PRACTICAL FILTER DESIGN & IMPLEMENTATION LAB
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1 1 of 7 PRACTICAL FILTER DESIGN & IMPLEMENTATION LAB BEFORE YOU BEGIN PREREQUISITE LABS Itroductio to Oscilloscope Itroductio to Arbitrary/Fuctio Geerator EXPECTED KNOWLEDGE Uderstadig of LTI systems. Laplace trasform for circuit aalysis Trasfer Fuctios aalysis ad sythesis Practical filterig desig with MATLAB EQUIPMENT Workstatio with MATLAB 6. Digital Multimeter Programmable Power Supply AFG3000 Series Arbitrary/Fuctio Geerator TDS3034B Digital Phosphor Oscilloscope MATERIALS Complete set of resistors i ECE Toolkit Complete set of capacitors i ECE Toolkit Set of Operatioal Amplifiers Solderless Breadboard Oscilloscope Probes 3 ½ floppy Speaker OBJECTIVES After completig this lab, you should be able to desig ad build a circuit that implemets a practical filter give a set of filter specificatios. INTRODUCTION Filters ca be defied by a set of five parameters: passbad gai (A) measured i db, passbad ripple (Ap) measured i db, pass-bad edge frequecies (ωp) measured i rad/sec, stopbad edge frequecies (ωs) measured i rad/sec, ad stop-bad atteuatio (As) measured i db. You should kow how to use MATLAB to fid a trasfer fuctio of several types of filters that meet these specificatios. Specifically, you should be familiar with Butterworth, Chebyshev, ad Elliptical filters. I this lab, you will desig ad build a Chebyshev type audio filter.
2 of 7 Figure 1. Graphical example of lowpass filter specificatios. PRELAB THE CHEBORD COMMAND The chebord fuctio ca be used to determie the order ad atural frequecy for a Chebyshev type filter. The sytax for chebord is: where [N, W] = chebord(wp, Ws, Rp, Rs, s ) Wp is the passbad frequecy i rad/sec. Ws is the stopbad frequecy i rad/sec. Rp is the passbad ripple i db. Rs is the stopbad atteuatio i db. s idicates the filter is cotiuous time, ot discrete-time
3 3 of 7 THE CHEBY COMMAND The cheby fuctio returs the polyomial coefficiets for the umerator ad deomiator of the trasfer fuctio for a Chebyshev type filter, based o the order ad atural frequecy supplied from the chebord fuctio. The sytax for cheby is: where [B, A] = cheby(n, Rs, W, type, s ) N is the order supplied from chebord. Rs is the stopbad ripple i db. W is the atural frequecy i rad/sec. type is the type of filter. e.g. high, low s idicates radia mode. Type help cheby at a MATLAB prompt for more specific iformatio o how to use this fuctio. Give the followig filter parameters, you will use MATLAB to determie the trasfer fuctio, ad pole ad zero locatios for a Chebyshev type filter. Passbad Gai: 1 db i passbad Fp: 1 khz Fs: 700 Hz Passbad ripple: 1 db Stopbad atteuatio: 40 db Aswer questio 1. To fid the order ad cutoff frequecy of our filter, we will use the chebord fuctio i MATLAB. Use this iformatio ad the cheby fuctio to determie the polyomial coefficiets for the trasfer fuctio of the filter. With these polyomial coefficiets, the trasfer fuctio ca be created usig the tf fuctio. Aswer questios 5. Before you ca sythesize this trasfer fuctio, you must determie the zeros ad poles. Remember that zeros are defied as the roots of the umerator of the trasfer fuctio ad poles are defied as the roots of the deomiator of the trasfer fuctio. These ca be determied usig the roots commad i MATLAB. Solve for the poles ad zeros of your trasfer fuctio. Store them to variables z ad p respectively, so that they will be easier to use later. Aswer questio 6. Notice that the zeros ad poles that are retured are complex values. You may also otice that they are i complex cojugate pairs. For sythesis, you will eed to combie the pairs of complex zeros ad poles ito secod-order polyomials with real coefficiets. As a result, each complex-cojugate pair of zeros ad poles will be implemeted usig oe secod-order stage of
