which drop off more smoothly at the edges."[2]

The Whittaker–Shannon interpolation formula describes how to use a perfect low-pass filter to reconstruct a continuous signal from a sampled digital signal. Real digital-to-analog converters use real filter approximations.

Continuous-time low-pass filtersThe gain-magnitude frequency response of a first-order (one-pole) low-pass filter. Power gain is shown in decibels (i.e., a 3 dB decline reflects an additional half-power attenuation). Angular frequency is shown on a logarithmic scale in units of radians per second.

There are many different types of filter circuits, with different responses to changing frequency. The frequency response of a filter is generally represented using a Bode plot, and the filter is characterized by its cutoff frequency and rate of frequency rolloff. In all cases, at the cutoff frequency, the filter attenuates the input power by half or 3 dB. So the order of the filter determines the amount of additional attenuation for frequencies higher than the cutoff frequency.

A first-order filter, for example, will reduce the signal amplitude by half (so power reduces by 6 dB) every time the frequency doubles (goes up one octave); more precisely, the power rolloff approaches 20 dB per decade in the limit of high frequency. The magnitude Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency, and a diagonal line above the cutoff frequency. There is also a "knee curve" at the boundary between the two, which smoothly transitions between the two straight line regions. If the transfer function of a first-order low-pass filter has a zero as well as a pole, the Bode plot will flatten out again, at some maximum attenuation of high frequencies; such an effect is caused for example by a little bit of the input leaking around the one-pole filter; this one-pole–one-zero filter is still a first-order low-pass. See Pole–zero plot and RC circuit.

A second-order filter attenuates higher frequencies more steeply. The Bode plot for this type of filter resembles that of a first-order filter, except that it falls off more quickly. For example, a second-order Butterworth filter will reduce the signal amplitude to one fourth its original level every time the frequency doubles (so power decreases by 12 dB per octave, or 40 dB per decade). Other all-pole second-order filters may roll off at different rates initially depending on their Q factor, but approach the same final rate of 12 dB per octave; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote. See RLC circuit.

Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order-n all-pole filter is 6n dB per octave (i.e., 20n dB per decade).

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (the asymptotes of the function), they will intersect at exactly the "cutoff frequency". The frequency response at the cutoff frequency in a first-order filter is 3 dB below the horizontal line. The various types of filters – Butterworth filter, Chebyshev filter, Bessel filter, etc. – all have different-looking "knee curves". Many second-order filters are designed to have "peaking" or resonance, causing their frequency response at the cutoff frequency to be above the horizontal line. See electronic filter for other types.

The meanings of 'low' and 'high' – that is, the cutoff frequency – depend on the characteristics of the filter. The term "low-pass filter" merely refers to the shape of the filter's response; a high-pass filter could be built that cuts off at a lower frequency than any low-pass filter – it is their responses that set them apart. Electronic circuits can be devised for any desired frequency range, right up through microwave frequencies (above 1 GHz) and higher.

References

1. ̂ Sedra, Adel (1991). Microelectronic Circuits, 3 ed.. Saunders College Publishing. p. 60. ISBN 0-03-051648-X.

2. ̂ Mastering Windows: Improving Reconstruction

High-pass filterA high-pass filter, or HPF, is an LTI filter that passes high frequencies well but attenuates (i.e., reduces the amplitude of) frequencies lower than the filter's cutoff frequency. The actual amount of attenuation for each frequency is a design parameter of the filter. It is sometimes called a low-cut filter or bass-cut filter.[1]

First-order continuous-time implementationThe simple first-order electronic high-pass filter shown in Figure 1 is implemented by placing an input voltage across the series combination of a capacitor and a resistor and using the voltage across the resistor as an output. The product of the resistance and capacitance (R×C) is the time constant (τ); it is inversely proportional to the cutoff frequency fc, at which the output power is half the input power. That is,

here fc is in hertz, τ is in seconds, R is in ohms, and C is in farads.

Figure 2: An active high-pass filter

Figure 2 shows an active electronic implementation of a first-order high-pass filter using an operational amplifier. In this case, the filter has a passband gain of -R2/R1 and has a corner frequency of

Because this filter is active, it may have non-unity passband gain. That is, high-frequency signals are inverted and amplified by R2/R1.

[edit] Discrete-time realizationFor another method of conversion from continuous- to discrete-time, see Bilinear transform.

Discrete-time high-pass filters can also be designed. Discrete-time filter design is beyond the scope of this article; however, a simple example comes from the conversion of the continuous-time high-pass filter above to a discrete-time realization. That is, the continuous-time behavior can be discretized.

From the circuit in Figure 1 above, according to Kirchoff's Laws and the definition of capacitance:

where Qc(t) is the charge stored in the capacitor at time t. Substituting Equation (Q) into Equation (I) and then Equation (I) into Equation (V) gives:

This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenly-spaced points in time separated by ΔT time. Let the samples of Vin be

represented by the sequence , and let Vout be represented by the sequence which correspond to the same points in time. Making these substitutions:

And rearranging terms gives the recurrence relation

That is, this discrete-time implementation of a simple continuous-time RC high-pass filter is

By definition, . The expression for parameter α yields the equivalent time constant RC in terms of the sampling period ΔT and α:

If α = 0.5, then the RC time constant equal to the sampling period. If , then RC is

significantly smaller than the sampling interval, and .

