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Review of Complex Numbers

Review of Complex Numbers. Introduction to Complex Numbers. Complex numbers could be represented by the form Where x and y are real numbers Complex numbers are denoted: N = {x}+j{y}, where x is considered the REAL part and Y is considered the IMAGINARY part

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Review of Complex Numbers

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  1. Review of Complex Numbers

  2. Introduction to Complex Numbers • Complex numbers could be represented by the form • Where x and y are real numbers • Complex numbers are denoted: N = {x}+j{y}, where x is considered the REAL part and Y is considered the IMAGINARY part • If x = 0, N is considered an IMAGINARY NUMBER • If y = 0, N is considered a REAL number

  3. Properties of Complex Numbers • The sum of two complex numbers is a complex number: • (x1 + jy1) + (x2 + jy2) = (x1 + x2) + j(y1+ y2); • Example, Express the following complex numbers in the form x + iy, x, y real: • (−3 + i)(14 − 2i) • The product of two complex numbers is a complex number: • (x1 + jy1)(x2 + jy2) = x1(x2 + jy2) + (jy1)(x2 + iy2) • = x1x2 + x1(jy2) + (jy1)x2+ (jy1)(jy2) • = x1x2 + ix1y2 + iy1x2 + i2y1y2 • = (x1x2 + {−1}y1y2) + i(x1y2 + y1x2) • = (x1x2 − y1y2) + i(x1y2 + y1x2)

  4. Calculating with complex Numbers • Example 1: solve the system • (1 + i)z + (2 − i)w = 2 + 7i • 7z + (8 − 2i)w = 4 − 9i. • The determinant of the coefficient matrix is • = (1 + i)(8 − 2i) − 7(2 − i) • = (8 − 2i) + i(8 − 2i) − 14 + 7i • = −4 + 13i .

  5. Calculating with complex Numbers • Applying Cramer’s rule: Solve for w!

  6. Calculating with complex Numbers • Class exercise: solve the system: • (1 + i)z + (2 − i)w = −3i • (1 + 2i)z + (3 + i)w = 2 + 2i.

  7. Calculating with complex Numbers • Example 2: solve the system: z2= 1 + i. • Let z = x + iy. • >> (x + iy)2 = x2 − y2 + 2xyi = 1 + i, • >> x2− y2 = 1 and 2xy = 1. • >> x ≠0 and y = 1/(2x) • >> • >> 4x4− 4x2 − 1 = 0 • >> • >> • >> >>

  8. Calculating with complex Numbers • Class exercise: solve the system: z2= 1 + i√3

  9. Cartesian and polar representation of a complex number •  Every complex number z = x+iy can be represented by a point on the Cartesian plane known as complex plane by the ordered pair (x, y).

  10. Cartesian and polar representation of a complex number • The Cartesian coordinate pair (x, y) is also equivalent to the polar coordinate pair (r,θ), where r is the (nonnegative) length of the vector corresponding to (x, y), and θis the angle of the vector relative to positive real line. • x = r cosθ • y = r sin θ • Z = x + jy = r cosθ + j rsinθ = r (cosθ + j sin θ) • |z| = r = √(x2 + y2) • tanθ = (y/x) • θ = arctan(y/x)+ (0 or Π) (Πis added iffx is negative)

  11. The Euler Formula • ej θ = cosθ + j sin θ • Z = x+ jy = r cosθ + j rsinθ = r (cosθ + j sin θ) = r ej θ • R is the distance of the point z from the origin • 1/Z = 1/ r ej θ=( 1/ r) e-j θ

  12. Conjugate of a complex number • Let z = x + jy • The complex conjugate of z is the complex number defined by z* = x − jy. • Geometrically, the complex conjugate of z is obtained by reflecting z in the real axis • z* = x − jy = r e-j θ • z + z* = (x + jy) + (x – jy) = 2x = 2Re(z) • zz* = (x + jy) (x – jy) = x2+y2 = |z|2

  13. Some useful identities • 1e±j Π=-1 ; e±jnΠ=-1 for n odd integer • e±j2nΠ=1 for n integer • ej Π/2= j • E-j Π/2= -j

  14. Examples • Express the following numbers in polar form (also sketch the geometric representation): • 2+j3 • 1 – j3 • Use the MATLAB function cart2pol to convert the above numbers to polar form

  15. Examples • Express the following numbers in polar form (also sketch the geometric representation): • 2+j3 • r = |z| = √(22+32) = √13 • Θ = tan-1(3/2) = 56.30 • 2+j3 = √13ej56.3º

  16. Examples • Represent the following numbers in the complex plane and express them in Cartesian form: • 2 ej Π/3 • 4 e- j3Π/4 • Use the MATLAB function pol2cart to convert the above numbers from polar to Cartesian form

  17. Examples • Represent the following numbers in the complex plane and express them in Cartesian form: • 2 ej Π/3 • = 2cos(Π/3) + 2jsin(Π/3) • =2(1/2) +2 j(√3/2) • =1+j√3

  18. Examples • Determine z1z2 and z1/z2 for • z1 = 3 + j4 = 5ej53.1º • z2 = 2 + j3 = √13 ej56.3º • Solve this problem in both polar and Cartesian forms • Solve this problem using MATLAB

  19. Determine z1z2 and z1/z2 for • z1 = 3 + j4 = 5ej53.1º • z2 = 2 + j3 = √13 ej56.3º • Polar: • z1z2 = (3+j4)(2+j3) = (6-12)+j(8+9) = -6+j17 • z1/z2 = (3+j4)(2-j3) /(22+32) = (18/13) – j(1/13) • Cartesian: • z1z2 = (5ej53.1º )(√13 ej56.3º )= 5√13 ej(53.1º+ 56.3º ) =5√13 ej(109.4º) • z1/z2 = (5ej53.1º )/(√13 ej56.3º )=(5/√13) ej(53.1º- 56.3º ) =(5/√13) ej(-3.2º) Examples

  20. Examples • Consider X(ω), a complex function of a real variable ω: • X(ω) = (2 + j ω)/(3 + j4 ω) • Express X(ω) in Cartesian form, and find its real and imaginary parts. • Express X(ω) in polar form and find its magnitude and angle.

  21. Examples • Consider X(ω), a complex function of a real variable ω: • X(ω) = (2 + j ω)/(3 + j4 ω) • Express X(ω) in Cartesian form, and find its real and imaginary parts. • X(ω) = ((2 + j ω)(3 - j4 ω) )/(32 + 42 ω2) • (6+4ω2)/(9+16 ω2) - j5ω/9+ω2) • Express X(ω) in polar form and find its magnitude and angle. • X(ω) =[√(4 + ω2) ejarctan(w/2)]/ [√(9 + 16ω2) ejarctan(4w/3)] • √((4 + ω2)/ √(9 + 16ω2)) ej(arctan(w/2)-arctan(4w/3))

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