ELEC3002-complex1

ELEC3002-complex1 - ELEC3002/ELEC7005 Lecture 1 Complex...

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ITEE complex 1/1 ELEC3002/ELEC7005 ELEC3002/ELEC7005 Lecture 1 Complex Analysis (Chapter 1 in Glyn James) This lecture covers section 1.2 A/Prof. Nick Shuley, Room 73-535

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ITEE complex 1/2 What in this section? What in this section? All about the use of complex analysis and its application to Electrical Engineering. All of chapter 1 of Glyn James. Assignment 2 is not on this section for 2008. Additional references: search the library titles under “Complex Analysis”. There are heaps.
ITEE complex 1/3 Why complex analysis? Why complex analysis? Extensively used in engineering applications (especially in EM theory, Control, DSP, Comms). Makes use of Laplace, Fourier and Z-transforms where complex analysis plays a key role. Key role in partial differential equations. Generic material for electrical engineers. Not just complex numbers!

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ITEE complex 1/4 A little revision on complex numbers A little revision on complex numbers first. first. a complex number: z x jy = + note that engineers use ‘j’, mathematicians and physicists use ‘i’. Addition, subtraction, division and multiplication are all defined complex conjugate z z x jy = = - real and imaginary parts can be extracted according to: ( 29 ( 29 1 1 Re ; Im 2 2 z x z z z y z z j = = + = = - powers of complex numbers are naturally defined using j 2 = -1 etc. roots of complex numbers need a bit more explanation. ? j =
ITEE complex 1/5 Polar form Polar form The polar form is of most use to engineers ( 29 cos sin z r j θ = + r is the magnitude (absolute value, modulus) θ is the phase (argument) Note that the use of amplitude and phase are the main parameters in what engineers call the frequency domain Here, 2 2 1 arg tan z r x y zz y z x - = = + = = =

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ITEE complex 1/6 the argument the argument The argument needs clarification. Generally, θ is measured in radians in the positive direction (counter clockwise) from the x-axis in the Argand diagram (complex plane). θ z Re Im The principal value of the argument is defined for by the unambiguous interval arg z π - N.B. Kreysig uses Arg (with a capital) for the principal value.
ITEE complex 1/7 complex operations complex operations Multiplication and division using the complex representation generally follows the rules of: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 1 1 1 2 2 2 2 cos sin ; arg arg arg ; arg arg arg z z rr j z z z z z z z z z z z z z z z z θ = + + + = = + = = - De Moivre’s theorem comes in handy for integer powers ( 29 cos sin n n z r n j n = +

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ITEE complex 1/8 Roots Roots If z = w n there is only one value of z for each value of w , but for the converse w = n z there are exactly n values of w for each z . In other words the square root (or n’th root) function is a
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ELEC3002-complex1 - ELEC3002/ELEC7005 Lecture 1 Complex...

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