Chapter 1. Network Graphs and
Tellegen’s Theorem
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
1.
The concept of a graph
2.
Cut sets and Kirchhoff's current law
3.
Loops and Kirchhoff's voltage llaw
aw
4.
Tellegen’s theorem
5.
Applications
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
2
1. The concept of a graph
o In this course, we want to develop systematic procedures to analyze and
establish properties of any network of any complexity.
o “Network” has same meaning ass “circuit”:
“circuit”:
¾ i.e., an interconnection of elements.
ellementts.
o Network usually carries idea of comp
complexity.
plexity.
o A network is a circuit with
wiith many
many eelements.
lements.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
3
1. The concept of a graph
‰ KVL and KCL do not make any assumption about nature of network
elements.
‰ We can overlook nature of eleme
elements
ents tto
o reduce network to a graph.
‰ The first section of this chapter develops
develop
ps concept
concept of a graph.
‰ Graph-theoretic ideas are used to formulate
formulate KVL and KCL.
‰ Then, we derive Tellegen's
Tellege
en's theorem
theorem to
to prove
prove several
several properties of
networks.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
4
1. The concept of a graph
™ Consider any physical network, e.g., an 80-element lumped network.
™ Suppose we consider only those frequencies which permit us to model
physical network as a connection
n of
of lumped
lumped elements.
™ When we say “network N unde
under
er consideration”,
consid
deration”, we mean this model.
™ In this chapter, network N may be linear
linear or nonlinear, active or passive,
time-invariant.
time-varying or time-in
nvarian
nt.
™ We disregard n
nature
ature o
off e
elements
lements ssince
ince K
Kirchhoff's
irchhoff's laws
laws d
do
o not depend on
elements.
nature of element
ts.
™ We replace each element
elemen
nt of
of network
network N b
byy a b
branch
ranch (a line segment).
™ At ends of each branch we draw black dots called nodes.
™ Some authors use “edge” for branch and “vertex” or “junction” for node.
™ Result of this process is a graph.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
5
1. The concept of a graph
¾ In Fig. 1.1b, even though
2 inductors are mutually
coupled, graph does not
indicate M.
¾ M pertains to nature of
branches and iss not a
property of graph.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
6
1. The concept of a graph
o By graph we mean:
¾ a set of nodes together with a set of branches with condition that
¾ each branch terminates at each
eaach end
end into a node.
o Definition of graph includes spe
special
ecial casee in
in which a node has no branch
connected to it.
o Since start and end nodes
nodees of
of a branch
branch are
are not required
required to be distinct, a
include
self-loop:
graph may inclu
udee a self
f-lloop:
¾ i.e., a loop con
consisting
nsiisting of
of a ssingle
inggle branch.
braanch.
o In this course, we shall not encounter such graphs.
o Although they appear in engineering work, e.g., in flow graphs.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
7
1. The concept of a graph
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
8
1. The concept of a graph
o G1 is a subgraph of G if:
¾ G1 is itself a graph,
¾ every node of G1 is a node of G
G,, aand
nd
¾ every branch of G1 is a branc
branch
ch o
off G.
G.
o We obtain G1 by deleting from G some
som
me branches and/or some nodes.
o A subgraph with only on
one
ne n
node
ode is
is ccalled
alled a d
degenerate
egenerate subgraph.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
9
1. The concept of a graph
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
10
1. The concept of a graph
‰ We adopt reference directions for branch voltages and branch currents
that are called associated:
‰ Arrowhead that specifies curr
current
rent rreference
eference direction always points
toward minus terminal for vvoltage
oltage
e reference
reference direction.
‰ Branch voltage and current of kthh b
branch
ranch are denoted by vk and jk.
‰ In this chapter, we alway
always
ys use associated
asso
ociated re
reference
eference directions.
‰ We need only iindicate
ndicate aarrow
rro
ow o
off ccurrent.
urrent.
‰ We omit plus and mi
minus
inus signs
signs for
for vvoltage.
oltage.
‰ We usually use:
‰ letter j to designate branch currents,
‰ letter i to designate loop currents or mesh currents.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
11
1. The concept of a graph
™ Such a graph is called an oriented graph.
™ We may number nodes and branches.
™ Branch 4 is incident with node 2 and
and 3.
3.
™ Branch 4 leaves node 3 and enters
entters node
node 2.
