Circuits and Systems I

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1 Circuits and Systems I LECTURE #2 Phasor Addition lions@epfl Prof. Dr. Volkan Cevher LIONS/Laboratory for Information and Inference Systems

2 License Info for SPFirst Slides This work released under a Creative Commons License with the following terms: Attribution The licensor permits others to copy, distribute, display, and perform the work. In return, licensees must give the original authors credit. Non-Commercial The licensor permits others to copy, distribute, display, and perform the work. In return, licensees may not use the work for commercial purposes unless they get the licensor's permission. Share Alike The licensor permits others to distribute derivative works only under a license identical to the one that governs the licensor's work. Full Text of the License This (hidden) page should be kept with the presentation

3 Outline - Today Today <> Section 2-6 Lab 1! Next week <> Section 3-1 Section 3-2 Section 3-3 Section 3-7 Section 3-8 Recommended self-study next week + Appendix A: Complex Numbers read Appendix B: MATLAB read

4 Lecture Objectives Phasors = Complex Amplitude Complex Numbers represent Sinusoids jωt z( t) = Xe = ( Ae j ϕ ) e jωt Develop the ABSTRACTION: Adding Sinusoids = Complex Addition PHASOR ADDITION THEOREM

5 Lecture Objectives Phasors = Complex Amplitude Complex Numbers represent Sinusoids Develop the ABSTRACTION: Adding Sinusoids = Complex Addition PHASOR ADDITION THEOREM CSI Progress Level:

6 Do You Remember The Complex Numbers? To solve: z 2 = -1 z = j Math and Physics use z = i Complex number: z = x + j y Lab 2 warm-up y z x Cartesian coordinate system

7 Polar Form Vector Form Length =1 Angle = θ Common Values j has angle of 0.5π -1 has angle of π - j has angle of 1.5π also, angle of -j could be -0.5π = 1.5π -2π because the PHASE is AMBIGUOUS

8 Polar <> Rectangular Relate (x,y) to (r,θ) r r θ 2 = x 2 + = Tan 1 2 y ( y ) x Most calculators do Polar-Rectangular x y = = θ x r cos r sin θ θ y Need a notation for POLAR FORM

9 Euler s Formula Complex Exponential Real part is cosine Imaginary part is sine Magnitude is one θ e j = cos( θ ) + j sin( θ ) θ re j = rcos( θ ) + jr sin( θ )

10 Complex Exponential e j ω t = cos( ω t) + j sin( ω t) Interpret this as a Rotating Vector θ = ωt Angle changes vs. time ex: ω=20π rad/s Rotates 0.2π in 0.01 secs θ e j = cos( θ ) + j sin( θ )

11 cos = Real Part jωt Real Part of Euler s cos( ωt) = Re{ e } General Sinusoid x( t) = Acos( ω t + ϕ) So, Acos( ωt + ϕ ) = Re{ Ae j( ωt+ ϕ ) } = Re{ Ae jϕ e jωt }

12 Real Part Example Acos( ωt + ϕ) = R { j t } e Ae ϕ e j ω Evaluate: x( t) = Re { j t } 3 je ω Answer: x( t) = Re { j } ( 3 ) ωt e j e { j0.5 } 3 π jωt e e = 3cos( ωt 0.5π ) = R

13 Complex Amplitude General Sinusoid x( t) = Acos( ωt + ϕ) = Re { j t } Ae ϕ e j ω Complex AMPLITUDE = X Then, any Sinusoid = REAL PART of Xe jωt x( t) = Re jωt jϕ z ( t) = Xe X = Ae { j t } { j j t } Xe ω = Re Ae ϕ e ω

14 Z DRILL (Complex Arith)

15 How to AVOID Trigonometry Algebra, even complex, is EASIER!!! Can you recall cos(θ 1 +θ 2 )? Use: real part of e j (θ 1 +θ 2 ) = cos(θ1 +θ 2 )

16 Recall Euler s FORMULA Complex Exponential Real part is cosine Imaginary part is sine Magnitude is one θ e j = cos( θ ) + j sin( θ ) e j ω t = cos( ω t) + j sin( ω t)

17 Real & Imaginary Part Plots PHASE DIFFERENCE = π/2

18 Complex Exponential e j ω t = cos( ω t) + j sin( ω t) Interpret this as a Rotating Vector θ = ωt Angle changes vs. time ex: ω=20π rad/s Rotates 0.2π in 0.01 secs e jθ = cos( θ) + jsin( θ)

19 See Demo on CD-ROM Chapter 2 Rotating Phasor

20 Cos = REAL PART Real Part of Euler s cos(ωt) = Re{ e jω t } General Sinusoid So, x(t) = Acos(ωt +ϕ) Acos(ω t +ϕ) = Re Ae { j (ω t +ϕ) } = Re{ Ae jϕ e jω t }

21 Complex Amplitude General Sinusoid x(t) = Acos(ωt +ϕ) = Re{ Ae jϕ e jωt } Sinusoid = REAL PART of (Ae jφ )e jωt x(t) = Re { Xe jωt }= Re{ z(t) } Complex AMPLITUDE = X z(t) = Xe jωt X = Ae jϕ

22 POP QUIZ: Complex Amp Find the COMPLEX AMPLITUDE for: x( t) = 3 cos(77π t + 0.5π ) Use EULER s FORMULA: x( t) = Re { } 3e j(77πt+ 0.5π ) = Re { j j t } 3e 0.5π e 77π X = 3e j0.5π

23 Want to Add Sinusoids ALL SINUSOIDS have SAME FREQUENCY HOW to GET {Amp,Phase} of RESULT?

24 Add Sinusoids Sum Sinusoid has SAME Frequency

25 Phasor Addition Rule Get the new complex amplitude by complex addition

26 Phasor Addition Proof

27 POP QUIZ: Add Sinusoids ADD THESE 2 SINUSOIDS: x 1 ( t) = cos(77 π t) x 2 ( t) = 3 cos(77 π t π ) COMPLEX ADDITION: j0 1e + 3e j0.5π

28 POP QUIZ (answer) COMPLEX ADDITION: 1 + j 3 = 2e jπ /3 j 3 = 3e j0.5π 1 CONVERT back to cosine form: x 3 ( t) = 2cos(77π t + π ) 3

29 Add Sinusoids Example x 1 ( t) t m1 x 2 ( t) t m2 x 3( t) = x1( t) + x2( t) t m3

30 Convert Time-Shift to Phase Measure peak times: t m1 = , t m2 = , t m3 = Convert to phase (T=0.1) φ 1 =-ωt m1 = -2π(t m1 /T) = 70π/180, φ 2 = 200π/180 Amplitudes A 1 =1.7, A 2 =1.9, A 3 =1.532

31 Phasor Add: Numerical Convert Polar to Cartesian X 1 = j1.597 X 2 = j sum = X 3 = j Convert back to Polar X 3 = at angle π/180 This is the sum

32 Add Sinusoids X 1 VECTOR (PHASOR) ADD X 3 X 2

33 Next week <> Section 3-1 Section 3-2 LAB TIME NOW! Section 3-3 Section 3-7 Section 3-8

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