ASC_1_2009

ANALYSIS & SYNTHESIS OF

CIRCUITS

2008-2009

CONTENTChapter 1. Topology of circuits

Chapter 2. Flowgraphs

Chapter 3. State-equations

Chapter 4. Stability analysis

Chapter 5. Description of electric circuits

Chapter 6. Passive one-ports

Chapter 7. Passive two-ports

Chapter 8. Impedance matching circuits

Chapter 9. Passive filters

Chapter 10. Synthesis of circuits

Prof. dr. ing. Marina TOPA

Components of the mark (100 points = mark 10)

During the semester (50 points+10 points)→ tests regarding theoretical knowledge TC:3 tests x 5p=15p→ tests regarding solution of problems (applications) TS:3 tests x 5p=15p→ tests regarding practical skills (labs) NL: 10p→ participation at courses AC: 10p

! An additional test (TC4+TS4)Final exam (50 points) E:→ theory 10p→ multiple-choice questions 20p→ problems 20p

Final markN=(TC+TS+NL+AC+E)/10

Tests regarding theoretical knowledge

1. TC1(C1, C2) 23.03.09

2. TC2 (C3, C4) 06.04.09

3. TC3 (C5,C6, C7) 04.05.09

4. TC4 (C8, C9) 18.05.09

Prof. dr. ing. Marina TOPA

References-for lecture-

• V. Popescu, Semnale, circuite şi sisteme, Partea I. Teoriasemnalelor, Editura Casa Cartii de Stiinta, Cluj-Napoca, 2001.

• Marina Topa, Semnale, circuite şi sisteme, Partea a II-a. Teoria sistemelor, Editura Casa Cartii de Stiinta, Cluj-Napoca, 2002.

• Gh. Cartianu, M. Săvescu, I. Constantin, D. Stanomir, Semnale, circuite şi sisteme, Editura didactică şi pedagogică, Bucureşti, 1980.

• Adelaida Mateescu, N. Dumitriu, L. Stanciu, Semnale şi sisteme. Aplicaţii în filtrarea semnalelor, Editura Teora, Bucureşti 2001.

• A.V. Oppenheim, A. S. Willsky, I. T. Young, Signals andSystems, Prentince-Hall, 1983.

• A. D. Poularikas, S. Seely, Signals and Systems, PWS Publishers, Boston, 1985.

References-for applications-

• Adelaida Mateescu, D. Stanomir (coordonatori), Probleme de analiza şi sinteza circuitelor, Editura tehnică, Bucureşti, 1976.

• M. Săvescu, T. Petrescu, S. Ciochină, Semnale, circuite şi sisteme. Probleme. Editura didactică şi pedagogică, Bucureşti, 1981.

• Ioana Popescu, Victor Popescu, Erwin Szopos, Marina Ţopa, Semnale, circuite şi sisteme. Îndrumător de laborator IV, Editura Casa Cărţii de Ştiinţă, Cluj-Napoca, 2003.

References- for the applications -

• Adelaida Mateescu, D. Stanomir (coordonatori), Probleme de analiza şi sinteza circuitelor, Editura tehnică, Bucureşti, 1976.

• M. Săvescu, T. Petrescu, S. Ciochină, Semnale, circuite şi sisteme. Probleme. Editura didactică şi pedagogică, Bucureşti, 1981.

• Ioana Popescu, Victor Popescu, Erwin Szopos, Marina Ţopa, Semnale, circuite şi sisteme. Îndrumător de laborator IV, Editura Casa Cărţii de Ştiinţă, Cluj-Napoca, 2003.

CONTENTCourse 1. Topology of circuits

Course 2. Flowgraphs

Course 3. State-equations

Course 4. Stability analysis

Course 5. Description of electric circuits

Course 6. Applications I

Course 7. Passive one-ports

Course 8. Passive two-ports

Course 9. Impedance matching circuits

Course 10. Applications II

Course 11. Passive filters

Course 12. Passive filters & Synthesis

Course 13. Synthesis of circuits

Course 14.Applications IIIProf. dr. ing. Marina TOPA

Course 1. Topology of electric circuits

1.1 The approach of topology

1.2 Topological graphs

1.3 Matrix description of topological graphs

1.4 Matrix equations of electric circuits

Prof. dr. ing. Marina TOPA Analysis & Synthesis of Circuits -1


Network topology deals with those properties of lumped networks which are related to the interconnection of branches only.

