082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

348 Chapter 9 Complex numbers
Engineeringapplication9.2
Characteristicimpedance
In a later application we shall explain the concept of the characteristic impedance
ofanelectricaltransmissionlineinmoredetail.Fornow,wecanassumethatitisan
important electricalparameter represented by the equation
√
R+jωL
Z 0
=
G+jωC
whereR,L,GandCaretransmissionlineparametersand ωistheangularfrequency,
all of which are real numbers. If R and G are zero, as they would be for an ideal
transmissionline, thenZ 0
iswholly real because:
√ √
jωL L
Z 0
=
jωC = C
Alternatively, ifGorRis included then the equation involves taking the square root
of a complex number.
Given values ofR,G,L,C and ω we can write
R+jωL
G+jωC
in Cartesian form (see Example 9.9) and hence in polar form (see Section 9.5).
Application of De Moivre’s Theorem will then allow us to calculate the required
square roots tofindZ 0
.
Another application of De Moivre’s theorem is the derivation of trigonometric
identities.
Example9.20 UseDeMoivre’s theoremtoshow that
cos3θ =4cos 3 θ−3cosθ
and
sin3θ =3sinθ−4sin 3 θ
Solution We know that
(cosθ +jsinθ) 3 =cos3θ +jsin3θ
Expanding the l.h.s. wefind
cos 3 θ+3jcos 2 θsinθ−3cosθsin 2 θ−jsin 3 θ =cos3θ+jsin3θ
Equating the real parts gives
cos 3 θ −3cos θ sin 2 θ = cos3θ (9.10)
and equating the imaginarypartsgives
3cos 2 θ sinθ −sin 3 θ = sin3θ (9.11)