2.161 Signal Processing: Continuous and Discrete Fall 2008


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1 MT OpenCourseWare Signal Processing: Continuous and Discrete Fall 00 For information about citing these materials or our Terms of Use, visit:
2 MASSACHUSETTS NSTTUTE OF TECHNOLOGY DEPARTMENT OF MECHANCAL ENGNEERNG.6 Signal Processing Continuous and Discrete OpAmp mplementation of Analog Filters. ntroduction Practical realizations of analog filters are usually based on factoring the transfer function into cascaded secondorder sections, each based on a complex conjugate polepair or a pair of real poles, and a firstorder section if the order is odd. Any zeros in the system may be distributed among the second and firstorder sections. Each first and secondorder section is then implemented by an active filter and connected in series. For example the thirdorder Butterworth highpass filter would be implemented as s 3 H(s) s 3 0s 00s 000 s s H(s) s 0s 00 s 0 as shown in Fig.. The design of each loworder block can be handled independently.!! " & & " > Figure : A thirdorder Butterworth filter (a) as a single thirdorder section, and (b) as a secondorder and firstorder section cascaded. Statevariable active filters The statevariable filter design method is based on the block diagram representation used in the socalled phasevariable description of linear systems that uses the outputs of a chain of cascaded integrators as state variables. Consider a secondorder filter block with a transfer function Y (s) b s b s b 0 H(s) () U(s) s a s a 0 D. Rowell October, 00
3 and split H(s) into two subblocks representing the denominator and numerator by introducing an intermediate variable x and rewrite X(s) H (s) () U(s) s a s a 0 H (s) Y (s) b s b s b 0 (3) X(s) so that H(s) H (s)h (s). The differential equations corresponding to Eqs. () and (3) are d x dx a a 0 x u () dt dt and d x dx y b b b 0 x. (5) dt dt Rewrite Eq. () explicitly in terms of the highest derivative d x dt dx a a 0 x u. (6) dt Consider a pair of cascaded analog integrators with the output defined as x(t), as shown in Fig., so that the derivatives of x(t) appear as inputs to the integrators. Note that Eq. J N J Figure : Cascaded integrators with output x(t). gives an explicit expression for the input to the first block in terms of the outputs of the two integrators and the system input, and therefore generates the block diagram for H (s) (Eq. (5)) shown in Fig. 3. K J N J Figure 3: State variable realization of H (s) X(s)/U(s).
4 O J > > > K J N J Figure : Full secondorder state variable realization. Equation (5) shows that the output y(t) is a weighted sum of x(t) and its derivatives, leading to the complete secondorder state variable filter block shown in Fig.. This basic structure may be used to realize the four basic filter types by appropriate choice of the numerator. Figure 5 shows how the output may be selected to achieve the following transfer functions: H lp (s) Y (s) a 0 U(s) s a s a 0 a unity gain lowpass filter (7) H bp (s) Y (s) a s U(s) s a s a 0 a unity gain bandpass filter () H hp (s) Y 3 (s) s U(s) s a s a 0 a unity gain highpass filter (9) H bs (s) Y (s) s a 0 U(s) s a s a 0 a unity gain bandstop filter (0) ; " J F ;! D EC D F ; F 7 : : : ; M F Figure 5: State variable implementation of various filter types. 3
5 ". Opamp Based StateVariable Filters Electronic implementation of the block diagram structure of Fig. 5 involves weighted summation and integration. These two operations can de achieved by the two opamp circuts shown in Fig. 6. For the summer in Fig. 6a the output is B L L L E L K J E L K J > Figure 6: Elementary opamp circuits: (a) a summer, and (b) an integrator. R f R f v out v v R R and for the integrator in Fig. 6b t v out (t) v in (t)dt. R in C 0 and we note. Common opamp summing and integrating circuits involve a sign inversion.. Opamp integrators implicitly have a nonunity gain (unless R in C ).. A Three Opamp State Variable Filter Circuit! D EC D F K JF K J E # L L! )! ) ) L M F K JF K J $ F K JF K J Figure 7: A three opamp implementation of a secondorder statevariable filter. Figure 6 shows a common implementation of the secondorder statevariable filter using three opamps. Amplifiers A and A are integrators with transfer functions H (s) and H (s). R C s R C s
