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Converting Between Delta and Wye Configurations

Because both the Delta and Wye configurations are used often throughout electronics, they appear in the design of new circuits as well as in older ones. Therefore, it is important to be able to convert back and forth between the two. Usually it is enough to know the formulas or equations required to perform these conversions. However, for complete understanding, it is a good idea to know and understand how these formulas are derived and why they work.

To do this, we will first determine the initial equations describing both configurations. Then we will be able to solve them simultaneously to determine the conversion formulas.

### Deriving the Initial Equations

We derive the initial equations by noting that for the two configurations to appear the same externally, it is necessary for the externally measured resistances, RXY, RXZ, and RYZ, to be the same for either configuration. Therefore, we can determine these and set them equal to each other. This gives us our three initial simultaneous equations:

 (1) RXY = R1 + R2 = (RA + RB) × RC = RA×RC + RB×RC (RA + RB) + RC RA + RB + RC (2) RXZ = R1 + R3 = (RA + RC) × RB = RA×RB + RB×RC (RA + RC) + RB RA + RB + RC (3) RYZ = R2 + R3 = (RB + RC) × RA = RA×RB + RA×RC (RB + RC) + RA RA + RB + RC

### Solving For R1, R2, and R3

From the basic equations above, it looks much easier to solve for the numbered resistors, so we'll do that first. That in turn may make it easier to solve these equations in the other direction. We'll isolate R1 by subtracting Equation (3) from Equation (1) to get R1 - R3, and then add Equation (2) to that result. The expressions for R2 and R3 can be derived the same way, and will be quite similar.

 (1) - (3) = R1 + R2 - (R2 + R3) = R1 - R3 (1) - (3) + (2) = R1 - R3 + R1 + R3 = 2R1

Applying these to the lettered resistors, we get:

 R1 - R3 = RA×RC + RB×RC - RA×RB + RA×RC RA + RB + RC RA + RB + RC = RB×RC - RA×RB RA + RB + RC 2R1 = RB×RC - RA×RB + RA×RB + RB×RC RA + RB + RC RA + RB + RC = 2(RB×RC) RA + RB + RC R1 = RB×RC RA + RB + RC R2 = RA×RC RA + RB + RC R3 = RA×RB RA + RB + RC

### Solving for RA, RB, and RC

To solve these expressions for the lettered resistors, we first note that the equation for R1 above contains only a single instance of RA. Therefore we will rearrange that equation and solve it for RA. Then we will substitute that value for RA in the denominator of the equation for R2, and simplify the result as much as possible. This will give us simplified relationships that we can more easily apply to these expressions.

 R1 = RB × RC RA + RB + RC R1(RA + RB + RC) = RB × RC RA + RB + RC = RB × RC R1 RA = RB × RC - RB - RC R1 R2 = RA × RC RA + RB + RC R2(RA + RB + RC) = RA × RC R2( RB × RC - RB - RC + RB + RC) = RA × RC R1 R2 × RB × RC = R1 × RA × RC [ R3 × RC = ] R2 × RB = R1 × RA RB = R1 × RA R2 RC = R1 × RA R3

Now we can replace RB and RC with very simple expressions involving RA, so that we will be able to solve for RA in terms of only numbered resistors. RB and RC can then be found in the same way, and will have similar expressions.

 RA = RB × RC - RB - RC R1 R1 × RA R1 × RA × RA = R2 R3 - R1 × RA - R1 × RA R1 R2 R3 RA = R1 × RA × R1 × RA - R1 × RA - R1 × RA R1 × R2 × R3 R2 R3 1 = R1 × RA - R1 - R1 R2 × R3 R2 R3 R1 × RA = 1 + R1 + R1 R2 × R3 R2 R3 R1 × RA = R2 × R3 + R1 × R3 + R1 × R2 RA = R2 × R3 + R3 + R2 R1 RB = R1 × R3 + R1 + R3 R2 RC = R1 × R2 + R1 + R2 R3

You don't really need to know these derivations, although understanding them will help you with your understanding of electronics in general. However, you should know the conversion formulas themselves, and be able to apply them when designing or analyzing electronic circuits.

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