Saturday, 12 January 2013

Low-pass filter - Wiki(Description)

http://en.wikipedia.org/wiki/Low-pass_filter

Low-pass filter

A low-pass filter is an electronic filter that passes low-frequency signals and 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. 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), anti-aliasing filters for conditioning signals prior to analog-to-digital conversion, 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.

An optical filter could correctly be called low-pass, but conventionally is described as "longpass" (low frequency is long wavelength), to avoid confusion.

[hide]

[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 dividerdiscussed in more detail below.

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

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 analoguesynthesisers. See subtractive synthesis.

[edit]Ideal and real filters

The sinc function, the impulse responseof 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 arectangular 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 frequencyand 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 a factor of 4), or 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 theirQ 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- $\scriptstyle n$ all-pole filter is $\scriptstyle 6n$ dB per octave (i.e., $\scriptstyle 20n$ dB per decade).

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (theasymptotes 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 filterfor 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

$\frac{\text{Output}}{\text{Input}} = K \frac{1}{1 + s \tau}$

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 resistorin 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 the time constant of the filter $\scriptstyle \tau \;=\; RC$ (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:

$f_\mathrm{c} = {1 \over 2 \pi \tau } = {1 \over 2 \pi R C}$

or equivalently (in radians per second):

$\omega_\mathrm{c} = {1 \over \tau} = {1 \over R C}$

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 $\scriptstyle V_\mathrm{out}$ (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:

$f_{\text{c}} = \frac{1}{2 \pi R_2 C}$

or equivalently (in radians per second):

$\omega_{\text{c}} = \frac{1}{R_2 C}$

The gain in the passband is −R₂/R₁, and the stopband drops off at −6 dB per octave (that is −20 dB per decade) as it is a first-order filter.

[edit]Discrete-time realization

For 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 thendiscretizing 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:

$v_{\text{in}}(t) - v_{\text{out}}(t) = R \; i(t)$

(V)

$Q_c(t) = C \, v_{\text{out}}(t)$

(Q)

$i(t) = \frac{\operatorname{d} Q_c}{\operatorname{d} t}$

(I)

where $Q_c(t)$ is the charge stored in the capacitor at time $\scriptstyle t$ . Substituting equation Q into equation I gives $\scriptstyle i(t) \;=\; C \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}$ , which can be substituted into equation V so that:

$v_{\text{in}}(t) - v_{\text{out}}(t) = RC \frac{\operatorname{d}v_{\text{out}}}{\operatorname{d}t}$

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 $\scriptstyle \Delta_T$ time. Let the samples of $\scriptstyle v_{\text{in}}$ be represented by the sequence $\scriptstyle (x_1,\, x_2,\, \ldots,\, x_n)$ , and let $\scriptstyle v_{\text{out}}$ be represented by the sequence $\scriptstyle (y_1,\, y_2,\, \ldots,\, y_n)$ which correspond to the same points in time. Making these substitutions:

$x_i - y_i = RC \, \frac{y_{i}-y_{i-1}}{\Delta_T}$

And rearranging terms gives the recurrence relation

$y_i = \overbrace{x_i \left( \frac{\Delta_T}{RC + \Delta_T} \right)}^{\text{Input contribution}} + \overbrace{y_{i-1} \left( \frac{RC}{RC + \Delta_T} \right)}^{\text{Inertia from previous output}}.$

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

$y_i = \alpha x_i + (1 - \alpha) y_{i-1} \qquad \text{where} \qquad \alpha \triangleq \frac{\Delta_T}{RC + \Delta_T}$

By definition, the smoothing factor $\scriptstyle 0 \;\leq\; \alpha \;\leq\; 1$ . The expression for $\scriptstyle \alpha$ yields the equivalent time constant $\scriptstyle RC$ in terms of the sampling period $\scriptstyle \Delta_T$ and smoothing factor $\scriptstyle \alpha$ :

$RC = \Delta_T \left( \frac{1 - \alpha}{\alpha} \right)$

If $\scriptstyle \alpha \;=\; 0.5$ , then the $\scriptstyle RC$ time constant is equal to the sampling period. If $\scriptstyle \alpha \;\ll\; 0.5$ , then $\scriptstyle RC$ is significantly larger than the sampling interval, and $\scriptstyle \Delta_T \;\approx\; \alpha RC$ .