Graph theory = mathematical discipline which deals with network topology !



→ Oriented (directed) graphs→ Nonoriented (undirected) graphsThe directed graph Gd associated with the given circuit C is obtained by replacing each two-terminal element by a line segment, called edge with an arrow in the same direction as the assumed positive current through that edge.


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1.2 Topological graphsPath = a set of edges b1, b2, …, bn in Gn is called a path between two nodes Vl and Vk if the edges can be labeled such that:1.Consecutive edges bi and bi+1 always have a common endpoint.2.No node of Gn is the endpoint of more than two edges in the set.3.Vj is the endpoint of exactly one edge in the set, and so is Vk.A path = a route between two nodes.


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1.2 Topological graphsA directed graph Gn is said to be connected if there exists a path between any two nodes of the graph.


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If a directed graph Gn is not connected, it consists of a number of connected subgraphs. Notations: l=nr of edges in the circuitn=number of nodes of the circuit;s=number of connected subgraphs for a not connected graph).

1.2 Topological graphs

Loop = a subgraph Gs of a graph Gn where :1. Gs is connected.2. Every node of Gs has exactly two incident edges of Gs.


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1.2 Topological graphsTree = a subgraph Gs of a connected graph Gn where:1. Gs is connected.2. Gs contains all nodes of Gn.3. Gs has no loops.Tree branches = edges in a tree.Links (chords) = edges which do not belong to a tree.Cotree = all the links of a given graph.


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1.2 Topological graphsCutset = a set of branches of a connected graph Gn where:1. The removal of the set of branches (but not their endpoints) results

in a graph that is not connected. 2. After the removal of the set of edges, the restoration of any branch

from the set will result in a connected graph again.


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1.2 Topological graphsFundamental loop = a loop which contains only one link

Fundamental cutset = a cutset which contains only one tree branch


(2)

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1

2(1)

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Σ2

Σ5

Σ3

b6

(2)

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1

2(1)

3(3)

(4)

6b4

b1

1.3 Matrix description of topologyIncidence matrix = a (n x l) matrix for a directed graph Gd with nnodes and l edges:


[ ] where:

1 if edge is incident at node and the arrow is pointing away from node i

1 if edge is incident at node and the arrow is pointing toward node i

0 if edge is not inci

ij

ij

ij

ij

a

a j i

a j i

a j

=

=

= −

=

aA

dent at node i

1 0 0 1 1 01 1 1 0 0 00 1 0 1 0 10 0 1 0 1 1

2 3 5 1 4 61234

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥− −⎢ ⎥

− −⎣ ⎦

aA

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1.3 Matrix description of topologyReduced incidence matrix = the incidence matrix after deletion of slines, where s is the number of the connected parts. Obs.: for a connected graph s=1


1 0 0 1 1 01 1 1 0 0 00 1 0 1 0 1

2 3 5 1 4 6123

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥− −⎢ ⎥⎣ ⎦

A

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1.3 Matrix description of topologyLoop (circuit) matrix B = a (nl x l) matrix for a directed graph Gdwith nl nodes and l edges:


[ ] where:

1 if edge j is in loop i and their directions agree

1 if edge j is in loop i and their directions oppose

0 if edge j is not in loop i

ij

ij

ij

ij

b

b

b

b

=

=

= −

=

B

1 0 0 1 0

1 2 3

1

4 5 6

ib⎡ ⎤

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

B

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(1) bi

1.3 Matrix description of topologyFundamental (basic) loop matrix Bf = a submatrix of B that consists of the maximum number of independent rows of B; it shows the relationship fundamental loops-edges. Bf is a (l-n+s) x l matrix.