[edit] Algorithmic implementation

The filter recurrence relation provides a way to determine the output samples in terms of the input samples and the preceding output. The following pseudocode algorithm will simulate the effect of a high-pass filter on a series of digital samples:

// Return RC high-pass filter output samples, given input samples, // time interval dt, and time constant RC function highpass(real[0..n] x, real dt, real RC) var real[0..n] y var real α := RC / (RC + dt) y[0] := x[0] for i from 1 to n y[i] := α * y[i-1] + α * (x[i] - x[i-1]) return y

The loop which calculates each of the n outputs can be refactored into the equivalent:

for i from 1 to n y[i] := α * (y[i-1] + x[i] - x[i-1])

However, the earlier form shows how the parameter α changes the impact of the prior output y[i-1] and current change in input (x[i] - x[i-1]). In particular,

A large α implies that the output will decay very slowly but will also be strongly influenced by even small changes in input. By the relationship between parameter α and time constant RC above, a large α corresponds to a large RC and therefore a low corner frequency of the filter. Hence, this case corresponds to a high-pass filter with a very narrow stop band. Because it is excited by small changes and tends to hold its prior output values for a long time, it can pass relatively low frequencies. However, a constant input (i.e., an input with (x[i] - x[i-1])=0) will always decay to zero, as would be expected with a high-pass filter with a large RC.

A small α implies that the output will decay quickly and will require large changes in the input (i.e., (x[i] - x[i-1]) is large) to cause the output to change much. By the relationship between parameter α and time constant RC above, a small α corresponds to a small RC and therefore a high corner frequency of the filter. Hence, this case corresponds to a high-pass filter with a very wide stop band. Because it requires large (i.e., fast) changes and tends to quickly forget its prior output values, it can only pass

relatively high frequencies, as would be expected with a high-pass filter with a small RC.

[edit] Applications

[edit] Audio

High-pass filters have many applications. They are used as part of an audio crossover to direct high frequencies to a tweeter while attenuating bass signals which could interfere with, or damage, the speaker. When such a filter is built into a loudspeaker cabinet it is normally a passive filter that also includes a low-pass filter for the woofer and so often employs both a capacitor and inductor (although very simple high-pass filters for tweeters can consist of a series capacitor and nothing else). An alternative, which provides good quality sound without inductors (which are prone to parasitic coupling, are expensive, and may have significant internal resistance) is to employ bi-amplification with active RC filters or active digital filters with separate power amplifiers for each loudspeaker. Such low-current and low-voltage line level crossovers are called active crossovers.[1]

Rumble filters are high-pass filters applied to the removal of unwanted sounds near to the lower end of the audible range or below. For example, noises (e.g., footsteps, or motor noises from record players and tape decks) may be removed because they are undesired or may overload the RIAA equalization circuit of the preamp.[1]

High-pass filters are also used for AC coupling at the inputs of many audio amplifiers, for preventing the amplification of DC currents which may harm the amplifier, rob the amplifier of headroom, and generate waste heat at the loudspeakers voice coil. One amplifier, the professional audio model DC300 made by Crown International beginning in the 1960s, did not have high-pass filtering at all, and could be used to amplify the DC signal of a common 9-volt battery at the input to supply 18 volts DC in an emergency for mixing console power.[2] However, that model's basic design has been superseded by newer designs such as the Crown Macro-Tech series developed in the late 1980s which included 10 Hz high-pass filtering on the inputs and switchable 35 Hz high-pass filtering on the outputs.[3] Another example is the QSC Audio PLX amplifier series which includes an internal 5 Hz high-pass filter which is applied to the inputs whenever the optional 50 and 30 Hz high-pass filters are turned off.[4]

Mixing consoles often include high-pass filtering at each channel strip. Some models have fixed-slope, fixed-frequency high-pass filters at 80 or 100 Hz that can be engaged; other models have 'sweepable HPF'—a high-pass filter of fixed slope that can be set within a specified frequency range, such as from 20 to 400 Hz on the Midas Heritage 3000, or 20 to 20,000 Hz on the Yamaha M7CL digital mixing console. Veteran systems engineer and live sound mixer Bruce Main recommends that high-pass filters be engaged for most mixer input sources, except for those such as kick drum, bass guitar and piano, sources which will have useful low frequency sounds. Main writes that DI unit inputs (as opposed to microphone inputs) do not need high-pass filtering as they are not subject to modulation by low-frequency stage wash—low frequency sounds coming from the subwoofers or the public address system and wrapping around to the stage. Main indicates that high-pass filters are commonly used for directional microphones which have a proximity effect—a low-frequency boost for very close sources. This low frequency boost commonly causes problems up to 200 or 300 Hz, but Main notes that he has seen microphones that benefit from a 500 Hz HPF setting on the console.[5]

[edit] Image

High-pass and low-pass filters are also used in digital image processing to perform transformations in the spatial frequency domain.[citation needed] The so-called Unsharp Mask used in most of the image editing software is a high pass filter.

