Complex Numbers

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ComplexNumbers
Math Review with Matlab
Sinusoidal Addition

• S. Awad, Ph.D.
• M. Corless, M.S.E.E.
• E.C.E. Department
• University of Michigan-Dearborn

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Sinusoidal Addition

• A useful application of complex numbers is the
  addition
• of sinusoidal signals having the same frequency

• General Sinusoid
• Eulers Identity
• Sinusoidal Addition Proof
• Phasor Representation of Sinusoids
• Phasor Addition Example
• Addition of 4 Sinusoids Example

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General Sinusoid

• A general cosine wave, v(t), has the form

M Magnitude, amplitude, maximum
value w Angular Frequency in radians/sec
(w2pF) F Frequency in Hz T Period in seconds
(T1/F) t Time in seconds q Phase Shift,
angular offset in radians or degrees
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Eulers Identity

• A general complex number can be written in
  exponential polar form as

• Eulers Identity describes a relationship between
  polar form complex numbers and sinusoidal signals

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Useful Relationship

• Eulers Identity can be rewritten as a function
  of general sinusoids

• Resulting in the useful relationship

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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
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Complex Representation

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

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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

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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

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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

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Phasor Addition

• As shown previously, two sinusoidal voltages of
  the same frequency can easily be added using
  their phasors

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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
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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

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Rectangular Addition

• To perform addition by hand, the Phasors must be
  written in rectangular (conventional) form

• Now the Phasors can be added

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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

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Addition Verification

• Matlab can be used to verify the complex addition

V12exp(j0) V23exp(-jpi/2)
V3V1V2 V3 2.0000 - 3.0000i
M3abs(V3) M3 3.6056 theta3
angle(V3)180/pi theta3 -56.3099
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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 TT2T Two periods t
0TT/256TT Time Vector v12cos(2pift)
v23sin(2pift) v3v1v2
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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_12cos(2000\pit)') title('Sinus
oidal Addition') subplot(3,1,2)
plot(t,v2) grid on axis( 0 TT -4
4) ylabel('v_23sin(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')

• v_1 prints v1

• \pi prints p

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Plot Results

• Plots show addition of time domain signals

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Verification Code

• Plot the added signal, v3, and the hand derived
  signal to verify that they are the same

v_hand3.6056cos(2pift-56.3059pi/180) subpl
ot(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) gri
d on ylabel('3.6cos(2000\pit -
56.3\circ)') xlabel('Time')
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Graphical Verification

• The results are the same

• Thus Phasor addition is verified

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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
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Enter in Phasor Form

• Transform signals into phasor form

• Create phasors as Matlab variables in polar form

V1 1exp(j0) V2 2exp(-jpi/6) V3
3exp(-jpi/3) V4 4exp(-jpi/2)
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Add Phasors
V5 V1 V2 V3 V4 M5 abs(V5) M5
8.6972 theta5_rad angle(V5) theta5_deg
theta5_rad180/pi theta5_deg -60.8826

• Add phasors then extract Magnitude and Phase

• Convert back into Time Domain

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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 0T/256T Time Vector v11cos(2
pift) v22cos(2pift-pi/6) v33cos(2pif
t-pi/3) v44cos(2pift-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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Signals to be Added
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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 M5cos(2pifttheta5_rad) subplot(2
,1,2)plot(t,v5_phasor) grid on ylabel('From
Phasor Addition') xlabel('Time')
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Compare Results

• The results are the same

• Thus Phasor addition is verified

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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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Summary

• Reviewed general form of a sinusoidal signal

• Used Eulers 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