Laboratory of the circuits and signals Laboratory work No. 4 ACTIVE FILTERS Objectives: to get acquainted with active filter circuits and parameters, design methods, build and investigate active LPF, HPF and BPF. Introduction Main Concepts A filter is a circuit that passes certain frequencies easily, and attenuates all other frequencies. An active filter is a circuit that includes an RC filter network followed by an op-amp to provide gain and impedance characteristics. A low-pass filter passes low frequencies from DC up to the cutoff frequency f c. At f c the response is down 3 db or 0.707 from the response in the band-pass. The passband of a filter is that group of frequencies easily passed. The bandwidth of a low-pass filter is equal to f c. BW = f c A high-pass filter rejects all frequencies below f c, and passes all frequencies above f c. A roll-off rate of -6 db/octave or 20 db/decade occurs for a single-chain filter stage. A single chain stage consists of one resistor and one capacitor. A band-pass filter passes all frequencies between an upper and lower f c. All other frequencies above or below these frequencies are attenuated. The bandwidth is found by BW = f c2 f c1. The center frequency (f 0 ) is the mean of the two cutoff frequencies and is found by The bandwidth of a band-pass filter is BW = f 0 /Q. A band-stop filter rejects all frequencies between two cutoff frequencies and passes all others. Other popular names for a band-stop filter are notch filter, band-reject filter, and band-elimination filter. Filter Response Characteristics A very common active filter is the Butterworth. A Butterworth filter provides a very flat response in the passband and a roll-off rate of 20 db/decade. The Butterworth filter does have a phase shift, and for this reason it is not often used in pulse application. Overshoot will occur with pulses applied. Another filter, the Chebyshev, has the characteristic of a very rapid roll-off rate greater than 20 db/decade/chain. This filter will have ripples in the response within the passband. The Bessel filter is used for filtering pulse waveforms. There is little phase shift, so pulse waveforms are not distorted.
The Elliptical (Courer) filter will have ripples in the response in the passband and within passband The cutoff frequency f c of an active filter is determined by the values of R and C.A first-order (single-chain) filter has f c = 1/2π RC. The number of chains determines the roll-off rate. Each chain adds a roll-off rate of 6 db/octave or 20 db/decade. Active Low-Pass Filters An active low-pass filter consists of an RC filter network followed by an op-amp amplifier. Low-pass filters can be cascaded to obtain a greater roll-off rate. For example, a two-chains filter followed by another two-chains filter will produce a roll-off rate of 80 db/decade. Active High-Pass Filters A high-pass active filter follows the same theory as a low-pass filter. The positions of the resistors and capacitors are reversed. High-pass filters can be cascaded in a similar manner to low-pass filter. Active Band-Pass Filters An active band-pass filter can be made using cascaded high-pass and low-pass filters. Another band-pass filter is the multiple-feedback filter. This filter uses a high-pass and a low-pass section providing two feedback paths. A state-variable filter is often used as a band-pass filter. State-variable filters use a summing amplifier and two integrators to provide the band-pass output. Active Band-Stop Filters Band-stop filters can use a state-variable filter or a multiple-feedback filter. The order of the low pass filter is calculated from: log (10 n α min / 10 α max/10 [ 1) /(10 1) ] 2log( ω p / ω s ) here α min - attenuation is pass-band, in db; α max attenuation is stop-band, in db; ω s - stop-band boundary frequency, ω p pass-band boundary frequency. The order of the high pass filter is calculated from: log (10 n α min / 10 α max/10 [ 1) /(10 1) ] 2log( ω s / ω p )
2nd Order Opamp Filters and their diagrams The figures below illustrate using opamps as active 2nd order filters. Three 2nd order filters are shown, low pass, high pass, and bandpass. Each of these filters will attenuate frequencies outside their passband at a rate of 12dB per octave or 1/4 the voltage amplitude for each octave of frequency increase or decrease outside the passband. First order low or high pass cutoff frequency (-3dB point) = 1/(2π *R*C) 2nd order low or high pass cutoff frequency (-3dB point) = 1/2π (R1*R2*C1*C2)^.5 Linear network Magnitude scaling by scale factor K m : 1. Multiply all resistances and inductances by K m. 2. Divide all capacitances by K m. 3. For r m type controlled source (i.e, CCVS) multiply the parameter r m by K m. 4. For g m type controlled source (i.e, VCCS) divide the parameter g m by K m. 5. Parameters of VCVS and CCCS remain unchanged. 6. Ideal OpAmp remain unchanged. Linear network Frequency scaling by scale factor K f : 1. Divide all inductances and capacitances by K f. 2. All resistance values and controlled sources parameters remain unchanged. 3. Ideal opamp remain unchanged. RC common chains, build on 1 Ω resistor and 1 F capacitor, has cutoff frequency For the low or high pass filter RC chain it is possible suppose that cutoff frequency passband boundary frequency ω p. ω C = 1 rad/sec. ωc is the same as