[edit] See also DSL filter Band-stop filter Band-pass filter Bias tee Low-pass filter

[edit] References1. ^ a b c Watkinson, John (year=1998). The Art of Sound Reproduction. Focal Press.

pp. 268, 479. ISBN 0240515129. http://books.google.com/books?id=01u_Vm5i5isC&pg=PA479. Retrieved March 9, 2010.

2. ̂ Andrews, Keith; posting as ssltech (January 11, 2010). "Re: Running the board for a show this big?". Recording, Engineering & Production. ProSoundWeb. http://recforums.prosoundweb.com/index.php/m/462291/0/. Retrieved 9 March 2010.

3. ̂ "Operation Manual: MA-5002VZ". Macro-Tech Series. Crown Audio. year=2007. http://www.crownaudio.com/pdf/amps/128313.pdf. Retrieved March 9, 2010.

4. ̂ "User Manual: PLX Series Amplifiers". QSC Audio. 1999. http://media.qscaudio.com/pdfs/plxuser.pdf. Retrieved March 9, 2010.

5. ̂ Main, Bruce (February 16, 2010). "Cut 'Em Off At The Pass: Effective Uses Of High-Pass Filtering". Live Sound International (Framingham, Massachusetts: ProSoundWeb, EH Publishing).

[edit] External links Common Impulse Responses ECE 209: Review of Circuits as LTI Systems  – Short primer on the mathematical

analysis of (electrical) LTI systems. ECE 209: Sources of Phase Shift  – Gives an intuitive explanation of the source of

phase shift in a high-pass filter. Also verifies simple passive LPF transfer function by means of trigonometric identity.

Low-pass filterFrom Wikipedia, the free encyclopediaJump to: navigation, search

A low-pass filter is a filter that passes low-frequency signals but attenuates (reduces the amplitude of) 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 when used in audio applications. A low-pass filter is the opposite of a high-pass filter, and a band-pass filter is a combination of a low-pass and a high-pass.

Low-pass filters exist in many different forms, including electronic circuits (such as a hiss filter used in audio), digital filters for smoothing sets of data, acoustic barriers, blurring of images, and so on. The moving average operation used in fields such as finance is a particular kind of low-pass filter, and can be analyzed with the same signal processing techniques as are used for other low-pass filters. Low-pass filters provide a smoother form of a signal, removing the short-term fluctuations, and leaving the longer-term trend.

Contents[hide]

1 Examples of low-pass filters o 1.1 Acoustic o 1.2 Electronic

2 Ideal and real filters 3 Continuous-time low-pass filters

o 3.1 Laplace notation 4 Electronic low-pass filters

o 4.1 Passive electronic realization o 4.2 Active electronic realization

5 Discrete-time realization o 5.1 Algorithmic implementation

6 See also 7 References 8 External links

[edit] Examples of low-pass filters

[edit] Acoustic

A stiff physical barrier tends to reflect higher sound frequencies, and so acts as a low-pass filter for transmitting sound. When music is playing in another room, the low notes are easily heard, while the high notes are attenuated.

[edit] Electronic

In an electronic low-pass RC filter for voltage signals, high frequencies contained in the input signal are attenuated but the filter has little attenuation below its cutoff frequency which is determined by its RC time constant.

For current signals, a similar circuit using a resistor and capacitor in parallel works in a similar manner. See current divider discussed in more detail below.

Electronic low-pass filters are used to drive subwoofers and other types of loudspeakers, to block high pitches that they can't efficiently broadcast.

Radio transmitters use low-pass filters to block harmonic emissions which might cause interference with other communications.

The tone knob found on many electric guitars is a low-pass filter used to reduce the amount of treble in the sound.

An integrator is another example of a single time constant low-pass filter.[1]

Telephone lines fitted with DSL splitters use low-pass and high-pass filters to separate DSL and POTS signals sharing the same pair of wires.

Low-pass filters also play a significant role in the sculpting of sound for electronic music as created by analogue synthesisers. See subtractive synthesis.

[edit] Ideal and real filters

The sinc function, the impulse response of an ideal low-pass filter.

An ideal low-pass filter completely eliminates all frequencies above the cutoff frequency while passing those below unchanged: its frequency response is a rectangular function, and is a brick-wall filter. The transition region present in practical filters does not exist in an ideal filter. An ideal low-pass filter can be realized mathematically (theoretically) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with its impulse response, a sinc function, in the time domain.

However, the ideal filter is impossible to realize without also having signals of infinite extent in time, and so generally needs to be approximated for real ongoing signals, because the sinc function's support region extends to all past and future times. The filter would therefore need to have infinite delay, or knowledge of the infinite future and past, in order to perform the convolution. It is effectively realizable for pre-recorded digital signals by assuming extensions of zero into the past and future, or more typically by making the signal repetitive and using Fourier analysis.

