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Node Voltage MethodThis simplest among many circuit analysis methods is
applicable only for connected circuits N made of linear 2-terminal resistors and current sources. The only variables in the linear equations are the n-1 node voltages e1, e2 , …, en-1 for an n-node circuit.
Step 1. Choose an arbitrary datum node and label the remaining nodes consecutively , , …, , andlet e1, e2 , …, en-1 be node-to-datum voltages.
Step 2. Express the current of each resistor Rj via Ohm’s law in terms of 2 node-to-datum voltages:
Step 3. Apply KCL to each node , , …, with each resistor current ij expressed in terms of and .
Step 4. Solve the (n-1) independent linear equations for e1
, e2 , …, en-1.
Step 5. Solve for the resistor currents via Eq.(1).
1 2 n-1
ji
+ -jR Ω
je+je−
jj
j
jei
Re+ −−
= (1)
1 2 n-1
je+je−
5A2A
+
-
6v
6i
2e 3e
1e
+- 3v
4v+
-5v +
-8Ω 6Ω
4Ω2Ω
+
-1v 2v
+
-
35 6i e
=3i
11
2
2ei e−
=1
23
4ei e−
=
24 8
i e=
1
2 3
KCL at : 1 31 2 ( )( ) 52 4
e ee e −−+ =1 (2)
KCL at : 1 2 2( ) 22 8
e e e− −+ = −2 (3)
KCL at : 1 3 3( ) 24 6
e e e− −+ =3 (4)
matrix form : 1
2
3
3 1 1
4 2 41 5
02 81 5
04 12
52
2
eee
− −
−
−
= −
(5)NodevoltageEquations
1.EXAMPLE
Recast Eqs. (2), (3), and (4) in
3 1 1
4 2 41 5
02 81 5
04 1 2
2 0d e t3 8 4
− −
−
−
∆ =
1(a) T o solve for , replace colum n 1 of m atrix from (5) and calculatee
1
1 15
2 45
2 08
52 0
1 2
1 1 5d e t9 6
− −
−
∆ =
C ram er's R ule ⇒
11
115 20 2396 384
Ve ∆ = = = ∆ 2(b) T o solve for , replace colum n 2 of m atrix from (5) and calculatee
22
76 20 15.296 384
e V∆ = = = ∆
3(c) T o solve for , replace colum n 3 of m atrix from (5) and calculatee
33
93 20 18.696 384
e V∆ = = = ∆
2
3 15
4 41
2 021 5
24 1 2
7 6d e t9 6
−
− −
−
∆ =
3
3 15
4 21 5
22 81 2
04 1 2
9 3d e t9 6
−
− −
−
∆ =
( 6 )
( 7 )
( 8 )
( 9 )
(1 0 )
(1 1 )
Solving eq. (5) by C ram er's R ule (or any other m ethod) :
C ram er's R ule ⇒
C ram er's R ule ⇒
1 2 3O nce the node-to-datum voltages , , are found,e e eall resistor currents and voltages are found trivially via K V L, K C L, and O hm 's law :
1 21 17.8 7.8 , 3.92
v V i Ae e= − = = =
1 32 24.4 4.4 , 1.14
v V i Ae e= − = = =
33 32 3.4 , 2v Ve e i A= − = = −
24 415.2 15.2 , 1.9
8v V ie A= = = =
35 518.6 18.6 , 3.1
6v V ie A= = = =
16 6 23 , 5v V ie A= − = − =
V erification of SolutionT ellegen's T heApp oly :rem
6
1 1 2 2 3 3 4 41
( ) ( ) ( ) ( )j jj
v i v i v i v i v i=
= + + +∑
(7.8)(3.9) (4.4)(1.1) (3.4)( 2)= + + −
5 5 6 6( ) ( )v i v i+ +
(15.2)(1.9) (18.6)(3.1) ( 23)(5)+ + + −
?0=
Modified Node Voltage MethodCircuit N + v1e1 e2
6V2A3Ω
4Ω
i4
i2
i1
i3
_v3v4 +-
+
_+
_v2
+1
3
2
_1
12
4ei e−
=
12 3
i e=
KCL at : 1 1 2( ) 23 4e e e−
+ =1 (1)
KCL at : 1 23
( ) 04
ie e−− + =2 (2)
For each voltage source , add an equation .j jjs j svev e+ −− =
2 6e = (3)
S u b stitu tin g (3 ) in to (1 ), w e o b ta in :
1 11
( 6) 2 63 4e e Ve−
+ = ⇒ = (4)
S u b stitu tin g (4 ) in to (2 ), w e o b ta in :
3 0i = (5)
1 2When the circuit contains " " voltages , , , , uses s sv v v
αα
1 2their associated currents , , , when applying KCL.s s si i i
α
Step 1.
