solved01 - Then z = j*(3/2). iii) The modulus of z is the...

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S 1.1 (P 1.9) _____________ Cartesian form: Numerator: a = (1 + j*sqrt(3))*(1 - j) = (1+sqrt(3)) + j*(-1+sqrt(3)) Denominator b = (sqrt(3) + j)*(1 + j) = (-1+sqrt(3)) + j(1+sqrt(3)) b' (complex conjugate) = (-1+sqrt(3)) - j(1+sqrt(3)) |b|^2 = (sqrt(3)-1)^2 + (sqrt(3)+1)^3 = 8 a*b' = 4 - j*4*sqrt(3) a/b = a*b'/|b|^2 = 1/2 - j*sqrt(3)/2 Polar form: 1+j*sqrt(3): modulus = 2, angle = pi/3 1-j: modulus = sqrt(2), angle = -pi/4 sqrt(3) + j: modulus = 2, angle = pi/6 1+j: modulus = sqrt(2), angle = pi/4 a: modulus = 2*sqrt(2), angle = pi/12 b: modulus = 2*sqrt(2), angle = 5*pi/12 a/b: modulus = 1, angle = pi/12 - 5*pi/12 = -pi/3 (Note that |a/b|=1, which follows directly from the identities |z1*z2| = |z1|*|z2| and |z1/z2| = |z1|/|z2| .) S 1.2 _____ z = (1+j*2)/(3-j*b). Multiplying numerator and denominator by 3+j*b, we obtain z = (1+j*2)(3+j*b)/(9+b^2) = (3-2*b)/(9+b^2) + j*(6+b)/(9+b^2) i) z is purely real if its imaginary part is zero, i.e., if b=-6. Then z = 1/3. ii) z is purely imaginary if its real part is zero, i.e., if b = 3/2.
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Unformatted text preview: Then z = j*(3/2). iii) The modulus of z is the ratio of the moduli of 1+2*j and 3-j*b. Thus |z|^2 = 5/(9+b^2) If |z|^2 = 1/2, then 9+b^2 = 10, i.e., b = +1 or -1. S 1.3 _____ We first express z in polar form: |z| = r = sqrt( 9/25 + 81/100 ) = 1.081665 angle(z) = t = atan(9/6) = 0.9827934 radians Then |z^17| = r^17 = 3.798 angle(z^17) = 17*t = 16.707, which reduces to 4.141 by subtracting 4*pi from it. S 1.4 _____ i) |z-1+j|=1 represents a circle centered at z0 = 1-j = (1,-1) and having radius equal to 1. From elementary geometry, the point on the circle closest to the origin will be z = (c,-c) where c = 1-sqrt(2)/2 = 0.2929. The modulus of z equals sqrt(2)-1 = 0.4142. ii) |z-2| = |z-3*j| represents the perpendicular bisector of the linear segment joining (2,0) and (0,3). The distance of the perpendicular bisector from the origin can be found geometrically by considering two similar triangles. It equals 5/(2*sqrt(13)) = 0.6934...
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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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solved01 - Then z = j*(3/2). iii) The modulus of z is the...

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