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Complex Numbers

Math Review with Matlab:. Complex Numbers. Sinusoidal Addition. S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn. Sinusoidal Addition. A useful application of complex numbers is the addition of sinusoidal signals having the same frequency.

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Complex Numbers

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  1. Math Review with Matlab: ComplexNumbers Sinusoidal Addition S. Awad, Ph.D. M. Corless, M.S.E.E. E.C.E. Department University of Michigan-Dearborn

  2. Sinusoidal Addition A useful application of complex numbers is the addition of sinusoidal signals having the same frequency • General Sinusoid • Euler’s Identity • Sinusoidal Addition Proof • Phasor Representation of Sinusoids • Phasor Addition Example • Addition of 4 Sinusoids Example

  3. General Sinusoid • A general cosine wave, v(t), has the form: M = Magnitude, amplitude, maximum value w = Angular Frequency in radians/sec (w=2pF) F = Frequency in Hz T = Period in seconds (T=1/F) t = Time in seconds q = Phase Shift, angular offset in radians or degrees

  4. Euler’s Identity • A general complex number can be written in exponential polar form as: • Euler’s Identity describes a relationship between polar form complex numbers and sinusoidal signals:

  5. Useful Relationship • Euler’s Identity can be rewritten as a function of general sinusoids: • Resulting in the useful relationship:

  6. Sinusoidal Addition Proof • Show that the sum of two generic cosine waves, of the same frequency, results in another cosine wave of the same frequency but having a different Magnitude and Phase Shift (angular offset) Given: Prove:

  7. Complex Representation • Each cosine function can be written as the sum of the real portion of two complex numbers

  8. Complex Addition • ejwt is common and can be distributed out • The addition of the complex numbers M1ejq1 and M2ejq2 results in a new complex number M3ejq3

  9. Return to Time Domain • The steps can be repeated in reverse order to convert back to a sinusoidal function of time • We see v3(t) is also a cosine wave of the same frequency as v1(t) and v2(t), but having a different Magnitude and Phase

  10. Time Domain Voltage: v(t) Complex Domain Phasor: V(jw) Phasors • In electrical engineering, it is often convenient to represent a time domain sinusoidal voltages as complex number called a Phasor • Standard Phasor Notation of a sinusoidal voltage is:

  11. Time Domain Complex Domain Time Domain Phasor Addition • As shown previously, two sinusoidal voltages of the same frequency can easily beaddedusing their phasors

  12. Phasor Addition Example • Example: Use the Phasor Technique to add the following two 1k Hz sinusoidal signals. Graphically verify the results using Matlab. Given: Determine:

  13. Phasor Transformation • Since Standard Phasors are written in terms of cosine waves, the sine wave must be rewritten as: • The signals can now be converted into Phasor form

  14. Rectangular Addition • To perform addition by hand, the Phasors must be written in rectangular (conventional) form • Now the Phasors can be added

  15. Transform Back to Time Domain • Before converting the signal to the time domain, the result must be converted back to polar form: • The result can be transformed back to the time domain:

  16. Addition Verification • Matlab can be used to verify the complex addition: » V1=2*exp(j*0); » V2=3*exp(-j*pi/2); » V3=V1+V2 V3 = 2.0000 - 3.0000i » M3=abs(V3) M3 = 3.6056 » theta3= angle(V3)*180/pi theta3 = -56.3099

  17. Time Domain Addition • The original cosine waves can be added in the time domain using Matlab: f =1000; % Frequency T = 1/f; % Find the period TT=2*T; % Two periods t =[0:TT/256:TT]; % Time Vector v1=2*cos(2*pi*f*t); v2=3*sin(2*pi*f*t); v3=v1+v2;

