Chapter 3. Loop and Cut-set Analysis
By: FARHAD FARADJI, Ph.D.
Assistant Professor,
Electrical Engineering,
K.N. Toosi University of Technology
http://wp.kntu.ac.ir/faradji/ElectricCircuits2.htm
References:
Basic Circuit Theory, by Ch. A. Desoer and E. S. Kuh, 1969
Chapter Contents
0. Introduction
1. Fundamental theorem of graph theory
2. Loop analysis
3. Cut-set analysis
4. Comments on loop and cut-set analysis
anaalysis
between
5. Relation betwe
een
n B and
dQ
6. Summary
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
2
0. Introduction
o In previous chapter, we have learned to perform systematically:
¾ node analysis of any LTI network,
¾ mesh analysis for any planar LTI
LTI network.
network.
o Here, we briefly, discuss 2 generalizations
gene
eralizatio
ons (variations) of these methods:
¾ cut-set analysis,
¾ loop analysis.
o There are 2 reasons
reaasons for
for sstudying
tudying lloop
oop aand
nd ccut-set
ut-set analysis:
analysis:
1.
These methodss aare
re much
much more
more flexible
flexible than
than mesh
mesh and node analysis,
2.
They use concepts tthat
hat are
are useful
useful for
for writing
writing state equations.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
3
Chapter Contents
0. Introduction
1. Fundamental theorem of graph theory
2. Loop analysis
3. Cut-set analysis
4. Comments on loop and cut-set analysis
anaalysis
between
5. Relation betwe
een
n B and
dQ
6. Summary
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
4
1. Fundamental theorem of graph theory
‰ Let G be a connected graph and T a subgraph of G.
‰ T is a tree of connected graph G if:
9 T is a connected subgraph,
9 it contains all nodes of G, an
and
nd
9 it contains no loops.
‰ Given a connected graph
h G and
and a ttree
ree TT::
9 branches off T are
are called
called tree
tree branches,
branches,
9 branches of G nott in
in T are
are called
called links
links ((cotree
cottree branches,
branches, or chords).
‰ If a graph has nt nodes and
and has
has a ssingle
ingle branch
branch connecting
connecting every pair of
nodes, then it has ࢔࢚࢔࢚ି૛ trees:
¾ when nt = 5, there are 125 trees,
¾ when nt = 10, there are 108 trees.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
5
1. Fundamental theorem of graph theory
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
6
1. Fundamental theorem of graph theory
THEOREM
™ Given a connected graph G of nt nodes and b branches, and a tree T of G:
1.
There is a unique path alongg ttree
ree between
between any pair of nodes.
2.
There are nt-1 tree branches
branch
hes and b
b-n
-n
nt+1 links.
3.
Every link of G and unique tree path
patth between its nodes constitute a
unique loop (fundamental lo
loop
oop associated
asssociated with link).
4.
Every treee branch
branch o
off T ttogether
ogether with
with some
so
ome links
links defines
deffines a unique cut
set of G ((fundamental
fundamental cut
cut set
set associated
associated
d with
with tree
tree branch).
branch).
™ For proof, please refer to
to pages
pagees 478
478 and
and 479
479 of
of b
book.
ook.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
7
1. Fundamental theorem of graph theory
COROLLARY
™ Suppose:
9 G has nt nodes, b branches, and
an
nd s separate
separate parts,
9 T1, T2, ..., Ts are trees of each
h separate
separaate part:
¾
Set {TT1, T2, ..., Ts} is called a forest
fore
est of G.
¾
Forest has nt-s branches,
braanches, G has
ha s b
b-n
-n
nt++ss links.
lin
nks.
¾
Remaining
Remainin
ng statements
statemen
nts o
off ttheorem
heorem aare
re ttrue.
rue.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
8
Chapter Contents
0. Introduction
1. Fundamental theorem of graph theory
2. Loop analysis
3. Cut-set analysis
4. Comments on loop and cut-set analysis
anaalysis
between
5. Relation betwe
een
n B and
dQ
6. Summary
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
9
2. Loop analysis
2.1. Two basic facts of loop analysis
¾ Consider a connected graph with b branches and nt nodes.
¾ Pick an arbitrary tree T.
¾ There are n = nt-1 tree branchess and
and I = b-n
b-n links.
¾ Number branches as follows:
9 links first from 1 to ll,,
9 tree branches
branch
hes next
nextt from
from l+1
l+1 tto
o b.
b.
