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 Skip Navigation LinksMath Help > Physics > Electricity

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

At any instant the sum of all the currents flowing into any circuit node is equal to the sum of all the currents flowing out of that node:

Iin = Iout

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

I = 0


Kennelly's Star-Delta Transformation

A star network of three impedances ZAN, ZBN and ZCN connected together at common node N can be transformed into a delta network of three impedances ZAB, ZBC and ZCA by the following equations:

ZAB = ZAN + ZBN + (ZANZBN / ZCN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZCN
ZBC = ZBN + ZCN + (ZBNZCN / ZAN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZAN
ZCA = ZCN + ZAN + (ZCNZAN / ZBN) = (ZANZBN + ZBNZCN + ZCNZAN) / ZBN

Similarly, using admittances:

YAB = YANYBN / (YAN + YBN + YCN)
YBC = YBNYCN / (YAN + YBN + YCN)
YCA = YCNYAN / (YAN + YBN + YCN)

In general terms:

Zdelta = (sum of Zstar pair products) / (opposite Zstar)
Ydelta = (adjacent Ystar pair product) / (sum of Ystar)

 


Kennelly's Delta-Star Transformation

A delta network of three impedances ZAB, ZBC and ZCA can be transformed into a star network of three impedances ZAN, ZBN and ZCN connected together at common node N by the following equations:

ZAN = ZCAZAB / (ZAB + ZBC + ZCA)
ZBN = ZABZBC / (ZAB + ZBC + ZCA)
ZCN = ZBCZCA / (ZAB + ZBC + ZCA)

Similarly, using admittances:

YAN = YCA + YAB + (YCAYAB / YBC) = (YABYBC + YBCYCA + YCAYAB) / YBC
YBN = YAB + YBC + (YABYBC / YCA) = (YABYBC + YBCYCA + YCAYAB) / YCA
YCN = YBC + YCA + (YBCYCA / YAB) = (YABYBC + YBCYCA + YCAYAB) / YAB

In general terms:

Zstar = (adjacent Zdelta pair product) / (sum of Zdelta)
Ystar = (sum of Ydelta pair products) / (opposite Ydelta)

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

Zleft = 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

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

Then the total resistance across the network is calculated as

1/(1/Ztop + 1/(Zleft + Zright))

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.