Star Delta Conversion and Delta Star Conversion

Besides the series or parallel connection, we can connect the resistors or other circuit elements also in STAR and DELTA connection.

Star to Delta Transformation

Let R1 R2 and R3 are three resistors. These resistors form a star connection with terminals 1, 2 and 3 respectively. Also, we consider the other three resistors Ra Rb and Rc. Again, these resistors form a delta between terminals 1 & 2, 2 & 3 and 3 & 1 respectively.  Now, we imagine that the star and delta are equivalent to each other. We have shown these in the figures below.

Star Delta Connections
Star Delta Connections

Note: The word equivalent means that the delta network between the terminals 1, 2 and 3 replaces the corresponding star network between the same set of terminals and vise versa.

Concept for Star Delta or Delta Star Derivation

MAIN CONCEPT FOR THE DERIVATION: Resistance seen by the same pair of terminals must be equal in both connections. WHY? Because during the conversion the terminals remain same hence the resistance seen by any two terminals remains the same.

Step 1: Find resistances seen by each pair of terminals in both networks

Terminal Resistances for Star Network

Where,

R12 = resistance seen by terminals 1 & 2

R23 = resistance seen by terminals 2 & 3

R31 = resistance seen by terminals 3 & 1

For writing equations (i) to (iii), imagine a battery connected between terminals 1 and 2 (since we want to find R12). Now, the resistance seen by this battery is R1 + R(since terminal 3 is open circuited now). For R23, connect the battery between terminals 2 and 3 and similarly for R31.

Terminal Resistances for Delta Network

For writing equation (iv) to (vi), again imagine the battery connected between 1 & 2, Now the battery encounters two parallel paths one containing Ra while another path contains  Rb & Rc. Hence, resistance seen by terminals 1 and 2 in Delta connection i.e. R12 is Ra∥(Rb + Rc). In the same manner, find R23 and R31.

Step 2: Manipulate equations to find R1 R2 and R3

Equating equation (i) and (iv), gives

Again, equating (ii) & (v) and  (iii) & (vi) respectively gives

Now, by adding equation (vii), (viii) and (ix), we get,

Finally, subtracting equation (viii) from equation (x) to get R1

In a similar way,

Step 3:  Find the value of R1R2 + R2R3  + R3 R1

Then, putting the values of R1, R2 and R3 from equation (xi), (xii) and (xiii) into R1R2 + R2R3+ R3 R1 and manipulating gives,

Step 4: The final step

Finally, divide equation (xiv) by equation (xi) to get, Ra

Now, we get Rb and Rc in the same way

TRICKs to Learn the Formulae of Star Delta Transformation

For Ra notice that Ra is the resistance connected between terminals 1 and 2 of DELTA network. So, look at the terminals 1 and 2 of STAR network

First, write the sum of the products of R1, R2 and R3 taken two at a time (i.e. R1R2+R2R3+R3R1) and then divide it by the resistor not connected between the terminals 1 and 2 i.e. by R3 (see figure).

In the same way, for Rb,  terminal 2 and 3 are considered and the resistance not connected between terminals 2 and 3 i.e. R1.

Similarly, for  Rc,  terminal 3 and 1 are considered and the resistance not connected between terminals 3 and 1 i.e. R2.

Remember, the numerator is the same in all the three formulae.

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