Real filters for real-time applications approximate the ideal filter by truncating and windowing the infinite impulse response to make a finite impulse response; applying that filter requires delaying the signal for a moderate period of time, allowing the computation to "see" a little bit into the future. This delay is manifested as phase shift. Greater accuracy in approximation requires a longer delay.

An ideal low-pass filter results in ringing artifacts via the Gibbs phenomenon. These can be reduced or worsened by choice of windowing function, and the design and choice of real filters involves understanding and minimizing these artifacts. For example, "simple truncation [of sinc] causes severe ringing artifacts," in signal reconstruction, and to reduce these artifacts one uses window functions "which drop off more smoothly at the edges."[2]

The Whittaker–Shannon interpolation formula describes how to use a perfect low-pass filter to reconstruct a continuous signal from a sampled digital signal. Real digital-to-analog converters use real filter approximations.

[edit] Continuous-time low-pass filters

The gain-magnitude frequency response of a first-order (one-pole) low-pass filter. Power gain is shown in decibels (i.e., a 3 dB decline reflects an additional half-power attenuation). Angular frequency is shown on a logarithmic scale in units of radians per second.

There are many different types of filter circuits, with different responses to changing frequency. The frequency response of a filter is generally represented using a Bode plot, and the filter is characterized by its cutoff frequency and rate of frequency rolloff. In all cases, at the cutoff frequency, the filter attenuates the input power by half or 3 dB. So the order of the filter determines the amount of additional attenuation for frequencies higher than the cutoff frequency.

A first-order filter, for example, will reduce the signal amplitude by half (so power reduces by 6 dB) every time the frequency doubles (goes up one octave); more precisely, the power rolloff approaches 20 dB per decade in the limit of high frequency. The magnitude Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency, and a diagonal line above the cutoff frequency. There is also a "knee curve" at the boundary between the two, which smoothly transitions between the two straight line regions. If the transfer function of a first-order low-pass

filter has a zero as well as a pole, the Bode plot will flatten out again, at some maximum attenuation of high frequencies; such an effect is caused for example by a little bit of the input leaking around the one-pole filter; this one-pole–one-zero filter is still a first-order low-pass. See Pole–zero plot and RC circuit.

A second-order filter attenuates higher frequencies more steeply. The Bode plot for this type of filter resembles that of a first-order filter, except that it falls off more quickly. For example, a second-order Butterworth filter will reduce the signal amplitude to one fourth its original level every time the frequency doubles (so power decreases by 12 dB per octave, or 40 dB per decade). Other all-pole second-order filters may roll off at different rates initially depending on their Q factor, but approach the same final rate of 12 dB per octave; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote. See RLC circuit.

Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order-n all-pole filter is 6n dB per octave (i.e., 20n dB per decade).

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (the asymptotes of the function), they will intersect at exactly the "cutoff frequency". The frequency response at the cutoff frequency in a first-order filter is 3 dB below the horizontal line. The various types of filters – Butterworth filter, Chebyshev filter, Bessel filter, etc. – all have different-looking "knee curves". Many second-order filters are designed to have "peaking" or resonance, causing their frequency response at the cutoff frequency to be above the horizontal line. See electronic filter for other types.

The meanings of 'low' and 'high' – that is, the cutoff frequency – depend on the characteristics of the filter. The term "low-pass filter" merely refers to the shape of the filter's response; a high-pass filter could be built that cuts off at a lower frequency than any low-pass filter – it is their responses that set them apart. Electronic circuits can be devised for any desired frequency range, right up through microwave frequencies (above 1 GHz) and higher.

[edit] Laplace notation

Continuous-time filters can also be described in terms of the Laplace transform of their impulse response in a way that allows all of the characteristics of the filter to be easily analyzed by considering the pattern of poles and zeros of the Laplace transform in the complex plane (in discrete time, one can similarly consider the Z-transform of the impulse response).

For example, a first-order low-pass filter can be described in Laplace notation as

where s is the Laplace transform variable, τ is the filter time constant, and K is the filter passband gain.

[edit] Electronic low-pass filters

[edit] Passive electronic realization

Passive, first order low-pass RC filter

One simple electrical circuit that will serve as a low-pass filter consists of a resistor in series with a load, and a capacitor in parallel with the load. The capacitor exhibits reactance, and blocks low-frequency signals, causing them to go through the load instead. At higher frequencies the reactance drops, and the capacitor effectively functions as a short circuit. The combination of resistance and capacitance gives you the time constant of the filter τ = RC (represented by the Greek letter tau). The break frequency, also called the turnover frequency or cutoff frequency (in hertz), is determined by the time constant:

or equivalently (in radians per second):

One way to understand this circuit is to focus on the time the capacitor takes to charge. It takes time to charge or discharge the capacitor through that resistor:

At low frequencies, there is plenty of time for the capacitor to charge up to practically the same voltage as the input voltage.

At high frequencies, the capacitor only has time to charge up a small amount before the input switches direction. The output goes up and down only a small fraction of the amount the input goes up and down. At double the frequency, there's only time for it to charge up half the amount.

Another way to understand this circuit is with the idea of reactance at a particular frequency:

Since DC cannot flow through the capacitor, DC input must "flow out" the path marked Vout (analogous to removing the capacitor).

