Introduction to Electric Circuits

3.9 Delta-star transformation 57

and for the resistance between terminals 3 and 1"

R3 + R1 = R3,[R,2 + Rz3]/(R12-+- R23 + R3,) (3.19)

Subtract Equation (3.18) from Equation (3.17) to give, after expanding the

brackets,

e 1 - e 3 -- (e12R23 -Jr- e12R31- e23R12- e23R31)/(e12 Jr- R23 -']- e31 )

el- R3 = (e12R31- e23R31)/(e12 + R23 + e31) (3.20)

Adding Equations (3.19) and (3.20), we obtain

2R1 = (R3~R12 + R31R23 + R~zR3,- Rz3R3~)/(R~2 + R23 + R3~)

= 2R31R12/(R12 + R23 + R31 )

So

R~ = R3~R12/(R~2 + R23 + R31) (3.21)

If we now subtract Equation (3.19) from Equation (3.18) and add the result to

Equation (3.17) we obtain

R2 = R12R23/(R12 + R23 + R31 ), (3.22)

Finally, by subtracting Equation (3.17) from Equation (3.19) and adding the

result to Equation (3.18):

R3 = Rz3R3~/(Ra2 + R23 + R3a) (3.23)

An easy way to remember the rules for changing from delta to star is to draw

the star set inside the delta set as shown in Fig. 3.31.

Figure 3.31

A

R31 R12

C

Any star equivalent resistor is then given as 'the product of the two delta
resistors on either side divided by the sum of all three delta resistors'.

Example 3.9

Determine the input resistance (i.e. the resistance between terminals A and B)

in the circuit of Fig. 3.32.