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Design your next filter the easy way  this article by Tim Orr shows you how. THERE ARE THREE main types of filter  lowpass, bandpass, and highpass. Each does more or less what its name implies. A low pass filter passes all frequencies below the socalled ' roll off' point and increasingly blocks all frequencies above this point. A bandpass filter passes all frequencies above a lower 'rolloff' point and below a higher rolloff point. A highpass filter passes all frequencies above the rolloff point. Firstly, consider the simple lowpass filter shown in Fig. 1a. The frequency response ( shown in Fig 1b) is nearly flat until the break point shown as fb. Above this point the response rolls off at 6 dB/octave. The break point is defined as the frequency where the resistance equals the capacitive reactance. At this point the output is attenuated to 0.707 (3 dB) of the input. Although the resistance equals the capacitive reactance, the output is not half of the input. It is the vector sum of the two and hence is 0.707 of the input. As the frequency response is a complex curve it is commonly approximated by a straight line. Such a line is called an asymptote ( Fig. 1c). Note the frequency response graph uses logarithmic scales, octave or decades along the frequency axis, and dBs along the vertical axis representing output voltage divided by input voltage. Phase shift with respect to frequency is often plotted as in Fig. 1d. Phase and frequency response plots are also known as Bode diagrams and are most useful in showing a filter's performance. Note that for the lowpass filter of Fig. 1a, phase shift starts at 0°, is 45° at fb and approaches 90° as frequency approaches infinity. This is not an active filter. It is made up from passive components and its output cannot be loaded substantially without changing its performance.
Fig. 1c. Approximation to response. Fig. 1b. Frequency response Fig. 1d Phase shift y Frequency plot. Fig. 1e. Active filter to perform the same task as the passive circuit of Fig. 1a. Summary of low pass filter of Fig. 1 Figure le shows the same filter in active form, the op amp being used as a voltage follower serving only to isolate the filter's output. This configuration is known as a first order filter  the expression 'first order' being an indication of the rolloff slope. When a steeper slope is required, a higher order filter ( that is, one with more elements) must be used. These are dealt with later. Passing Highs The simple highpass filter shown in Fig 2a is the complement of the lowpass filter  the elements have simply been interchanged. Hence the complementary curves of Fig 2b. Note the break point and rolloff slope are similar. Passing bands A simple bandpass filter is shown in Fig 3a. Although it uses an inductor this is only to illustrate the bandpass theory. The frequency response ( Fig 3b) is symmetrical, rolling off at 6 dB/octave on either side of its peak. This filter is called a second order filter because it has two reactive sections ( L and C). The C produces the +6 dB/octave portion of the slope, the L the 6 dB portion. The response of the filter peaks, and the slopes become much steeper where these two slopes meet. The sharpness of the peak determines the quality of the filter ( Q). Resonance occurs at the frequency known as the centre frequency  shown on our drawing as fc. Fig. 2a. Simple highpass active filter. The bandpass filter is socalled because it passes signals within a certain bandwidth. This bandwidth is defined as being the frequency range contained between the two points that are 3 dB below the resonant peak. Thus there is a fixed relationship between centre frequency (fc), bandwidth ( fbw), and Q. The centre frequency is fc = 1/277VLC. Fig. 2b. Frequency response (above) and phase response (left) of the high pass filter. Summary of the high pass filter of Fig. 2 Filter type High pass Filter order First order Roll off slope 46dB/ ocme Break point fb fb= 1 / 2 pi R Hz Phase shift at fb 45° TABLE 2 Fig. 3b. Bandpass frequency response. Fig. 3c Bandpass phase response. Fig. 3d. Effect of varying 0 on the frequency and phase characteristics This is only approximate as it assumes that the value of R is relatively low. As R decreases, the Q increases. Thus R has the effect of damping the resonances, and as it approaches zero ohms, Q approaches infinity. The phase shift is shown in Fig 3c. As this filter is a second order structure the total phase movement will be twice that of a first order structure, i.e. 180°. Figure 3d shows the phase and frequency responses for different values of Q. Note that a high Q has a very rapid rate of change of phase, a low Q has only