Since AC flows very well through the capacitor — almost as well as it flows through solid wire — AC input "flows out" through the capacitor, effectively short circuiting to ground (analogous to replacing the capacitor with just a wire).

The capacitor is not an "on/off" object (like the block or pass fluidic explanation above). The capacitor will variably act between these two extremes. It is the Bode plot and frequency response that show this variability.

[edit] Active electronic realization

An active low-pass filter

Another type of electrical circuit is an active low-pass filter.

In the operational amplifier circuit shown in the figure, the cutoff frequency (in hertz) is defined as:

or equivalently (in radians per second):

The gain in the passband is −R2/R1, and the stopband drops off at −6 dB per octave as it is a first-order filter.

Sometimes, a simple gain amplifier (as opposed to the very-high-gain operational amplifier) is turned into a low-pass filter by simply adding a feedback capacitor C. This feedback decreases the frequency response at high frequencies via the Miller effect, and helps to avoid oscillation in the amplifier. For example, an audio amplifier can be made into a low-pass filter with cutoff frequency 100 kHz to reduce gain at frequencies which would otherwise oscillate. Since the audio band (what we can hear) only goes up to 20 kHz or so, the frequencies of interest fall entirely in the passband, and the amplifier behaves the same way as far as audio is concerned.

[edit] Discrete-time realizationFor another method of conversion from continuous- to discrete-time, see Bilinear transform.

The effect of a low-pass filter can be simulated on a computer by analyzing its behavior in the time domain, and then discretizing the model.

A simple low-pass RC filter

From the circuit diagram to the right, according to Kirchoff's Laws and the definition of capacitance:

  

   (V)

  

   (Q)

  

   (I)

where Qc(t) is the charge stored in the capacitor at time t. Substituting equation Q into

equation I gives , which can be substituted into equation V so that:

This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenly-spaced points in time separated by ΔT time. Let the samples of vin be

represented by the sequence , and let vout be represented by the sequence which correspond to the same points in time. Making these substitutions:

And rearranging terms gives the recurrence relation

That is, this discrete-time implementation of a simple RC low-pass filter is the exponentially-weighted moving average

By definition, the smoothing factor . The expression for α yields the equivalent time constant RC in terms of the sampling period ΔT and smoothing factor α:

If α = 0.5, then the RC time constant is equal to the sampling period. If , then RC is

significantly larger than the sampling interval, and .

[edit] Algorithmic implementation

The filter recurrence relation provides a way to determine the output samples in terms of the input samples and the preceding output. The following pseudocode algorithm will simulate the effect of a low-pass filter on a series of digital samples:

// Return RC low-pass filter output samples, given input samples, // time interval dt, and time constant RC function lowpass(real[0..n] x, real dt, real RC) var real[0..n] y var real α := dt / (RC + dt) y[0] := x[0] for i from 1 to n y[i] := α * x[i] + (1-α) * y[i-1] return y

The loop which calculates each of the n outputs can be refactored into the equivalent:

for i from 1 to n y[i] := y[i-1] + α * (x[i] - y[i-1])That is, the change from one filter output to the next is proportional to the difference between the previous output and the next input. This exponential smoothing property matches the exponential decay seen in the continuous-time system. As expected, as the time constant RC increases, the discrete-time smoothing parameter α decreases, and the output samples

respond more slowly to a change in the input samples  – the system will have more inertia.

[edit] See also

A Class D amplifier with an integral low pass filter, intended for powering subwoofersElectronics portal

Baseband Digital filter : Another realization of a low-pass filter High-pass filter Band-stop filter

[edit] References1. ̂ Sedra, Adel (1991). Microelectronic Circuits, 3 ed.. Saunders College Publishing.

p. 60. ISBN 0-03-051648-X.2. ̂ Mastering Windows: Improving Reconstruction

[edit] External links Low-pass filter Low Pass Filter java simulator ECE 209: Review of Circuits as LTI Systems  – Short primer on the mathematical

analysis of (electrical) LTI systems. ECE 209: Sources of Phase Shift  – Gives an intuitive explanation of the source of

phase shift in a low-pass filter. Also verifies simple passive LPF transfer function by means of trigonometric identity.

Cutoff frequencyFrom Wikipedia, the free encyclopediaJump to: navigation, search This article is about signal processing. For cutoff in theoretical physics, see Cutoff. For other uses, see Cutoff (disambiguation).

Magnitude transfer function of a bandpass filter with lower 3dB cutoff frequency f1 and upper 3dB cutoff frequency f2

A bode plot of the Butterworth filter's frequency response, with corner frequency labeled. (The slope −20 dB per decade also equals −6 dB per octave.)

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 (attenuated or reflected) rather than passing through.

Typically in electronic systems such as filters and communication channels, cutoff frequency applies to an edge in a lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a boundary between a passband and a stopband. It is sometimes taken to be the point in the filter response where a transition band and passband meet, for example as defined by a 3 dB corner, a frequency for which the output of the circuit is -3 dB of the nominal passband value. Alternatively, a stopband corner frequency may be specified as a point where a transition band and a stopband meet: a frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB.

