In one line
Star-delta transformation replaces three resistors connected in a star (Y) with an equivalent set of three in a delta (triangle), or the reverse, so that networks that are neither series nor parallel can be reduced.
Explained simply
Some circuits look like a tangle where no two resistors are simply in series or parallel, such as a bridge. It is like a road map you cannot simplify by just merging straight roads. Swapping a triangle of roads for a three-pointed junction (or the reverse) suddenly makes simple series and parallel roads appear.
Technically, a star (three resistors meeting at a common centre point) and a delta (three resistors forming a closed triangle) connected between the same three terminals A, B, C are equivalent if the resistance measured between every pair of terminals is the same in both.
How it works
- Identify three terminals forming a delta (triangle) or a star (meeting at one internal node).
- Delta to star: each star arm = product of the two delta resistors touching that terminal ÷ sum of all three delta resistors.
- Star to delta: each delta resistor between two terminals = sum of the two star arms at those terminals + their product ÷ the third arm.
- Redraw the circuit and reduce using series and parallel rules.
Key formula
Delta to star: \( R_A = \dfrac{R_{AB}R_{CA}}{R_{AB}+R_{BC}+R_{CA}} \), and similarly for \( R_B, R_C \).
Star to delta: \( R_{AB} = R_A + R_B + \dfrac{R_AR_B}{R_C} \), and similarly for \( R_{BC}, R_{CA} \).
For equal resistors: \( R_Y = R_\Delta/3 \) and \( R_\Delta = 3R_Y \). \( R_A, R_B, R_C \) = star arms at terminals A, B, C (\( \Omega \)); \( R_{AB} \) etc. = delta resistors between the named terminals (\( \Omega \)).
Quick example
Three 9 \( \Omega \) resistors in delta: each star arm \( = 9\times 9/27 = 3\,\Omega \). Check backwards: \( 3 + 3 + (3\times3)/3 = 9\,\Omega \). Real use: finding the resistance of an unbalanced Wheatstone bridge between two terminals.
Common mistakes
- Using the delta-to-star formula when going star-to-delta (the products go in different places).
- Taking the wrong pair of delta resistors for a star arm: use the two that touch that terminal.
- Forgetting that the internal star node disappears in a delta, so nothing else may be connected to it.
How MU asks it
Asked as "Derive the star-delta transformation equations" (5–8 marks) or "Find the equivalent resistance between A and B" using the transformation (5–8 marks). For a derivation, equate resistance between each pair of terminals in both forms (with the third terminal open), get three equations and solve. In numericals, show each transformed circuit and write the formula before substituting.