In many circuits, resistors are neither in series nor in parallel, so the rules for series or parallel circuits cannot be applied. For these circuits, it may be necessary to convert from one circuit form to another to simplify the solution. Two typical circuit configurations that often have these difficulties are the wye (Y) and delta ( D ) circuits. They are also referred to as tee (T) and pi ( P ) circuits, respectively.
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 (∆) = Rb || (Ra + Rb)
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.
Wye to delta conversion
For conversion to wye network to delta network adding equations (v), (vi) and (vii) we get,
R1R2 + R2R3 + R3R1 = RaRbRc(Ra +Rb + Rc)/(Ra + Rb + Rc)2
= RaRbRc/(Ra + Rb + Rc) —————— (ix)
Dividing equation (ix) by each of the equations (v), (vi) and (vii) we get,
Ra = R1R2 + R2R3 + R3R1/ R1
Rb = R1R2 + R2R3 + R3R1/ R2
Rc = 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 = RY, Ra = Rb = Rc = R∆
For these condition conversion formulas become,
RY = R∆/3
Or R∆ = 3RY
Here Y connection is like series connection. And Del is like parallel connection.
