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 Math 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

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)

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

- 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

### 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).

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