¾ Every link defines a ffundamental
un
ndamen
ntall loo
loop:
op:
9 loop formed by link and
d un
unique
niq
que tree
e path
patth between
between nodes of that link.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
10
2. Loop analysis
2.1. Two basic facts of loop analysis
o b = 8, nt = 5, n = 4, and l = 4.
o To apply KVL to each fundamental
fundamentaal lloop,
oop, we adopt a reference direction for
reference
direction
loop which agrees with referenc
ce d
irecttion of link which defines that
fundamental loop:
¾ fundamental loop 1 has
has same orientation
orientaatio
on as link 1.
o KVL equations can
can
n be
be written
writtten for
for 4 ffundamental
undam
mental
branch
loops in terms off b
raanch vvoltage.
oltaage.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
11
2. Loop analysis
2.1. Two basic facts of loop analysis
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
12
2. Loop analysis
2.1. Two basic facts of loop analysis
‰ l linear homogeneous algebraic equations in v1, v2, …, vb obtained by
applying KVL to each fundamental
fundamentaal lloop
oop constitute a set of l linearly
independent equations.
‰ B is an I b matr
matrix
rix ccalled
alleed ffundamental
undamental lloop
oop m
matrix.
atrix.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
13
2. Loop analysis
2.1. Two basic facts of loop analysis
™ Since:
¾ each fundamental loop includes
includ
des o
one
ne link only,
¾ orientations of loop and link
k aare
re picked
piccked to be same,
¾ number of links are 1, 2, ... , I,
¾ tree branches are I+
I+1,
+1, l+2,
l+2, ...
... , b
b,,
matrix B has form:
fo
orm:
™ 1l designates a unit matrix of order l.
™ F designates a rectangular matrix of I rows and n columns.
™ Rank of B is l.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
14
2. Loop analysis
2.1. Two basic facts of loop analysis
¾ KCL implies that any current that comes to a node must leave this node.
¾ Call i1, i2, ... , il currents in l links of
of tree
tre
ee T.
¾ Each of these currents flowing iin
n its
its respective
respective fundamental loop.
¾ Each tree branch current is superposition
superp
position of one or more loop currents.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
15
2. Loop analysis
2.1. Two basic facts of loop analysis
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
16
2. Loop analysis
2.1. Two basic facts of loop analysis
o Mesh analysis is not always a special case of
loop analysis.
o This will be the case if for each mesh
mesh current
cu
urrent
there is one branch that is traversed by
only that mesh current.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
17
2. Loop analysis
2.2. Loop analysis for LTI networks
o For simplicity, we consider networks with resistors.
o Extension to general case is exac
exactly
ctly tthe
he same
same as generalization discussed in
previous chapter.
o Branch equations are written in m
matrix
atrix form as:
o R is a diagonall b
branch
ranch rresistance
esistance m
matrix
atrix of
of dimension
dim
mension b
b..
o vs and js are voltage source
so
ourcce and
an
nd current
curre
ent source
so
ourcee vectors.
vecctors.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
18
2. Loop analysis
2.2. Loop analysis for LTI networks
‰ Zl is loop impe
impedance
edance matrix
matrix o
off o
order
rder l.l.
‰ es is loop voltage sou
source
urce vvector.
ector.
‰ Loop impedance matrix has
has properties
pro
opeertties similar
similar tto
o those of mesh impedance
matrix discussed in previous chapter.
‰ R is symmetric.
‰ Zl is symmetric.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
19
2. Loop analysis
2.2. Loop analysis for LTI networks
Example:
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
20
2. Loop analysis
2.2. Loop analysis for LTI networks
Example:
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
21
2. Loop analysis
2.3. Properties of the loop impedance matrix
™ Analysis of a resistive network and sinusoidal steady-state analysis of a
similar network are very closely rrelated.
elated.
™ Main difference is in the appearance
appeaarance off phasors
phasors and impedances.
1.
If network ha
has
as n
no
o ccoupling
oupling eelements:
lements:
¾ branch impedance
impedancce matrix
matrix Zb((jʘ)
jʘ) is
is d
diagonal,
iagonal,
¾ loop impedance matrix
mattrix Zl((jʘ)
jʘ) is
is symmetric.
syymmetric.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
22
2. Loop analysis
2.3. Properties of the loop impedance matrix
2.
If network has no coupling elements, Zl(j
((jʘ)
ʘ) can be written by inspection:
a.
zii is equal to sum of impedances
impedaances iin
n loop i.
zii is called self-impedance
e of
of loop
loo
op i.i.
b.
zik is equal to plus or minus sum off impedances of branches common
to loop i and to loop k.
Plus sign applies if,
iff, in
in branches
branches common
common to
to loop
loop i and loop k, loop
referencee directions
directtions agree.
agree.
opposite.
Minus sign applies
app
plies when
when they
they aare
re o
ppossite.
3.
If all current sources ar
are
re cconverted,
onverted, b
byy TThevenin's
hevenin's theorem, into voltage
sources, forcing term esi is algebraic sum of all source voltages in loop i.