6 Let τ R C and τ R C. Because of the gain factors in the integrators and the sign inversions we have dv d v v (t) τ and v 3 (t) τ τ. () dt dt Amplifier A 3 is the summer. However, because of the sign inversions in the opamp circuits we cannot use the elementary summer of Fig. 6a. Applying Kirchoff s Current Law at the noninverting and inverting inputs of A 3 gives V in v v v v 3 v v v 0 and 0. () R 5 R 6 R R Using the infinite gain approximation for the opamp, we set v v and R 3 R 5 R R 6 v 3 v v V in, R 3 R R 5 R 6 R 3 R R 5 R 6 and substituting for v and v 3 from Eq. () we generate a differential equation in v d v R /R 3 dv R R /R 3 v V in (3) dt τ ( R 6 /R 5 ) dt R 3 τ τ τ τ ( R 5 /R 6 ) which corresponds to a lowpass transfer function with K lp a 0 H(s) () s a s a 0 where a 0 a K lp R R 3 τ τ R /R 3 R 6 /R 5 τ R 3 /R R 5 /R 6 A BandPass Filter: the transfer function where A HighPass Filter: the transfer function where Selection of the output as the output of integrator A generates K bp a s H bp (s) τ sh lp (s) s a s a 0 R 6 K bp R5 (5) Selection of the output as the output of the summer A 3 generates H hp (s) τ τ s H lp (s) R /R 3 K hp R5 /R 6 K hp s (6) s a s a 0 5
7 ! " E # )! L! $ ) L ) L % ' & HA A? J K J F K J ) " Figure : A bandstop secondorder statevariable filter. A BandStop Filter: A bandstop characteristic requires a pair of conjugate zeros on the imaginary axis as defined in Eq. (0). This may be done by including an additional summing amplifier A as shown in Fig.. The output is R 9 R 9 V o (s) V (s) V 3 (s) R R 7 R 9 R 9 V (s) τ τ s V (s) R R 7 f R 7 R and R 3 R, the filter transfer function simplifies to V o (s) V o (s) V (s) K bs (s a 0 ) H bs (s) V in (s) V (s) V in (s) s a s a 0 where R 9 K bs. ( R 5 /R 6 )R.3 A Simplified Two Opamp Based Statevariable Filter: f the required filter does not require a highpass action (that is, access to the output of the summer A ) the summing operation may be included at the input of the first integrator, leading to a simplified circuit using only two opamps shown in Fig. 9. With the infinite gain assumption for the opamps, that is V V, and with the assumption that no current flows in either input, we can apply Kirchoff s Current Law (KCL) at the node designated (a) in Fig. 0: i i f i 3 0 (V in v a ) sc (v v a ) v a 0 (7) R R 3 Using assumption above, v a V out, and realizing that the second stage is a classical opamp integrator with transfer function V out (s) v (s) R C s 6
8 E! ) K J ) Figure 9: Two opamp implementation of a statevariable secondorder active lowpass filter. (V in V out ) R sc ( R C sv out V out ) V out R 3 0 () E B L E E E!! ) K J Figure 0: Feedback summation at the input of the first integrator. Eq. () may be rewritten (V in V out ) sc ( R C sv out V out ) V out 0 (9) R R 3 which may be rearranged to give the secondorder transfer function which is of the form where V out (s) /τ τ (0) V in (s) s (/τ )s ( R /R 3 )/τ τ K lp a 0 H lp (s) () s a s a 0 a 0 ( R /R 3 ) τ τ () a τ (3) K lp R /R 3 () 7
9 . FirstOrder Filter Sections: Single pole lowpass filter sections with a transfer function of the form KΩ 0 H(s) s Ω0 may be implemented in either an inverting or noninverting configuration as shown in Fig.. The inverting configuration (Fig. (a)) has transfer function! E E K J K J > Figure : Firstorder lowpass filter sections (a) inverting, and (b) noninverting. V out (s) Z f R /R C V in (s) Z in R s /R C where Ω 0 /R C and K R /R. The noninverting configuration of Fig. b is a firstorder RC lag circuit buffered by a noninverting (high input impedance) amplifier with a gain K R 3 /R. ts transfer function is V out (s) R 3 /R C V in (s) R s /R C.5 Summary of Features of the Statevariable Filters Statevariable filters are capable of lowpass, bandpass, highpass and bandstop functions. They are capable of realizing both overdamped and lightly damped pole pairs. They are relatively insensitive (compared to other designs) to variation in component values. They do not require a wide range of component values. The coefficients in the transfer function may be set independently. Other designs may require fewer opamps.
10 3 Design Examples: The following two examples involve allpole lowpass filters, and are therefore suitable for the two opamp circuit. We will use the following procedure to determine the component values. Given a filter with a unitygain pole pair described by K lp a 0 H(s) s a s a 0 where a 0, a, and K lp are as defined in Eqs. () through (). The circuit components are chosen as follows: (a) We note that a /τ /R C, and therefore choose a convenient value for C and let R /a C. (b) We arbitrarily let R 3 R. setting K lp 0.5. (c) With this condition a 0 /τ τ a /R C, so we may choose a convenient value for C and then determine R a /(a 0 C ), which also defines R 3. The design is then complete. 3. Example mplement a secondorder Butterworth filter with a3db cutoff frequency of 000 rad/sec (59 Hz). The transfer function of the Butterworth filter is Following the above procedure 0 6 H(s) s s 0 6 (a) Let C 0.7 μf (a common value). Then R 0 6 /( 0.7) 50 Ω. (b) Let C 0.7 μf. Then R R 3 /( ) 607 Ω and the final filter is shown in Fig. 3. The common 7 opamp has been specified in this case. 3. Example Design a fifthorder Chebyshev Type lowpass filter, with a cutoff frequency of 000 rad/s, and allowing db of ripple in the passband. The Matlab commands [z,p,k] cheby(5,,000, s ) filter zpk(z,p,k) 9
11 " %. " %. E $ % 9 $ % 9 % " # " 9 % " K J Figure : Secondorder Butterworth design example. generate the following filter transfer function H(s) (s 6.s 9300)(s 7.9s 9300)(s 9.5) s 6.s 9300 s 7.9s 9300 s 9.5 We implement the filter as two cascade secondorder sections (each with a gain of K lp 0.5) as above, and a single firstorder noninverting section with a gain of. We will use the two opamp circuit Let all capacitors have a value of 0.7 μf. () For the first section a 6., b 9300: R, 5 Ω, ac a 6. R R 3, 6 Ω bc 9, () For the second section a 7.9, b 9, 300: R, 93 Ω, ac a 7.9 R R Ω bc 9, () For the firstorder section K Ω c 9.5: R 7, 39 Ω, Ω c C Let R, 500 Ω, R 3 (K )R 3, 500, 500 Ω and the design is complete. The final circuit is shown in Fig. 3. 0
12 " % " % E " $ " " $ " " # " % % " % " % % % & '! # " # %! " ' " % K J Figure 3: Fifthorder Chebyshev Type lowpass design example. All resistor values are in ohms, all capacitor values are in microfarads. 3.3 Example 3 Design a secondorder bandstop filter to reject 60 Hz interference, with a bandwidth of 0Hz. We start with a firstorder lowpass prototype filter with a cutoff frequency of rad/sec. (Note that the lowpass to bandstop transformation will generate a secondorder filter) The Matlab commands H lp (s). s [num,dden]lpbs(,[ ],*pi*60,*pi*0) filter tf(num,den) generate the following filter transfer function s 00 H(s) s 5.7s 00
13 We use the design equations described for bandstop filters in Section.3 with the circuit shown in Fig., that is where V o (s) V o (s) V (s) K bs (s a 0 ) H bs (s) V in (s) V (s) V in (s) s a s a 0 a 0 a K bs τ τ R 6 /R 5 τ R 9 ( R 5 /R 6 )R under the constraints that R 3 R and R 7 R in Fig.. (a) Let τ τ / a 0 / Then (arbitrarily) let C C 0.7 μf, so that R R 0.007/ Ω. (b) Since a, R 6 /R 5 τ R R 5 a τ We let R Ω and R Ω. (c) We let R 3 R R 7 R 0000 Ω. (d) We set K bs so that R 9 0.5K bs ( R 5 /R 6 )R 6000 Ω. Which completes the design, as shown in Fig.. Note that this filter inverts the signal, so if the application requires maintaining the sign of the input an extra opamp inverter should be used. E " % " % $ # $ " " L # $ " " L! L )! ) K J # ) ) " Figure : Secondorder bandstop filter design example. All resistor values are in ohms, all capacitor values are in microfarads.
