Solutions to Magnetic Bearing Lab #3 Notch Filtering of Resonant Modes

Transcription

1 Solutions to Magnetic Bearing Lab #3 Notch Filtering of Resonant Modes by Löhning, Matthias University of Calgary Department of Mechanical and Manufacturing Engineering 25 July 2004

2 Preliminary note: For this solution, the same set-up is used as in lab2. For more information, please see the preliminary note in lab2 solution. The used average filter is the same as described in lab2 solution exercise 2. Exercise 1: The resonance frequency should be about 784 Hz. This is the value of the resonance peak of lab2 solution in figure 5. Exercise 2: The chirp signal parameters are chosen as follows: - As a result of a wider resonance peak, the chosen start frequency is 480 Hz; this corresponds to a usable start frequency of 540 Hz (see figure 1). - The target time is set to 26 seconds to get a better frequency resolution in spite of the high frequency range. - The end frequency is set to 1000 Hz; this corresponds to a usable end frequency of 1400 Hz (see figure 1). Figure 1: Quanser output signal - Power spectrum The Gain is set to 0.2. A higher value would cause too much noise and a lower value would result in a too low resonance amplitude. Page 2

3 In figure 2, the read off resonance frequency is 790 Hz. Figure 2: Quanser input signal - Power spectrum Figure 3: Quanser input signal Bode Diagram Page 3

4 fn2 gives the frequency f 2 of f Hz. In figure 3, the read magnitudes A1 and A2 are and Thus A A2 Abase 1 is calculated as Figure 3 gives Apeak of In figure 3, f1 is set to Hz. The equation f 2 Exercise 3: The solution of this exercise is: rad wn 2 * * f n 4964 sec Apeak Q Abase Exercise 4: With the help of a matlab m-file (see Appendix B), RG and RQ are chosen so that RG, RQ, RF 1 and RF 2 are in the range of 1 kω and 1 MΩ. Given the assumption RG RQ 5k and the equation Q RG RQ 1 ( 2.5 *10 4 ) 2 RG RQ R RF , the solution is F RF 2 RF 2 Exercise 5: ( rad / sec) 2 and RF 1 RF 2 give the solution RF 1 RF k. Thus RF 1 is k. 2 The equations wn RF 2 Exercise 6: For convenience, the notch filter is built in matlab as a digital filter since the measuring signal already exists. Furthermore, this digital filter design has a better signal-to-noise ratio and no component tolerance. It also avoids any problems with the wires and adjustment to the desired characteristics. The disadvantage of a limited upper frequency is avoided in this case. Exercise 7 to 9: The resistance adjustments are not necessary because of the digital filter design. Page 4

5 Exercise 10: For a better result of the notched signal, both parameters f n and Q are adjusted to 746 Hz and 3. One reason for the smaller value of f n compared to exercise 2 is the real resonance frequency of the open loop system of about 772 Hz (see lab2 solution exercise 9). Another reason is the non-symmetry of the measured signal. The different value of Q can be explained by measuring and reading accuracy of the resonance frequency and the different value of f n. The Bode Diagram of the theoretical notch filter can be seen in figure 4. Figure 5 and 6 show a simulation result of simulink. The result of the notch filter adjustment can be seen in figure 7 and 8. Figure 4: theoretical notch filter transfer function Bode Diagram Page 5

6 Figure 5: input signal to the notch filter (chirp signal) Bode Diagram Figure 6: output signal from the notch filter Bode Diagram Page 6

7 Figure 7: unnotched signal Bode Diagram Figure 8: notched signal Bode Diagram Page 7

8 Appendix A: Simulink model of the notch filter Simulink model to adjust the notch filter: Appendix B: Matlabcode Exercise 4 and 5: Q=58.9/2.8955; fn=790; RG=5*1000; RQ=5*1000; eta=(((q-0.5)*rg*rq)/(2.5*10^4*(rg+rq)))^2; RF2=sqrt(10^18/eta)/(2*pi*fn); RF1=eta*RF2; disp(['rg = ', num2str(rg/1000), ' kohm']); disp(['rq = ', num2str(rq/1000), ' kohm']); disp(['rf1 = ', num2str(rf1/1000), ' kohm']); disp(['rf2 = ', num2str(rf2/1000), ' kohm']); Page 8

