Assignment # 6 solution

Transcription

1 Faculty of Engineering Electrical Engineering Dep. Assignment # 6 solution Question (1) a) Find the steady state sinusoidal response of the system? Since all the poles of G(z) are located in the unit circle, so the system is stable, and hence we can find the steady state response of the system using the following equation: y UG T ω G T U0 = 5, T = 1sec (always 1 if not given) and ω = rad/sec, Then: Ge T Ge = And hence y..u Since our sinusoidal input is cosine. b) Find all the response "transient and steady state" of the system? First we have to make partial fraction expansion as the following: Yz Gz Uz,Uz.. so Yz z 5z 3z z z 0.45z 0.1z z z 1

2 Faculty of Engineering Electrical Engineering Dep. After Partial Fraction Expansion, Y(s) becomes Yz z k1 z0.3 k2 z 0.25 k3 z0.5 α z j Find the values of the constants (k1, k2, k3, α and α ) α z j k1 = Yz and the same for the other constants Then Y(z) becomes: Yz z z0.3 z z z z z j z z j To find the inverse z transform of the two conjugate terms: z from a then: za z z j e j e. e e. the other term will be e. and the sum of two terms is cos108k cos 3 k To find y(k), then we have to find the inverse Z-transform for Y(z) as the following: yk cos Uk Transient Response Steady State Response

3 Faculty of Engineering Electrical Engineering Dep. Note that the steady state response is the same as in part (a). c) After how many samples, will the transient response vanishes? The transient response will approximately go to zero after N T samples where 4.6 N T and p is largest magnitude of all the poles. ln( p ) N T ln( 0.5 ) Thus the transient response will vanish approximately after 7 samples of the system output. d) Draw the Bode plot of the system analytically? To draw the bode plot, we substitute in G(e T ) and T=1sec to find the following values: Ge Gz 20logGe Ge Ge 0 Inf db 0 Ge j db Ge. 3 j db Ge j db Ge j db Ge j db Ge j db

4 Faculty of Engineeringg Electrical Engineering Dep. Then we draw these values to get the bode plot: Magnitude Phase: Note thatt the graphs are not accurate small. since the number of points is too

5 Faculty of Engineeringg Electrical Engineering Dep. e) From the previous part, find the bandwidth and the margins (Gain and Phase) of the system? The bandwidth B of a discretee system is defined as the frequency B at which jbt 1 j0t 1 G( e ) G( e ) where 20log -3dB, i.e. the bode plot is 3dB 2 2 down from its value at zero frequency. To find the margins: Gain Margin 20log,wher re wcisthe frequency in rad/sec at which G the phase equals to From the above graph wc = 2rad/sec then 20log = db Phase Margin Ge 180, where wg is the frequency in rad/sec at which the magnitude equals to 0 db. Note that there is no such frequency so the phase margin = inf. Or there is no phase margin. The graph of the bode plot using MATLAB will be: Note that since the discrete bode plot graph is periodic and not like the continuous bode plot (aperiodic) it can be drawn over 1 period only. (pi/t) and T = 1sec.

6 Faculty of Engineeringg Electrical Engineering Dep. Question (2) Obtain a Nyquist plot of L(z) and find the gain and phase margins, a) L( z) 0.5 z z 1.5 ( z 1) Solution: 1 From graph max negative real axis. GM = 20 log ( is the point on the Nyquist plot that intersects the max max ) db is the gain margin GM that often expressed in db: GM = 20 log ( max ) db = 20 log ( ) db = db. Phase margin PM: the angle at which the unit circle intersects the Nyquist plot (after subtracting from the intersection angle). From the above graph we can say that PM = Do the same for branches (b) and (c).

7 Faculty of Engineeringg Electrical Engineering Dep. Question (3) a) Nyquist Plot Thus the upper gain margin is 20 log ( PM = max ) db = 20 log (3.0211) = db

8 Faculty of Engineeringg Electrical Engineering Dep. b) Bode Plot Gain Margin 20log the phase equals to G,wher re wcisthe frequency in rad/sec at which From the above graph wc = 0.45rad/ sec then GM 20log Phase Margin Ge 180, where wg is the frequency in rad/secc at which the magnitude equals to 0 db. From the above graph wg = 0.05 PM = Ge G. = 9.65 db We note thatt the results must be the same either using bode or Nyquist plot.

