EE nd Order Time and Frequency Response (Using MatLab)
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1 EE nd Order Time and Frequency Response (Using MatLab) Objective: The student should become acquainted with time and frequency domain analysis of linear systems using Matlab. In addition, the student should develop an enhanced understanding of 2 nd order systems. Prelab: For a prelab, we want you to acquaint yourself with some additional Matlab capabilities. Complete this tutorial. Time required is about 3060 minutes. Complex arithmetic» % define a complex constant» y=2j y = i» % note that this is the same as y = 2*j. However, j2 is interpreted as named variable!» z=3j z = i» % create a third constant which depends upon the two existing ones» x=z/y x = i» % obtain the magnitude and phase of this new constant» m=abs(x)» ph=angle(x) m = ph = » % by default, angles are in radians. To convert to degrees:» deg=ph*180/pi deg = Vector (onedimensional array) operations» % create a 1x3 row vector» v1=[x y z] v1 = i, i, i» % create a 3x1 column vector» v2=[x; y; z] v2 = i, i, i» % The semicolons made the difference!»» % multiply the two vectors» vectprod=v1*v2 vectprod = i» % transpose v2 into a row vector with conjugated imaginary parts» v2t=v2' v2t = i, i, i» % the command v2' (without the period) produces the conjugate of v2t. To transpose without conjugation, command v2.» % multiply two identically dimensioned vectors, term by term» termprod=v1.*v2t termprod = » % note how we needed.* to denote term by This laboratory originally written by Rick Brown, April 6, Most recent update: April 8, 2001 by J. B. Burl.
2 term arithmetic operations between two identically dimensioned arrays. We were here multiplying complex values by there conjugates, producing the squares of the magnitudes of the complex values» % perform a square root operation on every term in the array» termsqrt=sqrt(termprod) termsqrt = Generating test point sets» t1=linspace(0,1,5) t1 = » % start at zero, stop at 1, there are 5 values» t2=0:1:5 t2 = » % start at zero, increment by 1, stop at 5» t3=logspace(1,1,21) t3 =» % create a vector of 21 logarithmically spaced timedependent values» val=cos(t3) val = Columns 1 through Columns 8 through Columns 15 through » % plot val versus t3» plot(t3,val),grid» % replot the data on a semilog time scale» semilogx(t3,val),grid Columns 1 through Columns 8 through Columns 15 through » % here logspace takes on 21 logarithmically spaced values between 10^1 and 10^1. While you can use decimal powers of ten, you probably don t mean to.
3 Working with polynomials» % coefficients of descending polynomial powers of s, such as 1s^3 0s^2 02s2, are inserted into a defining coefficient array» p1=[ ] p1 = » % to solve for roots of the polynomial:» r=roots(p1) r = i i» % define a row vector for s^22s1» p2=[1 2 1] p2 = 1 2 1» % multiplying the two polynomial vectors together» p3=conv(p1,p2) p3 = » % that is, p3 = s^52s^4s^32s^24s2» roots(p3) ans = i i » % three of the five roots are the same, since p1 is a factor of p3» % The Laplace transform of an output function will be a ratio of two polynomials in s, N(s)/D(s). Here, Y(s)=(10)/(s625s10)» n=[10 0];» d=[1 5 10];» % the roots may be real or complex:» roots(d) ans = i i» % matlab defaults to a 4 decimal place display» to change the display» format long» roots(d) ans = i i» % every time you type format you toggle between the short and long form» format» roots(d) ans = i i Time and frequency response» % define a transfer function, here H(s) = Num(s)/Den(s)=s/(s^2s4s)» num=[1 0];» den=[1 1 4];» % generate an impulse response» impulse(num,den)» % as written, the command automatically generates a sensible output plot,» % but if you want to control the timesample vector» t=0: 0.1: 5;» impulse(num,den,t)» % generate a unit step response» step(num,den,t)» % to plot both reponses on one axis» % define two named vectors, y1 and y2» y1=impulse(num,den,t); y2=step(num,den,t);» plot(t,y1,t,y2),grid» % if you want, add a plot title» title('impulse resp=blue, step resp=green')» % generate a frequency response for a new transfer function. Here is one way:» n=[20 80]; d=[1 4 16];» w=logspace(1,2,101);» Gain=freqs(n,d,w);» mag=abs(gain)» db=20*log10(mag)» ph=angle(gain)*180/pi» subplot(211), semilogx(w, db), grid» subplot(212), semilogx(w,ph), grid
4 Fourier series manipulations» k=5: 1: 5;» k k = » ck=exp(j*pi*k/3).*sin(k*pi/3)./(k*pi); Warning: Divide by zero.» % matlab doesn t crash when division by zero is attempted, but it reports the value as NaN (not a number)» % we need to investigate what happened» ck ck = Columns 1 through i i i i Columns 5 through i NaN NaNi i i Columns 9 through i i i» % when k=0, the quotient is 0/0 using L Hopital s rule and differentiating with respect to k, shows» ck(6)=1/3; lim (pi/3)cos(k*pi/3)./(k*pi) = 1/3 k 0 so we manually reset the dc term to 1/3» % plot the magnitude spectra» stem(k,abs(ck)), grid Matlab state variable solutions» % define a second order state model» A = [0 1 ; 2 3]; B=[0; 1]; C=[4 5]; D=[0];» % generate an equivalent transfer function» [num,den] = ss2tf(a,b,c,d) num = den = 1 3 2» % determine the system roots from the A matrix» copy=poly(a);» r=roots(copy) r = 2 1» % generate time responses» t = 0: 0.05 : 10» % the state variable model allows for multiple inputs (the Uvector). For a singleinput system, specify input 1:» y3=impulse(a,b,c,d,1,t);» y4=step(a,b,c,d,1,t);» % to produce several graphs on one sheet use subplot» subplot(2,2,1),plot(t,y3),grid,title( title1 )» subplot(2,2,2),plot(t,y4),grid,title( title2 )» % here is an alternate set of commands for a frequency response db plot» w=logspace(1,1,41);» [mag,phase]=bode(a,b,c,d,1,w); % notice that phase is returned in degrees using bode. This didn t happen with freqs» subplot(2,2,3), semilogx(w,20*log10(mag)),grid, ylabel('db'), title( title3')» subplot(2,2,4), semilogx(w,phase), grid, title( title4') Viewing tabular data» % to dump mag and phase versus omega» [w, mag, phase ]» % it was necessary to transpose the row vectors for a neat presentation. Since all the data was real, either or. suffices. Should your data be too sparse to use, consider rerunning your commands with a new logspace set
