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.
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
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 + R2 (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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