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Unformatted text preview: WorksheetActive LP Chebeyshev Filter Design (Revised) The motivation for this design example is to deal with the design of active circuits when 2nd order filter
2
kwo transfer functions are of the form H NLP(S) = , k < 1. This includes as a special case the co
s2 +Js+w§ 2nd order Butterworth 3dBNLP transfer function of the form 1
HMBNLP (s) = —2—\/_— which is not all that interesting in terms of teaching design principles.
5 + 2s+1 So what we do is design a 2““l order Chabeyshev filter which has some ripple in the passband unlike the
Butterworth filter. The ripple allows the filter to transition out of the pass band into the stop band more
quickly than Butterworth. The ripple is like the rocking motion one uses to rock a car out of the mud or
snow when stuck. The Butterworh transition is like spinning your tires—it works but not all that well at
least some of the time. OBJECTIVE: Design a 2“d order Chebeyshev filter with passband edge frequency fp = 1500 Hz with a
corresponding Amax over the passband of Amax = 2 dB. STEP 1. Let MATLAB do the working. >>wp = 2*pi*1500 wp = 9.4248e+03 >>Amax = 2; >>n = 2; »% Chebeyshev Design
>>[z,p,k] = cheblap(n,Amax)
Z = p : —4.0191e—01 + 8.1335e—01i
—4.0l91e—01 — 8.1335e—01i k = 6.537 8e—01 >>num = poly(z) num = L 1 >>den = real(poly(p)) den = 1.0000e+00 8.0382e—01 8.2306e—01 REMARK: at this point: H NL 10(5) _—. ~L— = ﬂ... .
2 a’o 2 s2 + 0.80382s+ 0.82306 STEP 2. Graph magnitude response to see if MATLAB output makes sense.
»w = 020.0123; »h = freqs(k*num,den,w); >>subplot(2,1,1), plot(w,abs(h)) » grid »ylabel(‘Gain') »subplot(2,1,2), plot(w, —20*log10(abs(h))) » grid >>ylabel('Loss in dB')
»xlabel('Normalized rad1frequency') U 0.5 1 1.5 2 2.5 i] a as 1 _ 15 2 2:5 :1
Normalized rad frequency STEP 3. "Discover" a candidate circuit. The Sallen & Key Saraga design of the active circuit shown 4 . 3 k 0.65378
below Is: H . (s): 7: H (s)=—~————=——————
NLPW 2 1 NLP 2 we 2 52 +0.803825+0.82306 S +—S+1 s +#—S+a)0 STEP 4. Find wO and Q of the MATLAB transfer function.
»WO 2 sqrt(den(3)) wO = 9.0723e—01 >>Q = wO/den(2) Q = 1.1286e+00 4
Step 4. Realize the circuit Transfer function: H NL P cir (s) = 3 Note that R can be chosen
s2 + s + 1
independently of the rest so R = 19 is convenient.
R1 : R2 : C1 = C2 :
k 0.65378 .
Step 5. Realize HNLP(S) = ———~————— — —————~—— In two steps. a) _ 2
s2+—0s+wg s +0.80382s+ 0.82306 (i) Adjust gain using input attenuation. Compute RA and RB as depicted below. .J Design equations: (1) Parallel combination of RA and RB = (2) DC gain of circuit transfer function is: (3) DC gain of MATLAB transfer function is: R
(4) Voltage division requires: ———B—— =
RA + RB
R R R
(5) A=RA{B—]=RA>< Hence
RA + RB RA + RB (ii) Realize the filter designed in MATLAB by frequency scaling by (no. This will yield the
filter transfer function computed in MATLAB. NOTE: ALL Resistances remain the same as in above
step. Cln = __— an = ______ (subscript n = new) Label the circuit at this point. Step 6. Find Km so that the smallest capacitor is 10 nF. Compute all remaining parameter values of your LP circuit. Use a different magnitude scale factor for R, say K m2 = Rn=——— RAf:—«— RBf_——— RZf: C1 f = C2 f = (f indicates final value) Label the final circuit: ...
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 Spring '06

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