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Converting Between Delta and Wye Configurations
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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:
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<=====>
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(1) RXY = R1 + R2 =
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(RA + RB) × RC
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=
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RA×RC + RB×RC
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(RA + RB) + RC
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RA + RB + RC
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(2) RXZ = R1 + R3 =
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(RA + RC) × RB
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=
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RA×RB + RB×RC
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(RA + RC) + RB
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RA + RB + RC
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(3) RYZ = R2 + R3 =
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(RB + RC) × RA
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=
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RA×RB + RA×RC
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(RB + RC) + RA
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RA + RB + RC
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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.
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(1) - (3)
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=
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R1 + R2 - (R2 + R3)
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=
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R1 - R3
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(1) - (3) + (2)
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=
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R1 - R3 + R1 + R3
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=
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2R1
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Applying these to the lettered resistors, we get:
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R1 - R3
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=
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RA×RC + RB×RC
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-
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RA×RB + RA×RC
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RA + RB + RC
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RA + RB + RC
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=
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RB×RC - RA×RB
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RA + RB + RC
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2R1
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=
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RB×RC - RA×RB
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+
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RA×RB + RB×RC
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RA + RB + RC
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RA + RB + RC
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=
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2(RB×RC)
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RA + RB + RC
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R1
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=
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RB×RC
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RA + RB + RC
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R2
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=
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RA×RC
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RA + RB + RC
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R3
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=
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RA×RB
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RA + RB + RC
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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.
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R1
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=
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RB × RC
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RA + RB + RC
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R1(RA + RB + RC)
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=
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RB × RC
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RA + RB + RC
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=
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RB × RC
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R1
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RA
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=
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RB × RC
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- RB - RC
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R1
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R2
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=
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RA × RC
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RA + RB + RC
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R2(RA + RB + RC)
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=
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RA × RC
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R2(
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RB × RC
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- RB - RC + RB + RC)
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=
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RA × RC
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R1
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R2 × RB × RC
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=
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R1 × RA × RC
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[ R3 × RC = ] R2 × RB
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=
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R1 × RA
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RB
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=
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R1 × RA
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R2
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RC
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=
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R1 × RA
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R3
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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.
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RA
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=
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RB × RC
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- RB - RC
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R1
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R1 × RA
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R1 × RA
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×
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RA
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=
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R2
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R3
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-
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R1 × RA
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-
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R1 × RA
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R1
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R2
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R3
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RA
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=
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R1 × RA × R1 × RA
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-
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R1 × RA
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-
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R1 × RA
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R1 × R2 × R3
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R2
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R3
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1
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=
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R1 × RA
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-
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R1
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-
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R1
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R2 × R3
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R2
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R3
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R1 × RA
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=
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1
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+
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R1
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+
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R1
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R2 × R3
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R2
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R3
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R1 × RA
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=
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R2 × R3
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+
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R1 × R3
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+
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R1 × R2
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RA
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=
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R2 × R3
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+
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R3
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+
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R2
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R1
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RB
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=
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R1 × R3
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+
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R1
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+
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R3
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R2
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RC
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=
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R1 × R2
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+
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R1
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+
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R2
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R3
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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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