UNIVERSITY OF MASSACHUSETTS DARTMOUTH
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
ECE 201
CIRCUIT THEORY I
DELTA–TO-WYE (PI–TO–TEE) EQUIVALENT CIRCUITS
BACKGROUND
DELTA (OR PI) CONNECTION
The analysis of the Wheatstone bridge circuit introduces us to a resistor connection that is neither
series, parallel, or series-parallel, as shown below.
The resistors Ra, Rb, and Rc in the circuit shown below on the left are connected between nodes
a, b, and c in such a way as to resemble the Greek letter delta Δ. This configuration can be redrawn into a shape that resembles the Greek letter π without making or changing any
connections.
WYE (OR T) CONNECTION
The resistors R1, R2, and R3 in the circuit shown above on the right appear to be connected in a
configuration that resembles the letter Y. It turns out that this connection can also be re-drawn
into a shape that resembles the letter T without disturbing any connection(s).
THE Δ TO Y TRANSFORMATION
The Δ-connected resistor circuit can be replaced by the “equivalent” Y-connected resistor circuit
through mathematical “transformation”. The two circuits are said to be “equivalent” because
when 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 the
a, b, and c terminals. These resistances are computed on the following page.
R (R + R )
c a
b = R +R
R =
ab R + R + R
1
2
a
c
b
R (R + R )
a b
c
R =
= R +R
bc R + R + R
2
3
a
c
b
R (R + R )
c
a
R = b
= R +R
ca R + R + R
1
3
a
c
b
The resistor values in the equivalent Y-connected circuit can be determined from the resistor
values of the Δ-connected circuit as
R R
b c
R =
1 R +R +R
a
c
b
R R
c a
R =
2 R +R +R
a
c
b
R R
a b
R =
3 R +R +R
a
c
b
The resistor values of the equivalent Δ-connected circuit can be expressed in terms of the resistor
values in the Y-connected circuit as
R R +R R +R R
2 3
3 1
R = 1 2
a
R
1
R R +R R +R R
2 3
3 1
R = 1 2
b
R
2
R R +R R +R R
2 3
3 1
R = 1 2
c
R
3
2
PRELAB CALCULATIONS
Three resistors are connected in a Δ-configuration as shown below.
Determine the resistor values for the equivalent Y-connected circuit. Show some sample
calculations and then draw 2 complete circuit diagrams. Be sure to do all of the work in your lab
notebook, and have copies ready to hand in at the class before the lab session.
220Ω
330Ω
470Ω
PROCEDURE/RESULTS
1. Construct both of the circuits on your breadboard using resistors of the the closest standard
value. Using your digital multimeter, measure the resistance between the a-b, b-c, and c-a
terminals of each configuration and compare the results with the expected values. Summarize
your 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 a
time) and measure the current drawn with your digital multimeter. Create a table of the
measurements in your lab notebook, and have a copy ready to be handed in when you leave the
lab.
3. Are the two circuits equivalent as far as the 5 Volt source is concerned? Why or why not?
3