October 1966 RadioElectronics
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics.
See articles from RadioElectronics,
published 19301988. All copyrights hereby acknowledged.

Anyone who has taken
classes in circuit analysis is familiar with calculating deltawye conversions,
from delta to wye, and from wye to delta. My introduction was not for resistor
circuits, but for transformers used in electric power distribution systems. That
was way back in the mid 1970s whilst in a high school
electrical vocational program, where rather than take a full day of
traditional classes, half of each day was spent engaging in a mix of theory
handson practical experience, doing motor control, security wiring, household
and industrial wiring, service entrance and load center installations, etc. This
"Solving Delta and Wye Networks by Transformation" article appearing in a 1966
issue of RadioElectronics magazine presents a nice introduction to
technique. No fancy mathematics are involved. A useful example is worked out
using a Wheatstone bridge circuit.
Solving Delta and Wye Networks by Transformation
Fig. 1  Using deltawye transformations to sold circuit problems.
What if a circuit is not simply series or parallel?
By S. R. Simmons
Most technicians and engineers feel confident about solving simple series and
parallelresistance problems. But sometimes you encounter circuits that cannot be
classified as either series or parallel. They are combinations of both. Delta and
wye circuits are examples of these more complex networks.
Many delta and wye circuits have capacitively or inductively reactive components;
solutions to problems with those require the use of complex values. Circuits of
a purely resistive nature, however, aren't so difficult. Once you understand how
to handle these, you may want to tackle the tougher ones, keeping in mind that reactive
circuit problems are solved in the same manner while using complex numbers.
One resistivenetwork problem appears in Fig. 1. Suppose you, as an engineer,
need to know what voltage will be developed across resistor R5. At first glance,
you might conclude that resistors R3 and R5 are the ones to be concerned about.
But careful study reveals that, because resistor R2 is connected to the junction
of resistors R3 and R5, the entire circuit is involved. This type of problem can
be solved by using simultaneous equations, but chances for error are rather high.
A simpler method is a form of calculation known as deltawye transformation.
In Fig. 2a is a network known as a delta connection. Let's transform it to a
wye connection, as shown in Fig. 2b. To accomplish this, draw the wye circuit inside
the delta (Fig. 3). This helps you to see the relationship between the two networks.
Label all components and connection points.
Determine the values of resistors R_{a}, R_{b}, and R_{c}.
Skipping the derivations till later (box), the formulas are:
Fig. 2  Performing a deltawye transformation.
These formulas are a form of the familiar productoversum formula for parallel
resistances. The denominators are identical in all three formulas, but the numerators
vary in value. You can figure out what the numerator will be in each formula by
noticing the points where the resistors connect. That is, if you solve for resistor
R_{a}, the numerator will be composed of the two resistors in the delta
that connect to point A. In like manner, the numerator for resistor R_{b}
contains resistors R2 and R3. Finally, the resistors that determine the numerator
in solving for R_{c} are R3 and R1. Bear this relationship in mind when
you perform other transformations.
You develop an easily understood equivalent circuit for the network in Fig. 1
by transforming the upper half (RR2R3) into a wye circuit. The equivalent circuit
then appears as in Fig. 4, which is a simple seriesparallel circuit. From that
point on, solving for the voltage across resistor R5 is easy. By Ohm's law, you
should get 230 volts for an answer.
Fig. 4
A study of transformations would not be complete without an explanation of how
to transform a wye circuit back to a delta. The procedure is similar except for
the formulas.
Begin the transformation by enclosing the wye network with the delta equivalent,
as in Fig. 5. Again minus the derivations, the formulas are:
In these formulas, It is the numerators that are identical, while the denominators
vary in value. The critical thing to remember is which resistor to use as the denominator
for each formula. Rule of thumb is to use the wye resistor directly opposite the
delta resistor that you are solving for.
Thus, the resistor opposite R2 will be R_{c}, and that value is the denominator.
Likewise, resistor R_{a} is opposite R3, and R_{b} is opposite R1.
Look through any electronic text, manual or magazine and you'll see such complicatedlooking
circuits as bridges, filters, attenuators, and voltage dividers. If you look closely,
you'll see that these circuits are usually one form or another of delta or wye.