In the case of a waveguide or an antenna, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.

Cutoff frequency can also refer to the plasma frequency.

Contents

[hide]

1 Electronics 2 Communications 3 Waveguides

o 3.1 Mathematical analysis 4 See also 5 References 6 External links

[edit] ElectronicsIn electronics, cutoff frequency or corner frequency is the frequency either above or below which the power output of a circuit, such as a line, amplifier, or electronic filter has fallen to a given proportion of the power in the passband. Most frequently this proportion is one half the passband power, also referred to as the 3dB point since a fall of 3dB corresponds

approximately to half power. As a voltage ratio this is a fall to of the passband voltage.[1]

However, other ratios are sometimes more convenient. For instance, in the case of the Chebyshev filter it is usual to define the cutoff frequency as the point after the last peak in the frequency response at which the level has fallen to the design value of the passband ripple. The amount of ripple in this class of filter can be set by the designer to any desired value, hence the ratio used could be any value.[2]

[edit] CommunicationsIn communications, the term cutoff frequency can mean the frequency below which a radio wave fails to penetrate a layer of the ionosphere at the incidence angle required for transmission between two specified points by reflection from the layer.

[edit] WaveguidesThe cutoff frequency of an electromagnetic waveguide is the lowest frequency for which a mode will propagate in it. In fiber optics, it is more common to consider the cutoff wavelength, the maximum wavelength that will propagate in an optical fiber or waveguide. The cutoff frequency is found with the characteristic equation of the Helmholtz equation for electromagnetic waves, which is derived from the electromagnetic wave equation by setting the longitudinal wave number equal to zero and solving for the frequency. Thus, any exciting frequency lower than the cutoff frequency will attenuate, rather than propagate. The following derivation assumes lossless walls. The value of c, the speed of light, should be taken to be the group velocity of light in whatever material fills the waveguide.

For a rectangular waveguide, the cutoff frequency is

where are the mode numbers and a and b the lengths of the sides of the rectangle.

The cutoff frequency of the TM01 mode in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by

where r is the radius of the waveguide, and χ01 is the first root of J0(r), the bessel function of the first kind of order 1.

For a single-mode optical fiber, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2.405.

[edit] Mathematical analysis

The starting point is the wave equation (which is derived from the Maxwell equations),

which becomes a Helmholtz equation by considering only functions of the form

ψ(x,y,z,t) = ψ(x,y,z)eiωt.

Substituting and evaluating the time derivative gives

The function ψ here refers to whichever field (the electric field or the magnetic field) has no vector component in the longitudinal direction - the "transverse" field. It is a property of all the eigenmodes of the electromagnetic waveguide that at least one of the two fields is transverse. The z axis is defined to be along the axis of the waveguide.

The "longitudinal" derivative in the Laplacian can further be reduced by considering only functions of the form

where kz is the longitudinal wavenumber, resulting in

where subscript T indicates a 2-dimensional transverse Laplacian. The final step depends on the geometry of the waveguide. The easiest geometry to solve is the rectangular waveguide. In that case the remainder of the Laplacian can be evaluated to its characteristic equation by considering solutions of the form

Thus for the rectangular guide the Laplacian is evaluated, and we arrive at

The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular geometry crossection with dimensions a and b:

where n and m are the two integers representing a specific eigenmode. Performing the final substitution, we obtain

which is the dispersion relation in the rectangular waveguide. The cutoff frequency ωc is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber kz is zero. It is given by

The wave equations are also valid below the cutoff frequency, where the longitudinal wave number is imaginary. In this case, the field decays exponentially along the waveguide axis.

[edit] See also Angular frequency Full width at half maximum High-pass filter Low-pass filter Time constant Miller effect

[edit] References1. ̂ Van Valkenburg, M. E.. Network Analysis (3rd edition ed.). pp. 383–384. ISBN 0-

13-611095-9.

http://www.amazon.com/Network-Analysis-Mac-Van-Valkenburg/dp/0136110959. Retrieved 2008-06-22.

2. ̂ Mathaei, Young, Jones Microwave Filters, Impedance-Matching Networks, and Coupling Structures, pp.85-86, McGraw-Hill 1964.

 This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C" (in support of MIL-STD-188).

[edit] External links Calculation of the center frequency with geometric mean and comparison to the

arithmetic mean solution Conversion of cutoff frequency f c and time constant τ Mathematical definition of and information about the Bessel functions

DC-controlled low-pass filter has variable breakpoint

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DC-controlled low-pass filter has variable breakpointBy Contributing Author

December 01, 1997

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When maximum performance is demanded from communications systems, test equipment, or any other frequency-sensitive systems, it’s imperative to get the point where the filter response is −3 dB (the breakpoint of the low-pass filter) placed exactly right. Placing the breakpoint correctly ensures minimum distortion in the passband while yielding maximum attenuation of unwanted frequencies in the stopband. This is very hard to accomplish in multiple frequency systems because when a break frequency is placed correctly for one task it is, usually by definition, wrong for another task.