Step 2.
Step 3.1 21 2 1Solve the ( -1) + equations for , , , , , .n s s sn ie e e i i
αα −
Explicit matrix form of Node Voltage Equations
Assumption: Circuit N contains only linear resistors and independent current sources which do not form cut sets.
Step 1. Delete all “β” current sources from N and draw the reduced digraph G of the remaining pure resistor circuit. Assume G has “n” nodes and “b” branches.
Step 2. Pick a datum node and label the node-to-datum voltages e1, e2 , … , en-1 , and derive the reduced-incidence matrix A. Define the “branch admittance matrix” Yb and independent current source vector is as follow:
Step 3. Form the Node voltage equation
bY
1
2
1 1 1
2 2 2
0 0 00 0 0
(1) , (2)
0 0 0
s
ss
b b b s
ii Y vii Y v
i Y v iβ
=
i
i v1where , resistance of branch .j j
j
Y R jR
=
current= algebraic of all sources node su , mmsi entering
1, 2, , -1.m n= …m
n s=Y e i (3)
where is called the .Tn b node - admittance matY AY A rix
Deriving the Node Voltage Equation
The “β ” current sources can be deleted since they can be trivially accounted for by representing their net contribution at each node m by the algebraic sum of all current sources entering node m , m = 1, 2, …, n-1. The KCL equations therefore takes the “augmented” form
Substituting (1) for i in (4), we obtain
Substituting KVL
for v in (5), we obtain
s=A i i
b s=A Y v i
T=v A e
( )Tb s=A Y A e i
nY
(4)
(5)
(6)
(7)
( )n s=Y e i
Writing Node-Admittance Matrix Yn By Inspection
nY
(8)
:Node Votage Equation1
2
1
11 12 1, 1
21 22 2, 1
1,1 1,2 1, 1
1
2
1 n
sn
sn
n n n snn
iY Y YiY Y Y
Y eY
e
Y i
e
−
−
−
− − − − −
=
e si
Diagonal Elements of nY1 = sum of adm ittances of all resistorsm m j
j
Y YR
connected to node , 1, 2, , -1m m n= …
Off-Diagonal Elements of nY1 = sum of adm ittances of all resistorsjk j
j
Y YR
- (connected across node and node j k )
Symmetry Property:sym m etric m atri is a , i .e ., xn jk k jY Y=Y
P roof :
( )TT T T T Tn b b b n= = = =Y A Y A A Y A A Y A Y
S in ce in (1 ) is a d iag o n a l m atrix , Tb b b=Y Y Y
5A2A
+
-
6v
6i
2e 3e
1e
+- 3v
4v+
-5v +
-8Ω 6Ω
4Ω2Ω
+
-1v 2v
+
-
35 6
ei =3i
1 21 2
e ei −=
1 32 4
e ei −=
24 8
ei =
1
2 3
KCL at : 1 31 2 ( )( ) 5 42 4
e ee e −−+ = −1 (2)
KCL at : 1 2 2( ) 4 2 32 8
e e e− −+ = − −2 (3)
KCL at : 1 3 3( ) 2 14 6
e e e− −+ = +3 (4)
Matrix Form :
1
2
3
3 1 1
4 2 41 5
02 81 5
04 12
11
3
eee
− −
−
−
= −
(5)NodeEquations
EXAMPLE
1A
4A
3A
Mesh Current MethodThe next simplest among many circuit
analysis methods is applicable only for connected planar circuit N (with a planar digraph) made of 2-terminal linear resistors and voltage sources. The only variables in the equations are “ l ” conceptual mesh currents
circulating in the “ l ” meshes in a clockwise direction (by convention) :
1 2
ˆ ˆ ˆ, , ,lm m ml l l
1R 10i
11i 13i
12i
3i
7i
2i
4i
5i
6i
9i8i
1i
2R
3R
4R
5R
6R
7R
8R
9R
10R
11R
12R
13R
1i2i
5i4i 6i
3i
ji
jR
ci
ai bi
di
ˆ ˆa bj i ii = −
Mesh Current MethodThe next simplest among many circuit
analysis methods is applicable only for connected planar circuits made of 2-terminal linear resistors and voltage sources. The only variables in the associated “mesh-current equations” are “ l ” mesh current
which we define to be circulating in the “ l ” meshes in a clockwise direction (by convention). Unlike node-to-datum voltages in the node voltage method which are physical in the sense they can be measured by a volt meter, the “mesh” currents are abstract variables introduced mathematically for writing a set of equations whose solution can be used to find each resistor current ij trivially via
Where is the circulating current flowing through Rj in the same (resp., opposite) direction as the reference current ij.
