2025-L03 - Bring Calculator to Recitation Practice Complex...

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Unformatted text preview: ! Bring Calculator to Recitation Practice Complex arithmetic Help Sessions (during the semester) 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 2 " ! # ! Check the Forums & Announcements daily SP First URL: www.Rose-Hulman.edu/DSPFirst Username = gt, password = student Be prepared for Lab Work the Pre-Lab section Even better, do some or all of the Exercises for Verification Lab Verifications: Turn in at end of Lab Counts as part of the Lab Report score Lectures are being posted PDF format (4 per page) Lab Report due one week later Learn your Lab TA's format requirements Lab FAQs are posted, but relate to old labs Bring your laptop to lab if you want to use your own environment. 8/25/2008 EE-2025 Fall-2008 jMc-BHJ Get PDF files of Labs & HWs from t-square 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 3 4 $ %& This Lecture: '' &%( % ' )*+ , ' Phasors = Complex Amplitude Complex Numbers represent Sinusoids Chapter 2, Section 2-6 Other Reading: Appendix A: Complex Numbers Appendix B: MATLAB Next Lecture: start Chapter 3 8/25/2008 5 z (t ) = Xe jt = ( Ae j )e jt Develop the ABSTRACTION: Adding Sinusoids = Complex Addition PHASOR ADDITION THEOREM EE-2025 Fall-2008 jMc-BHJ 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 6 -$ . / 0 1 ,) $ 2 Algebra, even complex, is EASIER !!! Can you recall cos(1+2) ? Use: real part of ej(1+2) = cos( + ) 1 2 e j (1 + 2 ) = e j1 e j 2 = (cos1 + j sin 1 )(cos 2 + j sin 2 ) = (cos1 cos 2 - sin 1 sin 2 ) + j (...) 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 7 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 8 3 ) ( 4 2 Complex Exponential Real part is cosine Imaginary part is sine Magnitude is one e j = cos( ) + j sin( ) e jt = cos( t ) + j sin( t ) 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 9 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 10 )( 5 5 )% % e j t = cos( t ) + j sin( t ) Interpret this as a Rotating Vector t = Angle changes vs. time ex: =20 rad/s Rotates 0.2 in 0.01 secs e j = cos( ) + j sin( ) 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 11 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 12 6 7 )( 5 ( $ cos( t) = e{ j t } e x(t) = Acos( t + ) A cos( t + ) = e{ j ( t + ) } Ae = e{ Ae e j 8/25/2008 EE-2025 Fall-2008 jMc-BHJ x(t) = Acos( t + ) = e{Ae j e j t } ! j " $ Ae # e#jt x(t) = e { j t }= e{z(t)} Xe X also where the phasor sits at t=0 j t } 13 8/25/2008 z(t) = Xe j t Fall-2008 jMc-BHJ X = Ae j 14 EE-2025 ) 8 -9 / 0 / : % $$ ' % ') $' Find the COMPLEX AMPLITUDE for: x(t ) = 3 cos(77 t + 0.5 ) Use EULER's FORMULA: When all SINUSOIDS have SAME freq.... HOW to GET {Amp,Phase} of RESULT ? x(t ) = e 3e j ( 77 t +0.5 ) j 0.5 { = e{ 3e EE-2025 Fall-2008 e j 77 t } } 15 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 8/25/2008 X = 3e j 0.5 jMc-BHJ 16 $$ ' % ') $' Sum Sinusoid has SAME Frequency ; ') $$ )% %& ' (& ) ' 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 17 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 18 ' !' " 62 !' ! A1e j1 A1e j1 e jt = Ae j e jt 2 1 A1e j1 e jt + A2 e j 2 e jt Ae j = A1e j1 + A2 e j 2 A2 e j 2 8/25/2008 EE-2025 Fall-2008 jMc-BHJ A2 e j 2 e jt 19 Phasor addition 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 20 ) 8 -9 ' ) 8 -. < 1 ADD THESE 2 SINUSOIDS: COMPLEX ADDITION: x1 (t ) = cos(77 t ) x2 (t ) = 3 cos(77 t + 0.5 ) COMPLEX ADDITION: 1 + j 3 = 2e j / 3 j 0.5 j 3 = 3e 1 CONVERT back to cosine form: 1e j 0 + 3e j 0.5 8/25/2008 EE-2025 Fall-2008 jMc-BHJ x3 (t ) = 2 cos(77 t + ) 3 21 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 22 $$ ' % ') $' 5 ( = ' ! Measure peak times: tm1=-0.0194, tm2=-0.0556, tm3=-0.0394 Convert to phase (T=0.1) 1=-tm1 =-2(tm1 /T)=70/180, 2= 200/180 Amplitudes A1=1.7, A2=1.9, A3=1.532 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 23 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 24 9% Convert Polar to Cartesian X1 = 0.5814 + j1.597 X2 = -1.785 - j0.6498 sum = X3 = -1.204 + j0.9476 6 $$ ' % ') $' Convert back to Polar X3 = 1.532 at angle 141.79/180 This is the sum 8/25/2008 EE-2025 Fall-2008 jMc-BHJ * ! "+ $ 25 8/25/2008 EE-2025 Fall-2008 jMc-BHJ 26 ' !' " 62 !' A1e j1 A1e j1 e jt = Ae j e jt 2 1 A1e j1 e jt + A2 e j 2 e jt Ae j = A1e j1 + A2 e j 2 A2 e j 2 8/25/2008 EE-2025 Fall-2008 jMc-BHJ A2 e j 2 e jt 27 ...
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