1. Three Phase Circuits
    1. Three Phase Circuit Connections
    2. There are two basic schemes for connecting a load or generator in a three-phase circuit. The Y or "wye" connection joins neutrals of each phase at a common junction. The or "delta" connection is a triangle whose vertices form the buses, and there is no neutral bus. Examples of each connection scheme are shown in Figure 1.


    3. Balanced Three-Phase Circuits
    4. In the three phase circuit arrangements displayed in Figure 1, each leg is shown with an impedance. If the impedances are identical on each phase, then the load is said to be balanced. A three-phase source is balanced when each leg produces equal magnitude voltage, with phase difference between any two phases, as shown in Figure 2.


      By convention, the three phases are designated a, b, and c. If a phase is chosen as reference, then the balanced line-to-neutral voltages in a wye-connected generator will be

      This phase sequence is called abc or positive sequence, in which leads by , and leads by , as shown in Figure 3 on the left. An alternative arrangement is acb, or negative sequence, shown in Figure 3 on the right.

      The delta-connected generator does not have line-to-neutral voltages, of course. The voltages between phases, or line-to-line voltages, form a closed path around buses a, b, and c, and normally sum to zero, in the case of a balanced system. In a wye-connected generator, line-to-line voltages can be calculated from line-to-neutral voltages. The line-to-line voltages are greater in magnitude by a factor of , and they lead the line-to-neutral voltages by :


      This is illustrated in Figure 4, a phasor diagram of positive-sequence voltages for a balanced wye-connected generator.



      Example Problem B3.3.1

      Find the positive sequence line-to-line voltages in the balanced wye-connected generator if the line-to-neutral voltages have a magnitude of 12kV, and phase a is the reference.


      Line currents in a three-phase system can be calculated from phase voltages and impedances. For a balanced system with a wye-connected load, the line current equations can be written as


      Figure 5 illustrates the line currents in a balanced wye-connected load. is the impedance per phase between the source and the neutral.




      Load currents in a balanced delta-connected system are calculated using this equation:



      Line currents always sum to zero in a delta-connected system because there is no neutral wire. Figure 6 illustrates the currents in a delta-connected load.

      Example Problem B3.3.2

      Given a wye-connected load of impedance and balanced positive sequence source with , find the load currents.


    5. Power Factor Correction

A capacitor is considered a generator of positive reactive power, and its function in the power system is to supply reactive power needed by inductive loads. It is desirable to operate with loads that are not highly inductive. By compensating such loads with a shunt capacitor, the source current and apparent power decrease, resulting in more efficient operation and better voltage regulation. Also, reduced current permits the use of smaller conductors, so a significant savings in equipment and wiring costs may be realized by keeping power factor close to unity. The practice of improving a lagging power factor by installing capacitors in parallel with an inductive load is called power factor correction.

Example Problem B3.3.3

Draw the power triangle for a single-phase source which delivers 80 kW to a load at 0.7 pf lagging. If the source voltage is 480 V, calculate the current required by this load.



      1. Example Problem B3.3.4
      2. Size the shunt capacitor required to reduce the power factor in Example Problem 3 to 0.9 lagging.


        1. The original load impedance found in Example Problem 3 is


This impedance contains resistive and inductive elements:





After power factor correction, load impedance will be






Note that a change in load impedance from to is accomplished by adding a capacitive reactance of . The corresponding capacitance is


Now, examine the power triangle after power factor correction.


The result is significantly lower reactive power requirement, with a reduction in load current.