2
ˆ ˆ, ,lm ml l
1
ˆ ,ml
ˆ ˆj a bi i i= −
ˆ ˆ (resp., )a bi i
1
2
3
ˆ6 0 2 2ˆ0 14 8 2ˆ2 8 10 5
iii
− − = − − −
(1)
Then calculate :
1 3 13ˆ ˆ
10i i i= − =
1 2 313 8 19ˆ ˆ ˆ, ,20 20 20
i i i= = =
2 113ˆ20
i i= =
3 1 25ˆ ˆ20
i i i= − =
4 3 211ˆ ˆ20
i i i= − =
5 28ˆ20
i i= =
6 319ˆ20
i i= =
1 162
10v i= =
2 21345
v i= =
3 2v = −
4 42285
v i= =
5 51265
v i= =
6 5v = −
MeshCurrent
Equations
Verification by Tellegen's Theorem :6
j=1j jv i =∑ ( ) ( )6 3 13 13 5 22 11 12 8 192 5
10 10 5 20 20 5 20 5 20 20 + + − + + + −
?0=
,
,
,
,
,
,
(2)
(3)
2V
+
-
6v
6i
+- 3v
4v+
-5v +
-8Ω 6Ω
4Ω2Ω
+
-1v 2v
+
-
5i3i
2i
4i
1i
3i1i
2i
+
1 3 1ˆ ˆi i i= −
2 1i i=
3 1 2ˆ ˆi i i= −
4 3 2ˆ ˆi i i= −
5 2i i=
( )1 3 1ˆ ˆ2v i i= −
2 14v i=
3 2v = −
( )4 3 2ˆ ˆ8v i i= −
5 26v i=
6 5v = −
,
,,
,,,
6 3i i=
Loop equation around mesh 1:1 2 3 3 1 1
ˆ ˆ ˆ0 2( ) 4 2 0v v v i i i− + + = ⇒ − − + − =
1 3ˆ ˆ6 2 2i i⇒ − =
Loop equation around mesh 2:3 5 4 2 3 2
ˆ ˆ ˆ0 ( 2) 6 8( ) 0v v v i i i− + − = ⇒ − − + − − =
2 3ˆ ˆ14 8 2i i⇒ − = −
5V +-
Loop equation around mesh 3:6 1 4 3 1 3 2
ˆ ˆ ˆ ˆ0 5 2( ) 8( ) 0v v v i i i i+ + = ⇒ − + − + − =
1 2 2ˆ ˆ ˆ2 8 10 5i i i⇒ − − + =
(1)
( 2 )
(3 )
5V2V
+
-
6v
6i
+- 3v
4v+
-5v +
-8Ω 6Ω
4Ω2Ω
+
-1v 2v
+
-
5i3i
2i
4i
1i
3i1i
2i
1 3ˆ ˆM esh 1 : 6 2 2i i− =
2 3ˆ ˆM esh 2 : 14 8 2i i− = −
1 2 3ˆ ˆ ˆM esh 3 : 2 8 10 5i i i− − + =
1 1 31 2ˆ ˆ ˆS olving from (1)3 3
i i i⇒ = −
2 2 34 1ˆ ˆ ˆS olving from (2)7 7
i i i⇒ = −
S ubstitu ting (4 ) and (5) in to (3) ⇒
(5 ) and (6) ⇒
319ˆ12
i A=
28ˆ
20i A=
113ˆ20
i A= (4 ) and (6) ⇒
(1 )
(2 )(3 )
(4 )
(5 )
(6 )
(7 )
(8 )
+- +
1i
2i
4i
6i
3i5i
We can redraw this circuit so that there are no intersecting branches.
1
2
3
4
1i
2i
4i
3i5i
1
2
3
46i
Hence the above circuit is planar and it is possible to formulate mesh current equations.
1i
2i
4i
6i
3i5i
We can redraw this circuit so that there are no intersecting branches.
