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.