Voltage sources whose reference direction pushes current in ith loop
reference direction are assigned a positive sign, others a negative sign.
4.
If network is resistive and if all its resistances are positive, det(Zl) > 0.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
23
Chapter Contents
0. Introduction
1. Fundamental theorem of graph theory
2. Loop analysis
3. Cut-set analysis
4. Comments on loop and cut-set analysis
anaalysis
between
5. Relation betwe
een
n B and
dQ
6. Summary
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
24
3. Cut-set analysis
3.1. Two basic facts of cut-set analysis
¾ Cut-set analysis is dual of loop analysis.
¾ First, we pick a tree (T).
¾ Next we number branches;
9 as before, links range from 1 to
o ll,, ttree-branches
ree-branches range from I+1 to b.
¾ We know that every tree
eb
branch
ranch defines
defines ((for
for ggiven
iven tree) a unique
fundamental cut
cut set.
set.
¾ That cut set is made u
up:
p:
¾ of links and
¾ of one tree branch (which defines cut set).
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
25
3. Cut-set analysis
3.1. Two basic facts of cut-set analysis
o Let us number cut sets as follows:
¾ cut set 1 is associated with tre
tree
ee branch
branch 5,
¾ cut set 2 with tree branch 6,, etc.
etc.
o For each fundamental cut set, we a
adopt
dopt a
reference direction for ccut
ut set
set which
which
agrees with tha
that
at of
of tree
e branch
branch
defining cut set.
seet.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
26
3. Cut-set analysis
3.1. Two basic facts of cut-set analysis
o Under these conditions, if we apply KCL to 4 cut sets:
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
27
3. Cut-set analysis
3.1. Two basic facts of cut-set analysis
‰ n linear homogeneous algebraic equations in j1, j2, ... , jb obtained by
applying KCL to each fundamenta
fundamental
al cut
cut set
set constitute a set of n linearly
independent equations.
‰ KCL equations are of form:
‰ Q = [qik] is an n b m
matrix
attrix aand
nd called
called fundamental
fundamental cut-set
cut-sset m
matrix:
atrix:
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
28
3. Cut-set analysis
3.1. Two basic facts of cut-set analysis
™ E is an appropriate n l matrix with e
elements
lements -1, +1, 0.
™ 1n is n n unit matrix.
matrix.
™ Q has a rank n.
™ n fundamental cut-set e
equations
quations iin
n terms
terms o
off b
branch
ranch currents are linearly
independent.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
29
3. Cut-set analysis
3.1. Two basic facts of cut-set analysis
¾ From KVL, each branch voltage can be expressed as a linear combination
of tree-branch voltages.
¾ For convenience, let us label tree-branch
treee-b
brancch voltages by e1, e2, ... , en.
¾ Node analysis is not always a special case of cut-set analysis.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
30
3. Cut-set analysis
3.2. Cut-set analysis for LTI networks
o For case of LTI resistive networks, branch equations are easily written in
matrix form:
o G is diagonal branch conductance matrix
matrix of dimension b.
o js and vs are source vectors.
vecttors.
o KCL and KVL are:
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
31
3. Cut-set analysis
3.2. Cut-set analysis for LTI networks
o Yq is cut-set admittance matrix.
o is is cut-set current source
sourrce vector.
vector.
o In scalar form, cut-set
cut-se
et eequations
quations are:
are:
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
32
3. Cut-set analysis
3.3. Properties of cut-set admittance matrix
‰ For sinusoidal steady-state analysis, cut-set admittance matrix Yq is:
1.
If network has no coupling eleme
elements:
ents:
¾ branch admittance matrix
matrix Yb((jʘ)
jʘ) iiss d
diagonal,
iagonal,
¾ cut-set admittance
adm
mittance matrix
matrix Yq((jʘ)
jʘ) iiss ssymmetric.
ymmetric.
2.
If network has no coupling
cou
upliing eelements,
lementts, Yq((jʘ)
jʘ) ccan
an be
be written by
inspection:
a.
yii is equal to sum of admittances of branches of the ith cut set.
b.
yik is equal to sum of all admittances of branches common to cut set i
and cut set k when, in branches common to their 2 cut sets, their
reference directions agree; otherwise, yik is negative of that sum.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
33
3. Cut-set analysis
3.3. Properties of cut-set admittance matrix
3.
If all voltage sources are transformed to current sources, isk is algebraic
sum of all current sources in cutt set
set k.
k.
Current sources whose reference
direction
refereence dir
rection is opposite to that of kth cut
set reference direction are assigned
asssigned a positive
positive sign.
All others are assigned a negative
negativve ssign.
ign.
4.
