
In order to get started in Mathematics, you must accept a set of axioms as "given"  that is, not needing proof. You should expect there to be .
At any instant the sum of all the voltage sources in any closed circuit is equal to the sum of all the voltage drops in that circuit:
∑E = ∑IZ
Similarly, at any instant the algebraic sum of all the voltages around any closed circuit is zero:
∑E  ∑IZ = 0
∑I_{in} = ∑I_{out}
Similarly, at any instant the algebraic sum of all the currents at any circuit node is zero:
∑I = 0
Z_{AB} = Z_{AN} + Z_{BN} + (Z_{AN}Z_{BN} / Z_{CN}) = (Z_{AN}Z_{BN} + Z_{BN}Z_{CN} + Z_{CN}Z_{AN}) / Z_{CN}
Z_{BC} = Z_{BN} + Z_{CN} + (Z_{BN}Z_{CN} / Z_{AN}) = (Z_{AN}Z_{BN} + Z_{BN}Z_{CN} + Z_{CN}Z_{AN}) / Z_{AN}
Z_{CA} = Z_{CN} + Z_{AN} + (Z_{CN}Z_{AN} / Z_{BN}) = (Z_{AN}Z_{BN} + Z_{BN}Z_{CN} + Z_{CN}Z_{AN}) / Z_{BN}
Similarly, using admittances:
Y_{AB} = Y_{AN}Y_{BN} / (Y_{AN} + Y_{BN} + Y_{CN})
Y_{BC} = Y_{BN}Y_{CN} / (Y_{AN} + Y_{BN} + Y_{CN})
Y_{CA} = Y_{CN}Y_{AN} / (Y_{AN} + Y_{BN} + Y_{CN})
In general terms:
Z_{delta} = (sum of Z_{star} pair products) / (opposite Z_{star})
Y_{delta} = (adjacent Y_{star} pair product) / (sum of Y_{star})
Z_{AN} = Z_{CA}Z_{AB} / (Z_{AB} + Z_{BC} + Z_{CA})
Z_{BN} = Z_{AB}Z_{BC} / (Z_{AB} + Z_{BC} + Z_{CA})
Z_{CN} = Z_{BC}Z_{CA} / (Z_{AB} + Z_{BC} + Z_{CA})
Similarly, using admittances:
Y_{AN} = Y_{CA} + Y_{AB} + (Y_{CA}Y_{AB} / Y_{BC}) = (Y_{AB}Y_{BC} + Y_{BC}Y_{CA} + Y_{CA}Y_{AB}) / Y_{BC}
Y_{BN} = Y_{AB} + Y_{BC} + (Y_{AB}Y_{BC} / Y_{CA}) = (Y_{AB}Y_{BC} + Y_{BC}Y_{CA} + Y_{CA}Y_{AB}) / Y_{CA}
Y_{CN} = Y_{BC} + Y_{CA} + (Y_{BC}Y_{CA} / Y_{AB}) = (Y_{AB}Y_{BC} + Y_{BC}Y_{CA} + Y_{CA}Y_{AB}) / Y_{AB}
In general terms:
Z_{star} = (adjacent Z_{delta} pair product) / (sum of Z_{delta})
Y_{star} = (sum of Y_{delta} pair products) / (opposite Y_{delta})
What is the resistance across this network, which looks something like an unbalanced Wheatstone Bridge?
The solution is obtained by using a Kennely StarDelta transformation. The star consisting of R1, R2, and R5 is changed to a delta, with the following resistances:
The resistance of the two resistors in parallel in the lower left part of the diagram is given by
Z_{left} = 1/(1/R_{4} + 1/(R_{1}+R_{2}+R_{1}R_{2}/R_{5}))
The resistance of the two resistors in parallel in the lower right part of the diagram is given by
Z_{right} = 1/(1/R_{3} + 1/(R_{2}+R_{5}+R_{2}R_{5}/R_{1}))
Then the total resistance across the network is calculated as
1/(1/Z_{top} + 1/(Z_{left} + Z_{right}))
1/(1/(R_{1}+R_{5}+R_{1}R_{5}/R_{2}) + 1/(1/(1/R_{4} + 1/(R_{1}+R_{2}+R_{1}R_{2}/R_{5})) + 1/(1/R_{3} + 1/(R_{2}+R_{5}+R_{2}R_{5}/R_{1}))))
This expression can be simplified as
(R_{1}R_{2}R_{3} + R_{4}R_{2}R_{5} + R_{1}R_{2}R_{4} + R_{3}R_{2}R_{5} + R_{1}R_{3}R_{4} + R_{5}R_{3}R_{4} + R_{3}R_{1}R_{5} + R_{4}R_{1}R_{5}) /
(R_{4}R_{5} + R_{1}R_{3} + R_{1}R_{5} + R_{3}R_{4} + R_{5} R_{2} + R_{1}R_{2} + R_{4} R_{2} + R_{3} R_{2})
 Ohm's Law
 Kirchhoff's Laws
 Th�venin's Theorem
 Norton's Theorem
 Th�venin and Norton Equivalence
 Superposition Theorem
 Reciprocity Theorem
 Compensation Theorem
 Millman's Theorem
 Joule's Law
 Maximum Power Transfer Theorem
 StarDelta Transformation
 DeltaStar Transformation
Back to the Basic Principles section
Upper Bound  definition of "upper bound" and "least upper bound" of either sets or functions
Definition of Interval  a subset satisfying certain properties of a totally connected set such as the set of real numbers. "Totally connected" means there exists a "≤" operator that is reflexive (a property in all sets), and antisymmetric, reflexive, and total (three of the thirteen properties listed above).
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