# ( ) 2. 1) Bode plots/transfer functions. a. Draw magnitude and phase bode plots for the transfer function

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1 ECSE CP7 olution Spring 5 ) Bode plot/tranfer function a. Draw magnitude and phae bode plot for the tranfer function H( ). ( ) ( E4) In your magnitude plot, indicate correction at the pole and zero. Step : Find pole, zero Zero:, Pole: E4 Step : Define region and find either H(jω) or H() < H( ) db/dec lope. 4 lope end at log(.) 4 < < 4 > 4. H( ) db/dec lope. 4 6 ( ) log H( ).. db/dec lope Step 3: Make correction Correction at i 3db becaue it i a ZERO^! Correction at ^4 i -3db becaue it i a POLE^!

2 ECSE H ( ) ( )( ) CP7 olution Spring 5 Phae plot.ω c ω j j H( j). ( ) ( j E4) ( 9) ( ) 9 9 ω c 3 ω 3 ( ) (. j3 j 3 ) ( 3 j E4) H j 3 ( 9) ( 9)

3 ECSE CP7 olution Spring 5 ω 35.ω c 3 already done 8 ω c 5 ( ) (. j5 j 5 ) ( 5 j E4) H j 5 ω 4 ( ) (. j4 j 4 ) ( 4 j E4) H j 4 H ( ) ( )( ) ( 9) ( 9) ( 9) ( 9)

4 ECSE CP7 olution Spring 5 b. Determine the tranfer function for the following magnitude and phae Bode plot (the plot are for the ame circuit). Bode Diagram Magnitude (db) Phae (deg) Frequency (rad/). We have a db pa band o log( K) db K. We have 4 db rolloff on the right where ωoe6 rad/, critically damped low pa filter, double pole there ( 6 ) ω o 6 ω o ( ) 3. t order high pa filter, db rolloff, ωo^4 rad/ 4 ω o 4

5 ECSE CP7 olution Spring 5 Put all together H( ) ( 6 ) ( 4 ) ( 4 6 ) ( ) ) nd order Bandpa Bode plot L C E- E-6 Vin R V out - a. Determine the tranfer function for the above circuit, H( ) H( ) α α ω R L R L LC V R ( ) V in ( ) R : L : C : 6 b. Determine α, ω, and ζ (damping ratio). R 3 b) Band pa filter L α : ω : L C db. Aymptote take the form of an inverted V. Each ha a db rolloff 3. At ω, log ab (H(jω)) db alway 4. The point of the inverted V i log ab (H(jω)) away from log (ζ) Narrow or wide? ω 4 α ζ : ζ. ω 5

6 ECSE CP7 olution Spring 5 c. Draw the Bode-magnitude plot. Indicate the magnitude (db) of the tranfer function at the reonant frequency. Remember peak mut alway be at db at ωo, we have to figure out where the aymptote will be,- db/dec. α H( ) goe to H( ) db/dec H( ) goe to H( ) H( jω ) αjω ω αjω ω LC α -db/dec alway! atω ζ ω j log H ( ) ζ ) ( )( ) 6

7 ECSE CP7 olution Spring 5 4) Deign problem-multiple Stage Uing only firt order filter and opamp circuit for each tage, deign a filter that meet the pecification below. You need to pick value for any reitor, capacitor or inductor in your circuit. Your circuit mut conit of at leat one inductor and at leat one capacitor. a. Lowpa filter with a cutoff frequency of MHz (Note: thi value i given in Hz). b. In the paband, the gain mut be >db c. The aymptotic lope of the topband hould be -6dB/decade d. You circuit mut contain at leat one inductor and at leat one capacitor. ω c : π MHz ω c rad Gain log K > db Slope indicate a triple pole R 3 : 6.8kΩ R :.59kΩ C : L : 3 H R for inductor/reitor acro reitor Remember ω c R L L ω c for capacitor/reitor acro capacitor RC Gain R 4 : 9kΩ R 5 : kω R 4 Amplifier R 5 R C Note: Need to overcome the -9 db lo at corner frequency to get > db to meet pec. So that i ~db needed Ω log( ) Alo need to iolate firt order filter ue buffer 7

