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:
SE = SIZ
Similarly, at any instant the algebraic sum of all the voltages around any
closed circuit is zero:
SE - SIZ
= 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:
SIin = SIout
Similarly, at any instant the algebraic sum of all the currents at any
circuit node is zero:
SI = 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
- Thévenin's Theorem
- Norton's Theorem
- Thévenin 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).
