.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

Contents

Name	Description
ComplexType	matrix representation of complex number
conjC	Conjugate value of complex number
absC	Absolute value of complex number
detC	Determinant of complex number matrix
invC	Inverse of complex number
sumC	Sum of complex numbers
prodC	Product of complex numbers
expI	Exponential of imaginary number
expC	Exponential of complex number
powerC	Power of complex number
sqrtC	Square-root of complex number
logC	Logarithm of complex number
cosC	Cosine of complex number
sinC	Sine of complex number
tanC	Tangens of complex number
atanC	Arc-tangens of complex number
coshC	Hyperbolic cosine of complex number
sinhC	Hyperbolic sine of complex number
tanhC	Hyperbolic tangens of complex number
atanhC	Area-tangens-hyp of complex number