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r e t l i F e v i t Ac n g i s e D s r e n n i g e B r Fo RO UN DED GE 0 HB roughout h t d e s u y are widel le, we look at the s r e t l fi e Activ this artic lear away some n I . s c i n o electr y ers & c t l fi e v i t c g this ver a basics of ystery surroundin of the m nteresting topic. i By ELMO V. JANSZ A filter is one of the most common types of circuit used in electronic equipment. By definition, a filter passes some fre­quencies and suppresses or attenuates others. Filters can be active or passive, depending on their con­ struction. Passive filters use passive components such as resis­tors, capacitors and inductors, whereas active filters include an amplifying device, such as a transistor or operational ampli­fier, in addition to a number of passive components. The presence of the amplifier gives the filter very good isolation between its input and output and a certain amount of amplification as well. In this article, we shall learn how to design active filt­ers using simple calculations. Let us start by establishing a few basic ideas about active filters. Fig.1 shows the idealised amplitude response of a low-pass filter. A low-pass filter is one that passes all frequencies up to a point and heavily attenuates or suppresses Fig.1: idealised amplitude response of a low-pass filter. Fig.2: idealised amplitude response of a high-pass filter. frequencies beyond this point. The amplitude response is a plot of the gain of the filter against frequency. The gain is calculated by divid­ing the output voltage by the input voltage in the equation: G = 20 log10(Vo/Vi) where G is the gain expressed in decibels; Vo is the output voltage; and Vi is the input voltage. In Fig.1, the frequency fc is called the cut-off frequency while region AB in which the gain is constant is called the filter’s passband. Beyond fc, the gain drops rapidly and this region is called the stop-band. The rate at which the line BD falls is measured in dB/ octave or dB/decade. The is the “slope” of the filter. An octave is a doubling or halving of frequency; ie, for a frequency of 2kHz, octaves above are 4kHz, 8kHz and so on, while octaves below are 1kHz, 500Hz, etc. Decades are a ten-fold increase or decrease in frequency. For a January 1994  37 Fig.3: response characteristic of a practical lowpass filter. Fig.5: basic circuit for a first order low-pass Butterworth active filter. where G is the passband gain in decibels; W is the normalised angular frequency; and n is the order of the filter. The normalised frequency is given by W/Wc where W is the frequency in question and Wc is the cut off frequency Fig.4: a filter with ripples in the passband is called a Chebyshev filter. frequency of 2kHz, decades above are 20kHz, 200kHz and so on, while decades below are 200Hz, 20Hz, 2Hz, etc. We now come to another important definition, the “order” of a filter. This is the rate at which the line BD in Fig.1 falls off, or the filter’s ability to attenuate frequencies outside its passband. A “first order” filter has an attenuation outside its passband of 6dB/octave or 20dB/decade. The order of a filter is also referred to as its roll-off or fall-off. A “second order” filter has a roll-off of 12dB/octave or 40dB/decade; ie, twice that of the first order filter. A third order filter will have a roll-off of three times that of a first order filter and so on for higher order filters. A high-pass filter is the complement of a low-pass filter and will have an idealised response characteristic as shown in Fig.2. Notice that frequencies below fc are attenuated heavily. The roll-off has the same values as stated above but in this case will have the opposite sign. A practical low-pass filter will have the response charac­ teristic shown in Fig.3. The cut-off frequency in this case is not a sharp transition point as shown in Figs.1 & 2 but the frequency at which the gain is reduced by 3dB, from its passband value. A filter with a response as shown in Fig.3 – ie, one having a flat response in the passband – is called a Butterworth filter. A filter could also have a response as shown in Fig.4, with ripples in the passband. This is called a Chebyshev filter. The shape of the filter’s response is determined by a con­stant (alpha) called the Damping Factor. There are other filters called Cauer, Bessel and Thompson filters but in this article we shall confine ourselves to Butterworth filters, as they are the most popular due to