1

! Topology(Equations( (r equations)

(KCL: n-1 equations)

(KTL: r-n+1 equations)

! Element(Equations((r equations)The$number$of$these$equations$is$equal$to$the$number$of$the$branches$as$they$are$the$equation$modeling$each$element$of$the$circuit,$and$hence$any$branch.$

The$circuit$analysis$problem$is$described$by$2r$equations$in$2r$unknowns.$The$equations$are$the$topology$equations$and$the$element$equations.$The$unknowns$are$the$branch$tensions$and$the$branch$currents.

Circuit with n nodes and r branches

!!

=

=

m r

n r

0v0i

Methods of the Circuit AnalysisOverview

General method

! General'Method'of'the'Circuit'Analysis:(r'branches,*2r'unknowns,*2r'equations)

! Method'of'the'Tension'Substitution:(r'branches,*r'unknowns,*r'equations)

k0,kkk

m k

n k

ViRv

0v0i

+=

=

=

!!

These%equations%are%for%a%generic%element%with%a%resistors%and%a%tension%source

!!

=+

=

m k0,kk

n k

0ViR0i

(n-1) eq.s

(r-n+1) e q.s

r eq.s

(n-1) eq.s

(r-n+1) eq.s

Methods of the Circuit Analysis

Department*of*Electrical,*Electronic,*and*Information*Engineering*(DEI)*> University*of*Bologna

2

Superposition+Principle:As#a#consequence#of#the#linearity#of#the#equations#which#describe#the#circuit#the#solution#of#the#equations#of#the#Tension#Substitution#Method#is#given#by#the#branch#currents#expressed#by#a#linear#combination#of#the#independent#sources#of#the#circuit.

ir = Gr1V01 + Gr2V02 + … + GrlV0l + αr,l+1I01 + αr,l+2I02 +…+ αr,gI0g

We#must#stress#that#this#is#only#valid#in#the#linear#case.#In#order#to#be#in#this#case,#the#element#equations#must#be#linear.

V0k and#I0k are#the#input#of#the#circuit,#ir is#an#output.##Usually#the#source#voltages#and#source#currents#are#the#inputs#of#the#circuit.#The#branch#voltages#and#branch#currents#are#the#outputs.

The#superposition#principle#states#that#any#branch#current#is#the#algebraic#sum#of#the#currents#through#the#branch#due#to#each#independent#source#acting#alone#(the#same#statement#holds#for#the#branch#voltages#also).#

Methods of the Circuit Analysis

In Out

=Out$$F$ $In

Methods of the Circuit Analysis

The transfer function can be defined in the time domain [voltages and currents: v(t) and i(t)], in the frequency domain [voltages and currents: V and I ] or in the Laplace transform-domain.

Transfer FunctionIn#a#circuit#we#will#distinguish#between#input& and##output.#The#inputs#are#the#independent#current#and#voltage#sources,#also#said#excitations.#The#output#are#the#branch#currents#and#the#tensions#(branch#voltages,#node#voltages#or#any#potential#difference#between#two#nodes).

In#a#linear,#time#independent#circuit#for#an#input=output#pair#a#transfer&function&(or#network&function)#is#defined.#The#transfer#function#is#the#ratio#between#an#output#and#an#input#when#the#other#sources,#except#the#one#considered,#are#switched#off.##

Linear,,time.independent

network

Department&of&Electrical,&Electronic,&and&Information&Engineering&(DEI)&; University&of&Bologna

3

As#it#had#been#stated#by#the#superposition#principle,#in#a#linear#circuit#any#voltage#vrand#any#current#is can#be#expressed#as#a#linear#combination#of#the#p independent#tension#sources#and#the#q independent#current#sources#:

vr =# αr1 V01 +#αr2 V02 +#….#+#αrp V0p +#Rr1 I01 +#Rr2 I02 +#….#+#Rrq I0qis =#Gs1 V01 +#Gs2 V02 +#….#+#Gsp V0p +#βs1 I01 +#βs2 I02 +#….#+#βsq I0q

The#coefficients#αri,#Rrj,#Gsi,#βsj are#the#transfer#functions#of#the#r voltages#and#the#stensions#when##coupled#two#by#two#to#the#p tension#sources#and#the#q current#sources.#The#transfer#functions##αri and#βsj are#dimensionless.##The#Rrj have#the#dimension#of#a#resistance#or#an#impedance#(Ω).#The#Gsi have#the#dimension#of#a#conductance#or#an#admittance#(S#=#1/Ω).

Voltage Gain: αrp = vr

V0p V0i = 0 per i≠pI0j = 0 ∀ j

Transf. Imped.: Rrq = vr

I0q V0i = 0 ∀ iI0j = 0 per j ≠ q

Transf. Admitt.: Gsp = is

V0p V0i = 0 per i≠pI0j = 0 ∀ j

Current Gain: βsq = isI0q V0i = 0 ∀ i

I0j = 0 per j ≠ q

Transfer Function

Department)of)Electrical,)Electronic,)and)Information)Engineering)(DEI))6 University)of)Bologna

Methods of the Circuit Analysis

! The$n&1$node$voltages$are$determined$by$the$solution$of$a$linear$non&homo&geneous$system$of$n&1$equations.$As$the$node$voltages$are$known,$the$branch$currents$are$obtained$from$eq.$3$and$the$branch$voltages$are$derived$from$eq.$2.

