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 R_{1} R_{2} and R_{3} are three resistors. These resistors form a star connection with terminals 1, 2 and 3 respectively. Also, we consider the other three resistors R_{a} R_{b} and R_{c}. 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

Where,

**R _{12}** = resistance seen by terminals 1 & 2

**R _{23}** = resistance seen by terminals 2 & 3

**R _{31}** = 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 R_{12}). Now, the resistance seen by this battery is R_{1 }+ R_{2 }(since terminal 3 is open circuited now). For R_{23, }connect the battery between terminals 2 and 3 and similarly for R_{31.}

### 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 R_{a} while another path contains R_{b} & R_{c}. Hence, resistance seen by terminals 1 and 2 in Delta connection i.e. **R _{12}** is R

_{a}∥(R

_{b}+ R

_{c}). In the same manner, find

**R**and

_{23}**R**

_{31}.**Step 2: Manipulate equations to find ****R _{1} R_{2} and R_{3}**

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 R_{1}

In a similar way,

**Step 3: Find the value of ****R _{1}R_{2 }+ R_{2}R_{3 } + R_{3} R_{1}**

Then, putting the values of R_{1, }R_{2 }and R_{3 }from equation (xi), (xii) and (xiii) into R_{1}R_{2 }+ R_{2}R_{3}+ R_{3} R_{1 }and manipulating gives,

**Step 4: The final step**

Finally, divide equation (xiv) by equation (xi) to get, R_{a}

Now, we get R_{b} and R_{c} in the same way

### TRICKs to Learn the Formulae of Star Delta Transformation

For R_{a} notice that R_{a} 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 R_{1}, R_{2} and R_{3} taken two at a time (i.e. R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}_{) }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 **R**_{b,} terminal 2 and 3 are considered and the resistance not connected between terminals 2 and 3 i.e. R_{1.}

Similarly, for **R**_{c,} terminal 3 and 1 are considered and the resistance not connected between terminals 3 and 1 i.e. R_{2.}

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

- Star Delta Conversion and Delta Star Conversion
- Norton’s Theorem – Norton’s Current and Resistance
- Circuit Analysis or Network Analysis
- Kirchhoff’s Current Law and Kirchhoff’s Voltage Law
- Thevenin’s Theorem Thevenin’s Voltage and Resistance
- Maximum Power Transfer Theorem
- Compensation Theorem
- Superposition Theorem Statement and Theory
- Reciprocity Theorem Statement Explanation and Examples
- Maxwell’s Loop Current Method or Mesh Analysis
- Millman’s Theorem to Voltage and Current Sources
- Nodal Analysis Method with Example of Nodal Analysis
- Solving Equations with Matrix Method
- Ideal Voltage Source and Ideal Current Source
- Independent and Dependent Current and Voltage Sources
- Electrical Source Conversion (Voltage and Current Sources)
- Voltage Division Rule and Current Division Rule