Delta to wye and wye to delta conversion

Sometimes we are not sure in electric circuits that the resistors are neither in parallel nor in series.

See the circuit below here the relation between R₁ resistor and R₆ resistor is series or parallel we are not sure.

In that case we can simplify this circuit using three terminal equivalent networks. Three terminal networks are used in three phase connection, matching networks and electrical filters.

Figure 2 shows Wye (Y) and Tee (T) networks and figure 3 shows delta (∆) and pie (Π) network.

These networks are the equivalent of large network. I will discuss here how to transform wye to delta and delta to wye.

Delta to wye conversion

It is easy to work with wye network. If we get delta network, we convert it to wye to work easily. To obtain the equivalent resistance in the wye network from delta network we compare the two networks and we confirm that they are same. Now we will convert figure 3 (a) delta network to figure 2 (a) wye network.

From figure 3 (a) for terminals 1 and 2 we get,

R₁₂ (∆) = R_b || (R_a + R_b)

From figure 2 (a) for terminals 1 and 2 we get,

R₁₂(Y) = R₁ + R₃

Setting wye and delta equal,

R₁₂(Y) = R₁₂ (∆) we get,

Equations (v), (vi) and (vii) are the equivalent resistances for transforming delta to wye conversion. We do not need to memorize these equations. Now we create an extra node shown in figure 4 and follow the conversion rule,

Each resistor in the Y network is the product of the resistors in the two adjacent Del branches, divided by the sum of the three Del resistors.

Now we will solve a problem how to convert delta to wye network. It will give clear concept.

Problem: convert the delta network to wye network.

From equation (v), (vi) and (vii) we can find the equivalent resistance of wye network,

The equivalent Y network configuration is shown in figure 6.

Wye to delta conversion

For conversion to wye network to delta network adding equations (v), (vi) and (vii) we get,

 R₁R₂ + R₂R₃ + R₃R₁ = R_aR_bR_c(R_a +R_b + R_c)/(R_a + R_b + R_c)²

                                = R_aR_bR_c/(R_a + R_b + R_c) —————— (ix)

Dividing equation (ix) by each of the equations (v), (vi) and (vii) we get,

R_a = R₁R₂ + R₂R₃ + R₃R₁/ R₁

R_b = R₁R₂ + R₂R₃ + R₃R₁/ R₂

R_c = R₁R₂ + R₂R₃ + R₃R₁/ R₃

For Y to delta conversion the rule is followed below,

Each resistor in the delta network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.

Delta and Y are said to be equal or balanced when

R₁ = R₂ = R₃ = R_Y, R_a = R_b = R_c = R_∆

For these condition conversion formulas become,

           R_Y = R_∆/3

Or     R_∆ = 3R_Y

Here Y connection is like series connection. And Del is like parallel connection.