# 2.161 Signal Processing: Continuous and Discrete Fall 2008

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2 MASSACHUSETTS NSTTUTE OF TECHNOLOGY DEPARTMENT OF MECHANCAL ENGNEERNG.6 Signal Processing Continuous and Discrete Op-Amp mplementation of Analog Filters. ntroduction Practical realizations of analog filters are usually based on factoring the transfer function into cascaded second-order sections, each based on a complex conjugate pole-pair or a pair of real poles, and a first-order section if the order is odd. Any zeros in the system may be distributed among the second- and first-order sections. Each first- and second-order section is then implemented by an active filter and connected in series. For example the third-order Butterworth high-pass 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 low-order block can be handled independently.!! " & & " > Figure : A third-order Butterworth filter (a) as a single third-order section, and (b) as a second-order and first-order section cascaded. State-variable active filters The state-variable filter design method is based on the block diagram representation used in the so-called phase-variable description of linear systems that uses the outputs of a chain of cascaded integrators as state variables. Consider a second-order 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 sub-blocks 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 second-order 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 second-order 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 low-pass filter (7) H bp (s) Y (s) a s U(s) s a s a 0 a unity gain band-pass filter () H hp (s) Y 3 (s) s U(s) s a s a 0 a unity gain high-pass filter (9) H bs (s) Y (s) s a 0 U(s) s a s a 0 a unity gain band-stop filter (0) ; " J F ;! D EC D F ; F 7 : : : ; M F Figure 5: State variable implementation of various filter types. 3

5 ". Op-amp Based State-Variable 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 op-amp 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 op-amp 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 op-amp summing and integrating circuits involve a sign inversion.. Op-amp integrators implicitly have a non-unity gain (unless R in C ).. A Three Op-amp 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 op-amp implementation of a second-order state-variable filter. Figure 6 shows a common implementation of the second-order state-variable filter using three op-amps. 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 op-amp circuits we cannot use the elementary summer of Fig. 6a. Applying Kirchoff s Current Law at the non-inverting 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 op-amp, 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 low-pass 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 Band-Pass Filter: the transfer function where A High-Pass 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 band-stop second-order state-variable filter. A Band-Stop Filter: A band-stop 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 Op-amp Based State-variable Filter: f the required filter does not require a high-pass 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 op-amps shown in Fig. 9. With the infinite gain assumption for the op-amps, 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 op-amp integrator with transfer function V out (s) v (s) R C s 6

8 E! ) K J ) Figure 9: Two op-amp implementation of a state-variable second-order active low-pass 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 second-order 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 . First-Order Filter Sections: Single pole low-pass filter sections with a transfer function of the form KΩ 0 H(s) s Ω0 may be implemented in either an inverting or non-inverting configuration as shown in Fig.. The inverting configuration (Fig. (a)) has transfer function! E E K J K J > Figure : First-order low-pass filter sections (a) inverting, and (b) non-inverting. 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 non-inverting configuration of Fig. b is a first-order R-C lag circuit buffered by a non-inverting (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 State-variable Filters State-variable filters are capable of low-pass, band-pass, high-pass and band-stop 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 op-amps.

10 3 Design Examples: The following two examples involve all-pole low-pass filters, and are therefore suitable for the two op-amp circuit. We will use the following procedure to determine the component values. Given a filter with a unity-gain 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 second-order Butterworth filter with a3db cut-off 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 op-amp has been specified in this case. 3. Example Design a fifth-order Chebyshev Type low-pass filter, with a cut-off frequency of 000 rad/s, and allowing db of ripple in the pass-band. The Matlab commands [z,p,k] cheby(5,,000, s ) filter zpk(z,p,k) 9

11 " %. " %. E \$ % 9 \$ % 9 % " # " 9 % " K J Figure : Second-order 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 second-order sections (each with a gain of K lp 0.5) as above, and a single first-order non-inverting section with a gain of. We will use the two op-amp 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 first-order 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: Fifth-order Chebyshev Type low-pass design example. All resistor values are in ohms, all capacitor values are in microfarads. 3.3 Example 3 Design a second-order band-stop filter to reject 60 Hz interference, with a bandwidth of 0Hz. We start with a first-order low-pass prototype filter with a cut-off frequency of rad/sec. (Note that the low-pass to band-stop transformation will generate a second-order 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 band-stop 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 op-amp inverter should be used. E " % " % \$ # \$ " " L # \$ " " L! L )! ) K J # ) ) " Figure : Second-order band-stop filter design example. All resistor values are in ohms, all capacitor values are in microfarads.

14 Single Op-Amp Second-Order Filter Sections There are many op-amp active filter circuits that will generate a second-order transfer function using a single op-amp. n this section we briefly introduce the infinite gain multiple feedback (MFB) structure and show how it may be configured as a low-pass, high-pass and band-pass second-order 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 Low-pass 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 low-pass 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 infinite-gain Multiple Feedback low-pass filter. where a 0 a k R R 3 C C () C R R R3 R R. A High-pass MFB Filter A high-pass 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 infinite-gain Multiple Feedback high-pass filter. a 0 R R C C 3 a C C C 3 R C C 3 (9) k C C

16 .3 A Band-pass MFB Filter A band-pass 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 infinite-gain Multiple Feedback band-pass 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 th-order high-pass Butterworth filter with a -3dB cut-off 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 second-order 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 high-pass filter is shown in Fig. 9. \$! ' # & " E " \$ ' & K J Figure 9: An th-order Butterworth Multiple Feedback high-pass filter with f c 000 Hz. Capacitances are in μf, and resistances are in ohms. 6

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