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

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Complex Numbers:Sinusoidal Addition 2 A useful application of complex numbers is the addition of sinusoidal signals having the same frequency n General Sinusoid General Sinusoid n Eulers Identity Eulers Identity n Sinusoidal Addition Proof Sinusoidal Addition Proof n Phasor Representation of Sinusoids Phasor Representation of Sinusoids n Phasor Addition Example Phasor Addition Example n Addition of 4 Sinusoids Example Addition of 4 Sinusoids Example

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

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Complex Numbers:Sinusoidal Addition 4 Eulers Identity n A general complex number can be written in exponential polar form as: n Eulers Identity describes a relationship between polar form complex numbers and sinusoidal signals:

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Complex Numbers:Sinusoidal Addition 5 Useful Relationship n Eulers Identity can be rewritten as a function of general sinusoids: n Resulting in the useful relationship:

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Complex Numbers:Sinusoidal Addition 6 Sinusoidal Addition Proof n 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:

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Complex Numbers:Sinusoidal Addition 7 Complex Representation n Each cosine function can be written as the sum of the real portion of two complex numbers

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Complex Numbers:Sinusoidal Addition 8 Complex Addition e j t is common and can be distributed out The addition of the complex numbers M 1 e j and M 2 e j results in a new complex number M 3 e j

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Complex Numbers:Sinusoidal Addition 9 Return to Time Domain n The steps can be repeated in reverse order to convert back to a sinusoidal function of time n We see v 3 (t) is also a cosine wave of the same frequency as v 1 (t) and v 2 (t), but having a different Magnitude and Phase

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

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Complex Numbers:Sinusoidal Addition 11 Phasor Addition n As shown previously, two sinusoidal voltages of the same frequency can easily be added using their phasors Time Domain Time Domain Complex Domain

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

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

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Complex Numbers:Sinusoidal Addition 14 Rectangular Addition n To perform addition by hand, the Phasors must be written in rectangular (conventional) form n Now the Phasors can be added

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Complex Numbers:Sinusoidal Addition 15 Transform Back to Time Domain n Before converting the signal to the time domain, the result must be converted back to polar form: n The result can be transformed back to the time domain:

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Complex Numbers:Sinusoidal Addition 16 » V1=2*exp(j*0); » V2=3*exp(-j*pi/2); Addition Verification n Matlab can be used to verify the complex addition: » V3=V1+V2 V3 = i » M3=abs(V3) M3 = » theta3= angle(V3)*180/pi theta3 =

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Complex Numbers:Sinusoidal Addition 17 Time Domain Addition n 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;

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Complex Numbers:Sinusoidal Addition 18 Code to Plot Results n Plot all signals in Matlab using three subplots 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'); \pi prints v_1 prints v 1

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Complex Numbers:Sinusoidal Addition 19 Plot Results n Plots show addition of time domain signals

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Complex Numbers:Sinusoidal Addition 20 Verification Code n Plot the added signal, v 3, and the hand derived signal to verify that they are the same v_hand=3.6056*cos(2*pi*f*t *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 \circ)'); xlabel('Time');

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Complex Numbers:Sinusoidal Addition 21 Graphical Verification n The results are the same n Thus Phasor addition is verified

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Complex Numbers:Sinusoidal Addition 22 Four Cosines Example Example: Use Matlab to add the following four sinusoidal signals and extract the Magnitude, M 5 and Phase, 5 of the resulting signal. Also plot all of the signals to verify the solution. Given: Determine:

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

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Complex Numbers:Sinusoidal Addition 24 Add Phasors n Add phasors then extract Magnitude and Phase » V5 = V1 + V2 + V3 + V4; » M5 = abs(V5) M5 = » theta5_rad = angle(V5); » theta5_deg = theta5_rad*180/pi theta5_deg = n Convert back into Time Domain

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Complex Numbers:Sinusoidal Addition 25 Code to Plot Voltages n Plot all 4 input voltages on same plot with different colors 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');

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Complex Numbers:Sinusoidal Addition 26 Signals to be Added

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Complex Numbers:Sinusoidal Addition 27 Code to Plot Results n Add the original Time Domain signals n Transform Phasor result into time domain 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'); 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');

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Complex Numbers:Sinusoidal Addition 28 Compare Results n The results are the same n Thus Phasor addition is verified

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

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Complex Numbers:Sinusoidal Addition 30 Summary n Reviewed general form of a sinusoidal signal n Used Eulers identity to express sinusoidal signals as complex exponential numbers called phasors n Described how Phasors can be used to easily add sinusoidal signals and verified the results in Matlab n Explained phasor addition concepts are useful for sinusoidal analysis of electrical circuits subject to sinusoidal voltages and currents

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