# Star to Delta Conversion

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121 Dear reader, Today we will learn how to connect to the Star-to-Delta in simple language.

### Explain to Star/Delta Conversation:

Complicated Network Solutions General series parallel circuit rules do not apply in many cases. Many critical networks are solved by using Karshaf’s formula or Maxwell loop current method, but many times the number of branches or branches for the number of equations is too much that it is very difficult to solve them. In those cases, the network can easily be solved by transforming from Star to Delta and Delta.

### T-connected and Equivalent Star Network As we have already seen, we can redraw the Tresistor network above to produce an electrically equivalent Star or Υ type network. But we can also convert a Pi or πtype resistor network into an electrically equivalent Delta or Δ type network as shown below.

### Pi-connected and Equivalent Delta Network Having now defined exactly what is a Star and Delta connected network it is possible to transform the Υ into an equivalent Δ circuit and also to convert a Δ into an equivalent Υcircuit using a the transformation process. This process allows us to produce a mathematical relationship between the various resistors giving us a Star Delta Transformation as well as a Delta Star Transformation.

These circuit transformations allow us to change the three connected resistances (or impedances) by their equivalents measured between the terminals 1-2, 1-3 or 2-3 for either a star or delta connected circuit. However, the resulting networks are only equivalent for voltages and currents external to the star or delta networks, as internally the voltages and currents are different but each network will consume the same amount of power and have the same power factor to each other.

### Delta to Star Conversion

Compare the resistances between terminals 1and 2.  Resistance between the terminals 2 and 3. Resistance between the terminals 1 and 3. This now gives us three equations and taking equation 3 from equation 2 gives: Then, re-writing Equation 1 will give us: Adding together equation 1 and the result above of equation 3 minus equation 2 gives: From which gives us the final equation for resistor P as: Then to summarize a little about the above maths, we can now say that resistor P in a Star network can be found as Equation 1 plus (Equation 3 minus Equation 2) or  Eq1 + (Eq3 – Eq2).

Similarly, to find resistor Q in a star network, is equation 2 plus the result of equation 1 minus equation 3 or  Eq2 + (Eq1 – Eq3) and this gives us the transformation of Q as: and again, to find resistor R in a Star network, is equation 3 plus the result of equation 2 minus equation 1 or  Eq3 + (Eq2 – Eq1) and this gives us the transformation of R as: When converting a delta network into a star network the denominators of all of the transformation formulas are the same: A + B + C, and which is the sum of ALL the delta resistances. Then to convert any delta connected network to an equivalent star network we can summarized the above transformation equations as.

### Star to Delta Connection

The value of the resistor on any one side of the delta, Δ network is the sum of all the two-product combinations of resistors in the star network divide by the star resistor located “directly opposite” the delta resistor being found. For example, resistor A is given as: with respect to terminal 3 and resistor B is given as: with respect to terminal 2 with resistor Cgiven as: with respect to terminal 1. By dividing out each equation by the value of the denominator we end up with three separate transformation formulas that can be used to convert any Delta resistive network into an equivalent star network as given below.