KN Toosi University of Technology
Chapter 8. Network Theorems
By: FARHAD FARADJI, Ph.D.
Assistant Professor,
Electrical and Computer 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
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Chapter Contents
0. Introduction
1. The substitution theorem
2. The superposition theorem
3. Thevenin-Norton
equivalent
network
ton e
quivalent n
etwork ttheorem
heore
em
4. The reciprocityy theorem
Electric Circuits 2
Chapter 8. Network Theorems
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0. Introduction
o In this chapter, we study 4 very general and useful network theorems:
¾ substitution theorem,
¾ superposition theorem,
¾ Thevenin-Norton
Nortton e
equivalent
quivvale
ent n
network
etwork theorem,
theeorem,
¾ reciprocity
y ttheorem.
heorem.
o Principal assumption
mption underlying alll these
these network
network theo
theorems
orem is uniqueness
of solution forr netw
network
under
work u
nder consideration.
consideration..
o Substitution theorem:
heorem:
¾ holds for all networks with a unique solution,
¾ can be applied to:
y
y
linear and nonlinear networks,
time-invariant and time-varying networks.
o Other 3 theorems apply only to linear networks.
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Chapter 8. Network Theorems
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0. Introduction
‰ A linear network:
¾ by definition, consists of:
™
linear elements:
9 linear
r resistors,
9 linear
r iinductors,
nductors,
9 linear
r coup
coupled
pled inductors,
indu
uctors,
9 linear
r capacitors,
9 linear
r tran
transformers,
nsformers,
9 linear
r depe
dependent
endent sources
sources
™
independent sources which are inputs to network.
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Chapter 8. Network Theorems
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0. Introduction
™ Superposition theorem and Thevenin-Norton equivalent network
theorem apply to all linear networks:
¾ time-invariant or
¾ time-varying.
ng.
™ Reciprocity theorem
eorem applies
appliees to
to a more
mo
ore restricted
resttricted class
classs of
of linear
lin
networks:
¾ elements must be time-invariant.
¾ elements can not
not include:
include:
dependent
ndentt ssources,
ource
es,
™ independent sources, and
™ gyrators.
™
Electric Circuits 2
Chapter 8. Network Theorems
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Chapter Contents
0. Introduction
1. The substitution theorem
2. The superposition theorem
3. Thevenin-Norton
equivalent
network
ton e
quivalent n
etwork ttheorem
heore
em
4. The reciprocityy theorem
Electric Circuits 2
Chapter 8. Network Theorems
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1. The substitution theorem
SUBSTITUTION THEOREM
¾ Consider an arbitrary network containing some independent sources.
Suppose that for these sources and for ggiven initial conditions network
has a unique solution
branch
sollution
n for
for aallll iits
ts b
ranch voltages
voltagges and
and branch
branch currents.
Consider a particular
other
branches.
rticularr branch
branch
h (k),
(kk), which is not
not ccoupled
ouple
ed to
to o
t
currrent and
and voltagee waveforms
waveforrms of
of branch
branch kk.
Let jk and vk bee current
Suppose that branc
branch
byy either:
ch k iiss rreplaced
eplaced
db
an independent
pendent current source with waveform
waveeform jk or
an independent voltage source with waveform vk.
If modified network has a unique solution for all its branch currents and
voltages:
these branch currents and voltages are identical with those of
original network.
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Chapter 8. Network Theorems
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1. The substitution theorem
Example 1:
y Consider circuit shown.
y Problem is to find voltage
g
V and current I o
off
tunnel diode ggiven
iven E,
E, R,
R, and
an
nd
tunnel-diode characteristic.
y Solution is obtained
taineed ggraphically.
raphicallyy.
y According to substitution
ubstittuttion th
theorem,
heorem, we may replace
replaace tunnel diode either:
¾ by a current source I or
¾ by a voltage source V.
y In both cases, solutions are same as
that obtained originally.
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Chapter 8. Network Theorems
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1. The substitution theorem
Example 2:
o Consider any LTI network which:
¾ is in zero state at time 0
¾ has no independent
ep
pende
ent ssources.
ourcces.
o Assume that there
here are
are 2 acc
accessible
cessiible termin
terminals
nals
to network which
hich form a port.
o If we use substitution theorem:
Electric Circuits 2
Chapter 8. Network Theorems
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Chapter Contents
0. Introduction
1. The substitution theorem
2. The superposition theorem
3. Thevenin-Norton
equivalent
network
ton e
quivalent n
etwork ttheorem
heore
em
4. The reciprocityy theorem
Electric Circuits 2
Chapter 8. Network Theorems
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2. The superposition theorem
SUPERPOSITION THEOREM
‰ Let ॉ be a linear network:
each element is either an independent source or a linear element.
