Wye delta conversion

Δ and Y conversion

[4] Example 1

[3] Example 2

Calculating resistances

[4] Example 1

[3] Example 2

[4] Example 3

Series-parallel formula

References

[1] Wye-delta (Y-Δ) transform is a math technique to simplify electrical network analysis.

The name is from the circuits' shapes, that resemble respectively letter 'Y' and Greek letter delta Δ.

The circuit transformation theory is by Arthur Edwin Kennelly (1899).

It's often used in 3-phase electric power circuits.

[2] Many circuit applications have components connected in 1 of 2 ways to make a 3-terminal network.

A delta (Δ) network is the shape of a delta/triangle.
A wye (Y)-network is the shape of letter 'Y'.
A pi (π) network is the shape of a retangle.
A tee (T) network is the shape of letter 'T'.

[4] To find resistance from Terminals A to B, how do we simplify the network?

We cannot reduce this by series-parallel combination of resistors, as resistors are not in series nor in a parallel.

This is when the ‘Delta-Wye’ (Δ-Y) or ‘𝛑-T transformation is used, which are the same networks of different styles, as shown on the right.

We can transform a Δ configuration of resistors to a Y configuration or the other way with a set of transformation equations. This makes a network easier to simplify.

Δ and Y conversion

[2] Note: The convesion equations are also for ‘𝛑-T’ resistor networks.

Δ and Y networks are often in 3-phase AC power systems, but aare often balanced (all resistors equal in value) and conversion from one to another need not involve such complex calculations.

[4] Example 1

First label the resistors:

RA = 30 Ω
RB = 50 Ω
RC = 20 Ω

Then imagine and draw at the new branches at the network's center:

Use the Δ → Y conversion equations to get R1, R2, and R3's values.

Lastly, erase the original network and its equivalent Y-network is:

[3] Example 2

Transform the circuit from delta to Y.

Use the Delta to Y-network formula:

Ra = 10 Ω
Rb = 30 Ω
Rc = 60 Ω

Draw the Y-network at the center with resistances.

Erase the original network and its Y-network is:

Calculating resistances

[4] Example 1

Find the equivalent resistance between Terminals A and B--we reduce the given resistors' arrangement to one resistor.

Firt converting the inner Y network to its Δ equivalent.

All Y-network's resistors are all of 3Ω, so: R1 = R2 = R3 = 3Ω

The intermediate network looks like this:

Now put the resistors values in the Y → Δ conversion formula to get RA, RB, and RC.

All the network's three Δ resistors are 9Ω.

So the new network after the Y → Δ conversion is:

The Δ network's resistors are parallel to the outer Δ network.

18||19 = 6Ω

The 6Ω resistors between AC and CB are in series: 6+6=12Ω

Solving the parallel resistor combination between A and B:

RAB = 6||12 = 4Ω

[3] Example 2

Find Rab and i.

Answer:

This circuit has 2 Y-networks:

1st Y-network: 24Ω, 30Ω, and 30Ω
2nd Y-network: 10Ω, 50Ω and 30Ω

Both can find Rab and i.

Consider we use 2nd Y-network.

With the 2nd Y-network:

Convert the Y-network into delta network with 30 Ω, 10 Ω, and 50 Ω.

r1 = 10 Ω
r2 = 50 Ω
r3 = 30 Ω

Then redraw the circuit:

Now combine the circuits in parallels:

24||46=15.771
30||230=26.538

And redraw again.

Now add series connection: 14.069+26.538=40.61

Redraw the circuit again:

Now add parallel connections:
76.67||40.61 = 26.5481

Rab = 26.5481+13 = 40 Ω

i = V/Rab = 100/40 = 2.5 V

[4] Example 3

Find the equivalent resistance between the circuit's Terminals A and B at the start of this page.

This network has two Δ networks.

Take the upper Δ resistors arrangement and convert it to its Y equivalent:

RA = 50Ω
RB = 10Ω
RC = 40Ω

The new network is:

With the Y equivalent of the Δ network, replace this in to the original circuit.

Resistors are in series, but not in parallel as no ratio are the same in opposite: 1:4 (5Ω:20Ω) and 5:2 (25Ω:10Ω).

So resistors are in parallel:
5+25=30Ω

10+20=30Ω

RAB = 4+15 = 19Ω

Series-parallel formula

The resistance combination formula calculates resistance between mixed series-parallel resistors, often involving 3 resistors: One in parallel with another and in series with another at the same time.

Its formula is: