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Kirchhoff Laws
Pekik Argo Dahono
Kirchhoff Voltage LawFor all lumped connectedcircuits, for all choices of datum
node, for all timest, for all pairs ofnodeskandj
For all lumped connectedcircuits, for all closed node
sequences, for all times t, the
algebraic sum of all node-to-nodevoltages around the chosen
closed node sequence is equal tozero.
jkkj eev =
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Kirchhoff Voltage Law
00
524524
5145342312
=++
=++++
vvvvvvvv
D
22552
4224
45445
4334
3223
2112
15115
eeev
eev
eeev
eev
eev
eev
eeev
==
=
==
=
=
=
==
Kirchhoff Current Law
For all lumped circuits, for all gaussiansurfacesS, for all times t, the algebraic sumof all currents leaving the gaussian surfaceSat timet is equal to zero.
For all lumped circuits, for all times t, thealgebraic sum of the currents leaving anynode is equal to zero.
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Kirchhoff Current Law
00
0
11986543
6541
21
=+++
=+++
=+
iiiiiiiiiii
ii
11i
9i
8i
10i
3i1i
2S
12i
6S
4S
7i
4i
3S
1S
2i 5S
+
5i
6i
Three Important Remarks
KVL and KCL are the two fundamentalpostulates of lumped-circuit theory.
KVL and KCL hold irrespective of the nature ofthe elements constituting the circuit. Hence, wemay say that Kirchhoff laws reflect theinterconnection properties of the circuit.
KVL and KCL always lead to homogeneous
linear algebraic equations with constant realcoefficients, 0, 1, and -1, if written in the fashionas we have discussed.
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From Circuits to Graph
A graph is defined by a set of nodes and a set ofbranches.
If each branch is given an orientation, indicatedby an arrow on the branch, we call the graph isdirected, or simply digraph.
5
11
4
2
2
3
4
5
3 4
5
6 7
The Circuit Graph : Digraph
310
49
48
47
426
425
24
33
212
11
ev
ev
ev
ev
eev
eev
ev
ev
eev
ev
=
=
=
=
=
=
=
=
=
=
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Two-terminal and three-terminal elements
)()()( titvtp =
+
i
v
i1
2 2
i
1
1 2
1i 2i
32v
13v
1i 2i
3i
+ 2+
1
3
3
+
21v
Other representations of three-terminal element
+2i
1i+
2v1v
1i
3i
1i
3i
2i
3i
2i
3i
211 2
3 3
1 2 1 2
33
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Multiterminal elements
=
=
1
1
)()()(n
kkk titvtp
Two-Ports
1
1'
2
2'
1 2
2v
2211
22
11
'
'
iviv
ii
ii
=
=
=
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Multiports
+
+
+
1
'1
2
'2
3
'3
3
2
1
1
'1
Hinged Graphs
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Grounded Two-Ports
Cut Sets
Given a connected digraph, a set ofbranches is called cut set iff
a) The removal of all the branches of thecut set results in an unconnecteddigraph.
b) The removal of all but any one branchleaves the digraph connected.
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Cut Sets
i
45 i6
S2
S1
5i
S
i
3
7i1
i3
i4
3
2
2
S1
1
Examples of Cut set
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Cut Set Law
For all lumped circuits, for all time t, thealgebraic sum of the currents associatedwith any cut set is equal to zero.
2 3
5 6i i
iii
i i
4 5
1 2 3
6 7
0321 =++ iii
Independent KCL Equations
1
2 3
4
5
63
4
1 2
0
0
0
0
654
532
431
621
=+
=++
=+
=+
iii
iii
iii
iii
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Independent KCL Equations
( )
( )
( )
( )
=
0
0
0
0
111000
010110
001101
100011
4
3
2
1
6
5
4
3
2
1
i
i
i
i
i
i
amatrix Aincidencethecalledismatrix4x6The
Incidence Matrix
+
=
(i)nodetouchnotdoeskbranchif0
(i)nodeenterskbranchif1-
(i)nodeleaveskbranchif1
ika
0iAa =
If in Aa, the incidence matrix of the connected digraph, we delete the rowCorresponding to the datum node, we obtain the reduced incidence matrix A,
which is of dimension (n-1)b. The corresponding KCL equation read
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Independence Property of KCL Equations
For any connected digraph withnnodes, theKCL equations for any n-1 of these nodesform a set of n-1 linearly independentequations.
