Chapter 23: Revision of Complex Number
Introduction
Complex numbers can be used to represent anything that is periodic.[cite: 1214, 1216, 1218] They are used in Laplace transforms and Fourier transforms, and are used to analyze varying voltages and currents.[cite: 1218, 1220] They are used in control systems.[cite: 1219] Complex numbers are also extended into digital signal processing and digital image processing, utilizing the complex variant of Fourier analysis and wavelet analysis for the transmission, compression, restoration and processing of digital audio signals, still images and video.[cite: 1220, 1221, 1226] The study of complex numbers is essential for digital applications and many other engineering disciplines.[cite: 1222, 1227, 1229]
23.1 Argand Diagram
On an Argand diagram, the y-axis is the imaginary axis, and the x-axis is the real axis.[cite: 1231, 1232, 1234, 1242] The written form (for example, 2+j2) is said to be in cartesian or rectangular form.[cite: 1244, 1245, 1249]
Angle Changes
An anticlockwise change of direction is an increase in phase.[cite: 1251, 1255] Multiplying by j gives a change of phase (increase) of 90°.[cite: 1261, 1262, 1264] Multiplying by -j results in a clockwise change of phase of 90°.[cite: 1265] This can be seen by multiplying the number a+jb by j on an Argand diagram.[cite: 1270, 1271, 1273]
23.2 Operations Involving Cartesian Complex Numbers
Addition
Addition is straightforward enough.[cite: 1276]
Multiplication
The product of a complex number and its conjugate (having the same number but with the complex part having a minus sign, i.e., a-jb) is a2 + b2.[cite: 1282, 1284] This property is used when dividing complex numbers.[cite: 1285, 1286]
Division
Division is achieved by multiplying the numerator and denominator by the complex conjugate of the denominator.[cite: 1290, 1292, 1293] Rationalizing is the elimination of the imaginary part of the denominator.[cite: 1298]
23.3 Complex Equations
If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal.[cite: 1303, 1311, 1312] This property is useful when deriving balance equations from ac bridges.[cite: 1312]
23.4 The Polar Form of a Complex Number
This is usually abbreviated to z = r∠θ, which is called the polar form of a complex number.[cite: 1338, 1339]
r is called the modulus or magnitude of z and is written |z|.[cite: 1340, 1341] It is determined by Pythagoras’s theorem: |z| = r = √(x2+y2).[cite: 1345, 1347]
θ is called the argument and is written arg z.[cite: 1349, 1354] It is deduced from θ = \tan^{-1}(y/x).[cite: 1350, 1358]
23.5 Multiplication and Division using Complex Numbers in Polar Form
Multiplication
Examples:
- 3∠25° × 2∠32° = 6∠57°[cite: 1363]
- 4∠11° × 5∠-18° = 20∠-7°[cite: 1363]
Division
The P→R and R→P buttons on a calculator can be used to convert complex numbers from polar form to rectangular form and rectangular form to polar form.[cite: 1366, 1367]
23.6 De Moivre’s Theorem – Powers and Roots of Complex Numbers
This result is true for all values of n, negative, positive or fractional.[cite: 1371, 1372]
A square root of a complex number is:[cite: 1373]
It is however important to remember that a real number has roots which are 180° apart.[cite: 1375, 1376, 1377]
24.1 Notes on Circuit Phase Angles
The phase angle of the current is relative to the phase angle of the voltage.[cite: 1392, 1393]