[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 that 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 $\scriptstyle RC$ increases, the discrete-time smoothing parameter $\scriptstyle \alpha$ decreases, and the output samples $\scriptstyle (y_1,\, y_2,\, \ldots,\, y_n)$ respond more slowly to a change in the input samples $\scriptstyle (x_1,\, x_2,\, \ldots,\, x_n)$ ; the system will have moreinertia. This filter is an infinite-impulse-response (IIR) single-pole lowpass filter.

Active filter

http://en.wikipedia.org/wiki/Active_filter

Active filter

An active filter is a type of analog electronic filter that uses active components such as an amplifier. Amplifiers included in a filter design can be used to improve the performance and predictability of a filter,^[1] while avoiding the need for inductors (which are typically expensive compared to other components). An amplifier prevents the load impedance of the following stage from affecting the characteristics of the filter. An active filter can have complex poles and zeros without using a bulky or expensive inductor. The shape of the response, the Q (quality factor), and the tuned frequency can often be set with inexpensive variable resistors. In some active filter circuits, one parameter can be adjusted without affecting the others. ^[1]

Using active elements has some limitations. Basic filter design equations neglect the finite bandwidth of amplifiers. Available active devices have limited bandwidth, so they are often impractical at high frequencies. Amplifiers consume power and inject noise into a system. Certain circuit topologies may be impractical if no DC path is provided for bias current to the amplifier elements. Power handling capability is limited by the amplifier stages.^[2]

Active filter circuit configurations (electronic filter topology) include:

Sallen and Key, and VCVS filters (low dependency on accuracy of the components)
State variable and biquadratic filters
Dual Amplifier Bandpass (DABP)
Wien notch
Multiple Feedback Filter
Fliege (lowest component count for 2 opamp but with good controllability over frequency and type)
Akerberg Mossberg (one of the topologies that offer complete and independent control over gain, frequency, and type)

Active filters can implement the same transfer functions as passive filters. Common transfer functions are:

High-pass filter – attenuation of frequencies below their cut-off points.
Low-pass filter – attenuation of frequencies above their cut-off points.
Band-pass filter – attenuation of frequencies both above and below those they allow to pass.
Notch filter – attenuation of certain frequencies while allowing all others to pass.

Combinations are possible, such as notch and high-pass (in a rumble filter where most of the offending rumble comes from a particular frequency). Another example is an elliptic filter.

[edit]Design of active filters

To design filters, the specifications that need to be established include:

The range of desired frequencies (the passband) together with the shape of the frequency response. This indicates the variety of filter (see above) and the center or corner frequencies.
Input and output impedance requirements. These limit the circuit topologies available; for example, most, but not all active filter topologies provide a buffered (low impedance) output. However, remember that the internal output impedance of operational amplifiers, if used, may rise markedly at high frequencies and reduce the attenuation from that expected. Be aware that some high-pass filter topologies present the input with almost a short circuit to high frequencies.
Dynamic range of the active elements. The amplifier should not saturate (run into the power supply rails) at expected input signals, nor should it be operated at such low amplitudes that noise dominates.
The degree to which unwanted signals should be rejected.
- In the case of narrow-band bandpass filters, the Q determines the -3dB bandwidth but also the degree of rejection of frequencies far removed from the center frequency; if these two requirements are in conflict then a staggered-tuning bandpass filter may be needed.
- For notch filters, the degree to which unwanted signals at the notch frequency must be rejected determines the accuracy of the components, but not the Q, which is governed by desired steepness of the notch, i.e. the bandwidth around the notch before attenuation becomes small.
- For high-pass and low-pass (as well as band-pass filters far from the center frequency), the required rejection may determine the slope of attenuation needed, and thus the "order" of the filter. A second-order all-pole filter gives an ultimate slope of about 12 dB per octave (40dB/decade), but the slope close to the corner frequency is much less, sometimes necessitating a notch be added to the filter.
The allowable "ripple" (variation from a flat response, in decibels) within the passband of high-pass and low-pass filters, along with the shape of the frequency response curve near the corner frequency, determine the damping factor (reciprocal of Q). This also affects the phase response, and the time response to a square-wave input. Several important response shapes (damping factors) have well-known names:

Chebyshev filter – slight peaking/ripple in the passband before the corner; Q>0.7071 for 2nd-order filters
Butterworth filter – flattest amplitude response; Q=0.7071 for 2nd-order filters
Linkwitz–Riley filter – desirable properties for audio crossover applications; Q = 0.5 (critically damped)
Paynter or transitional Thompson-Butterworth or "compromise" filter – faster fall-off than Bessel; Q=0.639 for 2nd-order filters
Bessel filter – best time-delay, best overshoot response; Q=0.577 for 2nd-order filters
Elliptic filter or Cauer filter – add a notch (or "zero") just outside the passband, to give a much greater slope in this region than the combination of order and damping factor without the notch.

http://www.swarthmore.edu/NatSci/echeeve1/Ref/FilterBkgrnd/Filters.html

Frequency Response and Active Filters

This document is an introduction to frequency response, and an introduction to active filters (filters using active amplifiers, like op amps). You might also want to read a similar document from National Semiconductor, A Basic Introduction to Filters - Active, Passive, and Switched-Capacitor.

Frequency Response -- Background

Up to now we have looked at the time-domain response of circuits. However it is often useful to look at the response of circuits in the frequency domain. In other words, you want to look at how circuits behave in response to sinusoidal inputs. This is important and useful for several reasons: 1) if the input to a linear circuit is a sinusoid, then the output will be a sinusoid at the same frequency, though its amplitude and phase may have changed, 2) any time domain signal can be decomposed via Fourier analysis into a series of sinusoids. Therefore if there is an easy way to analyze circuits with sinusoidal inputs, the results can be generalized to study the response to any input.

To determine the response of a circuit to a sinusoidal signal as a function of frequency it is possible to generalize the concept of impedance to include capacitors and inductors. Consider a sinusoidal signal represented by a complex exponential:

where j=(-1)^1/2 (engineers use j instead of i, because i is used for current), w is frequency and t is time. It is a common shorthand to use "s" instead of "jw".

Now let us look at the voltage-current relationships for resistors capacitors and inductors.

For a resistor ohms law states:

where we define the impedance, "Z", of a resistor as its resistance "R".

For a capacitor we can also calculate the impedance assuming sinusoidal excitation starting from the current-voltage relationship:

Note that for a capacitor the magnitude of the impedance, 1/wC, goes down with increasing frequency. This means that at very high frequencies the capacitor acts as an short circuit, and at low frequencies it acts as an open circuit. What is defined as a high, or low, frequency depends on the specific circuit in question.

Likewise, for an inductor you can show that Z=sL.

For an inductor, impedance goes up with frequency. It behaves as a short circuit at low frequencies, and an open circuit at high frequencies; the opposite of a capacitor. However inductors are not used often in electronic circuits due to their size, their susceptibility to parisitic effects (esp. magnetic fields), and because they do not behave as near to their ideal circuit elements as resistors and capacitors..

A Simple Low-Pass Circuit

To see how complex impedances are used in practice consider the simple case of a voltage divider.

If Z₁ is a resistor and Z₂ is a capacitor then

Generally we will be interested only in the magnitude of the response:

Recall that the magnitude of a complex number is the square root of the sum of the squares of the real and imaginary parts. There are also phase shifts associated with the transfer function (or gain, V_o/V_i), thought we will generally ignore these.

This is obviously a low pass filter (i.e., low frequency signals are passed and high frequency signals are blocked).. If w<<1/RC thenwCR<<1 and the magnitude of the gain is approximately unity, and the output equals the input. If w>>1/RC (wCR>>1 ) then the gain goes to zero, asdoes the output. At w=1/RC, called the break frequency (or cutoff frequency, or 3dB frequency, or half-power frequency, or bandwidth), the magnitude of the gain is 1/sqrt(2)@0.71. In this case (and all first order RC circuits) high frequency is defined as w>>1/RC; the capacitor acts as a short circuit and all the voltage is across the resistance. At low frequencies, w<<1/RC, the capacitor acts as an open circuit and there is no current (so the voltage across the resistor is near zero).

If Z₁ is an inductor and Z₂ is a resistor another low pass structure results with a break frequency of R/L.