2.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
12
1. The concept of a graph
o We describe oriented graph by:
¾ listing all branches and nodes,
¾ indicating which branch is ent
entering
tering aand
nd leaving which node.
o This is done by writing down a matrix.
matrix.
o Suppose that oriented graph is made
maade up
up of b branches and nt nodes.
o Suppose also that we number
nu
umber arbitrarily
arbitrarily aallll branches
branches and nodes.
o We call node-to-branch
node-tto-branch iincidence
ncid
dence m
matrix
atrix Aa a rectangular
rectangular matrix of nt rows
whose
and b columns whos
se (i,k)
(i,k)tthh eelement
lement aikk iis:
s:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
13
1. The concept of a graph
‰ Each branch leaves a single node and enters a single node.
‰ Each column of Aa contains a single +1 and a single -1, with all other
elements equal to 0.
‰ Conversely, to any nt x b matrixx w
with
ith pro
property
operty that each one of its columns
contains a single +1, a single -1, and 0s,
0s, we can associate an oriented
graph of b branches and nt nodes.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
14
Chapter Contents
1.
The concept of a graph
2.
Cut sets and Kirchhoff's current law
3.
Loops and Kirchhoff's voltage llaw
aw
4.
Tellegen’s theorem
5.
Applications
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
15
2. Cut sets and Kirchhoff's current law
™ To express KCL systematically for any network, we now develop concept of
cut set.
™ KCL states that algebraic sum of a
allll ccurrents
urrents leaving a node is 0.
™ If we partition network nodes into
in
nto 2 set
sets
ts b
byy a closed Gaussian surface (one
set of nodes is inside surface and
d other outside), KCL implies that sum of
currents leaving Gaussian surface iiss 0.
0.
™ Collection of al
allll b
branches
ranche
es that
that cross
cross G
Gaussian
aussiian surface
surfacee is called a cut set.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
16
2. Cut sets and Kirchhoff's current law
¾ A graph is connected if there is at least one path (along branches and
disregarding branch orientations) between any 2 nodes.
¾ By convention, a graph consistingg of
of only
only one node is connected.
¾ A connected graph is also said to
to be
be off one
one separate part.
¾ Given an unconnected graph, its maximal
maximal connected subgraphs are also
called separate parts.
¾ An unconnected
unconnecteed graph
graph m
must
ust h
have
ave aatt least
least 2 sseparate
eparate p
parts.
arts.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
17
2. Cut sets and Kirchhoff's current law
o When we say “remove a branch”, we mean that:
¾ we delete line segment that joins nodes but
¾ we leave nodes remaining.
o Idea of a cut set is related to ide
idea
ea of cutting
cuttting a connected graph into 2
separate parts by removing branches.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
18
2. Cut sets and Kirchhoff's current law
o A set of branches of a connected graph G is called a cut set if:
1.
removal of all branches of set causes remaining graph to have 2
separate parts, and
2.
removal of all but any one
e of
of bra
branches
anches of set leaves remaining graph
connected.
o In case graph G has s separate part
parts,
ts, a ccut
ut set is defined to be a set of
branches such that:
1.
removal o
off aallll b
branches
ranches of
of set
set causes
causes rremaining
emaining graph
graph to have s+1
separate parts, aand
nd
2.
removal of all but aany
ny o
one
ne of
of b
branches
ranches of
of sset
et leaves remaining graph
with s separate parts.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
19
2. Cut sets and Kirchhoff's current law
‰ Branches of cut set are indicated
indiccated by
heavier lines.
‰ Idea of cutting connected graph
grraph into
into
2 separate parts is emphasized by
by
dashed line crossing all branches
bran
nche
es
of cut set (idea of Gaussian surface).
surface
e ).
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
20
2. Cut sets and Kirchhoff's current law
™ Kirchhoff's current law (KCL
(KCL):
L):
¾ For any lumped network,
networkk,
for any of its cut sets, an
and
nd at any
any time,
time,
algebraic sum of all branch
bran
nch currents
curren
nts
traversing cut-set branches
0..
branch
hes is
is 0
™ To apply KCL:
1.
we assign a reference
referen
nce d
direction
irection
namely,
to cut set,
sett, n
amely,
outside
off
from inside tto
oo
utsid
de o
Gaussian surface,
2.
we assign a plus sign to
branch currents whosee
reference direction agrees
with that of cut set and
d
vice versa.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
21
2. Cut sets and Kirchhoff's current law
¾ For cut sets shown, KCL gives:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
22
2. Cut sets and Kirchhoff's current law
o KCL, as stated above, is a direct consequence of node law.