9 Matlabcode Exercise 10: 1) theoretical notch filter s=tf('s'); g=(s^2+(746*2*pi)^2)/(s^2+(746*2*pi)/(3)*s+(746*2*pi)^2); % transfer function of the notch filter % bode diagram of the notch filter freq = logspace(1, 5, 10000); % generate logarithmically spaced vector "freq" in rad/sec % transfer to matlab system sys % calculate the numerator and denominator % calculate the magnitude and the phase sys=ss(g); [num,den] = tfdata(sys); [mag,phase]=bode(num,den,freq); figure; subplot(211); loglog(freq/(2*pi),mag); title('bode Diagram of the theoretical notch filter'); ylabel('magnitude (log scale)'); xlim([ ]); subplot(212); semilogx(freq/(2*pi),phase); xlabel('frequency (Hz)'); ylabel('phase (deg)'); xlim([ ]); 2) notch filter simulation % take the simulation signal from notchtest.mdl and calculate the fourier transformed samplefreq=1/ ; % the sample frequency of the simulink model simulation parameter (1/sampletime) % the test time in the simulink model simulation parameters % the number of sample values (round up) testtime=26; samplevalues=ceil(testtime*samplefreq); % length of dft i=1; while 2^i<samplevalues i=i+1; end i=i-1; warning('the used data with the dft is only (per cent)...') dft 100*2^i/samplevalues % the next 5 line only calculate the dft range % to know how much data us used below in the f=samplefreq*(0:2^(i-1))/2^i; % generate the frequency vector (first 2^(i-1)+1 points (the other 2^(i-1)-1 points are redundant)) startf=540; endf=1400; % start frequency of the plots % end frequency of the plots % % signal input to notch filter load notchtestin % load data (folderpath and file name has to be correct) % signal = notchin % dft of the time signal infft=fft(notchin(2,1:length(notchin)),2^i); % calculate the 2^i dft inabs = abs(infft); % magnitude of the input data Page 9

10 inphase = angle(infft)/pi*180; % phase angle of the input data in degree % plot the input signal (Bode Diagram) figure; subplot(211); loglog(f(1:2^(i-1)),inabs(2:2^(i-1)+1)); ylim([10^(-1) 10^3]); xlim([startf endf]); title('bode Diagram of the notch filter input signal'); ylabel('magnitude (log scale)'); subplot(212); semilogx(f(1:2^(i-1)),inphase(2:2^(i-1)+1)); xlim([startf endf]); xlabel('frequency log scale (Hz)'); ylabel('phase (deg)'); % % signal output to notch filter load notchtestout % load data (folderpath and file name has to be correct) % signal = notchout % dft of the time signal outfft=fft(notchout(2,1:length(notchin)),2^i);% calculate the 2^i dft outabs = abs(outfft); % magnitude of the output data outphase = angle(outfft)/pi*180; % phase angle of the output data in degree % plot the output signal (Bode Diagram) figure; subplot(211); loglog(f(1:2^(i-1)),outabs(2:2^(i-1)+1)); xlim([startf endf]); title('bode Diagram of the notch filter output signal'); ylabel('magnitude (log scale)'); subplot(212); semilogx(f(1:2^(i-1)),outphase(2:2^(i-1)+1)); xlim([startf endf]); xlabel('frequency log scale (Hz)'); ylabel('phase (deg)'); 3) notch filter adjustment % take the simulation signal from Quanser card and calculate the fourier transformed samplefreq=1/ ; % the sample frequency of the simulink model simulation parameter (1/sampletime) k=101; % k values of average filter were put together; it has to be odd % % signal input (from sytem output 11) load identnotchinu % load data (folder path and file name has to be correct) % time vector = plot_time; amplitude of the signal = identnotch_s_q_in_u load identnotchiny % load data (folder path and file name has to be correct) % time vector = plot_time; amplitude of the signal = identnotch_s_q_in_y Page 10

