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 .

Kirchhoff's Voltage Law

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

Kirchhoff's Current Law

∑I_in = ∑I_out

Similarly, at any instant the algebraic sum of all the currents at any circuit node is zero:

∑I = 0

Kennelly's Star-Delta Transformation

Z_AB = Z_AN + Z_BN + (Z_ANZ_BN / Z_CN) = (Z_ANZ_BN + Z_BNZ_CN + Z_CNZ_AN) / Z_CN
Z_BC = Z_BN + Z_CN + (Z_BNZ_CN / Z_AN) = (Z_ANZ_BN + Z_BNZ_CN + Z_CNZ_AN) / Z_AN
Z_CA = Z_CN + Z_AN + (Z_CNZ_AN / Z_BN) = (Z_ANZ_BN + Z_BNZ_CN + Z_CNZ_AN) / Z_BN

Similarly, using admittances:

Y_AB = Y_ANY_BN / (Y_AN + Y_BN + Y_CN)
Y_BC = Y_BNY_CN / (Y_AN + Y_BN + Y_CN)
Y_CA = Y_CNY_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)

Kennelly's Delta-Star Transformation

Z_AN = Z_CAZ_AB / (Z_AB + Z_BC + Z_CA)
Z_BN = Z_ABZ_BC / (Z_AB + Z_BC + Z_CA)
Z_CN = Z_BCZ_CA / (Z_AB + Z_BC + Z_CA)

Similarly, using admittances:

Y_AN = Y_CA + Y_AB + (Y_CAY_AB / Y_BC) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_AB) / Y_BC
Y_BN = Y_AB + Y_BC + (Y_ABY_BC / Y_CA) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_AB) / Y_CA
Y_CN = Y_BC + Y_CA + (Y_BCY_CA / Y_AB) = (Y_ABY_BC + Y_BCY_CA + Y_CAY_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)

Resistance across a network of resistors

What is the resistance across this network, which looks something like an unbalanced Wheatstone Bridge?

The solution is obtained by using a Kennely Star-Delta 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/R₄ + 1/(R₁+R₂+R₁R₂/R₅))

The resistance of the two resistors in parallel in the lower right part of the diagram is given by

Z_right = 1/(1/R₃ + 1/(R₂+R₅+R₂R₅/R₁))

Then the total resistance across the network is calculated as

1/(1/Z_top + 1/(Z_left + Z_right))

1/(1/(R₁+R₅+R₁R₅/R₂) + 1/(1/(1/R₄ + 1/(R₁+R₂+R₁R₂/R₅)) + 1/(1/R₃ + 1/(R₂+R₅+R₂R₅/R₁))))

This expression can be simplified as

(R₁R₂R₃ + R₄R₂R₅ + R₁R₂R₄ + R₃R₂R₅ + R₁R₃R₄ + R₅R₃R₄ + R₃R₁R₅ + R₄R₁R₅) /
      (R₄R₅  +  R₁R₃  +  R₁R₅  +  R₃R₄  +  R₅ R₂  +  R₁R₂  +  R₄ R₂  +  R₃ R₂)

Internet references

An assortment of electrical theorems, from BOWest Pty Ltd, an Electrical and Project Engineering Consultancy 

- 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
- Star-Delta Transformation
- Delta-Star Transformation

Kennelly's Delta-Star Transformation, from Circuit-Magic 

Related pages in this website

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).

The webmaster and author of this Math Help site is Graeme McRae.