Nodal Analysis (n"1$$equations,$n"1$unknowns)

Step$to$determine$the$node$voltages1.//Define/the/reference/node/and/assign/the/n71/node/voltages/which/are/the/voltage/of/the/non7reference/nodes/with/respect/to/the/reference/one.

2.///Apply/KCL/to/each/non7reference/node.3.///Apply/KTL/to/refer/the/node/voltages/to/the/branch/voltages.

4. Express/the/branch/currents/in/terms/of/the/node/voltages/through/the/element/equations/and/substitute/them/in/the/cur7rents/equations/given/by/KCL/in/step/2./

5.///Solve/the/resulting/simultaneous/equations/to/obtain/the/node/voltages.

k1h

32

uh

+&vr

h k

uk0

•

• •

• •

•

•

•

4

! The$method$is$based$on$the$node%voltages,$uk (k=$1,2,..,n61),$that$are$the$potential$differences$between$each$non6reference$node$and$the$reference$node$(ground).$Hence$each$node$voltage$is$the$voltage$of$that$node$with$respect$to$the$reference$node." KCL$$is$applied$to$each$non6reference$node$k,$k$=$1,2,…,$n61$(figure$above):

i1+i2+….+ih = 0 (1)" KTL$is$applied$to$relate$the$node$voltages$to$the$branch$voltages$(figure$below):

vr = uk-uh (2)

" The$currents$are$expressed$by$the$element$equations

(3)

" By$substituting$eq.$3$into$eq.$1$a$set$of$n61$equations$in$n61$unknowns$(uk,$for$k=1,2,…,$n61)$is$obtained.

Methods of the Circuit Analysis

This method is utilized in the AC regime where Ir , Uk and Zr replaces ir ,vr , and Rr .

r

hk

r

rr R

uuRvi !

==

Nodal Analysis (n"1$$equations,$n"1$unknowns)

k1h

32

uh

+6vr

h k

uk0

•

•

• •

•

•

• •

In#the#circuit#there#are#branches#that##contain#only#voltage#sources#(independent#or#controlled#sources).#For#these#branches#the#currents#cannot#be#expressed#in#terms#of#the#voltages.#A#branch#with#only#a#voltage#source#can#be##incorporated#into#a#closed#surface.#Thereafter#KCL#is#applied#to#this#surface.#This#branch#is#said#supernode.#As#it#results#from#KTL,#the#difference#of#the#node#voltages#at#the#terminals#of#the#source#branch,#is#given#by#the#source#voltage:

uk-uh = V0

!"

!#$

==+

=++

032

5432

541

Vu-u0 i-i -i i

0 i i i

• •

i4

i2

R4

R2

0

u2 u3

i3

R3

-+

V0u1 i5 R5

i1

R1

By expressing the 5 currents by means of the 3 node voltages, the system of equations is given by three equations with three unknowns which are the node voltages.

Nodal AnalysisSupernode

•

•••

Department-of-Electrical,-Electronic,-and-Information-Engineering-(DEI)-8 University-of-Bologna

5

Tellegen’s Theorem! The$Tellegen’s theorem$states$that$in$an$insulated$circuit$(not$connect$to$other$circuits$or$networks)$the$algebraic$sum$of$the$power$calculated$for$each$branch$is$equal$to$zero.

! Alternatively$it$can$be$stated$that$the$total$power$delivered$by$the$sources$is$equal$to$the$power$absorbed$by$the$loads.

0 iv 1k

kk =!=

r

Tellegen’s theorem is a consequence of the energy conservation principle. It fulfills the topology equations (KCL and KTL).

Methods of the Circuit Analysis

Department)of)Electrical,)Electronic,)and)Information)Engineering)(DEI))6 University)of)Bologna

Consider)an)independent)current)source)connected)to)the)port)AB)of)circuit)N.)Due)to)the)linearity)of)N)the)voltage)v is)given)by)the)linear)combination)of)the)p voltage)sources)V0i (i=1,..,p),)the)q current)sources)I0j (j=1,2,…,q))of N,)and)the)current)source)i. The)linear)combination)coefficients)are)the)transfer)functions:

j 0 Ii 0 Veq0 ieq

q

1j0jj

p

1i0iieqeqeq

q

1j0jj

p

1i0iieq

oj0i

iv R v V

I R V V V i R v

I R V i R v

!=!==

==

==

==

+=+="

++=

##

##

;or:

where $

$

Equivalent CircuitsThévenin’s Theorem

•

•

A

B

v i

Circuit'N •

Department)of)Electrical,)Electronic,)and)Information)Engineering)(DEI))6 University)of)Bologna

6

Therefore'the'following'relation'is'obtained:'

This'relation'is'the'element'equation'that'describes'the'series'between'an'independent'voltage'source'and'a'resistor.'It'defines'the'current'controlled'Thévenin equivalent-circuit.