1 2 3 4 5 61 1 1 1 0 0 04 0 1 0 1 1 06 0 0 1 0 1 1

f

bbb

− −⎡ ⎤= ⎢ ⎥

⎢ ⎥⎢ ⎥−⎣ ⎦

B

b6

(2)

5 4

1

2 (1)

3 (3)

(4)

6 b4

b1

If the branches and links are grouped, then: [ ]=f r cB B B2 3 5 1 4 6

1 1 0 1 0 011 0 1 0 1 040 1 1 0 0 16

f

bbb

⎡− − ⎤= ⎢ ⎥

⎢ ⎥⎢ ⎥−⎣ ⎦

Bc c=B 1

1.3 Matrix description of topologyCutset matrix Q = a (nc x l) matrix for a directed graph Gd with ncnodes and l edges:


[ ] where:

1 if edge j is in cutset i and their directions agree

1 if edge j is in cutset i and their directions oppose

0 if edge j is not in cutset i

ij

ij

ij

ij

q

q

q

q

=

=

= −

=

Q

1 1 0 0 1

1 2 3

1

4 5 6

iΣ −⎡ ⎤

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

−Q

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1.3 Matrix description of topologyFundamental cutset matrix Qf = a submatrix of Q that consists of the maximum number of independent rows of Q; it shows the relationship fundamental cutsets-edges.


1 2 3 4 5 62 1 1 0 1 0 03 1 0 1 0 0 15 0 0 0 1 1 1

f

Σ −⎡ ⎤= ⎢ ⎥Σ −⎢ ⎥

⎢ ⎥Σ −⎣ ⎦

Q

If the branches and links are grouped, then: [ ]=f r cQ Q Q2 3 5 1 4 6

1 0 0 1 1 020 1 0 1 0 130 0 1 0 1 15

f

⎡ − ⎤Σ= ⎢ ⎥−Σ ⎢ ⎥

⎢ ⎥−Σ ⎣ ⎦

Qr r=Q 1

(2)

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1

2 (1)

3 (3)

(4)

6

Σ2

Σ5

Σ3

1.3 Matrix description of topologyRelationships between the topological matrices


[ ] [ ];

; ; .

T T

Tr c

= =

= = = −f f

f r c f r c

B A 0 AB 0

B B 1 Q 1 Q B Q

Topological equations of circuits

I Kirchhoff( ) ; ( ) ;( ) ; ( ) ;

( ) ( ) ( )

( ) ( ); ( ) ( )

T

T T

t st s

t t t

t t s s

= == =

= − =

= =r

f f

r c c c

f c f c

A i 0 A I 0Q i 0 Q I 0

i Q i B i

i B i I B I

( ) ; ( ) ;

( ) ( ) ( )

( ) ( ); ( ) ( )T T

t t s st s

t t t

t t s s

= =

= − =

= =

f fT

c r r c r

f r f r

II Kirchhoffu( ) = Av( ); U( ) = AV( );B u 0 B U 0

u B u Q u

u Q u U Q U


( ) ( ) ( ) ( ) ( ) ;

( ) ( ) ( ) ( ) ( ) ;

ij ij ij ij ij

ij ij ij ij ij

U s E s Z s I s J s

I s J s Y s U s E s

⎡ ⎤+ = +⎣ ⎦⎡ ⎤+ = +⎣ ⎦

Ohm’s law

(i) (j)Zij

Eij

Jij

Iij

UZij

( ) ( ) ( )[ ( ) ( )]( ) ( ) ( )[ ( ) ( )]

s s s s ss s s s s

+ = ++ = +

U E Z I JI J Y U E

For linear circuits with passive elements and real sources, but no mutual inductances:

where Z(s) and Y(s) are diagonal.

1.3 Matrix equation for linear passive circuits I Elementary method


( ) ( )[ ( ) ( )]Ohm's law: ( ) ( ) ( )[ ( ) ( )]

( ) ( ) ( ) (

( ) ( )II Kirch

I Kirc

( )

hhoff: h

( ) (

off: (( )

) )

) ( )

)

(

n sc

TT

n

T

s s

s s s ss

ss

s s s ss s s s

s

s

s

s

s⎧

⇒ ⇒⎨= +⎩+ =

=

⇓

⇒ −

=

=

+

=

=UA

A VJ AY U E

I J Y U EAY A V

Y V J

U A V

AJ AY E

A I 0II Nodal method

I Kirchhoff: ( )(

( ) ( ) ( ( ) ( ) ( ))Ohm's law: ( ) ( ) ( )[ ( ) (

II Kirchhoff: (

( )( ) ( ( ) ( ) ( ))