9 Faculty of Engineering Electrical Engineering Dep. Using MATLAB: Continue Question (3) ek rk yk E(z) = R(z) Y(z), Y(z) = TF * R(z), Then: Ez Rz TFRz 1TFRz,TF GzCz 1CzGz Then SSE = lim Ez for unit step input Uz SSE = 0. for unit ramp input Uz SSE = for a parabolic input Uz SSE =, we investigate that the system type is one.

10 Faculty of Engineeringg Electrical Engineering Dep. Question (4) Consider a system with the following Bode plot a) Find the steady state sinusoidal output corresponding to the input, U(n) = 5 cos(2n)? the magnitude is about 8 db and the phase is about 110. Solution: Y ss (n) (5)(10^(8/20)) cos( 2n ) = cos(2n ) b) Find the bandwidth of the system? 1rad\sec. B c) Can this system track the following sinusoidal input, U( (n) = sin(5n)? no because its frequency (5rad/sec) is larger than (bandwidth 1rad/sec). d)findd the gain and phase margins of the system? Gain margin GM 3 db Phase margin PM =10 0 The Answers in this question are not accurate since it is taken from the graph.

11 Faculty of Engineering Electrical Engineering Dep. How to draw the magnitude bode plot using calculator, and then finding Bandwidth Example (5.1) Draw the bode plot and find the Bandwidth? آيف ترسم المعادلة السابقة باستخدام الا لة الحاسبة شغل الا لة الحاسبة واختار القاي مة الخاصة بالرسم وهي. Menu 3 اضغط F3 واختار نوع المعادلة =Y. اآتب المعادلة المظللة آما هي واستبدل w ب x ولكي تجد القيمة المطلقة اعمل التالي: اضغط على زر OPTIN لتظهر النافذة الا خرى. اضغط F2 لتختار القاي مة NUM ومنها اختر.abs ثم اضغط EXE لرسم المعادلة. بعد الرسم لتغيير الحدود بحيث تتناسب مع الرسم اللوغاريتمي اعمل التالي: اضغط على زر SHIFT ثم على زر OPTION غير = 0 Xmin و = 1000 Xmax و = 10 scale غير -100 = Ymin و = 10 Ymax و = 10 scale ثم ارسم مجددا فتظهر الرسمة بالشكل المطلوب. (١ (٢ (٣ (٤ (٥ ملاحظة// القيم السابقة ممكن أن تتغير من رسمة لا خرى وسوف تزداد خبرتك في اختيارها بالممارسة أآثر فا آثر للرسم.

12 Faculty of Engineering Electrical Engineering Dep. ٦) الا ن لكي نوجد ال BW اعمل التالي: اضغط F5(TABL) فتجد جدول يمثل أزواج مرتبة لقيم X مع Y. استبدل قيم X بالقيم التي تريدها فتقوم الا لة مباشرة با يجاد قيم Y المقابلة لها. الا ن لكي تتا آد من القيم التي قمنا بحسابها ضع قيم X آالتالي:,0.001,0.1,1,10, وهي القيم التي نختارها في الرسم اليدوي عادة. نعرف أن ال BW هو التردد المقابل (للقيمة عند الصفر db 3) على محور Y وبالتالي في هذا المثال هو التردد المقابل للقيمة -٣ حيث أن القيمة عند الصفر تساوي صفر. من خلال قيم X وقيم Y المقابلة لها تكون قد توقعت المدى الذي يمكن أن يوجد فيه قيمة على محور X تكون قيمتها المقابلة على محورY.-3dB ثم تختار قيم تجريبية حتى تصل للقيمة المطلوبة. وهي هنا تقريبا 3.8. rad/sec وفقكم االله لما يحب ويرضى