Citations and Corresponding Text Blocks

This document lists specific source citations mapped to their corresponding transcribed text blocks for Chapters 24 and 23. The layout uses a small table cell for the citation identifier and a much larger table cell for the transcribed content, with mathematical symbols styled accordingly.

24_JBECT_All.pdf (Chapter 24: Application of Complex Numbers to Series AC Circuits)

Citation	Corresponding Text Block
[1040-1042]	AC circuits may be analysed by using complex numbers for simplified phasor diagrams.
[1043-1044]	In pure resistance the circuit in polar equation is given by:
[1044]	Z = (VR∠0°) / (IR∠0°) = R
[1055-1059]	In pure inductance the current lags the applied voltage by 90°.
[1058-1064]	Z = (VL∠90°) / (I∠0°) = XL∠90° = jXL
[1075-1076]	The voltage lags the current by 90° in this circuit.
[1091]	Z = (VC∠-90°) / (IC∠0°) = XC∠-90° = -jXC
[1092-1093]	Where XC is 1 / ωC.
[1096-1098]	-jXC = -j / ωC = 1 / jωC
[1123-1125]	The current is said to be lagging even though the phase is +90° as the current is behind the voltage.
[1144-1146]	V = IZ, and VC = IXC
[1177-1178]	|Z| = √(R2 + (XL – XC)2), φ = tan-1((XL – XC) / R) Z = R + j(XL – XC) = |Z|∠φ
[1179-1184]	In an a.c. circuit containing several impedances connected in series, say Z1, Z2, Z3 … Zn, the total equivalent impedance is given by:
[1186]	ZT = Z1 + Z2 + Z3 + … + Zn
[1187-1191]	For a circuit containing parallel impedances Z1, Z2 and Z3, the potential difference is the same and equal to the supply voltage V.
[1204-1205]	ZT = (Z1Z2) / (Z1 + Z2)

23_JBECT_All.pdf (Chapter 23: Revision of Complex Number)

Citation	Corresponding Text Block
[1214-1221]	Complex numbers can be used to represent anything that is periodic. They are used in Laplace transforms and Fourier transforms, and are used to analyze varying voltages and currents.
[1243-1246]	The written form (for example, 2+j2) is said to be in cartesian or rectangular form.
[1251]	An anticlockwise change of direction is an increase in phase.
[1276]	Addition is straightforward enough.
[1278]	(a+jb)(c+jd) = (ac-bd) + j(ad+bc)
[1284-1286]	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. This property is used when dividing complex numbers.
[1296]	(1+2j)/(3-4j) × (3+4j)/(3+4j) = ((1×3) – (2×4)j) / (32+42) = (3-8j) / 25
[1303-1312]	If two complex numbers are equal, then their real parts are equal and their imaginary parts are equal.
[1337-1339]	z = x+jy = r\cos heta + j r\sin heta This is usually abbreviated to z = r∠θ, which is called the polar form of a complex number.
[1345-1347]	It is determined by Pythagoras’s theorem: |z| = r = √(x2+y2).
[1349-1350]	θ is called the argument and is written arg z. It is deduced from θ = an^{-1}(y/x).
[1362]	r1∠θ1 × r2∠θ2 = r1r2∠(θ1+θ2)
[1365]	(r1∠θ1) / (r2∠θ2) = (r1/r2)∠(θ1-θ2)
[1370]	(r∠θ)n = rn∠nθ
[1374]	√(r∠θ) = r1/2∠(½θ)