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Difference between Delta and Wye
WYE and Delta are terms used to describe commercial 3-phase wiring configuration for transformers used in the electric grid
Transformers
Transformers are used throughout the grid to accomplish different objectives. Transformers control the inverse relationship between voltage and amperage. Raise the voltage and the amperage goes down. Lower the voltage and the amperage goes up.
For example, wires leaving the power plant might carry 500,000 volts. The high voltage means low amps and low heat on the wire which allows economic  long-distance transmission of electricity. While high voltage is good for transmission, the switchgear and wiring needed for 500,000 volts is too large, expensive and dangerous for ordinary use inside a home or building.
Using transformers, the power company can change the volt-amp ratio. The result is homes and buildings receive low volt, high amp power. It works nicely because 120-240-208-277-480 volt power can be safely controlled by small switches, relays, cell phone chargers etc contained within steel and plastic enclosures, while the amperage (heat) is controlled by circuit breakers and then distributed to outlets, switches, motors, lights etc using correctly sized wire to match amp rating of breaker.

WYE and Delta wiring configurations are one method of using a transformer to reach a volt-amp objective.
Before going farther, it's important to describe how a transformer works.
Transformers work on a basic principle of magnetic induction where applying electricity to one coil of wire creates a magnetic flux that will energize the other coil of wire with electricity. How transformers work pdf
Inside a transformer are two coils of wire called the primary and secondary coils. Each coil is wrapped around a laminated iron core or more efficient amorphous metal core. The metal core is shared by both coils, but both coils of wire are 'insulated' from each other. They are electrically separated. There is no wire in common between the coils, only the metal core is shared.

Having different numbers of turns of wire, or varying the turns ratio, on each coil will raise or lower voltage. Different voltages can be achieved throughout the grid by varying the number of turns on the primary and secondary coils.

The power company connects 7,900 volts from the distribution lines to the primary coil. This produces voltage on the secondary coil. Different transformers are selected depending on the voltage specified for end user. The end user's wires are connected to the secondary coil and power is routed into the main breaker panel.

Fig 1-1 WYE

Fig 1-2 Delta
Figures 1-1 and 1-2 above are representation of wiring for WYE and Delta on the secondary side of three transformers.
Typical 3-phase service has 3 secondary coils. There are variations (not shown on this page) called Open WYE and Open Delta etc that have 2 coils.
The actual appearance of wiring is different than the representations above. For example, the transformer coils do not actually touch each other. Instead they are connected by a wire.

Figure 1-1 WYE represents each transformer coil as a zig-zag line.
WYE (star) transformer coils (windings) are wired together to form the Neutral wire. There is 1 common point shared by each of the 3 coils of wire as shown in Figure 1-1. The common point gives rise to the Neutral wire. The other end of each coil gives rise to the 3 Hot wires.
Measuring voltage from the center point to end of any coil is called the phase voltage. This can be seen in Fig 1-1 as 120 volts. Other WYE services can have different voltages depending on the turns ratio selected for the service.
The voltage measured between any 2 Hot wires is known as the line-to-line voltage or simply as the line voltage. This can be seen in Fig 1-1 as 208 volts.

In a wye connected system, the Hot wire to Hot wire line voltage is higher than the phase voltage by a factor of the square root of 3 (1.732). So 120 volts x 1.732 = 208 volts.
In a wye-connected system, phase amps (current) and line amp are the same.

Fig 1-2 Delta represents each transformer using four uuuu's.
Delta transformer coils are wired end to end. The Neutral wire if present is achieved by tapping into center of 1 coil.
'Line voltage and phase voltage are the same with Delta. If you measure Hot wire to Hot wire, you get same voltage as measuring both ends of one coil. However the amps vary between line voltage and phase voltage. Unlike Wye where phase and line voltage stay the same, with Delta, the line amperage is higher than the phase amperage. If phase amperage measure across one coil is 10 amps. the line amperage measured across two Hot wire is 10 amps x square root 3 or 1.732 = 17.32 amps.
As the generator spins, and voltage on each wire rises and falls, each Delta coil is energized at different times. At any given time, two of the three coils are energized. This adds extra current when measuring between the Hot wires.'

Delta has lower amps/ meaning less heat.
For example many transformer banks are wired Delta on the Power company or Primary coils, and Wye on the customer or secondary coils. The reason for this: the distribution voltage connected to primary side can be 4500-7900 volts, and using delta configuration means lower amps which means the manufacturer can use smaller wires on the primary side coils and save money, yet still deliver full power to the customer.
Materials used for the power grid are a balance of cost, safety, and function.
Star-Delta (WYE-delta) is also a configuration used for motor starter timers, such as a star-delta timer. With motors, they consume more amperage to start. To reduce the expense and heat, the motor is started with the star or wye configuration, and after the motor starts, the circuit is switched to delta for higher efficiency.

Why use 3-phase
"The horsepower rating of three-phase motors is about 150% greater than for single-phase motors with a similar frame size. The power delivered by a single-phase system pulsates, and the power falls to zero three times during each cycle. The power delivered by a three-phase circuit pulsates also, but it never falls to zero. In a three-phase system, the power delivered to the load is the same at any instant. This produces superior operating characteristics for three-phase motors."*

Resource:
*3 phase circuits with basic math .pdf
How to identify transformer wiring
What is 3-phase
How to wire 3-phase
3-phase timer


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