11 load identnotchinynotch % load data (folder path and file name has to be correct) % time vector = plot_time; amplitude of the signal = identnotch_s_q_in_y_notch [min,nin] = size(plot_time); % size of the data (mout=length of input signal) stoptime=min/samplefreq; samplevalues=min; % calculate the wincon test time % the number of sample values % dft of the time signal i=1; while 2^i<samplevalues i=i+1; end i=i-1; warning('the used data with the dft is only (per cent)...') 100*2^i/samplevalues f=samplefreq*(0:2^(i-1))/2^i; % the next 5 line only calculate the dft range % to know how much data us used below in the dft % generate the frequency vector (first 2^(i-1)+1 points (the other 2^(i-1)-1 points are redundant)) % calculate the 2^i dft % calculate the 2^i dft % calculate the 2^i dft % transfer function % transfer function of the notched signal inufft=fft(identnotch_s_q_in_u,2^i); inyfft=fft(identnotch_s_q_in_y,2^i); inynotchfft=fft(identnotch_s_q_in_y_notch,2^i); g=inyfft./inufft; gnotch=inynotchfft./inufft; % average filter and downsampling a=1; fnew=0; gaverage=0; gnotchaverage=0; for m = ((k-1)/2+1):k:2^(i-1)+1 fnew(a)=f(m); gaverage(a)=sum(g(m-(k-1)/2:m+(k-1)/2))/k; % to count the loops of the for loop % initialization of fnew % initialization % initialization % k values of g were put together as one value gaverage gnotchaverage(a)=sum(gnotch(m-(k-1)/2:m+(k-1)/2))/k; % k values of gnotch were put together as one value gnotchaverage a=a+1; end gabs=abs(g); gphase=angle(g)*180/pi; gaverageabs=abs(gaverage); gaveragephase=angle(gaverage)*180/pi; gnotchabs=abs(gnotch); gnotchphase=angle(gnotch)*180/pi; gnotchaverageabs=abs(gnotchaverage); gnotchaveragephase=angle(gnotchaverage)*180/pi; % absolute value of g % angle of g in degrees % absolute value of gaverage % angle of gaverage in degrees % absolute value of gnotch % angle of gnotch in degrees % absolute value of gnotchaverage % angle of gnotchaverage in degrees startf2=540; endf2=1400; % start frequency of the Bode plots % end frequency of the Bode plots % plot the combined measured and averaged unnotched signal (Bode Diagram) figure; subplot(211); loglog(f(13:2^(i-1)),gabs(14:2^(i-1)+1),'b',fnew,gaverageabs,'g'); xlim([startf2 endf2]); title('bode Diagram of the unnotched signal'); ylabel('magnitude (log scale)'); legend('original signal','averaged signal'); Page 11

12 subplot(212); semilogx(f(13:2^(i-1)),gphase(14:2^(i-1)+1),'b',fnew,gaveragephase,'g'); xlim([startf2 endf2]); xlabel('frequency log scale (Hz)'); ylabel('phase (deg)'); legend('original signal','averaged signal'); % plot the combined measured and averaged notched signal (Bode Diagram) figure; subplot(211); loglog(f(13:2^(i-1)),gnotchabs(14:2^(i-1)+1),'b',fnew,gnotchaverageabs,'g'); xlim([startf2 endf2]); title('bode Diagram of the notched signal'); ylabel('magnitude (log scale)'); legend('original signal','averaged signal'); subplot(212); semilogx(f(13:2^(i-1)),gnotchphase(14:2^(i-1)+1),'b',fnew,gnotchaveragephase,'g'); xlim([startf2 endf2]); xlabel('frequency log scale (Hz)'); ylabel('phase (deg)'); legend('original signal','averaged signal'); Page 12