network KN Toosi University of Technology
Chapter 8. Network Theorems
Assistant Professor,
Electrical and Computer Engineering,
K. N. Toosi University of Technology
References:
Basic Circuit Theory, by Ch. A. Desoer and E. S. Kuh, 1969
KNTU
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
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Chapter 8. Network Theorems
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0. Introduction
o In this chapter, we study 4 very general and useful network theorems:
&frac34; substitution theorem,
&frac34; superposition theorem,
&frac34; Thevenin-Norton
Nortton e
equivalent
quivvale
ent n
network
etwork theorem,
theeorem,
&frac34; 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:
&frac34; holds for all networks with a unique solution,
&frac34; 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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0. Introduction
 A linear network:
&frac34; 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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0. Introduction
 Superposition theorem and Thevenin-Norton equivalent network
theorem apply to all linear networks:
&frac34; time-invariant or
&frac34; time-varying.
ng.
 Reciprocity theorem
eorem applies
appliees to
to a more
mo
ore restricted
resttricted class
classs of
of linear
lin
networks:
&frac34; elements must be time-invariant.
&frac34; elements can not
not include:
include:
dependent
ndentt ssources,
ource
es,
 independent sources, and
 gyrators.

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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
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Chapter 8. Network Theorems
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1. The substitution theorem
SUBSTITUTION THEOREM
&frac34; 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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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:
&frac34; by a current source I or
&frac34; by a voltage source V.
y In both cases, solutions are same as
that obtained originally.
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1. The substitution theorem
Example 2:
o Consider any LTI network which:
&frac34; is in zero state at time 0
&frac34; 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:
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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
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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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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(&middot;) is a function of vector-input waveform:
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2. The superposition theorem
&frac34; If function f is linea
linear,
ar, then
then superposition
superposition theorem
theo
orem holds:
holds:
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2. The superposition theorem
y In general, superposition does not apply to nonlinear networks.
y If a particular network has:
&frac34; all linear elements
&frac34; 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:
&frac34; element values,
alues,
&frac34; source location,
ation
n,
&frac34; source waveform,
veform and
&frac34; response.
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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:
&frac34; superposition
tion ttheorem
heorem applies
applies iin
n particular
particu
ulaar to
to sinusoidal
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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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.
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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
etworrk ttheorem
heore
em
4. The reciprocityy theorem
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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
d, aand
nd
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,
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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3. Thevenin-Norton equivalent network theorem
THEVENIN-NORTON THEOREM
&frac34; 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.
&frac34; ॉ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
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
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
&frac34; 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
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Chapter 8. Network Theorems
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4. The reciprocity theorem
RECIPROCITY THEOREM
y Consider an LTI network ॉܴ which consists of:
&frac34; resistors,
&frac34; inductors,
&frac34; coupled inductors,
ductors,
&frac34; capacitors, and
&frac34; transformers
ers only.
on
nly.
y Elements below
w aree rules
ruless o
out:
ut :
&frac34; gyrators,
&frac34; dependent sources, and
&frac34; 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
&frac34; 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,
will
mmeter w
ill not
not change.
change.
9 Source and
d meter are zero-impedance
ed
devices.
evices.
&frac34; 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
&frac34; 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’
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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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:
&frac34; Terminal pairs 11’ and 22’ are obtained
by performing
pliers
mingg p
lie
ers entries.
en
ntrie
es.
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4. The reciprocity theorem
Example:
 Statement 2:
&frac34; 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:
&frac34; 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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