Balanced Three-Phase Circuits

Balanced Three-Phase Circuits 

The voltages in the three-phase power system are produced by a
synchronous generator. In a balanced system, each of the three
instantaneous voltages have equal amplitudes but are separated from the
other voltages by a phase angle of 120. The three voltages (or phases) are
typically labeled a, b and c. The common reference point for the three
phase voltages is designated as the neutral connection and is labeled as n.
We may define either a positive phase sequence (abc) or a negative phase
sequence (acb) as shown below. The three sources Van, Vbn and Vcn are
designated as the line-to-neutral voltages in the three-phase system.

LINE-TO-LINE VOLTAGES

An alternative way of defining the voltages in a balanced three-phase
system is to define the voltage differences between the phases. These
voltages are designated as line-to-line voltages. The line-to-line voltages
can be expressed in terms of the line-to-neutral voltages by applying
Kirchoff’s voltage law to the generator circuit, which yields

Inserting the line-to-neutral voltages for a positive phase sequence into the
line-to-line equations yields

If we compare the line-to-neutral voltages with the line-to-line voltages, we find the following relationships,

THREE-PHASE CONNECTIONS

The sources and loads in a three-phase system can each be connected in either a wye (Y) or delta ()) configuration. Note that the wye connections are line-to-neutral while the delta connections are line-to-line with no neutral. Also note the convention on the node designations (lowercase letters at the source connections and uppercase letters at the load

connections).

BALANCED WYE-WYE CONNECTION

The balanced three-phase wye-wye connection is shown below. Note that the line impedance for each of the individual phases in included in the circuit. The line impedances are assumed to be equal for all three phases. The line currents (IaA, IbB and IcC) are designated according to the source/load node naming convention. The source current, line current, and load current are all one in the same current for a given phase in a wye-wye connection.

Note that the line current magnitudes are equal and each line current lags the respective line-to-neutral voltage by the impedance phase angle 2Z. Thus, the balanced voltages yield balanced currents. The phasor diagram for the line currents and the line-to-neutral voltages is shown below. If we lay the line-to-neutral voltage phasors end to end, they form a closed triangle (the same property is true for the line currents). The closed triangle shows that the sum of these phasors is zero.

BALANCED DELTA - WYE CONNECTION

A balanced delta - wye system consists of a balanced delta connected source and a balanced connected load. There are also the line voltages as well as the phase voltages. We can generate an equation through getting a loop from the circuit and this may help us solved for the line currents. Like in other connections, We may also transform delta connected souce to wye connected source.

THREE-PHASE CONNECTIONS INVOLVING DELTA SOURCES OR LOADS
In addition to the wye-wye three-phase connection, there are three

other possible configurations of wye and delta sources and loads.

The most efficient way to handle three-phase circuits containing delta sources and/or loads is to transform all delta connections into wye connections.



I've learned that in a connected balanced source, Line currents and phase  currents are equal,while in a delta - connected balanced source, line voltage and phase voltage is equal also. It is easy to solve for the parameters if the system is in wye-wye connection. so for us to have an easy solving, we have to convert it into wye connection.