.PowerSystems.Utilities.Complex

Information

Auxiliary sub-package for didactic purposes showing the treatment of complex numbers with rotation matrices. Complex numbers are represented as 2x2 matrix, such that arithmetic operations on complex numbers become equivalent to arithmetic operations on their matrix representations.

Complex number (x + jy), considered as an element of an algebra, and represented by the real 2x2 matrix z:

  z = [x, -y]
      [y,  x]

The following relations hold for

- complex conjugate:   transpose(z)
- addition:            z1 + z2
- multiplication:      z1*z2
- inverse:             transpose(z)/det(z)
- absolute value:      sqrt(det(z))

where det(z) denotes the determinant of matrix z.

Example:

The linear differential equation with constant coefficients for a real scalar current I and voltage V

  L*dI/dt + R*I = V

  (R + j*omega*L)*i = v

  ComplexType Z =  R*re + omega*L*im (complex impedance)
  ComplexType i = i1*re + i2*im      ('current phasor')
  ComplexType v = v1*re + v2*im      ('voltage phasor')

  Z*i = v

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