8.5:
The Y-to-Delta Circuit
In electrical power grids, balanced Y-to-Delta circuits feature Y-connected voltage sources and delta-connected loads, typically without a neutral line.
Assuming a positive phase sequence, the phase voltages are expressed to estimate the line voltages.
These line voltages equal the voltages across the load impedances and facilitate the calculation of phase currents, which have similar magnitudes and are out of phase by 120 degrees.
Applying Kirchhoff's Current Law at the nodes of the delta-connected loads reveals the relationship between the line and phase currents.
The magnitudes of the calculated line currents are the square root of three times the magnitudes of the corresponding phase currents, and the line currents lag the phase currents by 30 degrees.
Another way of analyzing the Y-to-Delta system is to convert the delta-connected loads to their corresponding Y-configuration.
The resulting balanced Y-to-Y system can then be analyzed by considering its single-phase equivalent circuit and determining the line currents.
The magnitudes and phase angles of the phase currents are calculated from their corresponding line currents using appropriate Delta-to-Y transformations.
8.5:
The Y-to-Delta Circuit
A balanced wye-to-delta circuit comprises balanced Y-connected voltage sources and delta-connected loads with no neutral line connection.
The initial step in analyzing a wye-to-delta circuit is to assume a positive phase sequence. These phase voltages are then utilized to calculate the line voltages that occur directly across the delta-connected load impedances. Van, Vbn, and Vcn are the phase voltages in wye, and Vab, Vbc, and Vca are the line voltages for a delta circuit. The relation between the line voltages for a wye-to-delta circuit can be calculated as:
These voltages across the load impedances are subsequently used to calculate the phase currents using Ohm's law. Due to the balanced nature of the circuit, these phase currents have identical magnitudes but are separated in phase by 120 degrees.
Kirchhoff's Current Law (KCL) is applied at the nodes of the delta-connected loads to derive the relationship between line and phase currents.
In a balanced wye-to-delta system, the magnitude of the line currents is related to the magnitudes of the phase currents by a factor of the square root of 3. Line currents lag 30 degrees from their corresponding phase Currents in the delta configuration.
One alternate method of analyzing the wye-to-delta circuit is to transform the delta-connected loads to an equivalent wye configuration. This will convert the circuit into a balanced wye-to-wye system, which can be simplified into single-phase equivalent circuits for analysis. In this manner, the phase currents can be estimated from their corresponding line currents, considering the phase shift introduced by the transformation.
This detailed understanding of phase and line currents and the relationship between wye-connected sources and delta-connected loads is essential for designing and operating an efficient electrical system.