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UNIVERSITY OF MASSACHUSETTS DARTMOUTH

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

ECE 201

CIRCUIT THEORY I

DELTATO-WYE (PITOTEE) EQUIVALENT CIRCUITS

BACKGROUND

DELTA (OR PI) CONNECTION

The analysis of the Wheatstone bridge circuit introduces us to a resistor connection that is neitherseries, parallel, or series-parallel, as shown below.

The resistors Ra, Rb, and Rc in the circuit shown below on the left are connected between nodesa, b, and c in such a way as to resemble the Greek letter delta . This configuration can be re-drawn into a shape that resembles the Greek letter without making or changing anyconnections.

WYE (OR T) CONNECTION

The resistors R1, R2, and R3 in the circuit shown above on the right appear to be connected in aconfiguration that resembles the letter Y. It turns out that this connection can also be re-drawninto a shape that resembles the letter T without disturbing any connection(s).

THE TO Y TRANSFORMATION

The -connected resistor circuit can be replaced by the equivalent Y-connected resistor circuitthrough mathematical transformation. The two circuits are said to be equivalent becausewhen they are connected to an external source, or sources, they exhibit identical behavior.

In order for the two circuits to be equivalent, they must exhibit identical resistances between thea, b, and c terminals. These resistances are computed on the following page.

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R (R + R )c a bR = = R + R

ab 1 2R + R + Ra cb

R (R + R )a cbR = = R + R

bc 2 3R + R + Ra cb

R (R + R )c abR = = R + Rca 1 3R + R + R

a cb

The resistor values in the equivalent Y-connected circuit can be determined from the resistorvalues of the -connected circuit as

R Rcb

R =1 R + R + R

a cb

R R

c aR =2 R + R + R

a cb

R Ra bR =

3 R + R + Ra cb

The resistor values of the equivalent -connected circuit can be expressed in terms of the resistorvalues in the Y-connected circuit as

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R R + R R + R R1 2 2 3 3 1R =

a R1

R R + R R + R R1 2 2 3 3 1R =

b R2

R R + R R + R R1 2 2 3 3 1R =

c R3

PRELAB CALCULATIONS

Three resistors are connected in a -configuration as shown below.

Determine the resistor values for the equivalent Y-connected circuit. Show some samplecalculations and then draw 2 complete circuit diagrams. Be sure to do all of the work in your labnotebook, and have copies ready to hand in at the class before the lab session.

PROCEDURE/RESULTS

1. Construct both of the circuits on your breadboard using resistors of the the closest standardvalue. Using your digital multimeter, measure the resistance between the a-b, b-c, and c-aterminals of each configuration and compare the results with the expected values. Summarizeyour results in a table.

2. Apply 5 Volts DC to the a-b, b-c, and c-a terminals of each circuit (one pair of terminals at atime) and measure the current drawn with your digital multimeter. Create a table of themeasurements in your lab notebook, and have a copy ready to be handed in when you leave thelab.

3. Are the two circuits equivalent as far as the 5 Volt source is concerned? Why or why not?

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