Thévenin’s theorem:

! A#linear,#time-independent#circuit#N#with#an#highlighted#port#is#considered.#The#circuit#is#equivalent#to#an#independent#voltage#source#in#series#with#a#resistor.#The#voltage#of#the#source#is#the#open#circuit#voltage#between#A#and#B.#The#resistor#is#the#equivalent#resistor#seen#from#the#port#AB#when#all#independent#sources#of#N#are#switched#off.#

Thévenin’s Theorem

eqeq V i R v +=

Req

Veq•

• B

A

v

i

+-

Thévenin’s equivalentcircuit

•

•

A

B

v

Circuit'N

Department-of-Electrical,-Electronic,-and-Information-Engineering-(DEI)-> University-of-Bologna

j 0 Ii 0 Veq0 veq

q

1j0jj

p

1i0iieqeqeq

q

1j0jj

p

1i0iieq

oj0i

vi G i I

I V G I I vG i

I V G vG i

!=!==

==

==

==

+=+="

++=

##

##

;or

where $

$

Norton’s TheoremConsider)an)independent)voltage)source)connected)to)the)port)AB)of)circuit)N.)Due)to)the)linearity)of)N)the)current)i is)given)by)the)linear)combination)of)the)pvoltage)sources)V0i (i=1,..,p),)the)qcurrent)sources)I0j (j=1,2,…,q))of N,)and)the)by)the)voltage)source)v.)The)linear)combination)coefficients)are)the)transfer)functions:

•

•

A

B

v

i

+-

Circuit'N

Equivalent Circuits

Department)of)Electrical,)Electronic,)and)Information)Engineering)(DEI))6 University)of)Bologna

7

eqeq I vG i +=

Norton’s TheoremTherefore'the'following'relation'is'obtained:'

This'relation'is'the'element'equation'that'describes'the'parallel'between'an'independent'current'source'and'a'resistor.'It'defines'the'current'controlled'Norton&equivalent&circuit.

Norton’s&theorem.! A#linear,#time-independent#circuit#N#with#a#highlighted#port#is#considered.#The#circuit#N#is#equivalent#to#the#parallel#between#an#independent#current#source#and#a#resistor.#The#current#of#the#source#is#that##flowing#through#N#when#the#port#AB#is#short#circuited.#The#resistor#is#the#equivalent#resistor#seen#from#the#port#AB#when#all#independent#sources#of#N#are#off.#

Norton’s equivalent circuit

eq eqi G v I= +

GeqIeq

•

• B

A

v

i

Geq = 1/Req

•

•

A

B

i

v

Circuit'N

Department&of&Electrical,&Electronic,&and&Information&Engineering&(DEI)&= University&of&Bologna

Summary

(2) I vG i :circuit equivalent sNorton'

(1) V i R v:circuit equivalent sThévenin'

eqeq

eqeq

+=

+=

When%eq.%1%is%divided%by%Req,%eq.%2%is%obtained.%When%eq.%2%is%divided%by%Geqeq.%1%is%obtained.%%If%Veq of%Thévenin’s circuit%is%known,%Ieq of%Norton’s%circuit%can%be%derived%by%subtracting%eq.%1%from%eq.%2%multiplied%by%Req.%Alternatively%from%Ieq of%Norton’s%circuit,%Veq of%Thévenin’s circuit%can%be%derived.%

0G ( GI

V G1 R

0R ( RV

I R1 G

eqeq

eqeq

eqeq

eqeq

eqeq

eqeq

)ifnin n to Thévefrom Norto;

)ifton nin to Norfrom Théve;

!"==

!"==

Equivalent Circuits

Department)of)Electrical,)Electronic,)and)Information)Engineering)(DEI))6 University)of)Bologna

8

Maximum Power Transfer

Req

Veq

•

•B

A

RL

i

+-

RL=Req

P (RL)

RL

PMax

The$Thévenin equivalent$circuit$can$be$used$in$finding$the$maximum$power$which$a$linear$circuit$can$deliver$to$a$load.$

The&entire&circuit&is&replaced&by&the&Théveninequivalent&&except&for&the&load&which&is&an&adjustable&load&resistor&RL.&The&power&delivered&to&the&load&is

For&a&given&circuit,&Veq and&Req are&fixed.&&By&varying&RL,&the&power&delivered&to&the&load&varies.&The&power&is&small&or&large&for&small&or&large&values&of&RL.&The&maximum&power&transfer&theorem&states&that:

! Maximum$power$is$transferred$to$the$load$when$the$load$resistance$is$equal$to$the$Thévenin equivalent$resistance$.

2

eq2L L

eq L

Vp R i R

R R! "

= = # $# $+% &

•

•

Department&of&Electrical,&Electronic,&and&Information&Engineering&(DEI)&; University&of&Bologna