1. Math Review with Matlab:
4/1/2017
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

2. Sinusoidal Addition General Sinusoid Euler’s Identity
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. Phasors Time Domain Voltage: v(t) Complex Domain Phasor: V(jw)
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:
Time Domain
Voltage: v(t)
Complex Domain
Phasor: V(jw)

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

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 = i
» M3=abs(V3)
M3 =
3.6056
» theta3= angle(V3)*180/pi
theta3 =

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 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);
ylabel('v_2=3sin(2000\pit)');
subplot(3,1,3); plot(t,v3);
ylabel('v_3 = v_1 + v_2');
xlabel('Time');
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 *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)');

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

22. Four Cosines Example Given: Determine:
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 Add phasors then extract Magnitude and Phase
» V5 = V1 + V2 + V3 + V4;
» M5 = abs(V5)
M5 =
8.6972
» theta5_rad = angle(V5);
» theta5_deg = theta5_rad*180/pi
theta5_deg =
Add phasors then extract Magnitude and Phase
Convert back into Time Domain

25. Code to Plot Voltages
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');

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