If network is resistive aand
nd if
if all
all its
its resistances
resistances are
are positive, det(Yq) > 0.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
34
3. Cut-set analysis
3.3. Properties of cut-set admittance matrix
Example:
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
35
Chapter Contents
0. Introduction
1. Fundamental theorem of graph theory
2. Loop analysis
3. Cut-set analysis
4. Comments on loop and cut-set analysis
between
5. Relation betwe
een
n B and
dQ
6. Summary
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
36
4. Comments on loop and cut-set analysis
™ Both loop analysis and cut-set analysis start with choosing a tree for given
graph.
™ Since number of possible trees fo
for
or a ggraph
raph is usually large, 2 methods are
extremely flexible.
™ They are more general than mesh
h analysis
analyysis and node analysis.
™ Fundamental loops ffor
or particular
particular tree
tree coincide
coincidee with
with
4 meshes of graph.
™ Thus, mesh currents are identical with
fundamental loop currents.
™ Mesh analysis for this particular example is a
special case of loop analysis.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
37
4. Comments on loop and cut-set analysis
¾ Fundamental cut sets for particular tree coincide with sets of branches
connected to nodes 1, 2, 3, and 4.
¾ If node 5 is picked as datum node
node,
e, ttree-branch
ree-branch voltages are identical with
node-to-datum voltages.
¾ Node analysis for this particular example
examplle is a
special case of cut-set analysis.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
38
4. Comments on loop and cut-set analysis
o It should be pointed out that for this graph,
meshes are not special cases of fundamental loops.
o There exists no tree such that 5 meshes
meshes are
fundamental loops.
o Similarly,
y in this graph,
grap
ph, if
if n
node
ode 4 is
is p
picked
icked aass datum
datum
tree
which
node, there exists no tre
ee w
hich gives
gives tree-branch
tree-branch
voltages identical to node-to-datum voltages.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
39
4. Comments on loop and cut-set analysis
o As far as relative advantages of cut-set analysis and loop analysis,
conclusion is same as that between mesh analysis and node analysis.
o It depends:
9 on graph as well as
9 on kind and number of source
sources
es in network.
o For example, if number of
of ttree
re
ee branches
branches (n)
(n)) is much
much smaller than number
method
usually
more
of links (l), cut-set
cut-se
et m
eth
hod is
is u
suallly m
ore efficient.
efficient.
o In previous chapter, dual
duality
lity aapplied
pplied only
only to
to planar
planar graphs and planar
networks (node and mesh analysis).
o Now, duality extends to non planar graphs and networks:
9 For example, cut sets and loops are dual concepts.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
40
Chapter Contents
0. Introduction
1. Fundamental theorem of graph theory
2. Loop analysis
3. Cut-set analysis
4. Comments on loop and cut-set analysis
anaalysis
5. Relation between B and Q
6. Summary
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
41
5. Relation between B and Q
‰ If we:
¾ start with an oriented graph G,
¾ pick any one of its trees, say T
T,,
¾ write fundamental loop matrix
mattrix B and
an
nd fundamental
fundamental cut-set matrix Q,
we should expect to find a very close co
onnection between these matrices.
connection
‰ B tells us which branch is
is in
in which
which fundamental
fundamental loop.
loop.
‰ Q tells us whic
which
ch branch
branch is
is in
in which
which fundamental
fundamental cut
cut set.
set.
‰ Precise relation between
betw
ween B and
and Q iss sstated
tated in
in following
following theorem.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
42
5. Relation between B and Q
THEOREM
™ Call B fundamental loop matrix and Q fundamental cut-set matrix of same
oriented graph G, and let both matrices
matrices pertain to same tree TT; then:
If we number links from
m 1 tto
o l aand
nd number
number tree
tree branches
branches from I+1 to b,
then:
¾ Product of I b matrix B and b n matrix QT is l n zero matrix.
¾ Product of every row of B and every column of QT is 0.
¾ Product of every row of Q by every column of BT is 0.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
43
5. Relation between B and Q
9 Relation between B and Q is extrem
extremely
mely useful.
9 Whenever we k
know
no
ow one
e off tthese
hese m
matrices,
atrices, w
we
e ccan
an wr
write
rite
e other one by
inspection.
9 Both matrices B and Q are
e uniquely
un
niquelyy specified
speccified b
byy I n matrix F.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
44
Chapter Contents
0. Introduction
1. Fundamental theorem of graph theory
2. Loop analysis
3. Cut-set analysis
4. Comments on loop and cut-set analysis
anaalysis
between
5. Relation betwe
een
n B and
dQ
6. Summary
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
45
6. Summary
o Analogies between 4 methods of analysis deserve to be emphasized:
o Each one of “connection”
“connecttion” m
matrices
atrices A
A,, M,
M, Q,
Q, an
and
nd B iiss off ffull
ulll rank.
Electric Circuits 2
Chapter 3. Loop and Cut-set Analysis
46