  18. Code to Plot Results subplot(3,1,1); plot(t,v1); grid on; axis([ 0 TT -4 4]); ylabel('v_1=2cos(2000\pit)'); title('Sinusoidal Addition'); subplot(3,1,2); plot(t,v2); grid on; axis([ 0 TT -4 4]); ylabel('v_2=3sin(2000\pit)'); subplot(3,1,3); plot(t,v3); grid on; axis([ 0 TT -4 4]); ylabel('v_3 = v_1 + v_2'); xlabel('Time'); • Plot all signals in Matlab using three subplots • v_1 prints v1 • \pi prints p

  19. Plot Results • Plots show addition of time domain signals

  20. Verification Code • Plot the added signal, v3, and the hand derived signal to verify that they are the same v_hand=3.6056*cos(2*pi*f*t-56.3059*pi/180); subplot(2,1,1);plot(t,v3); grid on; ylabel('v_3 = v_1 + v_2'); xlabel('Time'); title('Graphical Verification'); subplot(2,1,2);plot(t,v_hand); grid on; ylabel('3.6cos(2000\pit - 56.3\circ)'); xlabel('Time');

  21. Graphical Verification • The results are the same • Thus Phasor addition is verified

  22. Four Cosines Example • Example: Use Matlab to add the following four sinusoidal signals and extract the Magnitude, M5 and Phase, q5 of the resulting signal. Also plot all of the signals to verify the solution. Given: Determine:

  23. Enter in Phasor Form • Transform signals into phasor form • Create phasors as Matlab variables in polar form » V1 = 1*exp(j*0); » V2 = 2*exp(-j*pi/6); » V3 = 3*exp(-j*pi/3); » V4 = 4*exp(-j*pi/2);

  24. Add Phasors » V5 = V1 + V2 + V3 + V4; » M5 = abs(V5) M5 = 8.6972 » theta5_rad = angle(V5); » theta5_deg = theta5_rad*180/pi theta5_deg = -60.8826 • Add phasors then extract Magnitude and Phase • Convert back into Time Domain

  25. Code to Plot Voltages f =1000; % Frequency T = 1/f; % Find the period t =[0:T/256:T]; % Time Vector v1=1*cos(2*pi*f*t); v2=2*cos(2*pi*f*t-pi/6); v3=3*cos(2*pi*f*t-pi/3); v4=4*cos(2*pi*f*t-pi/2); plot(t,v1,'k'); hold on; plot(t,v2,'b'); plot(t,v3,'m'); plot(t,v4,'r'); grid on; title('Waveforms to be added'); xlabel('Time');ylabel('Amplitude'); • Plot all 4 input voltages on same plot with different colors

  26. Signals to be Added

  27. Code to Plot Results • Add the original Time Domain signals v5_time = v1 + v2 + v3 + v4; subplot(2,1,1);plot(t,v5_time); grid on; ylabel('From Time Addition'); xlabel('Time'); title('Results of Addition of 4 Sinusoids'); • Transform Phasor result into time domain v5_phasor = M5*cos(2*pi*f*t+theta5_rad); subplot(2,1,2);plot(t,v5_phasor); grid on; ylabel('From Phasor Addition'); xlabel('Time');

  28. Compare Results • The results are the same • Thus Phasor addition is verified

  29. Sinusoidal Analysis • The application of phasors to analyze circuits with sinusoidal voltages forms the basis of sinusoidal analysis techniques used in electrical engineering • In sinusoidal analysis, voltages and currents are expressed as complex numbers called Phasors. Resistors, capacitors, and inductors are expressed as complex numbers called Impedances • Representing circuit elements as complex numbers allows engineers to treat circuits with sinusoidal sources as linear circuits and avoid directly solving differential equations

  30. Summary • Reviewed general form of a sinusoidal signal • Used Euler’s identity to express sinusoidal signals as complex exponential numbers called phasors • Described how Phasors can be used to easily add sinusoidal signals and verified the results in Matlab • Explained phasor addition concepts are useful for sinusoidal analysis of electrical circuits subject to sinusoidal voltages and currents

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