Work procedures: 1. Turn on your work place computer, load the Windows OS, find and load EWB program. 2. Data for your investigations are given in the table 1. Table 1. Initial data (Brickwall specification parameters and filter resistance values) No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 α,db 2 3 2 3 2 3 2 3 2 3 2 3 2 3 min α max,db 20 22 24 26 27 28 27 26 25 24 23 22 21 20 f s1, MHz 1 2 3 4 5 6 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8 f p1, MHz 2 4 6 8 10 12 3 5 7 9 11 13 15 16 f p2, MHz 4 8 12 16 20 24 5 10 14 18 22 30 25 30 f s2, MHz 8 16 24 32 40 48 10 20 28 36 44 60 50 60 R, k Ω 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Using initial data from the table 1 and scaling technique find LPF and HPF chains capacitances. Results put into table 2. Low pass passive filter High pass passive filter f c, MHz R, k Ω C, nf f c, MHz R, k Ω C, nf Table 2 3. Build passive LPF and HPF using single RC chains as shown in the figures 1 and 2 (do not forget change values of circuit components. Voltage source amplitude 1V, frequency 1 MHz). Measure in Bode plots cutoff frequencies and transient slope (roll-off rate). If measured cutoff frequencies differs from given (see table 1), scale capacitances to obtain necessary cutoff frequency. Results put into the table 3. Grab the screen shots of the build circuits and their Bode plots. Fig. 1. LPF passive RC chain and its magnitude response
Fig. 2. HPF passive RC chain and its magnitude response Low pass passive filter f c, MHz Roll-off, db/decade High pass passive filter C cor., nf f c, MHz Roll-off, db/decade C cor., nf Table 3 4. Build active LPF and HPF using single RC chains as shown in the figures 3 and 4 (do not forget change values of circuit components. Voltage source amplitude 1V, frequency 1 MHz, voltage source resistance 75 Ω, load resistance 75 Ω. Measure in Bode plots cutoff frequencies and transient slope (roll-off rate). If measured cutoff frequencies differs from given (see table 1), scale capacitances to obtain correct cutoff frequency. Results put into the table 4. Grab the screen shots of the build circuits and their Bode plots. Table 4 f c, MHz Low pass active filter Roll-off, db/decade High pass active filter C cor., nf f c, MHz Roll-off, db/decade C cor., nf Fig. 3. LPF active RC chain and its magnitude response
Fig. 4. HPF active RC chain and its magnitude response 5. Build active fifth order LPF and measure its frequency response (Fig. 5). Results put into table 5. Fig.5. Fifth order LPF active Butterworth filter and its magnitude response Low pass active fifth order filter f c, MHz Roll-off, db/decade Table 5
6. Start BPF filter design. At first calculate LPF and HPF necessary order. Initial data take from the table 1. Results put into the table 6. Low pass passive filter High pass passive filter n calculated n chosen n calculated n chosen Table 6 7. Build BPF as shown in the figure 6. Do not forget, that number of your filters chains may be is different then in shown diagram. Grab the screen shots of the build circuit and its Bode plot. Fig.6. Fourth order active BPF and its magnitude response 8. Sketch by hand brickwall specification of your BPF. If measured cutoff frequencies differs from given (see table 1), scale circuit capacitors and repeat measurements. Final results put into the table 7. Low pass transient section High pass transient section f c, MHz Slope, db/decade C cor., nf f c, MHz Slope, db/decade C cor., nf Table 7
Work report 1. Work objectives. 2. Initial data for investigations (table 2 and capacitance calculation description). 3. LPF and HPF passive single chains investigation (screen shots with EWB circuit diagrams and Bode plots, table 3). 4. LPF and HPF active single chains investigation (screen shots with EWB circuit diagrams and Bode plots, table 4). 5. BPF order calculation (formulas, table 4). 6. BPF investigation (screen shots with EWB circuit diagram and Bode plot, table 4). 7. Conclusions. Literature: Raymond A.DeCarlo/PEN-MIN LIN. Linear circuit analysis. Prentice Hall, Englewood Cliffs, New Jersey. Pp. 600-605, 738-762. Control questions 1. Filters classification. 2. Operational amplifiers connection types (sketch diagrams). 3. Explain, why opamp could be used as buffer. 4. Sketch examples of the first order active LPF and HPF. 5. Active filters advantages in comparison to passive filters. 6. Explain, why RC chains resistances in active filters must be at lest some kilohertz. 7. Butterworth type filters: circuit diagrams and magnitude response. 8. Magnitude and frequency scaling. Solve problem as in following example: given LPF passive chain with parameters: 9. L=1mH, C=1 nf, R 0 =1000 Ω, f c =40 khz. Using these parameters as basis build LPF chain with R 0 =100 Ω and f c =100 khz. 10. Explain Brickwall specification and sketch its example for LPF. 11. Explain, what is transient slope (roll-off) and how it depends on filter order. 12. Explain, what is ripples in the filters magnitude response. What kind of active filters suffers from them.