a slow rate of change. Time response Bandpass filters have a time response as well as a frequency response. When an impulse is applied to a bandpass filter it rings ( Fig 3e). The filter oscillates at the centre frequency ( fc), the amplitude of oscillations decaying exponentially with time. The ringing time Tr is the time taken for the oscillations to decay to 37% of their initial value. Ringing time is related to Q and fc by the following equation: Tr = Q/2 pi fc In practice it may prove difficult accurately to measure the Q of a highQ filter because the bandwidth is narrow. However if the filter can be made to ring a reasonably accurate measurement of Q can be obtained by measuring Tr and fc. Notch filters Another common type of filter is the bandreject or notch filter. There are many ways of building these, one way is shown in Fig 4. The input signal is subtracted from the bandpass output. By adjusting Ra with respect to R, complete cancellation can be obtained at fc. Thus the centre frequency of the bandpass filter is the centre frequency of the notch. The depth of this notch can be varied by altering the value of Ra. Very deep notches are possible: 50 dB being readily obtainable. As the Q of the bandpass filter is increased so is the Q of the notch filter. Note that Ra must be adjusted for each value of Q. Filter orders Consider the ideal filter shown in Fig 5a. Its response is flat right up to the break frequency with frequencies above fb attenuated to zero. It looks really... TABLE 3. Summary of bandpass filter Fig. 4. Notch filter using OpAmps. ... good but in real life filters like this don't exist! There is nevertheless a frequent need to design filters with very steep rolloff slopes. This is achieved by designing filters with lots of sections  thus increasing the filter order. Each reactive element in a filter increases the filter order by one, therefore a low pass active filter with three capacitors is known as a third order filter and will have an ultimate rolloff of three times 6 dB/octave i.e. 18 dB/octave. Unfortunately there is more to designing a third order lowpass filter than just sticking three first order circuits in line astern. That merely results in a very soggy curve! The filter should be flat in the pass band, then it should turn over and rapidly assume its rolloff slope. Examples of maximally flat filters are shown in Figs 5b and 5c. The effect of order number on a bandpass filter is shown in Fig 5d. Later in this article circuit diagrams and design charts are shown for various filter types and order numbers. It might seem that to persuade a filter to approach its ideal response all that is needed is to increase the order number. This is of course so  but there are problems in obtaining components of sufficient accuracy. An eighth order filter for example needs components having values within 1%  possible with resistors but virtually impossible for capacitors. Filter Shape The type of filter required to do a certain job will depend on what parameters are most important. Three basic characteristics must be considered ( high pass and lowpass only). 1. Good transient response 2. Maximum flatness within the pass band 3. Steep rolloff slope. Filters have been categorized into three basic types for simplicity. BESSEL FILTER: phase changes almost linearly with frequency, useful for systems where a good transient response is required  such as joining all the little pulses on the output of a digitalto analog convertor. Very poor initial roll off. BUTTERWORTH FILTER: This has the flattest pass band possible. Its two other parameters are a compromise  a reasonable overshoot and a fairly fast initial rolloff. CHEBYSHEV FILTER: This has a small amount of ripple in its pass band, a very fast initial roll off but a poor transient response. Rolling your own In all the examples which follow the filters have been designed for operation at 1 kHz. To change the operating frequency resistor/s RF must be scaled accordingly ( note: resistors RD are not changed). For example if the filter is required to operate at 250 Hz then RF must be multiplied by 1 ÷N . Figure 7 shows a first order low pass filter. Figure 8a, b, and c shows second, third and fourth order filters. The total design procedure is as follows: 1. Decide which type of filter is required  low, bandpass or high. 2. In the case of low or highpass decide which type of response is required, Bessel, Butterworth of Chebyshev. 3. Decide what filter order is needed. This will lead you to a particular order filter with components shown scaled for 1 kHz. 4. Scale the resistors RF accordingly. 5. Build and test the filter. As an example let us design an audio scratch filter having a break frequency of 7.5 kHz and an attenuation of more than 20 dB at 15 kHz. The first decision is type of response required. A rolloff of more than 20 dB/ octave is quite steep and so the Bessel filter is ruled out. The Chebyshev has poor transient response and we'd hear it ringing at 7.5 kHz. So we're left with the Butterworth. Next comes filter order. Third order gives  18 dB/octave: this is not enough. Fourth order gives  24dB/octave. So a fourth order Butterworth filter it is. Break frequency is 7.5 kHz so resistors RF and RF2 must be divided by 7.5. This gives the following rather funny values  RF1 = 1k42, RF2 1k42, RD1 = 5k9,