A Simple High-Pass Circuit

If Z₁ is a capacitor and Z₂ is a resistor we can repeat the calculation:

and

At high frequencies, w>>1/RC, the capacitor acts as a short and the gain is 1 (the signal is passed). At low frequencies, w<<1/RC, the capacitor is an open and the output is zero (the signal is blocked). This is obviously a high pass structure and you can show that the break frequency is again 1/RC.

If Z₁ is a resistor and Z₂ is an inductor the resulting circuit is high pass with a break frequency of R/L.

This concept of a complex impedance is extremely powerful and can be used when analyzing operational amplifier circuits, as you will soon see.

Active Filters

Low-Pass filters - the integrator reconsidered.

In the first lab with op-amps we considered the time response of the integrator circuit, but its frequency response can also be studied.

First Order Low Pass Filter with Op Amp

If you derive the transfer function for the circuit above you will find that it is of the form:

which is the general form for first-order (one reactive element) low-pass filters. At high frequencies (w>>w_o) the capacitor acts as a short, so the gain of the amplifier goes to zero. At very low frequencies (w<<w_o) the capacitor is an open and the gain of the circuit is H_o. But what do we mean by low (or high) frequency?

We can consider the frequency to be high when the large majority of current goes through the capacitor; i.e., when the magnitude of the capacitor impedance is much less than that of R₁. In other words, we have high frequency when 1/wC<<R₁, or w>>1/R₁C=w_o. Since R₁ now has little effect on the circuit, it should act as an integrator. Likewise low frequency occurs when w<<1/R₁C, and the circuit will act as an amplifier with gain -R₁/R₂= H_o.

High-Pass filters - the differentiator reconsidered.

The circuit below is a modified differentiator, and acts as a high pass filter.

First Order High Pass Filter with Op Amp

Using analysis techniques similar to those used for the low pass filter, it can be shown that

which is the general form for first-order (one reactive element) low-pass filters. At high frequencies (w>>w_o) the capacitor acts as a short, so the gain of the amplifier goes to H₀= -R₁/R₂. At very low frequencies (w<<w_o) the capacitor is an open and the gain of the circuit is H_o. For this circuit w₀=1/R₂C. Therefore this circuit is a high-pass filter (it passes high frequency signals, and blocks low frequency signals.

Band-Pass circuits

Besides low-pass filters, other common types are high-pass (passes only high frequency signals), band-reject (blocks certain signals) and band-pass (rejects high and low frequencies, passing only signal areound some intermediate frequency).
The simplest band-pass filter can be made by combining the first order low pass and high pass filters that we just looked at.

Simple Band Pass Filter with Op Amp

This circuit will attenuate low frequencies (w<<1/R₂C₂) and high frequencies (w>>1/R₁C₁), but will pass intermediate frequencies with a gain of -R₁/R₂. However, this circuit cannot be used to make a filter with a very narrow band. To do that requires a more complex filter as discussed below.

High Q (Low Bandwidth) Bandpass Filters.
For a second-order band-pass filter the transfer function is given by

where w_o is the center frequency, b is the bandwidth and H_o is the maximum amplitude of the filter. These quantities are shown on the diagram below. The quantities in parentheses are in radian frequencies, the other quantities are in Hertz (i.e. f_o=w_o/2p, B=b/2p). Looking at the equation above, or the figure, you can see that as w->0 and w->infinity that |H(s=jw)|->0. You can also easily show that at w=w_o that |H(s=jw_o)|=H₀. Often you will see the equation above written in terms of the quality factor, Q, which can be defined in terms of the bandwidth, b, and center frequency, w_o, as Q=w_o/b. Thus the Q, or quality, of a filter goes up as it becomes narrower and its bandwidth decreases.

If you derive the transfer function of the circuit shown below:

High-Q Bandpass Filter with Op Amp

you will find that it acts as a band-pass filter with:

and the center frequency and bandwidth given by:

Radian frequency Hertz

The notation R₁||R₂ denotes the parallel combination of R₁ and R₂,

.

