Three-Phase Electric Circuits: Balanced Wye-Wye Connection

We begin with the Y-Y system because any balanced three-phase system can be reduced to an equivalent Y-Y system.

Therefore, the analysis of this system should be regarded as the key to solving all balanced three-phase systems.

A balanced Y-Y system is a three-phase system with a balanced Y-connected source and a balanced Y-connected load.

Consider the balanced four-wire Y-Y system of Figure.(1), where a Y-connected load is connected to a Y-connected source.

We assume a balanced load so that load impedances are equal.

Although the impedance Z_Y is the total load impedance per phase, it may also be regarded as the sum of the source impedance Z_S, line impedance Z_l, and load impedance Z_L for each phase, since these impedances are in series.

As illustrated in Figure.(1), Z_S denotes the internal impedance of the phase winding of the generator; Z_l is the impedance of the line joining a phase of the source with a phase of the load; Z_L is the impedance of each phase of the load, and Z_n is the impedance of the neutral line.

Thus, in general

(1)

Z_S and Z_l are often very small compared with Z_L, so one can assume that Z_Y = Z_L if no source or line impedance is given. In any event, by lumping the impedances together, the Y-Y system in Figure.(1) can be simplified to that shown in Figure.(2).

Figure 1. A balanced Y-Y system, showing the source, line, and load impedances.

Figure 2. Balanced Y-Y connection

Assuming the positive sequence, the phase voltages (or line-to-neutral voltages) are

(2)

The line-to-line voltages or simply line voltages V_ab, V_bc, and V_ca are related to the phase voltages. For example,

(3a)

Similarly, we can obtain

(3b)

(3c)

Thus, the magnitude of the line voltages V_L is √3 times the magnitude of the phase voltages V_p, or

(4)

where

(5)

and

(6)

Also, the line voltages lead their corresponding phase voltages by 30◦. Figure.(3a) illustrates this. Figure.(3a) also shows how to determine V_ab from the phase voltages, while Figure.(3b) shows the same for the three-line voltages.

Notice that V_ab leads V_bc by 120◦, and V_bc leads V_ca by 120◦so that the line voltages sum up to zero as do the phase voltages.

Figure 3. Phasor diagrams illustrating the relationship between line voltages and phase voltages

Applying KVL to each phase in Figure.(2), we obtain the line currents as

(7)

We can readily infer that the line currents add up to zero,

(8)

so that

(9a)

or

(9b)

that is, the voltage across the neutral wire is zero. The neutral line can thus be removed without affecting the system.

In fact, in long-distance power transmission, conductors in multiples of three are used with the earth itself acting as the neutral conductor.

For easier understanding, you better read about balanced three-phase voltages first.

Power systems designed in this way are well grounded at all critical points to ensure safety.

While the line current is the current in each line, the phase current is the current in each phase of the source or load. In the Y-Y system, the line current is the same as the phase current.

We will use single subscripts for line currents because it is natural and conventional to assume that line currents flow from the source to the load.

Figure 4. A single-phase equivalent circuit

An alternative way of analyzing a balanced Y-Y system is to do so on a “per phase” basis.

We look at one phase, say phase a, and analyze the single-phase equivalent circuit in Figure.(4). The single-phase analysis yields the line current I_a as

(10)

From I_a, we use the phase sequence to obtain other line currents. Thus, as long as the system is balanced, we need only analyze one phase.

We may do this even if the neutral line is absent, as in the three-wire system.