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1 4/14/15 8:58 PM C:\Users\Harrn...\tlh2polebutter10rad see.rn 1 of 1 % Example 2pole butter tlh % Analog Butterworth filter design % design an 2-pole filter with a bandwidth of 10 rad/sec % Prototype H(s) 1 % wb = 1 rad/sec % s"'2 + 2"'(l/2)s +1 % [z,p,k) buttap(2); [b,a) = zp2tf(z,p,k); wb = 10; [b,a) = lp2lp(b,a,wb); % % 2 pole filter % convert the zeros and poles to polynomials % new bandwidth in rad/sec % transforms to the new bandwidth By hand replace s by s/wb % define the freq. in Hz for plotting f 0:15/200:100; w 2*pi*f; H = tf(b,a) % Continuous-time transfer function. % H 100 % wb 10 r/s % s"' s figure(l) bode(h,w) ; grid, title('bode Plot \omega_c 10 rad/sec') - S c+- ID~ S +- 1c>0 t e 0 - -I DO +- J too Ii. +loi> L t t}u') = _1T~ ~. --

-!D Bode Plotw = 10 rad/see c 0 -... -10 System: H Frequency (rad/s): 10.4-20 Magnitude (db): -3.

2 -!D Bode Plotw = 10 rad/see c System: H Frequency (rad/s): Magnitude (db): ::s- -30 Q) -g -40 :t:: c: g> -50 ::2: > Q) -45 (/) ro.c 0.. "0-Q) Frequency (rad/s)

Introduction 399 Of the four classical filter types based on magnitude specifications, the Butterworth filter is monotonic ill the passband and stopband, the Chebyshev I filter displays ripples in

3 Introduction 399 Of the four classical filter types based on magnitude specifications, the Butterworth filter is monotonic ill the passband and stopband, the Chebyshev I filter displays ripples in the passband but is monotonic in the stopband, the Chebyshev II filter is monotonic in the passband but has ripples in the stopband, and the elliptic filter has ripples in both bands. I REVIEW Magnitude Characteristics PANEL of Four Classical Filters Butterworth: Monotonic in both bands Chebyshev I: Monotonic passband, rippled stopband Elliptic: Rippled in both bands Chebyshev II: Rippled passband, monotonic stopband The design of analog filters typically relies on frequency specifications (passband and stopband edge(s)) and magnitude specifications (maximum passband attenuation and minimum stopband attenuation) to generate a minimum-phase filter transfer function with the smallest order that meets or exceeds specifications. Most design strategies are based on converting the given frequency specifications to those applicable to a lowpass prototype (LPP) with a cutoff frequency of 1 rad/s (typically the passband edge), designing the lowpassprototype, and converting to the required filter type using frequency transformations Prototype Transformations a; >JIll The lowpass-to-lowpass (LP2LP) transformation converts a lowpass prototype Hp(s) with a cutoff frequency of 1 rad/s to a lowpass filter H(s) with a cutoff frequency of Wx rad/s using the transformation s =? s/wx, as shown in Figure This is just linear frequency scaling. L\ \ J rr,(p.") - S+l S~ ~o "? I-- ~t> H-tol~)-: LP2LP transformation > S+LD s -- s/cox Figure 13.2 The lowpass-to-lowpasstransformation The lowpass-to-highpass (LP2HP) transformation converts a lowpass prototype Hp(s) with a cutoff frequency of 1 rad/s to a highpass filter H(s) with a cutoff frequency of Wx rad/s, using the nonlinear transformation s =? wx/ s. This is illustrated in Figure Highpass filter LP2HP transformation >.S--Olx/S Figure 13.3 The lowpass-to-highpass transformation The lowpass-to-bandpass (LP2BP) transformation is illustrated in Figure It converts a lowpass prototype Hp(s) with a cutoff frequency of 1 rad/s to a bandpass filter H(s) with a center frequency of

