Question

Show that, "Three unbalanced phasors of a 3- system can be resolved into three balanced system of phasors."

Answer 1

Consider an unbalanced phasor has shown in (a). The positive, negative, and zero sequence are shown in (b), (c), and (d) respectively.

Let's define the a-operator

$a=e^{j120\degree}=-\dfrac{1}{2}+j\dfrac{\sqrt{3}}{2}\\ a^2=e^{j240\degree}=-\dfrac{1}{2}-j\dfrac{\sqrt{3}}{2}\\$

from (b)

$V_{b1}=a^2V_{a1}\\ V_{c1}=aV_{a1}\\$

from (c)

$V_{b2}=aV_{a2}\\ V_{c2}=a^2V_{a2}$

from (d)

$V_{a0}=V_{b0}=V_{c0}$

$V_{a}=V_{a0}+V_{a1}+V_{a2}\\ V_{b}=V_{b0}+V_{b1}+V_{b2}\\ V_{b}=V_{a0}+a^2V_{a1}+aV_{a2}\\ V_{c}=V_{c0}+V_{c1}+V_{c2}\\ V_{c}=V_{a0}+aV_{a1}+a^2V_{a2}\\$