For reduced incidence matrix :
0Ai =
Independent KVL Equations
If node (4) is used as the reference (datumnode), then
Mev
or
=
=
=
=
+=
=
=
16
35
24
323
312
211
ev
ev
ev
eev
eev
eev
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Independent KVL Equations
The elements of the matrix are
It can be proved that M=AT
Thus, v=ATe
+
=
(i)nodetouchnotdoeskbranchif0
(i)nodeenterskbranchif1-
(i)nodeleaveskbranchif1
kim
Remarks Note that, in the digraph, (a) we choose current
reference directions, (b) we choose a datum node anddefine the reduced incidence matrix A, (c) we write KCLas Ai=0, (d) then we use associated reference directionsto find that KVL reads v=Ate. Thus whenever we invokethis last equation, we automatically use associatedreference directions for the branch voltages. We alsoassume the same datum node is used in writing KCLand KVL.
When we deal with digraphs which are not connected,
we could either use the concept of the hinged graph tomake the digraph connected or treat each separate partindependently. In the latter, each separate part will haveits own incidence matrix and datum node.
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Tellegen Theorem
Background
ips.relationshspecificno
havevoltagesandcurrentsthethatnotedbeshouldIt
thatverifytoeasyisIt
KVL.obeyvoltagesandKCLobeycurrentsthethatNote
hence
Let
hence
Let
=
=
===
===
===
===
6
1
321
654
654
321
0
112
654
413
321
k
kkiv
vvv
vvv
iii
iii
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Tellegen Theorem
( )
( )
( ) ( ) 0(Ai)eiAeieAiv
then
eAvand0AiSince
:Proof
0ivlyequivalentor
thenKVL,satisfyingvoltagesbranchof
setanybevandKCLsatisfying
currentsbranchofsetanybeiLet
TTTTTTT
T
T
====
==
==
=
=
=
b
k
kk
T
b
T
b
iv
vvv
iii
1
21
21
0
,,,
,,,
L
L
Remarks on Tellegen Theorem
The only requirement of Tellegen theoremis that the voltages and currents mustobey KVL and KCL, respectively.
No other assumptions are used. Thus,
vTi=0 vTi=0 vTi=0 vTi=0
Tellegen theorem depicts only theinterconnection properties of the circuit orthe topology of the digraph.
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Tellegen Theorem and Conservation of Energy
Tellegen theorem is much broader thanconservation energy theory because no timerestriction is used.
( )
0)()(0)()(
0)(
)(0)(
)(
0)()(0)()(
)0)
2
1
12
1
1
1
12
1
21
1221
112
==
==
==
==
=
=
==
==
tidt
tdvtidt
tdv
dt
tditv
dt
tditv
titvtitv
ttt
k
k
kk
k
k
b
k
kk
b
k
kk
kkkk
and
and
and
Thus,
eAv(andLet Ai(
b
1k
b
1k
T
The Relation Between Kirchhoff Lawand Tellegen Theorem
If, for all v satisfying KVL, vTi=0 then isatisfies KCL.
If, for all i satisfying KCL, vTi=0 then vsatisfies KVL.
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Example
=
=
=++
=++
N
k
kRkRk
N
kRkRk
RvvRvviv
iviviv
122211
12211
0//
0
=
=
=++
=++
N
k
kRkRk
N
kRkRk
RvvRvviv
iviviv
122211
12211
0//
0
22221
11 5.08.4// vRvvivRvv
N
k
kRkRk ===
22221
11 6// vRvvivRvv
N
k
kRkRk ===
volts4.2
5.08.46
2
22
=
=
v
vv
The End