o If we sum all expressions of KCL applied to nodes
inside Gaussian surface, we obtain
obtaiin ccut-set
ut-set law.
o Currents of branches joining 2 internal
in
nternall nodes
nodes
cancel out!
o This is easily verified:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
23
2. Cut sets and Kirchhoff's current law
o By adding KCL equations relative to cut sets I and II, we obtain that of cut
set III.
o 3 KCL equations are linearly depe
dependent.
endent.
o 3rd equation did not supply anyy information
informaation not contained in preceding
ones.
o In our general theory off network
network aanalysis,
nalysiis,
we shall have to
o select
sellect cut
cutt sets
seets
in such a way that
that each
each equation
equation
information.
supplies some new in
nformation.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
24
Chapter Contents
1.
The concept of a graph
2.
Cut sets and Kirchhoff's current law
3.
Loops and Kirchhoff's voltage law
4.
Tellegen’s theorem
5.
Applications
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
25
3. Loops and Kirchhoff's voltage law
o For our present systematic approach, we need a precise concept of a loop.
o Roughly speaking, a loop is a closed path.
o A subgraph L of a graph G is calle
called
ed a lloop
oop if:
1.
subgraph L is connected and
a nd
2.
precisely 2 branches of L are incident
incident with each node of L.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
26
3. Loops and Kirchhoff's voltage law
o A subgraph L of a graph G is called a loop if:
1.
subgraph L is connected and
2.
precisely 2 branches of L aree incident
incident with each node of L.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
27
3. Loops and Kirchhoff's voltage law
‰ Kirchhoff's voltage law (KVL):
¾ For any lumped network, for any of its loops, and at any time, algebraic
sum of branch voltages aroun
around
nd lloop
oop is 0.
‰ To apply KVL:
1.
we assign a reference direction
to loop,
2.
we assign a plus sign
siign tto
oab
branch
ranch
voltage when
when its
its branch
branch reference
reference
direction agrees
agree
es with
with that
that of
of loop
loop and
and
vice versa.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
28
Chapter Contents
1.
The concept of a graph
2.
Cut sets and Kirchhoff's current law
3.
Loops and Kirchhoff's voltage llaw
aw
4.
Tellegen’s theorem
5.
Applications
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
29
4. Tellegen’s theorem
o We introduce our first general network theorem, Tellegen's theorem.
o This theorem is extremely general.
o It is valid for any lumped networ
network
rk tthat
hat ccontains
ontains any elements
¾ linear or nonlinear,
¾ passive or active,
¾ time-varying or time
time-invariant.
e-invariant.
o This generalityy ffollows
ollows ffrom
rom tthe
he fact
fact that
that
¾ Tellegen's theore
theorem
em d
depends
e p e nds o
only
nly on
on 2 K
Kirchhoff’s
ircchhoff ’s laws.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
30
4. Tellegen’s theorem
‰ Consider an arbitrary lumped network with b branches and nt nodes.
‰ For convenience, choose associated reference directions for branch
voltages vk and branch currents jk·
‰ vk(t)j
)jk(t) is power delivered at ttime
ime t by
by network
network to branch k.
‰ Let us:
¾ disregard nature of b
branches
ranches aand
nd
¾ think of network
nettwork as
as an
an
G..
oriented graph
hG
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
31
4. Tellegen’s theorem
™ Tellegen's theorem asserts that:
™ Only requirement on vk is that they
they ssatisfy
atisfy all constraints imposed by KVL.
™ Similarly,
y jk must satisfy all constraints
constraiints imposed
imposed by KCL.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
32
4. Tellegen’s theorem
¾ Let us arbitrarily assign vk and jk,
¾ subject only to satisfying Kirchhoff's laws for all loops and nodes:
¾ KCL is satisfied since:
¾ KVL is satisfied since:
¾ To check Tellegen's theor
theorem:
rem:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
33
4. Tellegen’s theorem
o It is of crucial importance to realize that branch voltages v1, v2, …, vb are
picked arbitrarily subject only to KVL constraints.
o Similarly,
y branch currents j1, j2, ... , jb aare
re picked arbitrarily subject only to
KCL constraints.
o For example, suppose ‫ݒ‬ො 1, ‫ݒ‬ො 2, …, ‫ݒ‬ො ܾ and ଔƸ1, ଔƸ2, …, ଔƸܾ are other sets of
arbitrarily selected branch voltage
es and
and branch currents that obey same
voltages
KVL constraints and same
sam
me KCL
KCL cconstraints.
onstrraints.
o Then we may a
apply
pply Tellegen’s
Tellegen’s theorem
theorem and
and obtain:
ob
btain:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
34
4. Tellegen’s theorem
Proof of Tellegen's theorem:
o Let us assume, for simplicity, that:
¾ graph G is connected and
¾ has no branches in parallel ((only
onlyy 1 b
branch
ranch exists between any 2 nodes).
o Proof can be easily extended to ge
general
eneral case:
¾ If there are branchess iin
np
parallel,
arallel, rreplace
eplace tthem
hem by a single branch
current
off b
branch
whose curr
rent iiss ssum
um o
ranch
h currents.
curren
nts.
