ComplexNumbers
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
Sinusoidal AdditionA 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
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Complex Numbers: Sinusoidal Addition
General Sinusoid A general cosine wave, v(t), has the form:
)cos()( tMtv
M = Magnitude, amplitude, maximum value
= Angular Frequency in radians/sec (=2F)
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
Euler’s Identity A general complex number can be
written in exponential polar form as:
)sin()cos( jMMMe j
)sin(Im
)cos(Re
MMe
MMej
j
Euler’s Identity describes a relationship between polar form complex numbers and sinusoidal signals:
jMez
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Complex Numbers: Sinusoidal Addition
Useful Relationship Euler’s Identity can be rewritten as a function of general
sinusoids:
)sin()cos( tjMtMMe tj
)sin(Im
)cos(Re
tMMe
tMMetj
tj
tjMetM Re)cos(
Resulting in the useful relationship:
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Complex Numbers: Sinusoidal Addition
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)
)cos()(
)cos()(
222
111
tMtv
tMtv
)cos()(
)()()(
333
213
tMtv
tvtvtv
Given:
Prove:
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Complex Numbers: Sinusoidal Addition
Complex Representation
Each cosine function can be written as the sum of the real portion of two complex numbers
)cos()cos()( 22113 tMtMtv
21213 ReRe)( tjtj eMeMtv
21213 Re)( tjtj eMeMtv
21213 Re)( jtjjtj eeMeeMtv
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Complex Numbers: Sinusoidal Addition
Complex Addition ejt is common
and can be distributed out
The addition of the complex numbers M1ej and M2ej results in a new complex number M3ej 3
33 Re)( jj eMetv
213213
jjj eMeMeM
21213 Re)( jjtj eMeMetv
21213 Re)( jtjjtj eeMeeMtv
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Complex Numbers: Sinusoidal Addition
Return to Time Domain The steps can be
repeated in reverse order to convert back to a sinusoidal function of time
3
3
3
33
33
33
Re)(
Re)(
Re)(
tj
jtj
jtj
eMtv
eMetv
eMetv
)cos()( 333 tMtv
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
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Complex Numbers: Sinusoidal Addition
Phasors In electrical engineering, it is often convenient to
represent a time domain sinusoidal voltages as complex number called a Phasor
MMejV j)(Complex Domain
Phasor: V(j)
)cos()( tMtvTime Domain
Voltage: v(t)
Standard Phasor Notation of a sinusoidal voltage is:
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Complex Numbers: Sinusoidal Addition
Phasor Addition As shown previously, two sinusoidal voltages of the same
frequency can easily be added using their phasors
TimeDomain
)cos()cos()( 22113 tMtMtv
)cos()( 333 tMtv TimeDomain
ComplexDomain332211 MMM
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Complex Numbers: Sinusoidal Addition
Phasor Addition Example
Example: Use the Phasor Technique to add the following two 1k Hz sinusoidal signals. Graphically verify the results using Matlab.
ttv
ttv
)1000(2sin3)(
)1000(2cos2)(
2
1
)()()( 213 tvtvtv
Given:
Determine:
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Complex Numbers: Sinusoidal Addition
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
903)(902000cos3)(
02)()2000cos(2)(
22
11
jVttv
jVttv
22000cos3)2000sin(3)(2 tttv
902000cos3)(2 ttv
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Complex Numbers: Sinusoidal Addition
Rectangular Addition To perform addition by hand, the Phasors must be written
in rectangular (conventional) form
30903)(
0202)(
2
1
jjV
jjV
32)(
3002)(
)()()(
3
3
213
jjV
jjjV
jVjVjV
Now the Phasors can be added
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Complex Numbers: Sinusoidal Addition
Transform Back to Time Domain Before converting the signal to the time domain, the result
must be converted back to polar form:
23tan)3(2)(
32)(
1223
3
jV
jjV
3.566056.3)(3 jV
The result can be transformed back to the time domain:
)3.562000cos(6056.3)(3 ttv
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Complex Numbers: Sinusoidal Addition
» V1=2*exp(j*0);» V2=3*exp(-j*pi/2);
Addition Verification Matlab can be used to verify the complex addition:
903)(
02)(
2
1
jV
jV
3.566056.3)(3 jV
» V3=V1+V2V3 = 2.0000 - 3.0000i» M3=abs(V3)M3 = 3.6056
» theta3= angle(V3)*180/pitheta3 = -56.3099
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Complex Numbers: Sinusoidal Addition
Time Domain Addition The original cosine waves can be added in the time
domain using Matlab:
f =1000; % FrequencyT = 1/f; % Find the periodTT=2*T; % Two periodst =[0:TT/256:TT]; % Time Vector
v1=2*cos(2*pi*f*t);v2=3*sin(2*pi*f*t);v3=v1+v2;
tttv )1000(2sin3)1000(2cos2)(3
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Complex Numbers: Sinusoidal Addition
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);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 v1
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Complex Numbers: Sinusoidal Addition
Plot Results
Plots show addition of time domain signals
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Complex Numbers: Sinusoidal Addition
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');
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Complex Numbers: Sinusoidal Addition
Graphical Verification
The results are the same
Thus Phasor addition is verified
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Complex Numbers: Sinusoidal Addition
Four Cosines Example Example: Use Matlab to add the following four sinusoidal
signals and extract the Magnitude, M5 and Phase, 5 of the resulting signal. Also plot all of the signals to verify the solution.
90)1000(2cos4)(
60)1000(2cos3)(
30)1000(2cos2)(
)1000(2cos1)(
4
3
2
1
ttv
ttv
ttv
ttv
555
43215
)1000(2cos)(
)()()()()(
tMtv
tvtvtvtvtv
Given:
Determine:
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Complex Numbers: Sinusoidal Addition
Enter in Phasor Form Transform signals into phasor form
24904)(902000cos4)(
33603)(602000cos3)(
62302)(302000cos2)(
0101)(2000cos1)(
44
33
22
11
jVttv
jVttv
jVttv
jVttv
» V1 = 1*exp(j*0);» V2 = 2*exp(-j*pi/6);» V3 = 3*exp(-j*pi/3);» V4 = 4*exp(-j*pi/2);
Create phasors as Matlab variables in polar form
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Complex Numbers: Sinusoidal Addition
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/pitheta5_deg = -60.8826
Convert back into Time Domain
8826.606972.8)(5 jV
8826.60)1000(2cos6972.8)(5 ttv
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Complex Numbers: Sinusoidal Addition
Code to Plot Voltages Plot all 4 input
voltages on same plot with different colors
f =1000; % FrequencyT = 1/f; % Find the periodt =[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
Signals to be Added
t2000cos1
302000cos2 t
602000cos3 t
902000cos4 t
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Complex Numbers: Sinusoidal Addition
Code to Plot Results Add the original Time Domain signals
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
Compare Results The results
are the same
Thus Phasor addition is verified
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Complex Numbers: Sinusoidal Addition
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
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Complex Numbers: Sinusoidal Addition
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