The described low-pass filter has a breakpoint that’s continuously variable over a range of 20 to 1 by varying the dc control voltage. The gain stays constant regardless of the breakpoint setting. If digital control is advantageous, a DAC can easily be interfaced into the control voltage port because the control voltage ranges from 0 to 1.5 V.

The HA2841 op amp is the main amplification element in the circuit, and it was chosen because it has excellent dc characteristics coupled with high frequency response (see the figure). The input signal is amplified by the op amp, but only the dc portion of the output signal is fed directly back to the op-amp summing junction. This fixes the dc gain at −Rf/Rg. The ac portion of the output signal is passed through the HA2546 high-frequency multiplier before it’s fed back to the summing junction.

The HA2546 was chosen for this application because it’s extremely small time delay doesn’t introduce distortion. The feedback capacitor (C) blocks any dc multiplier errors. As Vx changes, the multiplier gain changes, so the apparent value of C changes. Consequently, the

2

breakpoint frequency is forced to change. In the equation for the multiplier, Voutm is the multiplier output voltage:

The equation for the complete circuit response is:

The control voltage (Vx), in conjunction with Rf and C, determine the breakpoint frequency, ω. Rf and C are used to center the frequency range, and Vx varies the frequency within this range. Both Rf and Rg set the gain, so there’s plenty of flexibility in the component selection. The component values shown yield a frequency range from 1.7 MHz when Vx = 0.1 V to 80 kHz when Vx = 1.25 V. The control input is similar to an op-amp input, thus it needn’t be driven by a low-impedance source. This input may be driven from a DAC to obtain digital control of the breakpoint, but the DAC output voltage must be level-shifted to 0 to 1.5 V.

Passive Low Pass Filter Navigation

Page: 2 of 8

Low Pass FiltersBasically, an electrical filter is a circuit that can be designed to modify, reshape or reject all unwanted frequencies of an electrical signal and accept or pass only those signals wanted by the circuits designer. In low frequency applications (up to 100kHz), passive filters are usually made from simple RC (Resistor-Capacitor) networks while higher frequency filters (above 100kHz) are usually made from RLC (Resistor-Inductor-Capacitor) components.

Simple First-order passive filters (1st order) can be made by connecting together a single resistor and a single capacitor in series across an input signal, (Vin) with the output signal, (Vout) taken from the junction of these two components. Depending on which way around we connect the resistor and the capacitor with regards to the output signal determines the type of filter construction resulting in either a Low Pass Filter or a High Pass Filter.

The function of any filter is to allow signals of a given band of frequencies to pass unaltered while attenuating or weakening the others that are not wanted. As there are two passive components within this type of filter design the output signal has a smaller amplitude than its corresponding input signal, therefore passive RC filters attenuate the signal and have a gain of less than one, (unity).

The Low Pass RC FilterA simple passive Low Pass Filter or LPF, can be easily made by connecting together in series a single Resistor with a single Capacitor as shown below. In this type of filter arrangement the input signal (Vin) is applied to the series combination (both the Resistor and Capacitor together) but the output signal (Vout) is taken across the capacitor only. This type of filter is known generally as a 1st order Filter, why 1st order?, because it has only "one" reactive component in the circuit, the Capacitor.

Low Pass Filter Circuit

As mentioned previously in the Capacitive Reactance tutorial, the reactance of a capacitor varies inversely with frequency, while the value of the resistor remains constant as the

Reset

frequency changes. At low frequencies the capacitive reactance, (Xc) of the capacitor will be very large compared to the resistive value of the resistor, R and as a result the voltage across the capacitor, Vc will also be large while the voltage drop across the resistor, Vr will be much lower. At high frequencies the reverse is true with Vc being small and Vr being large.

While the circuit above is that of an RC Low Pass Filter circuit, it can also be classed as a frequency variable potential divider circuit similar to the one we looked at in the Resistors tutorial. In that tutorial we used the following equation to calculate the output voltage for two single resistors connected in series.

We also know that the capacitive reactance of a capacitor in an AC circuit is given as:

Opposition to current flow in an AC circuit is called impedance, symbol Z and for a series circuit consisting of a single resistor in series with a single capacitor, the circuit impedance is calculated as:

Then by substituting our equation for impedance above into the resistive potential divider equation gives us:

So, by using the potential divider equation of two resistors in series and substituting for impedance we can calculate the output voltage of an RC Filter for any given frequency.

Example No1A Low Pass Filter circuit consisting of a Resistor of 4k7Ω in series with a Capacitor of C = 47nF is connected across a 10v DC supply. Calculate the output voltage (Vout) at a frequency of 100Hz and again at frequency of 10,000Hz or 10kHz.

At a frequency of 100Hz.

At a frequency of 10kHz.

Frequency ResponseWe can see above, that as the frequency increases from 100Hz to 10kHz, the output voltage (Vout) decreases from 9.9v to 0.718v. By plotting the output voltage against the input frequency, the Frequency Response Curve or Bode Plot function of the low pass filter can be found, as shown below.

Frequency Response of a 1st Order Low Pass Filter.