)

)])

;f

f

f

f

fs s s s ss s s s

s

ss s s

s

ss

s⎡ ⎤ ⎡ ⎤

⇒ =⎢ ⎥ ⎢ ⎥

= −

−⎣ ⎦ ⎣

+ =

==

=

⎦

⎧

⎪⎩+

⎪⇒ ⎨f B Z

Q I 0A I 0

A 0I

B Z B E Z J

I B E Z JU E Z I

B U 0J

Dimension : 2l

Dimension : n-s

1.3 Matrix equation for linear passive circuitsIII Loop (mesh) method

Prof. dr. ing. Marina TOPA Analysis & Synthesis of Circuits -1( ) (

( ) ( )[ ( ) ( )]Ohm's law:

I Kirchhoff: (

( )

( )

)

( ) ( )[ ( ) ( )]( ) ( ) ( ) ( ) ( )

( )II Kirchhoff: ( ) ( )

( )

)

f f

Tf f

Tf rT

r f f

s

f

r

f

r c

r

r

s s s ss s s s s

s s s s

s s

s

ss

ss

s

s⎧⎪⇒ ⇒⎨ = +⎪⎩+ = +

=

=⇓

=

==

⇒ −

U QQ J Q Y U E

I J Y U EQ Y Q U Q J

Y U J

UU Q

Y

U

Q E

Q I 0II Branch-based method

( ) ( ) ( ( ) ( ) ( ))Ohm's law: ( ) ( ) (

II Ki)[

rchh( ) ( )]

( ) ( ) ( )

I Kirchhoff: ( ) ( )( )

off: ( )

( ) ( ) (

( ) (

( )

)

)

Tf c T

f c

f f

Tf f c

c c

f

c

f

s s s s ss s s s

s

ss s

s

s

s

s s

s

s s

s s

= −+ = +

=⎧ =⎪⇒ ⎨⎪⎩

⇒

=

= −

⇓=

f

I B II B

B Z I B E Z JU E Z I J

B Z B I B

B U 0

Z

E

I E

B Z J

I

Dimension : n-s

Dimension : l-n+s

Questions1. Draw two circuits having the same topological graph.2. Define in your own words an edge and a node in a directed graph.3. Draw a graph composed of 7 edges and point out its paths (loops, cutsets).4. Draw a not connected directed graph having 2 connected subgraphs.5. Draw a graph composed of 7 edges and choose a tree. Point out the fundamental loops and cutsets.6. Draw a graph having at least a cutset that is not placed around a node.7. Draw a graph composed of 7 edges and choose a tree. Compute the reduced incidence matrix (fundamental loop matrix, fundamental cutset matrix).8. If the reduced incidence matrix is:

draw the corresponding topological graph.9. If the fundamental loop matrix is:

draw the corresponding topological graph.10. If the fundamental cutset matrix is:

draw the corresponding topological graph.Prof. dr. ing. Marina TOPA Analysis & Synthesis of Circuits -1

1 0 0 1 1 00 1 0 0 0 11 1 1 0 0 1

− −⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥− − −⎣ ⎦

A

1 0 0 1 0 0 00 1 0 0 1 0 00 1 1 0 0 1 01 0 1 0 0 0 1

f

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥− −⎣ ⎦

B

1 0 0 0 1 1 0 00 1 0 0 0 1 1 00 0 1 0 0 0 1 10 0 0 1 1 0 0 1

f

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥

−⎣ ⎦

Q

Questions10. Draw a circuit having 5 edges and find the topological matrices. 11.Write the impedance (admittance) matrix, the voltage and current sources vectors for the circuit in the figure below.12. Which of the matrix equations methods are more appropriate to be used for the circuit in the figure below?13. Write the nodal (elementary, mesh, branch-based) equations for the circuit in the figure below. 14. Find the current and voltages vectors considering the nodal (elementary, mesh, branch-based) equations of a circuit.


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