4 400 Chapter 13 Analog Filters Wo rad/s and a passband of B rad/s, using the nonlinear, quadratic transformation (13.1) Here, Wo is the geometric mean of the band edges WLand WH, with WLW H = w5, and the bandwidth is given by B = WH - WL. Any pair of geometrically symmetric bandpass frequencies Wa and Wb, with WaWb = w5, corresponds to the lowpass prototype frequency (Wb - wa) / B. The lowpass prototype frequency at infinity is mapped to the bandpass origin. This quadratic transformation yields a transfer function with twice the order of the lowpass filter. LP2BP transformation s2+0)~ s~ Bs > L-_-' ==-+O) Figure 13.4 The lowpass-to-bandpass transformation The lowpass-to-bandstop (LP2BS) transformation is illustrated in Figure It converts a lowpass prototype Hp(s) with a cutoff frequency of 1 rad/s to a bandstop filter H(s) with a center frequency of Wo rad/s and a stopband of B rad/s, using the nonlinear, quadratic transformation 8B :--~ 8 2 +W5 (13.2) Here B = WH - WL and w5 = WHWL. The lowpass origin maps to the bandstop frequency Woo Since the roles of the passband and the stopband are now reversed, a pair of geometrically symmetric bandstop frequencies Wa and Wb, with WaWb = w;, maps to the lowpass prototype frequency WLP = B/(Wb - wa). This quadratic transformation also yields a transfer function with twice the order of the lowpass filter. Bandstop filter Figure 13.5 The lowpass-to-bandstop transformation Lowpass Prototype Specifications Given the frequency specifications of a lowpass filter with band edges wp and w s, the specifications for a. lowpass prototype with a passband edge of 1 rad/s are "» = 1 rad/s and I/s = ws/w p rad/s. The LP2LP transformation is s ---t s/w p. For a lowpass prototype with a stopband edge of 1 rad/s, we would USe "» = ws/w p rad/s and I/s = 1 rad/s instead.

5 tion -w Butterworth Filters For the two-pole system with the transfer function ~11~N~ Section 8.6 Causal Filters 465 ~ B""C~ 3tti' ~ r 2- ~W~ cj.!i 1-wYt'yly 4 t a.. ).~ =vw...~f) it follows from the results in Section 8.5 that the system is a lowpass filter when C ;::: 1/V2. If C = 1/,V2, the resulting lowpass filter is said to be maximally flat, since the variation in the magnitude IH (w ) I is as small as possible across the passband of the filter. This filter is called the two-pole Butterworth filter. The transfer function of the two-pole Butterworth filter is lters are Factoring the denominator of H(s) reveals that the poles are located at,...,...: value of or which Wn s= --±]-- V2 Wn V2 (8.54) E NIl assband Note that the magnitude of each of the poles is equal to Wn- Setting s = jw in H(s) yields the magnitude function of the two-pole Butterworth filter: is down e offrerusal fil-. extent. yallowcharac- that.the irder of owever, er chart of the t causal ver, the ctly linm. The )f filter ~LJ\r~~ 2.:'D (8.55) From (8.55) it is seen that the 3-dB bandwidth of the Butterworth filter is equal to W n ; that is, IH(wn)ldB = -3 db. For a lowpass filter, the point where IH(w)ldB is down by 3 db is often referred to as the cutoff frequency. Hence.eo, is the cutoff frequency of the lowpass filter with magnitude function given by (8.55). For the case Wn = 2 rad/sec, the frequency response curves of the Butterworth filter are plotted in Figure Also displayed are the frequency response curves for the one-pole lowpass filter with transfer function H(s) = 2/(s + 2), and the two-pole

6 466 Chapter 8 Analysis of Continuous-Time Systems by Use of the Transfer Function Representation IH(w)1 ~? o.~ ~O~ ~~ LH(w) 111(;) : I One-pole filter H(s) = s ~ 2 If (.1) :. },1.J. Two-pole filter with s = 1 ji Two-pole filter with?: = 11'1/2 + ~ +--=~~==~===F==~~w o Passband (a) /.., IN EfI I P 5::)W =-J z, ( Hlz.) I;;. C::f~~il.- o ~--~--~--+_--+---~--~--~~r_~--_;--w {)" 2 One-pole filter H(s) = --- s+2 Two-pole filter with?: = I (b) Two-pole filter with s. = 11'1/2 FIGURE 8.33 Frequency curves of one- and two-pole lowpass filters: (a) magnitude curves; (b) phase curves. lowpass filter with l = 1 and with cutoff frequency equal to 2 rad/sec. Note that the Butterworth filter has the sharpest transition of all three filters. N-pole Butterworth filter. For any positive integer N, the N-pole Butterworth filter is the lowpass filter of order N with a maximally flat frequency response across the passband. The distinguishing characteristic of the Butterworth filter is that the poles lie on a semicircle in the open left-half plane. The radius of the semicircle is equal to We> where We is the cutoff frequency of the filter. In the third-order case, the poles are as displayed in Figure The transfer function of the three-pole Butterworth filter is.,....

3 Analog filters. 3.1 Analog filter characteristics

3 Analog filters. 3.1 Analog filter characteristics Chapter 3, page 1 of 11 3 Analog filters This chapter deals with analog filters and the filter approximations of an ideal filter. The filter approximations that are considered are the classical analog