Star, Delta circuits

Star, Delta circuits

Delta/Star transformation

When solving networks with considerable number of branches, sometimes we experiences a great difficulty due to a large number of unknown variable have to be find. Such complicated networks can be simplified by successively replacing delta meshes by equivalent star systems and vice versa.

Consider we have three resistances connected in delta fashion between terminals 1, 2 and 3 as shown in figure 15.1(a). These three resistances can be replaced by three star connected (or ‘Y’ connected) resistances as shown in figure 15.1(b).

If we can arrange these three resistances in such a way that both delta and star systems will show the same resistance between any pair of terminals we can replace any of these arrangement instead of the other one.  This means these two arrangements are electrically equivalent.

How to convert a delta connection to star connection?

In the delta connection, there are two parallel paths between terminals 1 & 2. One is having a resistance of R₁₂ and the other is having a resistance of (R₂₃+R₃₁).

So the resistance between terminals 1&2 = R₁₂ // (R₂₃+R₃₁)

In the star connection, the resistance between the same terminals = R₁ + R₂

For electrically equivalent arrangements these two values should be equal, so

R₁ + R₂ = R₁₂X (R₂₃+R₃₁) / (R₁₂+R₂₃+R₃₁) --------------------- (A)

Similarly for terminals 2 & 3 and terminal 3 & 1, we get

R₂ + R₃ = R₂₃X (R₃₁+R₁₂) / (R₁₂+R₂₃+R₃₁) --------------------- (B)

R₃ + R₁ = R₃₁X (R₁₂+R₂₃) / (R₁₂+R₂₃+R₃₁) ---------------------(C)

By solving these three equations,

R₁ = R₁₂R₃₁ / (R₁₂+R₂₃+R₃₁)

R₂ = R₂₃R₁₂ / (R₁₂+R₂₃+R₃₁)

R₃ = R₃₁R₂₃ / (R₁₂+R₂₃+R₃₁)

Star/Delta transformation

This can be easily done by using equations (A), (B) and (C)

1.       Multiply A&B , B&C and A&C

2.       Add them together and simplify them.

Then we get,

R₁₂ = R₁ + R₂ + (R₁R₂/R₃)

R₂₃ = R₂ + R₃ + (R₂R₃/R₁)

R₃₁ = R₃ + R1 + (R₃R₁/R₂)

*You do not need to hardly remember these equations. You must be intelligent enough to understand the pattern of these equations. We will discuss some solved problems in my next post.

My next post will be a “solved problems” post regarding everything we’ve learnt so far in network analysis

Pabindu lakshitha

B.Sc (Engineering Undergraduate)