Phase Diagrams

Author: Edmund A. Laport

These phase relations are confusing until set out in a separate diagram like Fig. 2.25, which shows all of the phases of the system. It is to be noted that the desired phase can be attained usually in two ways - by advancing phase or retarding it. The choice is important from the standpoint of network economy and minimum storage of energy in all the circuits of the system, a factor in its selectivity characteristic. The nearest rule that one can suggest for the choice is to go in the direction, positive or negative, that gives the smallest total phase shift in the networks of a branch circuit.

In the phase diagram for this problem (Fig. 2.25), I₂ is taken as reference phase, I₁ will lag by 45 degrees, and I₃ will lead by 45 degrees. The phase shift across A was found to be 77.5 degrees, the current I₂ leading the main line current I₀ by this amount. This brings I₀ at -77.5 degrees in the diagram. Next the phase lag in the secondary lines is shown with reference to I₀, so that the 135-degree line and A together total -212.5 degrees. An auxiliary vector is drawn for this.

Fig. 2.28 : Network C

Fig. 2.29 : Network D

It is seen that the smallest angle between this auxiliary vector and I₁ is counterclockwise, the direction of advance of phase. The angle is 167.5 degrees. Network B has been computed to have an advance of 55 degrees, leaving a phase advance of 112.5 degrees to be obtained in network D.

For the last branch of the circuit, it is noted that the angle between the auxiliary vector at 212.5 degrees and I₃ is -102.5 degrees (clockwise). If C is an L network, its phase lag is 66 degrees, and the values are as shown in Fig. 2.28. This leaves a lag of 36.5 degrees for network E.

Fig. 2.30 : Network E

After completing the synthesis of network E we have the circuits and values shown in Fig. 2.30.

Now we note that networks A, B, and C all have a parallel reactance directly across the main line. Thus, their combined parallel value can be obtained by using one reactance instead of three. There are values of j87, -j536, and j425 in parallel, which is equivalent to j83.5 ohms. Combining all the circuits for the system, we have the diagram of Fig. 2.31. See Chap. 5 for a simple method of network synthesis for problems of this type.