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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: Complex Number Theory

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Complex Numbers:Complex Number Theory 2 n General Complex Numbers General Complex Numbers n General Complex Numbers in Matlab General Complex Numbers in Matlab n Argand Diagrams Argand Diagrams n Exponential Form Exponential Form n Polar Form Polar Form n Polar Form in Matlab Polar Form in Matlab

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Complex Numbers:Complex Number Theory 3 Representing Complex Numbers The conventional representation of a complex number z is the sum of a real part x and an imaginary part y i is the imaginary unit where: ComplexRealImaginary n Conventional representation is also referred to as the rectangular form of a complex number

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Complex Numbers:Complex Number Theory 4 Functional Notation In different applications, the imaginary unit may be represented as i or j and may be placed before or after the imaginary part n In functional notation, it is sometimes convenient to write:

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Complex Numbers:Complex Number Theory 5 Matlab Complex Numbers n Matlab complex numbers consist of a real portion plus an imaginary portion n For example, use Matlab to find the square root of -1 » sqrt(-1) ans = i Real Portion = 0Imaginary Portion = -1

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Complex Numbers:Complex Number Theory 6 Entering Complex Numbers Both the variables i and j in the Matlab workspace are reserved for representing imaginary numbers Placing i or j after a number defines it as imaginary n Example: A complex number with real portion = 1 and imaginary portion = 2 can be entered in two ways: » z = 1+2i z = i » z = 1+2j z = i OR

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Complex Numbers:Complex Number Theory 7 Real and Imag Commands The real and imag functions extract the real and imaginary portion of a complex number respectively n Example: » z1= 2+4i z1 = i » z1_re = real(z1) z1_re = 2 » z1_im = imag(z1) z1_imag = 4

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Complex Numbers:Complex Number Theory 8 Argand Diagram n A complex number can be represented as a Point P(x,y) in the xy-plane (Cartesian plane) n Ordered Pair Notation n This representation is called an Argand Diagram n The xy-plane is often referred to as the Complex Plane or the z-plane

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Complex Numbers:Complex Number Theory 9 Feather Command The feather command draws complex numbers as arrows in the xy plane » z1=1+2j; » z2=3+3j; » z3=3+j; » feather(z1,'r'); » hold on » feather(z2,'b'); » feather(z3,'g'); » feather(z3,'k'); » ylabel('Imaginary'); » xlabel('Real');

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Complex Numbers:Complex Number Theory 10 Exponential Representation n Sometimes an exponential representation of a complex number is easier to manipulate than the rectangular sum of the real and imaginary part n The Complex Exponential Function e z will be used to represent a complex number whose Taylor Series Expansion is: n We can assume that e z will be valid for all z

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Complex Numbers:Complex Number Theory 11 Exponential Series If z = i, purely imaginary, where is real, the complex exponential function can be written using a Taylor Series Expansion: n Which can be separated into real and imaginary components since i raised to even powers of n will be -1

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Complex Numbers:Complex Number Theory 12 Derivation of Eulers Equation n Knowing that the Taylor Series Expansions for sine and cosine functions are: n Through substitution, Eulers Equation is derived:

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Complex Numbers:Complex Number Theory 13 Exponential Polar Form n Through Eulers Equation we see that: n This representation of a complex number is referred to as the Exponential Polar Form: n Sometimes Polar Form is written using the shortened notation:

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Complex Numbers:Complex Number Theory 14 Plotting Polar Form n Graphically depicting Eulers Equation in the First Quadrant of an Argand Diagram we see: r refers to the radius from z to the origin, commonly called the Magnitude of z is the Angle of z with respect to the Real Axis measured in degrees or radians

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Complex Numbers:Complex Number Theory 15 Magnitude n Through use of the Pythagorean theorem, the Magnitude (radius) of a complex number always has the relationship: n This relationship always holds true regardless of which quadrant of the Argand diagram that the number lies in The Magnitude of a complex number z=x+iy is denoted using Absolute Value notation

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Complex Numbers:Complex Number Theory 16 Angle n Determining the angle of a complex number depends on the quadrant of the Argand diagram The angle,, of a complex number z denoted by: Measuring the angle counter-clockwise from the x-axis gives a positive Measuring the angle clockwise from the x-axis gives a negative n Example: The angle of z=0+j can be represented in multiple ways

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Complex Numbers:Complex Number Theory 17 Angle of Quadrants I and IV Quadrant IQuadrant IV

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Complex Numbers:Complex Number Theory 18 Angle of Quadrants II and III Quadrant IIQuadrant III

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Complex Numbers:Complex Number Theory 19 Purely Real or Imaginary Numbers Rectangular Rectangular PolarPolar Short Long DegreesRadians i i j i -j i

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Complex Numbers:Complex Number Theory 20 Abs and Angle Commands The abs command in Matlab returns the magnitude of a complex number The angle command returns the angle of a complex number in radians n Remember that the conversion between radians and degrees is:

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Complex Numbers:Complex Number Theory 21 » z=1+i; » r=abs(z) r = » theta_rads=angle(z) theta_rads = » theta_degs=theta_rads*180/pi theta_degs = 45 Polar Example Use Matlab to find the magnitude and angle of z=1+i

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Complex Numbers:Complex Number Theory 22 Entering Polar Form n Complex numbers can be directly entered into Matlab using the exponential polar form » z=2*exp(i*pi/2) z = i Example: Create a complex number z with a magnitude of 2 and an angle of /2 radians

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Complex Numbers:Complex Number Theory 23 Polar Plotting Matlabs compass(z) or compass(x,y) command can be used to draw complex numbers on a polar plot n Note that z is a complex number in rectangular form » z1=3+3i; » compass(z1) » hold on » compass(4,-3) n Angles are displayed in degrees

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Complex Numbers:Complex Number Theory 24 Plotting Example n Example: Plot the following complex numbers by hand. Use Matlabs compass function to verify your results.

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Complex Numbers:Complex Number Theory 25 Matlab Verification z(1) = 2; z(2) = 3*exp(i*60*(pi/180)); z(3) = -(8^0.5) + j*(8^0.5); z(4) = 4*exp(-i*pi/2); z(5) = 2.5*exp(-i*150*(pi/180)); z(6) = 2i; compass(z)

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Complex Numbers:Complex Number Theory 26 Polar Plot

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Complex Numbers:Complex Number Theory 27 Summary n Representing complex numbers in rectangular, exponential, and polar forms n Using Eulers Equation to represent real and imaginary parts of complex numbers n Determining magnitude and angles of complex numbers n Graphing complex numbers using Argand Diagrams

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