5 Procedure: Consider this linear timeinvariant circuit 1H Annotate a polezero sketch of H(s) on the axes below: f(t) i(t) 6Ω v(t) 1/ 25 F To generate a describing equation, write a KVL expression in terms of the current, i Based upon the polezero sketch, predict the modes of the system s natural response Laplace transform the equation term by term For a unit step input, where F(s) = 1/s, write an s domain output expression for V(s). Generate the transfer function, I(s)/F(s) Via an inverse Laplace transform, determine an exact analytic expression for v(t). Produce a second transfer function, V(s)/I(s) Multiply the last two items to produce the desired transfer function, H(s) = V(s)/F(s)
6 1H Express the equations as a complete set of matrix state equations f(t) i(t) 1/ 25 F R v c v(t) The resistance has been decreased to an unknown value, 0 < R < 6Ω, to reduce damping in the circuit. Assign state variables, x1=i, x2=v c. Evaluate det [ si A ] as a function of R Write KVL in terms of i,v c, R, and f (if necessary) Write KCL at the node above the capacitor in terms of i,v c, R, and f (if necessary) Solve for the roots of your characteristic equation as R varies from 1Ω to 6Ω by steps of 1Ω Sketch and label the roots in the splane: Write an output equation for v in terms of i, v c, R, and f (if necessary)
7 Postlab (it may be necessary to finish this after lab): 1. Produce a matlab impulse response from your transfer function (for R=6Ω). Plot the response. consider the timedomain output equation y(t) = h(t)*u(t) y(t) = h(τ) u(tτ) dτ y(t) = h(τ) u((τt)) dτ since h(τ) is presumed to be zero for τ<0, we can change the lower limit of integration to zero. For τ>t, the product of the functions must be zero, so we can change the upper limit of integration to t and eliminate the need for the reversed unit step. t y(t) = h(τ) dτ 0 sketch the function u((τt)) on your impulse response plot, such that t corresponds to the first zeromagnitude value of h(t), other than at t=0. explain why this time is also the expected time for the peak overshot in the output. 2. Produce a matlab step response from your transfer function. Plot the response. from a step response data dump, estimate the rise time from the step response data dump, estimate the time and extent of the peak overshoot (as defined in last week s laboratory) 3. Produce a matlab db magnitude frequency response from your transfer function. Plot the response on a semilog scale using subplot (2,1,1) from a db response data dump, estimate the 3db bandwidth, ω c what frequency is associated with the peak in the frequency response, ω r? 4. Produce a matlab phase response from your transfer function (in degrees). Plot the response on a semilog scale using sublot (2,1,2) from a phase response data dump, what is the phase associated with low frequencies? with high frequencies? with the peak in the frequency response, ω r?
8 5. Use your state model to produce plots of the impulse response for R=1Ω through 6Ω. A set of suggested matlab commands is provided in the right column which allow you to interact with your plots. You may use these commands or those you device yourself. 6. Use your state model to produce plots of the step response for R=1Ω through 6Ω 7. Use your state model to produce frequency response db magnitude plots for R=1Ω through 6Ω 8. Use your state model to produce frequency response phase plots (in degrees) for R=1Ω through 6Ω 9. Comment on the effect of reducing the circuit damping. effect on peak overshoot in the step response effect on the rise time (A matlab command set to achieve the desire plots)» hold off % if you repeat the steps below, you thus start in a known setting» t=0:0.02:2; w=logspace(0,2,121);» for R=1:6; % set up a loop to repeat the plotting operation six times. Note that the cursor prompt» goes away in the loop» A=[ ]; B=[ ]; C=[ ]; D=[ ]; % insert the necessary element values yourself!» y3=impulse(a,b,c,d,1,t);» y4=step(a,b,c,d,1,t);» subplot(2,2,1),plot(t,y3),axis([ ]),grid on,» title('impulse response as R=1:6'); hold on; pause; % to label the plots intelligently you could develop this interactive procedure: % click the A box in the top line menu % locate the cursor appropriately and click % type an identifier (such as 1 for 1 Ohm) % click the mouse in the grey outside the graph % if you are unhappy with the location, click and drag as you wish % when ready to advance to the next plot strike any key» subplot(2,2,3),plot(t,y4),axis([ ]),grid on,» title('step response as R=1:6');» hold on;» pause; effect on the 3db bandwidth effect on phase at the peak in the frequency response, ω r? % repeat the labeling procedure for each plot as needed» [mag,phase]=bode(a,b,c,d,1,w);» subplot(2,2,2),semilogx(w,20*log10(mag))» grid on,» title('db plot for R=1:6'); hold on; pause;» subplot(2,2,4),semilogx(w,phase),grid on,» title('phase plot for R=1:6');» hold on;» pause;» end
9 Report: No formal report is required for this lab. You should turn in this lab handout fill all planks filled in and all questions answered. You should also turn in all plots and tabular data requested.
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