¾ If there are sever
several
ral separate
separate parts,
parts, p
proof
roof shows
shows ttheorem
heorrem holds for each
one of them.
over
Hence, it holds also when
when ssum
um ranges
ran
nges o
ver all
all branches of graph.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
35
4. Tellegen’s theorem
Proof of Tellegen's theorem:
‰ We first pick an arbitrary node as a reference node.
‰ We label it node 1.
‰ Thus, e1 = 0.
‰ Let eɲ and eɴ be voltages of ɲth and
d ɴtthh nodes.
‰ Once branch vo
voltages
olttages (v
(v1, v2, …,
…, vb ) are
are cchosen,
hosen,
node voltages ((ee1, …, eɲ, ... , eɴ, …)) are
are
uniquely specified by
by KVL.
KVL.
‰ Let us assume that branch
brancch k cconnects
o nn ect s
node ɲ and node ɴ.
‰ Let us denote current flowing in branch k
from node ɲ to node ɴby jɲɲɴɴ.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
36
4. Tellegen’s theorem
Proof of Tellegen's theorem:
™ If there is no branch join
joining
ning node
node ɲ ƚŽŶŽĚĞɴ͕
ƚŽŶ
ŶŽĚĞɴ
ɴ͕
we set:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
37
4. Tellegen’s theorem
Proof of Tellegen's theorem:
¾ For each fixed ɲ,
ɲ, ssum
um off all
all branch
braanch
h currents
cu
urrentts
leaving node ɲ iis:
s:
¾ &ŽƌĞĂĐŚĨŝdžĞĚɴ,
ɴ sum of all branch currentss
ůĞĂǀŝŶŐŶŽĚĞɴ is:
¾ By KCL, each one of these sums is 0.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
38
4. Tellegen’s theorem
Proof of Tellegen's theorem:
9 Hence:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
39
Chapter Contents
1.
The concept of a graph
2.
Cut sets and Kirchhoff's current law
3.
Loops and Kirchhoff's voltage llaw
aw
4.
Tellegen’s theorem
5.
Applications
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
40
5. Applications
5.1. Conservation of energy
o Considering an arbitrary network, for all t, we have:
o vk(t)j
)jk(t) is power delivered at timee t by
by network to branch k.
o At any time t, ssum
um
mo
off power
po
ower d
delivered
elive
ere
ed tto
o ea
each
ach branch
branch
h off network is 0.
o Sum of power delivered
dellive
ered by
by iindependent
ndependent ssources
ourcces to
to network
nettwork is equal to
absorbed
other
sum of power absorb
bed by
by alll o
ther branches
brancches of
of network.
network.
o This means that as far as lumped circuits are concerned, KVL and KCL imply
conservation of energy.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
41
5. Applications
5.1. Conservation of energy
‰ Tellegen's theorem has some astonishing consequences.
‰ For example, consider 2 arbitraryy llumped
umped networks whose only constraint
is to have same graph.
‰ Let us:
‰ choose same referen
reference
nce directions
directions and
a nd
‰ number branches
braanches in
in a similar
similar fashion.
fashion.
‰ Networks may be nonlinear
no
onlinear aand
nd ttime-varying
ime-varying aand
nd include
include independent
sources as well as dependent
depen
ndent ssources.
ources.
‰ Let:
‰ vk , jk be branch voltages and currents of 1st network and
‰ ‫ݒ‬ො ݇ and ଔƸ݇ be corresponding branch voltages and currents of 2nd
network.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
42
5. Applications
5.1. Conservation of energy
™ vk’s and ‫ݒ‬ො ݇ ’s satisfy same set of KVL constraints.
™ jk's and the ଔƸ݇ ’s satisfy same set o
off K
KCL
CL cconstraints.
onstraints.
™ Tellegen's theorem guarantees:
™ First 2 are expressions off conservation
conservation o
off e
energy.
nergy.
