11_14_Sinosoids - H 5‘ Nusfimw i 2 P4213353 1W{4/07:3 ®...

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Vador aa'lu'hcfi 59 m SUM 0-; W Jud-O 5cm has 4 VVMX 0F WAV 1..) i'HADUHD. ‘ _ ll/\L+/[email protected] Sinirsoids are easily expressed in turns of phasors, which are more convenient to work with than sine and cosine functions. , E$W% Tn D0 THE MW " F Y‘om A \m minis Phasors provide a simple means of analyzing linear circuits excited by ~1- a?“ sinusoidal sources; solutions of such circuits would be intractable other- ~ wise. The notion of solving ac circuits using phasors was first introduced mm by Charles Steinmetz in 1893. Before we completely define phasors and German-Amman mathematician and electrical ‘ apply them to circuit analysis, we need to be thoroughly familiar with ' engineer. _ complex numbers ' A complex number z can be written in rectangular form as . Appendix B presents a Short tutorial on 2— - x + j y , . (914a) tompiex numbers. Wherej =‘/-— ~—l;xistherea1partofz;yistheimaginarypartofz. The complex number z can also be written in polar or exponential form as 2 =rA¢3=requ . (914s) where r is the magnitude of z, and d: is the phase of 2. We notice that 2 can be represented in three ways: 2 = x + jy Rectangular form z = r d) Polar form (9.15) z = re” » ' Exponential form The relationship between the rectangular form and the polar form is shown in Fig. 9. 6, where the x axis represents the real part and the y . axis represents the' imaginary part of a complex number. Given 1: and y, Imaginary am we can get r and 45 as - z : r = t/xz + yz, a: tan—1 : ~ (9m) 2j .: y On the other hand, if we know r and ¢, we can obtain x and y as ‘1'. E x =rr cos (b, y = r sin¢ ' (9.16b) 0 I Real axis Thus, 2 may be written as _j - . .2} z=x+jy=rfl=r(cos¢+jsin¢) (9.17) Figure 9.6 Representation of a ' Addition and subtraction of complex numbers are better performed . °°mP1°x number 1 = x + U = ’13 in rectangular form; multiplication and. division are better done in polar form. Given the complex numbers “fl” " Addition: — .‘ .. 21 +22 = (x1+x2)+j(y1+ yz) Subtraction: , z: — z: =(x1— x2) + 1'01 — 9’2) Multiplication: _ ' _ . 2122 = r1r2g¢1+ 452 Division: - Z 7' , . _ —1 = —1 ¢1~¢2 plp'ionjugate: ' 22 r2 . . Rec' rocal: z‘=x‘j}’=r_/_—¢=re"¢ 1p 1 1 ¢ that from Eq. (9.18:), z _ r/—' 1 ‘ Square Root: 7 = _,- . _ J? = J; 45/2 ...
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