Elements mayy be
be time-varying.
time-vaaryin
ng.
We further assume
has
zero-state
sume tthat
hatt ॉ h
as a unique ze
ero-sstate response
respons to
independent source waveforms, whatever
be.
wh
hateverr tthey
hey may be
e.
Let response of ॉ be:
be:
either current
urrentt in
in a specific branch or
voltage across any specific node pair or
more generally any linear combination of currents and voltages.
Under these conditions, zero-state response of ॉ due to all independent
sources acting simultaneously is equal to:
sum of zero-state responses due to each independent source acting
one at a time.
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Chapter 8. Network Theorems
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2. The superposition theorem
™ When a voltage source is set equal to 0, it becomes a short circuit.
™ When a current source is set equal to 0, it becomes an open circuit.
™ Superposition theorem can be expressed
p
in terms of concept of linear
function.
™ Consider waveforms
efformss o
off iindependent
nde
epend
dent sour
sources
rces to
to be ccomponents
ompo
of a
vector which we call vector-input waveform.
waveforrm.
™ If vector-input waveform
wavveform is:
is:
™ Zero-state response v(·) is a function of vector-input waveform:
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Chapter 8. Network Theorems
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2. The superposition theorem
¾ If function f is linea
linear,
ar, then
then superposition
superposition theorem
theo
orem holds:
holds:
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Chapter 8. Network Theorems
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2. The superposition theorem
y In general, superposition does not apply to nonlinear networks.
y If a particular network has:
¾ all linear elements
¾ except for a few
few nonlinear
nonliinear ones
o nes
it is possible to
o make
makke superposition
su
uperrposiition hold by
by ccareful
areful sselection
electio of:
¾ element values,
alues,
¾ source location,
ation
n,
¾ source waveform,
veform and
¾ response.
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Chapter 8. Network Theorems
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2. The superposition theorem
o Superposition theorem has been stated exclusively in terms of zero-state
response of a linear network.
o Sinusoidal steady state is limiting condition (as ‫ ݐ‬՜ λ) of zero-state
nputt, it ffollows
ollo
ows that
t:
response to a sinusoidal in
input,
that:
¾ superposition
tion ttheorem
heorem applies
applies iin
n particular
particu
ulaar to
to sinusoidal
siinusoid steady state.
COROLLARY 1
o Let ॉ be an LTI
TI netw
network.
workk.
Suppose that all independent sources are sinusoidal (not necessarily of
same frequency).
Then steady-state response of ॉ due to all independent sources acting
simultaneously is equal to:
sum of sinusoidal steady-state responses due to each independent
source acting one at a time.
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Chapter 8. Network Theorems
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2. The superposition theorem
COROLLARY 2
‰ Let ॉ be an LTI network.
g across anyy node p
Let response be voltage
pair or current through any
branch of ॉ.
More specifically,
off zero-state
alllyy, call
caall X(s)
X(s) Laplace
Lapllace transform
transfo
orm o
zero
o-state response due to
all independent
nt sources acting simultaneously.
sim
multaneo
ously.
Then:
Ik(s), k = 1, 2, ..., m, are Laplace transforms of m inputs.
Hk(s), k = 1, 2, ..., m, are respective network functions from m
inputs to specified output.
Electric Circuits 2
Chapter 8. Network Theorems
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Chapter Contents
0. Introduction
1. The substitution theorem
2. The superposition theorem
3. Thevenin-Norton
equivalent
network
ton e
quivalent n
etworrk ttheorem
heore
em
4. The reciprocityy theorem
Electric Circuits 2
Chapter 8. Network Theorems
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3. Thevenin-Norton equivalent network theorem
THEVENIN-NORTON THEOREM
™ Let linear network ॉ, be connected by 2 of
its terminals 1 and 1’ to an arbitrary load.
Network elements
men
nts m
may
ay b
bee tim
time-varying.
me-varying.
We further assume
has
sume tthat
hatt ॉ h
as a unique solution:
so
olution:
when it is terminated by load
load,
d, aand
nd
when load
ad iss replaced
replaced by
by an
an iindependent
ndepend
dent ssource.
ource.
Let ॉ0 be network
work ob
obtained
btained
d from ॉ by setting:
all independent sources to 0 and
all initial conditions to 0.
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3. Thevenin-Norton equivalent network theorem
THEVENIN-NORTON THEOREM
Let eoc be open-circuit voltage of ॉ at
terminals 1 and 1’, as shown.