™ Last 2 expressions do not have an energy interpretation:
™ because they involve voltages of one network and currents of another.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
43
5. Applications
5.2. Conservation of complex power
¾ Consider a linear time-invariant network.
¾ For simplicity, let it have only one
e ssinusoidal
inu
usoidal source in branch 1.
¾ Suppose network is in sinusoida
sinusoidal
al ssteady
teady sstate.
tate.
¾ For each branch (still using associated
associaated reference
reference directions),
we represent vk by phas
phasor
sor Vk aand
nd j k b
byy p
phasor
hasor Jk·
¾ V1, V2, ... , Vb an
and
nd J1, J2, ... , Jb ssatisfy
atisfy all
all constraints
constraaints
imposed by KVL and KCL.
KCL.
¾ Conjugates ‫ܬ‬1ҧ , ‫ܬ‬2ҧ , ... , ‫ܾܬ‬ҧ ssatisfy
atisfy aallll KCL
KCL constraints.
constraints.
¾ By Tellegen's theorem:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
44
5. Applications
5.2. Conservation of complex power
o V1 is source voltage.
o J1 is source current (using associated
associaated reference
reference direction).
o ଵΤଶ ܸ1‫ܬ‬1ҧ is complex power delivered
delive
ered to
to b
branch
ranch 1 by rest of network.
o െ ଵΤଶ ܸ1‫ܬ‬1ҧ is complex power delivered
delive
ered by source to rest of network.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
45
5. Applications
5.2. Conservation of complex power
Theorem:
o Consider a linear time-invariant network,
network, which is:
o in sinusoidal steady state and
an
nd
o driven by several independentt
sources that are at ssame
ame ffrequency.
requency.
o Sum of complex
comple
ex power
power d
delivered
elivered by
by
each independentt so
source
ource to
to network
network is
is
equal to sum of complex
received
compllex power
power recei
ived
d
by all other branches of network.
network.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
46
5. Applications
5.3. The real part and phase of driving-point impedances
‰ Consider driving-point impedance Zin of linear time-invariant network.
‰ Let network be driven by a sinuso
sinusoidal
oidall ccurrent
urrent source at an angular
frequency ʘ.
‰ Source current is represented by p
phasor
hasor J1.
‰ Source voltage (using as
associated
ssociated rreference
eference d
direction)
irection) is represented by V1.
‰ Other branche
branches
es aare
re n
numbered
umbered from
from 2 tto
ob
b..
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
47
5. Applications
5.3. The real part and phase of driving-point impedances
™ All these impedancess a
are
re evaluated
evaluated at
at same
ssaame
angular frequency ʘ ((source
sourrce ffrequency).
requen
n c y ).
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
48
5. Applications
5.3. The real part and phase of driving-point impedances
¾ Case 1: Resistiv
Resistive
ve n
networks:
etworrks:
¾ Input impedance
imped
dan
nce
e iiss a positive
posiitive re
resistance.
esisttance.
¾ Case 2: RL networks:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
49
5. Applications
5.3. The real part and phase of driving-point impedances
¾ Case 3: RC networks:
¾ At any positive ʘ, dr
driving-point
riving-point
impedance
has
eh
as a phase
phase aangle
ngle between
between 0 and
and -90°.
-9 0 ° .
¾ Case 4: Lossless networks
networkks m
made
ade of
of C,
C, L ((including
including coupled inductors),
and/or ideal transformers:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
50
5. Applications
5.3. The real part and phase of driving-point impedances
¾ Case 5: RLC net
networks:
two
orks:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
51
5. Applications
5.4. Driving-point impedance, power dissipated, and energy stored
9 Consider a linear time-invariant RLC network
neetwork
driven by a sinusoidal current source.
sou
urce.
9 Network is in sinusoidal steadyy sstate.
tate.
9 Complex power delivered by sourc
source
ce
to network is:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
52
5. Applications
5.4. Driving-point impedance, power dissipated, and energy stored
o Average (over one period) of Riji2(t) is:
o Average of ଵΤଶLkjk2(t) is:
o Average of ଵΤଶClvl2((t)
t) is:
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
53
5. Applications
5.4. Driving-point impedance, power dissipated, and energy stored
‰ 1st term is average power dissipated (P
Pav).
‰ 2 terms in parentheses are:
‰ average magnetic energy stored
sttored EM
‰ average electric energy stored
d EE.
Electric Circuits 2
Chapter 1. Network Graphs and Tellegen’s Theorem
54