14 Single OpAmp SecondOrder Filter Sections There are many opamp active filter circuits that will generate a secondorder transfer function using a single opamp. n this section we briefly introduce the infinite gain multiple feedback (MFB) structure and show how it may be configured as a lowpass, highpass and bandpass secondorder filter. Figure 5 shows the configuration with passive elements (resistors and capacitors) represented by admittances. (Admittance is the reciprocal of impedance, and for a capacitor Y C sc, and for a resistor Y R /R.) ; " E ;! ; ; > ; # K J Figure 5: A General nfinite Gain Multiple Feedback Filter. For the circuit in Fig. 5 we can write node equations (Kirchoff s Current Law) at the node designated (a), and the summing junction (b): and eliminating V a gives the transfer function (Y Y Y 3 Y ) V a Y V in Y V out 0 Y 3 V a Y 5 V out 0 V out (s) Y Y 3 (5) V in (s) Y 5 (Y Y Y 3 Y ) Y 3 Y and the various filter forms may be created by appropriate substitution of resistors and capacitors for the five admittances.. A Lowpass MFB Filter f the circuit is configured as in Fig. 6 and we write Y G /R, Y sc, Y 3 G 3 /R 3, Y G /R, and Y 5 sc the resulting transfer function is G G 3 Vout (s) C C 5 (6) V in (s) s G G G 3 s G 3G C C C which can be written as a lowpass system similar to Eq. (7 ) V out (s) ka 0 H lp (s) (7) V in (s) s a s a 0 3
15 E! K J Figure 6: An infinitegain Multiple Feedback lowpass filter. where a 0 a k R R 3 C C () C R R R3 R R. A Highpass MFB Filter A highpass filter with a transfer function similar to Eq. (9), that is ks H hp (s) s a s a 0 may be formed by configuring the circuit as in Fig. 7, that is with Y sc, Y G /R, Y 3 sc 3, Y sc, and Y 5 G /R. Substitution into Eq. (6) gives E! K J Figure 7: An infinitegain Multiple Feedback highpass filter. a 0 R R C C 3 a C C C 3 R C C 3 (9) k C C
16 .3 A Bandpass MFB Filter A bandpass filter has a transfer function similar to Eq. (), that is ka s H hp (s) s a s a 0 f the circuit is configured as in Fig. and Y G /R, Y G /R, Y 3 sc, Y sc, and Y 5 G 3 /R 3 then E! K J Figure : An infinitegain Multiple Feedback bandpass filter. a 0 a k R 3 C C R R C C (30) R 3 C C R 3 C R (C C ). Example Design a thorder highpass Butterworth filter with a 3dB cutoff frequency of 000 Hz using cascaded MFB sections. The MATLAB commands [num, den] butter(, *pi*000, high, s ); filter tf(num, den) gives the transfer function s H(s) s 60s s 6. 0 s s s s 0s 3977 s 60s
17 mplement both secondorder systems according to Fig. 7, and let C C C 3 C 0. μf, so that Eqs. (30) become: a 0 R R C a 3 R C k or For the two sections 3 R and R a C a 0 R C s H (s) C C C 3 0.μF, R 639Ω, R 06Ω, s 0s 3977 and s H (s) C C C 3 0.μF, R 5Ω, R 90Ω. s 60s 3977 The complete highpass filter is shown in Fig. 9. $! ' # & " E " $ ' & K J Figure 9: An thorder Butterworth Multiple Feedback highpass filter with f c 000 Hz. Capacitances are in μf, and resistances are in ohms. 6
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