Balanced Wye-Wye Connection

The three phase voltage we get from a three phase synchronous generator is needed to be a balanced three phase voltage.

We will not study what is a three phase circuit and what is an AC circuit here, since they have been explained in other posts. But we need to remember that three phase circuits have four configuration:

• Balanced wye-wye connection
• Balanced wye-delta connection
• Balanced delta-delta connection
• Balanced delta-wye connection

We will learn the balanced wye-wye connection right now.

Balanced Wye-Wye Connection

From what we have learned in power factor correction, the higher the power factor, the better its power transfer efficiency and longer lifetime of our machine. But balanced voltage is not enough if the load is not balanced.

If the voltage is balanced but the load is not, the current will be imbalanced, thus causing harmonic distortion and generating bigger reactive power as a waste.

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

Wye configuration means the circuit has a connection where it forms a “Y” shape

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

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

Consider the balanced four-wire wye-wye system shown below, where a wye-connected load is connected to a wye-connected source. We assume a balanced load so that load impedances are equal.

Assume our circuit has no unbalanced three phase voltage and our load impedance is balanced.

We will use Z_Y as our equivalent impedances to represent the total impedance in  wye-connected loads.

From the illustration above:

• Z_S denotes the internal impedance of the phase winding of the generator as our source
• Z_l is the impedance of the line joining a phase of the source with a phase of the load, this is the impedance of our conductors
• Z_L is the impedance of each phase of the load
• Z_n is the impedance of the neutral line, connecting the neutral point from supply and load.

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.

Thus

Z_S and Z_l are often very small compared to load impedance Z_L so let’s remove them to make things simpler.

Since we remove Z_S and Z_I, it will be

And our wye connection diagram becomes simpler only with a balanced three phase source and a balanced three phase load.

And our three phase voltage with positive sequence (abc) is

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

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

where

From the equation above, we conclude that the line voltages lead their corresponding phase voltage by 30° as can be seen in a three phase voltage phasor diagram below.

Even if the line voltage leads the phase voltage by 30°, each line voltage still has a 120° phase shifted from other line voltages.

Applying Kirchhoff’s Voltage Law to each phase in the circuit above, we obtain the line currents as

Based on Kirchhoff’s Current Law, the algebraic sum of currents entering and leaving a node is zero, then

This concludes that the voltage across the terminal wire (Z_N) is zero. Then we can also remove this along with Z_S and Z_I to make things simpler again.

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 wye-wye system, the line current is the same as the phase current.

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

An alternative way of analyzing a balanced wye-wye 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 above. The single-phase analysis yields the line current Ia as

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.

Balanced Wye-Wye Connection Examples

To achieve a balance in a three phase circuit, we need to balance both the voltage and the load. This way also makes our electrical circuit have balanced three phase power.

Calculate the line currents in the three-wire wye-wye system of a circuit below.

Answer:

The three-phase circuit in above is balanced; we may replace it with its single-phase equivalent circuit such as explained before.

We obtain Ia from the single-phase analysis as

Where

Hence,

Since the source voltages are in positive sequence and the line currents are also in positive sequence,

Since the circuit is balanced, the power factor is close to unity thus its power transfer efficiency and lifetime are better for our machine.

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