RD2 = 48k7, C = 15 nF with component tolerances of 5%. We must now fit preferred values to our theoretically derived numbers. Resistor RD2 can be 47k. RD1 6k2 (just over the limit of tolerance). Resistors RF1 and RF2 are a bit of a problem. The solution is to use the nearest 1% tolerance resistor or use 1K5. This will lower the break frequency by 6% but as this is an audio filter it won't matter too much. Figure 9 provides design data for highpass filters. The design procedure is exactly the same as for the lowpass designs. A few problems may occur with the final results. One is that these filters have a voltage gain in their pass band  so you might find that although you have the required frequency response there is unexpected signal gain. This may cause some problems with op amp bandwidth. As a rule of thumb, op amps should have 10 to 100 times more bandwidth than the product of the filters' maximum operating frequency multiplied by the individual stage gain of each section. If the op amp runs out of bandwidth or introduces a phase shift the filter won't work properly. For the examples given if you use a 741 as the op amp the frequency limit is about 10 kHz. With an LM 318 the limit can go right out to 200 kHz. Another problem is the range of values of RF. If RF is too small large currents flow from the op amp and this may affect filter performance. If RF is too large there may be hum pickup, and dc offset voltages due to bias currents. So keep RF between 1 k and 100 k. If RF appears to need to exceed this range scale the capacitor accordingly.
Fig. 8b. Third order lowpass filter. To alter break frequency (in this case 1 k Hz)scale resistors accordingly. Fig. 8c. Fourth order lowpass filter.
Bandpass filters Several second order bandpass filters can be cascaded to produce a different response shape, which, like those discussed earlier for low and highpass filters, can be optimized to give maximum rolloff or maximum pass band flatness. Such filters do however tend to be difficult to design and so only second order filters will be discussed. Figure 10 shows a simple bandpass filter known as a multiple feedback circuit. This circuit can provide only low orders of Q ( up to about 5). It will probably oscillate if designed to provide higher Q. Note that a high Q implies a large gain at centre frequency. Care must therefore be taken to ensure that the op amp has sufficient bandwidth to cope. The design chart is shown in Fig 10  once again all values are shown for 1 kHz. The design procedure is to first choose the Q factor and then perform the frequency scaling. For instance if the centre frequency is to be 250 Hz then multiply both R1 and R2 by four. If a high C1 is required then you must use a multiple op amp circuit such as the 'state variable' or ' Bi  Quad' circuits shown later. Both circuits can achieve Qs as high as 500. Statevariable filter A state variable filter is shown in Fig 11. This filter has three main features: 1. It can provide a stable high Q 2. It is readily tuned 3. It is versatile, providing bandpass, lowpass and highpass outputs all at the same time. The Q of this filter is determined by the ratio of resistors RA and RB where RA/RB=3Q1. The resonant frequency fc is 1/2 pi 13C Note that there are two Cs and two RFs in the circuit. If the filter is to be tunable then both RFs should be changed by an equal amount ( the RFs can of course be a dual potentiometer). Note too that Q and fc are independent of each other so as the resonant frequency is changed Q remains constant  and vice versa. The requirements placed on the op amps in the statevariable filter are less than those for the multiple feedback circuit. The op amps need only have an open loop gain of 30 at the resonant frequency. Suppose we have a Q of 100 and an fc of 100 kHz. Then the open loop gain is 300, the frequency is 10 kHz and so the gain bandwidth product needed is 3 MHz. Fig. 1 lb. The state variable filter is called a universal filter because it can give band, low and highpass outputs  as shown above. Note that all these responses are second order in nature.