8 ECSE CP7 olution Spring 5 Solution : Lowpa filter Cutoff frequency of 6.8E6 rad/ Rolloff of 6dB/decade indicate a triple pole. Three firt order filter with amplifier tage for iolation. -9dB point relative to the paband at 6.8E6 rad/ Paband gain of db to meet pec, log K K 3 ωc H ( ) K ωc LPF tage, RL circuit: Chooe an RL circuit, R 6.8k, L E-3 LPF tage, RC circuit: Chooe an RC circuit, R.59k, C E- Amplifier tage: Chooe a non-inverting amplifier, R9k, Rk Iolation amplifier tage: unity gain Vac Vdc V R U.59k R C E- - OUT OPAMP.59k C E- U - R5 k OUT OPAMP R4 9k L E-3 VDB R3 6.8k KHz 3.KHz KHz 3KHz KHz 3KHz.MHz 3.MHz MHz 3MHz MHz DB(V(L:)) Frequency 8

9 ECSE CP7 olution Spring 5 If retriction of only firt order filter wa lifted...you could alo do... Solution : Second order LPF cacaded with a firt order LPF Ue an underdamped circuit with a damping ratio of.5. The gain term than need to be 3dB to atify the paband requirement, giving log K 3dB, K4.4\ Vac Vdc V R L U 68 E-4 R OUT.59k - OPAMP C.53E- R4 VDB C E- R3 k 44k H( ) ω LC ω c rad R T L LC choe L firt if you like L b : 4 ω L C ( ) L b.533 ω L C 4.53 need at leat a critically damped circuit, hould ue underdamped α ζ For better filter make ζ.5 ω ( ) α b : α R L R b : α b L b R b

10 ECSE CP7 olution Spring 5 Thi give -4 db roll off, with ω at 6.3*^6 and a very flat paband Still need another - db roll off o you can ue a firt order filter (like previou) which ha -3dB rolloff Gain For thi circuit we need over only overcome -3dB o that mean a total of 3 db log K Can ue a non inverting amplifer with a gain of 4.5 R 4b : R 3b : 35kΩ kω R 4b R 3b KHz 3.KHz KHz 3KHz KHz 3KHz.MHz 3.MHz MHz 3MHz MHz DB(V(C:)) Frequency

11 ECSE CP7 olution Spring 5 5) Deign Problem Deign a bandpa filter with the following pecification. Show your work and jutify your calculation. Include a chematic of your circuit. a. The paband i E4<ω<E6 b. The paband gain hould be 5dB c. The low frequency topband rolloff hould be db/decade d. The high frequency topband rolloff houdl be 6 db/decade e. The low frequency cutoff frequency, E4 [rad/] hould be -3dB relative to the pabnad. f. The high frequency cutoff frequency, E6 [rad/], hould be -3dB relative to the pabnad. g. You cannot ue any firt order circuit tage Build the circuit uing BPF LPF BPF ω clow 4 ω chigh 6 3correction 3correction db dec LPF ω c 6 ( double) db dec nd order overdamped circuit meaured acro the reitor R mut give ζ > need db correction at ^6 4 db dec rolloff nd order underdamped ζ.5 pa band gain log( K) 5db K.77 Non inverting amp chooe R R k R.77k.77kΩ kω R R.77 BPF α H( ) you know that α ω denominator mut be ( 4 ) 6 ( ) expand and α in numerator can be given by doing o

12 ECSE CP7 olution Spring 5 H( ) L 3. 6 ( 4 ) 6 ( ) Now find R/L you can pick R or L R R.k C 7 ω LC 3 7 LPF H( ) ω you know that denominator α ω mut be ( 6 ) ω c ω ω 6 α.5ω thi i the underdamped condition where correction i db α 5 5 pick L 3 ω LC C C 9 R α L R k

13 ECSE CP7 olution Spring 5 3

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