their design simplicity. The general equation for a Butterworth low-pass filter of order n is given by: Gain = 20 log [G/(1 + W2n)½ ] 38  Silicon Chip Design of a first order filter Let us now design a first order low-pass Butterworth active filter. The basic circuit is shown in Fig.5. The portion within the dotted line is a low-pass passive filer. The operational amplifier is connected in the non-inverting mode. The cut-off frequency (fc) and passband gain (G) are given by the following formulas: fc = 1/(2πRC) G = 1 + RB/RA Suppose we wish to construct a low-pass filter with a cut-off frequency of 2kHz. We start by selecting a value for C. Let this be .022µF. By using the formula fc = 1/(2πRC), we arrive at: R = 1/(2π x 2 x 103 x 0.022 x 10-6) = 3.617kΩ This would be selected as 3.6kΩ, using the closest value in the E24 (5%) range. Let us set the passband gain required equal to 2. There­ fore, using the formula for gain: RB/RA = G - 1 = 2 - 1 = 1 Therefore, we can make RA equal to RB and set both at 10kΩ. A 741 could be used for the operational amplifier and then you have your basic first order low pass filter. By interchanging C and R, you can produce the corresponding high pass filter. Second order low-pass filter The basic circuit of a second order low-pass filter is shown in Fig.6. Here again a network of passive components is placed around an op amp. Second order active Fig.6: basic circuit of a second order low-pass filter. Fig.7: the circuit for a unity gain low-pass active filter. filters are also often referred to as Sallen-Key filters. This circuit has two RC networks, hence it is a second order filter. The cut-off frequency fc for this filter is given by: fc = 1/2π(R1.R2.C3.C4)½ and the mid-band gain is given by: G = 1 + RB/RA In practice, two versions of this circuit are possible: either a filter with a passband gain of unity, or a filter with equal components; ie, R1 = R2 and C3 = C4. Unity gain For this example, it is customary to make R1 = R2 and then C3 and C4 are fixed in the ratio C3 = 2C4, in order to satisfy the damping factor (alpha) requirements for a Butterworth re­sponse. The required circuit is shown in Fig.7. Note that the op amp has been configured for unity gain, as a voltage follower, by connecting its inverting input to its output. Using the formula fc = 1/2π(R1.R2.C3.C4)½ and remembering that R1 = R2 = R and C3 = 2C4 (ie, if C4 = C then C3 = 2C), the above equation can now be written as: fc = 1/2π(R x R x 2C x C)½ = 1/2πCR√2 If we select R = 10kΩ and if a cut-off frequency of 1kHz is desired, C can be calculated from the above equation to give: C = 1/(2π x 103 x 10 x 103 x √2) = 0.01µF. Therefore, we can select C3 = 0.02µF and C4 = 0.01µF. The final design is now R1 = R2 = 10kΩ; C3 = 0.02µF; C4 = 0.01µF. Fig.9: unity gain second order high-pass filter. The passband gain for a Butterworth filter is defined by the equation: G=3-α and since α = √2, G = 1.586. Unfortunately, this is the only gain that will permit the circuit to operate correctly. By selecting R = 5kΩ and a cut-off frequency of 1kHz, the above equation gives C = .032µF. A .033µF polyester capacitor would be suitable. The gain of G = 1.58 can be satisfied by making RB = 27kΩ and RA = 47kΩ (using preferred values). The final circuit is shown in Fig.8. Second order high pass filters High pass filters can be set up by interchanging the R and C components of the low-pass circuit. Two versions of this cir­cuit are possible, as for the low-pass configurations – ie, a unity gain circuit and an equal component circuit. These are shown in Figs. 9 & 10. For Fig.9, if C1 = C2, then R4 = 2R3 in order to satisfy the damping requirements for a Butterworth response. Equal component filter If R1 = R2 = R and C3 = C4 = C, then the equation fc = 1/2π(R1.R2.C3.C4)½ becomes fc = 1/2πRC Fig.10: equal component high-gain Butterworth filter. Fig.8: equal component low-pass Butterworth filter. For the equal components version of Fig.10, if R3 = R4 and C1 = C2, then the gain is fixed by the equation: G=3-α With alpha = √2, this again fixes the gain at 1.586. Higher order filters can be obtained by cascading appro­ priate filter sections. For example, a fifth order filter can be produced by cascading two second order and one first order sec­tions. Filters can also be set up to pass a band of frequencies and so are called band-pass filters. A band-pass filter can be obtained by cascad­ing an appropriate high-pass and SC low-pass section. January 1994  39