When using a high Q, care must be taken with signal levels. The gain of the filter is +CI at resonance so if you are filtering a 1V signal with a Q of 100 you could expect a 100 V output signal! National Semiconductor manufactures an active filter IC which is a four .op amp network which can be used to make up statevariable filters with Qs of up to 500 and frequencies to 10 kHz. The device is designated AF 100. Bi  Quad filter Figure 12 shows a Bi  Quad active filter. It looks very similar to the statevariable filter but the minor changes cause it to behave quite differently. Unlike the statevariable filter it has only a bandpass and a highpass output. The resonant frequency is 1 fc  27rCR F V
Notch filters Two notch filters were shown earlier in this article. These worked by mixing a bandpass signal with the original or by mixing the low and highpass outputs. There are however many other methods of producing a notch response. The twinT circuit shown in Fig 14 is very interesting  a notch response is obtained using resistors and capacitors only. However as this is purely a passive device only a low Q can be obtained. The circuit is not used a great deal  perhaps because no less than six components determine its notch frequency. However it is of interest to note that when the twinT is placed in the feed back loop of a highgain amplifier a bandpass response will be obtained. If R is made variable it is possible to move the center frequency, although in doing so the Q varies. This effect has been exploited as the basis of many wahwah effects units!
Allpass filter Another way of obtaining a notch is the all pass filter shown in Fig 16. The frequency response might at first sight make this the most useless of all possible filters  it's flat! However the circuit produces a phase shift which goes from 180°, through 90° at fc – to 0 °. By cascading two such filters the phase shift is doubled: if we then mix the phasedelayed signal with the original a notch response is obtained. This is because at fc the two signals have the same magnitude but opposite phase and so cancel out. If the notch is to be tunable the RC time constants must be variable. Just one R may be varied if a small range is required  otherwise vary both Rs. Comb filter Several notches can be produced by cascading notch filters. This type of filter is called a comb filter. Every notch in the comb requires two all pass filters. The notches can be made to move up and down in frequency by making the Rs variable. This method is used to produce the 'phasing' effect used by rock groups. Figure 17 shows a small section of such a unit. Here a CMOS chip is used to provide a matched set of six MOSFETS. A common voltage controls the MOSFET's channel resistance. Thus as the control voltage varies so do the six MOSFET resistors  and the three notches move along the frequency axis in unison.
delay times. Another type of comb filter is shown in Fig 18. Here a time delay line is used instead of a phase delay line. This produces a large number of notches linearly spread along the frequency axis, their spacing being determined by the A bucketbrigade device may be used to implement the time delays (which can of course be made variable). This type of filter is known as a flanger and is used to generate high quality phasing effects. An even more impressive sound can be produced by adding feedback around the delay line: this produces a multi peak, highQ filter which makes very interesting 'musical' sounds when swept through its range.
Variable tuning A common requirement is for a variable centre or cutoff frequency. This causes problems with filters of orders more than two simply because it is difficult to obtain potentiometers with more than two ganged sections. Lots of companies make them but just try to buy one when it's needed! One solution is to use mark/space ratio modulation ( Fig 19). This has the advantage of offering continuously variable control over a range of about 100:1. Lots of sections may be used and they'll all track. As an example an eighth order four transmissiongates/ pack variable frequency filter can be made using a couple of CD4016s. Note though that:  1. The switching waveform must be several times higher in frequency than the highest frequency to be filtered. 2. More circuitry, to generate the switching waveform is required. 3. Switching noise tends to be generated. Audio circuits Active filters are widely used in equalizing audio signals in applications varying from simple tone controls in hifi systems to parametric equalizers in recording studios. Figure 20 shows a commonly used tone control with bass and treble functions. Cut and lift ranges are 20 dB. More flexible control can be obtained by using a multi band graphic equalizer such as that described as a constructional project in ETI's June 1977 issue. Testing Once the process of designing active filters has been reduced to a simple procedure, testing should be made as simple as possible too. The most basic method is to use a swept sinewave oscillator ( Fig 21). An XY oscilloscope is used to display frequencies logarithmically against amplitude (displayed linearly). The ideal display would be log amplitude but this is hard to do. The beauty of this method of testing is that the display is in real time so any changes appear instantly on the oscilloscope. If high Qs or rapid rolloffs at low frequencies are involved, then the sweep time will have to be reduced, otherwise the effects of ringing will 'time smear' the display. The harmonic distortion of the sinewave, can be quite large ( 0.5 to 2.0%) without causing too much of a display problem for most filter designs.
