Solving Delta and Wye Networks by Transformation
October 1966 Radio-Electronics

October 1966 Radio-Electronics [Table of Contents] Wax nostalgic about and learn from the history of early electronics. See articles from Radio-Electronics, published 1930-1988. All copyrights hereby acknowledged.

Anyone who has taken classes in circuit analysis is familiar with calculating delta-wye 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 hands-on 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 Radio-Electronics 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

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 parallel-resistance 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 resistive-network 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 delta-wye transformation.

In Fig. 2-a is a network known as a delta connection. Let's transform it to a wye connection, as shown in Fig. 2-b. 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:

These formulas are a form of the familiar product-over-sum 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 (R-R2-R3) into a wye circuit. The equivalent circuit then appears as in Fig. 4, which is a simple series-parallel 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 complicated-looking 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.