1.2 (Take II) To find I_ab, we just need to properly use Ohm's Law, V_ab = I_ab * Z_ab [Does the new forum software support subscripts and superscripts like the old one did? [sub][/sub] doesn't work. So I will use X_Y to denote X subscript Y. Likewise ^ means superscript, usually exponentiation.]

We are given Z_ab = Z = 88.8-j66.6. We aren't directly given V_ab, but note from the diagram that V_ab = V_an + V_nb, and that V_nb = - V_bn always. Negating a vector just means reflecting the vector through the origin. I.e. in polar form the magnitude is unchanged, and the angle is increased by 180 degrees (or decreased by 180 degrees, since a 360 degree change is no change at all). For polar form I'll use the notation a \ b to mean magnitude a and angle b. In other words, V = |V| \ ph V.

Let's call v = 34500 / (sqrt 3). So V_ab = v \ 0 + v \ 60 = v * (1\0 + 1\60). You can do the vector addition either in rectangular form [1\0 = 1 + j0, 1\60 = 0.5 + j sqrt(3)/2] or geometrially [tip to tail, the two vectors form an isoceles triangle with an obtuse 120 degree angle], but either way 1\0 + 1\60 = sqrt(3) \ 30. Thus V_ab = 34500 \ 30

Now we want to multiply (rather divide) two vectors (rather complex numbers), one which we have in polar form (V_ab) and one in rectangular form (Z_ab). We could do the computation in either polar or rectangular coordinates, but since the result is requested in polar form, let's use polar form. So we need to convert Z_ab to polar form. In general the vector x + jy has the polar form sqrt(x^2+y^2) \ arctan(y/x) [draw out the right triangle in the complex plane]. We get Z_ab = 111 \ (arctan(-3/4) = -36.9 degrees).

Division requires computing the multiplicative inverse, and to do that to a complex number in polar form, we have to take the reciprocal of the magnitude, and then negate the angle. [As seen in the usual complex polar form R*e^(ja), the reciprocal is R^(-1) * e^(-ja)]. So Z_ab^(-1) = 1/111 \ 36.9

Finally, we have I_ab = V_ab / Z_ab = V_ab * (Z_ab^(-1)) = 34500 \ 30 * 1/111 \ 36.9. To multiply in polar form, we just multiply the magnitudes and add the angles. That means I_ab = 34500/111 \ 66.9 = 311 \ 66.9 (amps).

Cheers, Wayne