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Three-Terminal Resistor Configurations

### The Wye ("Y") Configuration

Consider the schematic diagram to the right. Because the resistors are shown schematically in a way that resembles the letter "Y," this arrangement is known as a "Y" (or "Wye") configuration. At first glance, it may seem a bit strange, and perhaps not very useful. Is it a voltage divider with an extra resistor in series with the output? A kind of mixer with two inputs and one output? Or maybe a splitter with one input and two outputs? Or something else entirely?

In fact, this circuit is not generally used in any of the suggested ways. Nevertheless, it is found in a wide variety of devices and larger circuits, often with capacitors or inductors substituted for one or more of the resistors shown here.

Even with just resistors, this configuration is useful in a number of different ways. It is often used to provide a specific amount of signal attenuation in a transmission line, while still matching the characteristic impedance of the transmission line in both directions. It can also be used to match two transmission lines of different characteristic impedances, so that both transmission lines operate at maximum efficiency.

A key point to note about this configuration is that the resistance between any two of the three external connections will be the series combination of two of the three resistors. Thus, the resistance between points X and Y, which we can call RXY will be R1 + R2. As a consequence of this, the effective resistance of this circuit will always be substantially greater than the values of the individual resistors. This means relatively small resistor values can be used to obtain greater resistance effects.

It is often easier to understand this circuit if it is rearranged as shown in the second figure to the right. Now we see a clear input and output, although the actual electrical connections among the resistors are unchanged. This schematic arrangement also takes up much less room on paper or a Web page, so it is often preferred for display purposes. When it is displayed like this, it is generally identified as a "T" configuration rather than "Y." However, this does not change its behavior in any way.

### The Delta Configuration

A second configuration involving three resistors is shown in the schematic diagram to the right. Since the modern English/Latin alphabet has no symbol that resembles this triangle, we use the Greek letter "Delta" (Δ) to describe this configuration.

This configuration is just as important as the "Y" configuration above. An essential difference, however, is that in the Delta configuration, the resistance between any two points is a series-parallel combination of all three resistors. Therefore, the effective resistance of the circuit will be less than the values of the individual resistors involved. This can be very useful in situations where we want to be able to use larger resistance values than the circuit would normally require. Mathematically,

RXY = (RA + RB)RC/(RA + RB + RC)

As you might expect, it is usually convenient to redraw this circuit as shown in the second figure to the right. The circuit is electronically the same, but now has clear input and output connections. Because this layout resembles the Greek letter "pi" (π), it is usually identified as a "pi" configuration.

### Converting Between Wye and Delta Configurations

Although these two configurations look very different, it is quite possible for them to exhibit exactly the same resistance between each pair of external connections. Thus, RXY can be the same for both, as can RXZ and RYZ. In such a case, if you don't know the configuration, you have no way to tell which configuration is being used.

This is important, because sometimes it is necessary to convert from one configuration to the other. This can be because the calculated resistor values for one configuration are non-standard, or because the requirements of the circuit demand that resistance values be neither too large nor too small.

As a simple example, consider a case where such a circuit must use standard 5% resistors, and must also exhibit a resistance of 10K between any two connection points. Since this allows all three resistors to be the same, we can easily determine that the Wye confiuration would require three resistors of 5K each.

Unfortunately, this is a non-standard value. We could use 5.1K resistors instead, and hope that the result is close enough to be acceptble. But if we convert to a Delta configuration, we find that we can get the same result using three resistors of 15K each. This is a standard value, easily obtained. Therefore, we can get exactly the results we want without difficulty and without hoping we're "close enough."

Of course, the resistors in either configuration do not have to be of the same value (and usually aren't). Therefore, we need the appropriate mathematical equations to perform the conversions in either direction. For those interested in the derivation of these equations, I have done this separately, on this page. The end results, using the resistor designations in the figures above, are:

R1 = (RB×RC)/(RA + RB + RC)
R2 = (RA×RC)/(RA + RB + RC)
R3 = (RA×RB)/(RA + RB + RC)

RA = R2 + R3 + (R2×R3/R1)
RB = R1 + R3 + (R1×R3/R2)
RC = R1 + R2 + (R1×R2/R3)

You will encounter both configurations throughout the field of electronics, used in many different ways. You should be ready to recognize them when you see them.

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