Presentation is loading. Please wait.

Presentation is loading. Please wait.

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.

Similar presentations

Presentation on theme: "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."— Presentation transcript:


2 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

3 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

4 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

5 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:

6 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:

7 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:

8 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

9 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

10 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

11 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:

12 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

13 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:

14 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

15 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

16 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:

17 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 =

18 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;

19 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

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

21 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');

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

23 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:

24 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

25 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

26 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');

27 Complex Numbers:Sinusoidal Addition 26 Signals to be Added

28 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');

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

30 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

31 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

Download ppt "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."

Similar presentations

Ads by Google