1
2
3
4
1i
2i
4i
3i5i
1
2
3
46i
Hence the above circuit is planar and it is possible to formulate mesh current equations.
2Ω
4Ω
8Ω
6Ω5V2V
2Ω
5V
2V 6Ω
4Ω
8Ω
+
+
+
+-
+
All branch voltages and currents can be trivially calculated from e1 and i3.
121 1 1 0 , 0
4vv Ve i Ae= − = = =
12
2 2 6 , 23vv e V i A= = = =
23 3 6 , 0v V ie A= = =
14 4 6 , 2 Aev V i= − = − =
V erification of Solution T ellegen's T heorby :em
4
1 1 2 2 3 3 4 41
( ) ( ) ( ) ( )j jj
v i v i v i v i v i=
= + + +∑(0)(0) (6)(2) (6)(0) ( 6)(2)= + + + −
?0=
Note:The unknown variables in the
modified node voltage method consist of the usual n-1 node-to-datum voltages, plus the unknown currents associated with the voltage sources.
Hence, if there are “α” voltage sources, the modified node voltage method would consist of (n-1)+αindependent linear equations involving (n-1)+α unknown variables
1 21 2 1, , , , , , .n s s sie e ie iα−
(n-1) node-to-datumvariables
α currentvariables
Conservation of Electrical Energy
The algebraic sum of electrical energy flowing into all devices in a connected circuit is zero for all times .t > −∞
Proof .
Tellegen's Theorem ⇒
1( ) ( ) 0
b t
j jj
v t i t dt−∞
=
=∑∫
for all .t
1 1
0 0 0 0 0 1 1 0 10 0 0 0 0 1 0 1 01 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 01 0 0 1 0 0 0 0
00000000
00 0 0 0 1 0 0 0 00 1 0 0 0 4 0 0 00 0 1 0 0 0 3 0 00 0 0 1 0 0 0 0 00 0 0 0 0 0 0
60
t
2 1
de
−−
−=
−
∆
−
A
1A
4 4
0 0 0 0 0 1 1 0 10 0 0 0 0 1 0 1 01 1 1 0 0 0 0 0 01 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 01 0 0 0 1 0 0 0 00 0 1 0 0 4 0 0 00 0 0 0 0 0 3 0 00 0
0000000060 1 0 0 0 0 0
0 0 0 0 0 0 0 0 12
det
−−
−−
−
−−
= ∆A
4A
9 9
0 0 0 0 0 0 1 1 10 0 0 0 0 0 1 0 01 1 1 0 0 0 0 0 01 0 0 1 0 0 0 0 0
0 1 0 0 1 0 0 0 01 0 0 0 0 1 0 0 00 0 1 0 0 0 4 0 00 0 0 1 0 0 0 3 00 0 0 0 1
0000
0 0 0
00006 0
0 0 0
det
0 0 120 0 0
−−
∆
−−
−−
=
−
A
9A
11 11 21 21 31 31 41 41 51 51
61 61 71 71 81 81 91 91 10,1 10,1
1 a A a A a A a A a Aa A a A a A a A a A
+ + + + = + + + + +∆
( )91 91 10,1 10,1 11 21 811 , because 0a A a A a a a= + = = =∆
( ) ( )10,19191 10,1
AA a a= +∆ ∆
11k 12k
11 126 2k k= • + •
11 1 12 1s sk v k i= • + •
11e ∆=∆
Instantaneous Power of a 2-terminal device
D
+
-( )v t
( )i t1
2( ) ( ) ( )p t v t i t
Under Associated Reference Convention,
1( ) 0 atp t t T> =
1means ( ) Watts of power ent ersp T1.( )flow at s into t T=D
2( ) 0 atp t t T< =
2means ( ) Watts of power lea vesp T2.flows out of( ) at t T=D
1 2Energy entering from time to :T TD2
1 21
( ) ( )T
T T TW v t i t dt− = ∫
Instantaneous Power of an n-terminal device
1
1( ) ( ) ( )
n
j jj
p t v t i t−
=
=∑
1 2Energy entering from time to :T TD
2
1 21
1
1( ) ( )
n T
T T j jTj
W v t i t dt−
−=
=∑∫
D
k
n
1
2
n-1
1i2i
ki
1ni −
1v2v kv
1nv −
Tellegen’s Theorem has many deep
applications. For this course, it can be used to
check whether your answers in homework
problems, midterm and final exams are
correct.
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