Switched Capacitor Filters

There is a special type of active filter, the switched capacitor filter, that takes advantage of integration to achieve very accurate filter characteristics that are electronically tuneable

http://www.swarthmore.edu/NatSci/echeeve1/Ref/FilterBkgrnd/SwitchedCap.html

Switched Capacitor Circuits

�� In the last decade or so many active filters with resistors and capacitors have been replaced with a special kind of filter called a switched capacitor filter. The switched capacitor filter allows for very sophisticated, accurate, and tuneable analog circuits to be manufactured without using resistors.� This is useful for several reasons.� Chief among these is that resistors are hard to build on integrated circuits (they take up a lot of room), and the circuits can be made to depend on ratios of capacitor values (which can be set accurately), and not absolute values (which vary between manufacturing runs).

The Switched Capacitor Resistor

�� To understand how switched capacitor circuits work, consider the circuit shown with a capacitor connected to two switches and two different voltages.�

If S₂ closes with S₁ open, then S₁ closes with switch S₂ open, a charge (q is transferred from v₂ to v₁ with

If this switching process is repeated N times in a time (t, the amount of charge transferred per unit time is given by

Recognizing that the left hand side represents charge per unit time, or current, and the the number of cycles per unit time is the switching frequency (or clock frequency, f_CLK) we can rewrite the equation as

Rearranging we get

which states that the switched capacitor is equivalent to a resistor.� The value of this resistor decreases with increasing switching frequency or increasing capacitance, as either will increase the amount of charge transfered from v₂ to v₁ in a given time.

The Switched Capacitor Integrator

Now consider the integrator circuit.� You have shown (in a previous lab) that the input-output relationship for this circuit is given by (neglecting initial conditions):

We can also write this with the "s" notation (assuming a sinusoidal input, Ae^st, s=jw)

If you replaced the input resistor with a switched capacitor resistor, you would get

Thus, you can change the equivalent w' of the circuit by changing the clock frequency.� The value of w' can be set very precisely because it depends only on the ratio of C₁ and C₂, and not their absolute value.

The LMF100 Switched Capacitor Filter

In this lab you will be using the MF100, or LMF100 (web page, datasheet, application note). This integrated circuit is a versatile circuit with four switched capacitor integrators, that can be connected as two second order filters or one fourth order filter.� With this chip you can choose w' to either be 1/50 or 1/100 of the clock frequency (this is given by the ratio C₁/C₂ in the discussion above),. By changing internal and external connections to the circuit you can obtain different filter types (lowpass, highpass, bandpass, notch (bandreject) or allpass).

2^nd Order Filters

Filter Type Transfer Function

Low Pass

High Pass

Band Pass

Notch (Band Reject)

The pinout for the LMF100 is shown below (from the data sheet):

You can see that the chip, for the most part, is split into two halves, left and right. A block diagram of the left half ((and a few pins from the right half) is shown below.

The pins are described on page 8 of the datasheet. I will describe a few of them here:

50/100 - determines if the value of w' is w_CLK/100, or w_CLK/50.

CLK_A - is w_CLK.

INV_A - the inverting input to the op-amp

N/AP/HP_A - an intermediate output, and the non-inverting input to the summer. Used for Notch, All Pass or High Pass output.

BP_A - another intermediate output, the output of the first integrator. Used for Band Pass output.

LP_A - the output of the second integrator. Used for Low Pass output.

S1_A - an inverting input to the summer.

S_AB - determins if the switch is to the left or to the right. That is, this pin determines if the second inverting input to the summer is ground (AGND), or the low pass output.

The two integrators are switched capacitor integrators. Their transfer functions are given by,

where w' is w_CLK/100, or w_CLK/50, depending on the state of the 50/100 pin. Note that the integrator is non-inverting.

A Typical Circuit.

The diagram below shows one of the modes (mode 1) of operations (pages 13 through 20 of the datasheet).

Let's analyze this circuit and try to derive the filter specifications as given in the datasheet, and given below.

The low pass (LP_A) output is easily given in terms of the band pass output (BP_A), as well as the band pass output as a function of the summer (SUM, not labeled on diagram).

The summer output (SUM) is simply the output of the op amp (N_A) minus the lowpass output (LP_A). However we can see that the op amp is set up as the inverting summing circuit. So

Replace SUM on the left hand side using equation (2) from above, and LP_A using equation (1).

Rearranging brings

Equating this with the transfer function for a bandpass circuit

yields,

which is what we were trying to derive.

Similarly, the relationship between low pass and band pass, equation (1), can be used to find the low pass transfer function. The notch filter transfer function is derived in the same way.