Answer MET Question 12

Question: A. With the aid of delta and star connection diagrams, state the basic equation from which delta – star – delta conversion equation can be derived.

Answer: Delta and star connection diagrams

Derivation:

In Delta connection

$\displaystyle \small \mathrm{\frac{1}{R_{AB}}=\frac{1}{R_1}+\frac{1}{R_2+R_3}}$

In star connection

$\displaystyle \small \mathrm{R_{AB}=R_a + R_b}$

thus,

$\displaystyle \small \mathrm{R_a + R_b = \frac{R_1(R_2+R_3)}{R_1+R_2 + R_3}}$

similarly,

$\displaystyle \small \mathrm{R_b + R_c = \frac{R_2(R_3+R_1)}{R_1+R_2 + R_3}}$ 

$\displaystyle \small \mathrm{R_c + R_a = \frac{R_3(R_2+R_1)}{R_1+R_2 + R_3}}$

Adding the above three equations we get;

$\displaystyle \small \mathrm{R_a + R_b +R_c = \frac{R_1R_2+R_2R_3+R_3R_1}{R_1+R_2 + R_3}}$

Now;

$\displaystyle \small \mathrm{(R_a + R_b +R_c)-(R_b+R_c) = \frac{R_1R_2+R_2R_3+R_3R_1}{R_1+R_2 + R_3}-\frac{R_2(R_3+R_1)}{R_1+R_2 + R_3}}$

thus;

$\displaystyle \small \mathrm{R_a = \frac{R_3R_1}{R_1+R_2 + R_3}}$

Similarly ;

$\displaystyle \small \mathrm{R_b = \frac{R_1R_2}{R_1+R_2 + R_3}}$

$\displaystyle \small \mathrm{R_c = \frac{R_2R_3}{R_1+R_2 + R_3}}$

The above set of equations can be used for Delta-Star Transformation

Now;

$\displaystyle \small \mathrm{R_aR_b = \frac{R_1^2R_2R_3}{(R_1+R_2 + R_3)^2}}$

$\displaystyle \small \mathrm{R_bR_c = \frac{R_1R_2^2R_3}{(R_1+R_2 + R_3)^2}}$

$\displaystyle \small \mathrm{R_cR_a = \frac{R_1R_2R_3^2}{(R_1+R_2 + R_3)^2}}$

adding above three equations;

$\displaystyle \small \mathrm{R_aR_b+R_bR_c+R_cR_a = \frac{R_1R_2R_3}{R_1+R_2 + R_3}}$

dividing the above by
$\displaystyle \small \mathrm{R_c = \frac{R_2R_3}{R_1+R_2 + R_3}}$

We get;

$\displaystyle \small \mathrm{R_1 = \frac{R_aR_b+R_bR_c+R_cR_a}{R_c}}$

similarly

$\displaystyle \small \mathrm{R_2 = \frac{R_aR_b+R_bR_c+R_cR_a}{R_a}}$

$\displaystyle \small \mathrm{R_3 = \frac{R_aR_b+R_bR_c+R_cR_a}{R_b}}$

The above equations can be used for the transformation from Star to Delta