The Bode Plot shows the Frequency Response of the filter to be nearly flat for low frequencies and all of the input signal is passed directly to the output, resulting in a gain of nearly 1, unity until it reaches the Cut-off Frequency point ( ƒc ). This is because the reactance of the capacitor is high at low frequencies and blocks any current flow through the capacitor. After this point the response of the circuit decreases giving a slope of -20dB/ Decade or (-6dB/Octave) "roll-off" as signals above this frequency become greatly attenuated, until at very high frequencies the reactance of the capacitor becomes so low that it gives the effect of a short circuit condition on the output terminals resulting in zero output.

For this type of Low Pass Filter circuit, all the frequencies below this cut-off, ƒc point that are unaltered with little or no attenuation and are said to be in the filters Passband zone. This passband zone also represents the Bandwidth of the filter. Any signal frequencies above this point cut-off point are generally said to be in the filters Stopband zone and they will be greatly attenuated.

This "Cut-off", "Corner" or "Breakpoint" frequency is defined as being the frequency point where the capacitive reactance and resistance are equal, R = Xc = 4k7Ω. When this occurs the output signal is attenuated to 70.7% of the input signal value or -3dB (20 log (Vout/Vin)) of the input. Although R = Xc, the output is not half of the input signal. This is because it is equal to the vector sum of the two and is therefore 0.707 of the input. As the filter contains a capacitor, the Phase Angle ( Φ ) of the output signal LAGS behind that of the input and at the -3dB cut-off frequency ( ƒc ) and is -45o out of phase. This is due to the time taken to charge the plates of the capacitor as the input voltage changes, resulting in the output voltage (the voltage across the capacitor) "lagging" behind that of the input signal. The higher the input

frequency applied to the filter the more the capacitor lags and the circuit becomes more and more "out of phase".

The cut-off frequency point and phase shift angle can be found by using the following equation:

Cut-off Frequency and Phase Shift

Then for our simple example of a "Low Pass Filter" circuit above, the cut-off frequency (ƒc) is given as 720Hz with an output voltage of 70.7% of the input voltage value and a phase shift angle of -45o.

Low Pass Filter SummarySo to summarize, the Low Pass Filter has a constant output voltage from D.C. (0Hz), up to a specified Cut-off frequency, ( ƒc ) point. This cut-off frequency point is 0.707 or -3dB (dB = -20log Vout/Vin) of the voltage gain allowed to pass. The frequency range "below" this cut-off point ƒc is generally known as the Pass Band as the input signal is allowed to pass through the filter. The frequency range "above" this cut-off point is generally known as the Stop Band as the input signal is blocked or stopped from passing through. A simple 1st order low pass filter can be made using a single resistor in series with a single non-polarized capacitor (or any single reactive component) across an input signal Vin, whilst the output signal Vout is taken from across the capacitor. The cut-off frequency or -3dB point, can be found using the formula, ƒc = 1/(2πRC). The phase angle of the output signal at ƒc and is -45o for a Low Pass Filter.

The gain of the filter or any filter for that matter, is generally expressed in Decibels and is a function of the output value divided by its corresponding input value and is given as:

Applications of passive Low Pass Filters are in audio amplifiers and speaker systems to direct the lower frequency bass signals to the larger bass speakers or to reduce any high frequency noise or "hiss" type distortion. When used like this in audio applications the low pass filter is sometimes called a "high-cut", or "treble cut" filter.

If we were to reverse the positions of the resistor and capacitor in the circuit so that the output voltage is now taken from across the resistor, we would have a circuit that produces an output

frequency response curve similar to that of a High Pass Filter, and this is discussed in the next tutorial.

Time ConstantWe know from above, that the filters cut-off frequency (ƒc) is the product of the resistance (R) and the capacitance (C) in the circuit with respect to some specified frequency point and that by altering any one of the two components alters this cut-off frequency point by either increasing it or decreasing it. We also know that the phase shift of the circuit lags behind that of the input signal due to the time required to charge and then discharge the capacitor as the sine wave changes. This combination of R and C produces a charging and discharging effect on the capacitor known as its Time Constant (τ) of the circuit as seen in the RC Circuit tutorials.

This time constant, tau (τ), is related to the cut-off frequency ƒc as.

or expressed in terms of the cut-off frequency, ƒc as.

The output voltage, Vout depends upon the time constant and the frequency of the input signal. With an AC sinusoidal signal the circuit behaves as a simple 1st order low pass filter. But what if we where to change the input signal to that of a "square wave" shaped signal that has an almost vertical step input, the response of the circuit changes dramatically and produces a circuit known commonly as an Integrator.

The RC IntegratorThe Integrator is basically a low pass filter circuit that converts a square wave step response input signal into a triangular shaped waveform output as the capacitor charges and discharges. A Triangular waveform consists of alternate but equal positive and negative ramps. As seen below, if the RC time constant is long compared to the time period of the input waveform the resultant output waveform will be triangular in shape and the higher the input frequency the lower will be the output amplitude compared to that of the input.

The RC Integrator Circuit

This then makes this type of circuit ideal for converting one type of electronic signal to another for use in wave-generating or wave-shaping circuits.