Chapter 23: Revision of Complex Numbers
Date: 10-6-19
Introduction
Complex numbers can be used to represent everything that is periodic. Laplace transforms and Fourier transforms are used to analyse varying voltages and currents. They are used in control engineering, digital signal processing, and digital image processing. Complex numbers are also extended utilizing the complex version of Fourier analysis and wavelet analysis for the transmission, compression, restoration, and digital parsing of digital audio signals, still images, and video. The study of complex numbers is essential for digital analogue and many engineering disciplines.
23.1 Argand Diagrams
The y-axis is the imaginary axis, and the x-axis is the real axis. A number represented like z = x + jy is said to be in Cartesian or rectangular form.
Angle Changes
An anticlockwise change of direction is an increase in Angle, which corresponds to multiplication by j (a 90° change of phase). Conversely, a clockwise change of phase results from multiplication by -j.
23.2 Operations Involving Cartesian Complex Numbers
- Addition: Straightforward combination of real parts and imaginary parts.
- Multiplication: (a + jb)(c + jd) = (ac – bd) + j(ad + bc).
- Conjugate: The product of a complex number and its conjugate is (a + jb)(a – jb) = a2 + b2. This property is used when dividing complex numbers.
- Division: Performed by multiplying the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary part from the denominator.
23.3 Complex Equations
If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal. This property is useful when deriving balance equations from AC bridges.
23.4 The Polar Form of a Complex Number
The polar form of a complex number is written as z = r ∠ θ, where r is the modulus or magnitude of z (written as |z|) and θ is the argument (written as org z). Using Pythagoras’ theorem on an Argand diagram:
The argument is given by θ = tan-1(y/x).
23.5 Multiplication and Division Using Complex Numbers in Polar Form
- Multiplication: r1 ∠ θ1 imes r2 ∠ θ2 = (r1 r2) ∠( θ1 + θ2).
- Division: (r1 ∠ θ1) / (r2 ∠ θ2) = (r1 / r2) ∠( θ1 – θ2).
23.6 De Moivre’s Theorem – Powers and Roots of Complex Numbers
This result is true for all values of n (positive, negative, or fractional). For roots, a complex number has multiple roots that are spaced evenly (e.g., square roots are 180° apart).
24.1 Notes on Circuit Phase Angles
The phase Angle of the current relative to the voltage determines the power factor in AC circuits.