4 4 of 7 our filter. The real zero ad pole will be implemeted usig a first-order stage. These stages are described i detail below. LAB Cascadig Trasfer Fuctios x(t) H 1 (s) H (s) H 3 (s) H 4 (s) y(t) Figure. Example of cascaded trasfer fuctios with active outputs. Practical aalog filters are usually implemeted as a cascade of first or secod-order filters. Each of these simpler filters is called a stage. I order to cascade each stage, the output of the first stage becomes the iput of the ext stage, ad so o. This is equivalet to multiplyig the idividual trasfer fuctios for each stage with that of the ext. Figure gives a flow diagram of cascadig each stage. Each of the stages below has a egative leadig coefficiet due to the active outputs coected to a ivertig amplifier. If there are a eve umber of stages, these egative coefficiets will cacel ad should ot affect the trasfer fuctio of the overall cascaded circuit. First-Order Stage R f R i C v i Figure 3. First order highpass active filter. The first-order highpass filter show i Figure 3 should be used to implemet the stage that cotais the real pole ad the real zero. Note that the zero is at 0 rad/s. The trasfer fuctio of this circuit is give below.! k " s H ( s) = ( s! p) You should be able to show that the trasfer fuctio parameters k ad p have the followig relatioship to the circuit parameters. 1 R f p =! k =! R C R i i
5 5 of 7 Secod-Order Stage R R 1 R 1 C R C R 3 R 3 v i V BP V LP -V LP U1 U U3 R 5 R R R 4 V N U4 Figure 4. Secod-order otch filter with two imagiary zeros ad two complex poles. The circuit i Figure 4 represets the basic circuit that will be used i the other stages of your filter (oe stage for each pair of complex poles ad pair of imagiary zeros). It cosists of four parts. A badpass filter (U1), a itegrator (U), a iverter (U3), ad a summig amplifier (U4). The badpass filter, itegrator, ad iverter each have their ow output. The output of the circuit is labeled V N. The trasfer fuctio is give below.!! s + b1 s + b0!! 1 0 " k( s " z)( s " z*) s " ( z + z ) s + zz H ( s) = = " k = " k ( s " p)( s " p*) s " ( p + p ) s + pp s + a s + a Sice you wat the passbad gai to be 1 (0 db) for all of the filter stages, the gai deoted by k should be 1. Note that sice the zeros are purely imagiary, the coefficiet b 1=0. The relatioship of the trasfer fuctio coefficiets to the circuit parameters is described by the followig equatios. H ( s) = " k s! Q s s +! z + +! (1.1) 1 R!!! 1 5 = Q = z = 1" k = (1.) RC R R4Q R To chose the circuit values, you will eed to choose the complex cojugate pairs of zeros ad poles for each secod-order stage ad determie the value of Q for each stage. You should be able to easily show that these relatioships are give by the equatios below.!! = p! z = z Q = # " ( p + p ) R R
6 6 of 7 These equatios ca be implemeted usig the followig is the MATLAB code. Zeros >> abs(z) Poles >> abs(p) Q s >> -abs(p)./(*real(p)) Aswer questio 7. You may choose which pairs of imagiary complex-cojugate pairs of zeros are grouped with each pair of complex-cojugate pair of poles. Note, however, that the real pole ad zero should be implemeted by the first-order stage ad the zeros ad poles have uits of rad/sec. Idividual Stage Aalysis Oce you have decided which complex-cojugate pairs of zeros will be grouped with each complex-cojugate pair of poles, determie the trasfer fuctio for each stage. Simulate each trasfer fuctio i MATLAB ad geerate the bode plot, impulse respose, ad step respose for each trasfer fuctio. Aswer questios With values for the zero, pole, ad Q for each stage, you should be able to pick the circuit compoet values for each stage of the filter. Whe buildig the circuit, try to get as close as possible to your calculated compoet values, but you do ot have to be exact. Although this will itroduce some error i the expected respose ad the filter may ot satisfy the desig specificatios, the actual aalog filter respose should be close to the ideal respose modeled i MATLAB. Aswer questio Circuit Implemetatio ad Validatio Build the circuit with the compoet values that you calculated. Measure the output of the circuit with a siusoidal iput that has a 1 V amplitude. Record the amplitude of the output of each stage for may frequecies betwee 100 Hz ad 10 khz. Record may frequecies ear the trasitio zoe of the filter. Oce you have verified that each stage is workig correctly, repeat the process for the etire cascaded filter. Aswer questios Drive the circuit with a impulse ad a the a step from the Arbitrary/Fuctio Geerator (AFG). How does this compare with the output of MATLAB?
7 7 of 7 Aswer questios 0 1. Hook up a speaker to the output of the circuit ad apply a siusoidal sigal from the AFG that sweeps from 100 Hz ad 10 khz repeatedly. This sigal is called a chirp. Aswer questios 3.
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