Three phase sinusoidal waveforms can be represented as a rotating vector, which follows a circular path, in a complex Cartesian co-ordinate system. The mathematical equation for the vector x is given, where a is a complex exponent. On the vector diagram the instantaneous values of each of the three phases are summed with the resulting vector being x. By moving the red timeline the vector x can be seen to follow the circular track.

The second time plot is of the three phase x waveforms with and additional zero component, u0, added. This u0 value is added to all the three phases. Initially the u0 value is zero, therefore x’R,S,T(t) = xR,S,T(t) and the vector x’ = vector x.

If the sinusoidal zero-component is now selected by clicking on the red check box then the x’R,S,T(t) waveforms now have a third harmonic sinusoidal waveform added. The waveform shape of x’R,S,T(t) does not look the same as xR,S,T(t). The vector diagram now uses the x’R,S,T(t) waveforms to form the resultant vector. Even though the time waveforms have changed the resulting vector x’ still produces a circular track, thus indicating that the zero sequence component has had no influence on the fundamental component of the waveform.

This applet allows two other zero-sequence waveforms to be added, these are the rectangular and the random waveforms. Although these have totally different waveform shapes than the third harmonic sinusoid, the same circular track is produced by the vector x’. Thus indicating that the fundamental component of the waveform is not disturbed.

• Move the red timeline and observe how the individual phase vectors change and add up to the total vector x. Notice how the vector x tracks a circular path. 
• Select the SINUSOIDAL check box. Notice how a third harmonic sinusoid is added to all the three phase waveforms of x(t) to produce x’(t).
• Repeat the moving of the red timeline and observe that the vector x’ equals vector x and still tracks the same circular path.
• Select the other two types of waveforms, the RECTANGULAR and RANDOMIZED and observe how the waveform shapes of x’(t) are changed.
• Repeat the moving of the red timeline and observe that the vector x’ equals vector x and still tracks the same circular path.