Star and delta sums

SOLVED EXAMPLES ON STAR/DELTA TRANSFORMATION 

Q1). Determine the resistance between the terminals A&B and hence find the current through the voltage source. Refer figure 16.1

Answer:

See figure 16.1(a)

The resistors in between point 1, 2&3 are about to replace by a star connected system. Otherwise is difficult to find the total resistance.

So we have to use the delta to star transformation equations.

R₁ = R₁₂R₃₁ / (R₁₂+R₂₃+R₃₁)

R₁ = (60*40)/ (60+40+100)

R₁ = 12Ω

R₂ = R₂₃R₁₂ / (R₁₂+R₂₃+R₃₁)

R₁ = (100*60)/ 200

R₁ = 30Ω

R₃ = R₃₁R₂₃ / (R₁₂+R₂₃+R₃₁)

R₃ = (100*40)/ 200

R₃ = 20Ω

So we can redraw the network as shown in figure 16.2

Now we can easily find the total resistance between A&B terminals

R­_total = [(80+20)//(88+12)] + 30

R_total = 50 + 30

R_total = 80Ω

Applying ohm’s law to the total resistance,

I = V/R

I = 160v/80Ω

I = 2A

Q2) Find the total resistance between A&B terminals for the network shown in figure 16.3

Answer:

See figure 16.3(a)

We are about to replace the delta system by star system in between point 1, 2 &3

So we have to use the delta to star transformation equations.

R₁ = R₁₂R₃₁ / (R₁₂+R₂₃+R₃₁)

R₁ = (3*6)/ (3+6+9)

R₁ = 1Ω

R₂ = R₂₃R₁₂ / (R₁₂+R₂₃+R₃₁)

R₂ = (9*3)/18

R₂ = 1.5Ω

R₃ = R₃₁R₂₃ / (R₁₂+R₂₃+R₃₁)

R₃ = (6*9)/18

R₃ = 3Ω

So now we can replace the system as shown in figure 16.4

Now we can easily find the total resistance between A&B terminals

R_AB = (7Ω+3Ω) + (8.5Ω+1.5Ω) + 1Ω

R_AB = 6Ω

Q3). Find the total resistance between A&B terminals (R_AB) shown in figure 16.5

Answer:

You must understand that you have to use star/delta transformation for this problem. Unlike other problems, in this case it is not pointed out which system of resistance you must replace. So you yourself have to point it out.

This is very important. Though the tutorial problems guide you to find the replaceable systems, in practical level you will have to guide yourself manually. This means you must know how to choose the correct system to apply delta/star transformation.

See figure 16.6

See the circled systems in the figure. You have to replace these systems with delta systems. If you see it carefully, you’ll see that both systems are same (one is upside down of the other). So you don’t need to find two different sets of delta systems. See figure 16.7

This figure shows you the star to delta transformation. As the required equation for transformation are given in my previous post, I’ve directly put the values for the delta system shown in the above figure. Steps for this calculation are shown below.

R₁₂ = R₁ + R₂ + (R₁R₂/R₃)

R₁₂ = 3 + 2 + (3*2)/2

R₁₂ = 8Ω

R₂₃ = R₂ + R₃ + (R₂R₃/R₁)

R₂₃ = 2 + 2 + (2*2)/3

R₂₃ = 16/3Ω

R₃₁ = R₃ + R1 + (R₃R₁/R₂)

R₁₃ = 3 + 2 + (3*2)/2

R₁₃ = 8Ω

So we can redraw the network as shown in figure 16.8

Now we can easily find the total resistance between A&B terminals. For your better understanding I’ve simplified the network. See figure 16.9

So now it is simple.

R_AB = { [ (7+5)//8//8 ] + 5 } //8//4

R_AB = (3 + 5) // 8 // 4

R_AB = 4//4

R_AB = 2Ω

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)

                                                                               B.Sc. (Engineering Undergraduate)