Let isc be short-circuit
t-ccircuiit current
currrent of
of ॉ
nto 1
shown.
flowing out of 1 iinto
1’’ aass shown.
Under these conditions,
onditions,
whatever load
may
be,
d ma
ay b
e, vvoltage
oltage v
and current i remain
remaiin unchanged
unchanged
d
when networkk ॉ is replaced by
either its Thevenin equivalent or
by its Norton equivalent network.
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Chapter 8. Network Theorems
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3. Thevenin-Norton equivalent network theorem
THEVENIN-NORTON THEOREM
¾ Relaxed network ॉ0 is obtained
from ॉ by setting:
9 all its independent
pend
den
nt sources
sourrcess to
to 0,
0,
9 all initial conditions
onditions to
to 0.
0.
¾ ॉ0 could be referred
eferred to as
d zer
ro-sstate
zero-input and
zero-state
tworkk.
equivalent network.
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3. Thevenin-Norton equivalent network theorem
Example:
o Consider resistive circuit shown.
o Determine voltage
g across tunnel diode.
o We can:
9 use Thevenin
niin th
theorem
heoreem and
and
9 consider tunnel
unnel diode as load.
o First we determine
minee Thevenin
Thevenin equivalent
equivalent network
netw
work
of one-port faced
ced by tunnel
tunnell diode:
o Terminal characteristic of Thevenin equivalent
network is:
o Any intersection of characteristics gives one solution.
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3. Thevenin-Norton equivalent network theorem
COOLLARY
‰ Let linear “time-invariant” network ॉ be connected by 2 of its terminals, 1
and 1’, to an arbitrary load.
Let Eoc(s) be Lap
Laplace
place ttransform
ran
nsforrm o
off o
open-circuit
pen-circu
u it
voltage eoc(t) aatt terminals
terminals 1 and
and 1’
1’
(voltage when no current flows into ॉ thr
through
rough 1 and
d 1’).
1’).
Let Isc(s) be Laplace
aplacce ttransform
ransform of
of current
current isc((t)
t)
flowing out of 1 and
d into
into 1’
1’ when
when lload
oad
d is shorted.
shortted.
Let Zeq = 1/Yeq be impedance (seen between
terminals 1 and 1’) of network obtained
from ॉ by setting:
all independent sources to 0 and
all initial conditions to 0.
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3. Thevenin-Norton equivalent network theorem
COOLLARY
Under these conditions, whatever load may be,
V(s) across 1 and 1’ and I(s) through 1 and 1’
remain unchanged
by:
nged when network
netw
workk ॉ is
is replaced
replacced b
y:
either itss Thevenin
equivalent
network
Thevenin e
quivalent n
etworkk
or its Norton
orton eequivalent
quiivale
ent network.
Furthermore:
™ Above formula is especially useful in
computing Thevenin equivalent network
in circuits with dependent sources.
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3. Thevenin-Norton equivalent network theorem
¾ In general, Thevenin equivalent impedance can always be obtained by:
1.
2.
3.
setting to 0 all independent sources of ॉ,
connecting a “test” current source It(s) to terminals 1 and 1’, and
using node
de analysis
anallysis tto
o calculate
caalculate Vt(s), Laplace
Laplace ttransform
ransfo of zero-state
responsee tto
o It((s).
s ).
Then:
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Chapter 8. Network Theorems
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Chapter Contents
0. Introduction
1. The substitution theorem
2. The superposition theorem
3. Thevenin-Norton
equivalent
network
ton e
quivalent n
etwork ttheorem
heore
em
4. The reciprocityy theorem
Electric Circuits 2
Chapter 8. Network Theorems
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4. The reciprocity theorem
RECIPROCITY THEOREM
y Consider an LTI network ॉܴ which consists of:
¾ resistors,
¾ inductors,
¾ coupled inductors,
ductors,
¾ capacitors, and
¾ transformers
ers only.
on
nly.
y Elements below
w aree rules
ruless o
out:
ut :
¾ gyrators,
¾ dependent sources, and
¾ independent sources.
y ॉܴ is in zero state and is not degenerate.
y Connect 4 wires to ॉܴ thus obtaining 2 pairs of terminals 11’ and 22’.
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4. The reciprocity theorem
Statement 1
o Connect a voltage source e0 to terminals 11’ and
observe zero-state current response j2 in a short
circuit connected
22’..
ted to 22
o terminals
term
minals 22’
22’
Next, connect ssame
ame vvoltage
oltage ssource
ource e0 tto
ero-staate current
currrent response ݆෡1 iin
and observe zero-state
n a short
shorrt
ted to 11’.
circuit connected
eoreem asserts
asserts tthat:
hat:
Reciprocity theorem
and
nd
ology aand
nd e
lement values of ॉܴ an
whatever topology
element
whatever waveform e0 of source:
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4. The reciprocity theorem
Statement 2
‰ Connect a current source i0 to terminals 11’ and
observe zero-state voltage response v2 across
open-circuited
d terminals 22’.
22
2.
o tterminals
erm
minals 22’
22’
Next, connect same
same current
current source
source i0 tto
and observe zero-state
ero-staate voltage
volttage response ‫ݒ‬ෝ1 aacross
cross
d terminals 11’.
open-circuited
eoreem aasserts
sserts tthat:
hat:
Reciprocity theorem
whatever topology
element
ology aand
nd e
lement values of network
netwo
ork ॉܴ and
whatever waveform i0 of source:
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4. The reciprocity theorem
Statement 3
™ Connect a current source i0 to terminals 11’ and
observe zero-state current response j2 in a
short circuit connected
onnected to 22’.
22 .
o tterminals
erminaals 22’
22’
Next, connect a vvoltage
oltage ssource
ource e0 tto
and observe zero-state
ero-staate response
resp
ponse ‫ݒ‬ෝ1 acrosss
d terminals 11’.
open-circuited
eoreem aasserts
sserts tthat:
hat:
Reciprocity theorem
whatever topology
ology aand
nd element
element values of network
netwo
ork ॉܴ and
whatever waveform of source,
if i0(t) and e0(t) are equal for all t:
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4. The reciprocity theorem
¾ In Statement 1:
9 We observe short-circuit currents.
9 Assertion says that if voltage source e0 is
interchanged
zero-impedance
ed for a zero
o-im
mped
dancce ammeter,
amme
eter,
reading of aammeter
will
mmeter w
ill not
not change.
change.
9 Source and
d meter are zero-impedance
ed
devices.
evices.
¾ In Statement 2:
9 We observe open-circuit voltages.
9 Assertion says that if current source i0 is
interchanged for an infinite-impedance voltmeter,
readings of voltmeter will not change.
9 Source and meter are infinite-impedance devices.
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4. The reciprocity theorem
¾ In Statement 3:
9 For both measurements, there is:
o
o
an infinite impedance connected to 11’,
a zero impedance
mp
pedan
nce cconnected
onnected to
to 22’.
22’.
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4. The reciprocity theorem
Reciprocity in terms of network functions
Statement 1
y We define transfer admittance from 11’ to 22’ as:
We define transfer
11’
nsferr admittance
admittance ffrom
rom 22’
22’ to 1
1’ aas:
s:
Reciprocity theorem asserts that:
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4. The reciprocity theorem
Reciprocity in terms of network functions
Statement 2
o We define transfer impedance
p
from 11’ to 22’ as:
We define transfer
nsferr iimpedance
mpedance ffrom
rom 22’
22’ to 11’
11
1’ as:
as:
Reciprocity theorem asserts that:
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4. The reciprocity theorem
Reciprocity in terms of network functions
Statement 3
‰ We define transfer current ratio from 11’ to 22’ as:
We define transfer
11’
nsferr voltage
voltage ratio
ratio ffrom
rom 22’
22’ to 1
1’ aas:
s:
Reciprocity theorem asserts that:
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Chapter 8. Network Theorems
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4. The reciprocity theorem
o Any network which satisfies reciprocity theorem is called a reciprocal
network.
o Reciprocity theorem guarantees that any network made of LTI resistors,
capacitors, inductors,
ductors, coupled
coup
pled
d iinductors,
nd
ducttors, and
d ttransformers
ransformers is a reciprocal
network.
o It is also a fact that some LTI networks tha
that
at contain
contain d
dependent
e p e nd
sources are
reciprocal, whereas
others
ereaas oth
hers are not..
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4. The reciprocity theorem
Example:
™ Statement 1:
¾ Terminal pairs 11’ and 22’ are obtained
by performing
pliers
mingg p
lie
ers entries.
en
ntrie
es.
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Chapter 8. Network Theorems
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4. The reciprocity theorem
Example:
™ Statement 2:
¾ For the sake of variety, let us now pick soldering-iron entries.
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Chapter 8. Network Theorems
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4. The reciprocity theorem
Example:
™ Statement 3:
¾ For variety, let us pick 11’ to be defined by a soldering-iron entry,
pick
22’
be
defined
pliers
p
ick 2
2’ tto
ob
ed
efined as
as a p
liers eentry.
ntry
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Chapter 8. Network Theorems
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