CIRCUITS & SYSTEMS

UNIT-IV

IV SEMESTER

EEC-213

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Department of Computer Science and Engineering, BVCOE, New Delhi

Graph Theory

Introduction

• The graph theory or network topology deals with those properties of networks which do not change with the change in the shape of the networks.
• The graph theory concept eases the solution method for solving networks with a large number of nodes and branches.

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GRAPH OF A NETWORK

A linear graph (or simply a graph) is defined as a collection of points called nodes, and line segment called branches, the nodes being joined together by the branches.

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While drawing graph of a given network, the following rules are to be noted.

1. All passive elements between the nodes are represented by lines.
2. The independent current sources and voltage sources are represented by their internal impedances (i.e., current sources by open circuit and voltage sources by short circuit) if they are accompanied by passive element, viz., a shunt admittance in a current source and a series impedance in a voltage source.
3. If the sources are not accompanied by passive elements, an arbitrary impedance (say resistance R) or admittance is assumed to accompany the sources and finally, we find the results by letting the impedance R ® 0 or R ® µ as the case may be for the current or voltage sources.

Department of Computer Science and Engineering, BVCOE New Delhi

TERMINOLOGY

1. Node A node is a point in a circuit where two or more circuit elements join. Example a, b, c, d, e, f and g
2. Essential Node A node that joins three or more elements. Example b, c, e and g Network Topology (Graph Theory) 4.3
3. Branch A branch is a path that connects two nodes. Example v1, R1, R2, R3, v2, R4, R5, R6, R7 and I
4. Essential branch Those paths that connect essential nodes without passing through an essential node. Example cab, cde, cfg, be, eg, bg (through R7), and bg (through I )
5. Loop A loop is a complete path, i.e., it starts at a selected node, traces a set of connected basic circuit elements and returns to the original starting node without passing through any intermediate node more than once. Example abedca, abeg fca, cdebg f c, etc.
6. Mesh A mesh is a special type of loop, i.e., it does not contain any other loops within it. Example abedca, cdeg f c, gebg (through R7) and gebg (through I)

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

CONCEPT OF TREE

For a given connected graph of a network, a connected subgraph is known as a tree of the graph if the subgraph has all the nodes of the graph without containing any loop.

Twigs

The branches of tree are called twigs or tree-branches. The number of branches or twigs, in any selected tree is always one less than the number of nodes, i.e., Twigs = (n 1),

where n is the number of nodes of the graph. For this case, twigs = (4 1) = 3 twigs.

Links and Co-tree

If a graph for a network is known and a particular tree is specified, the remaining branches are referred to as the links. The collection of links is called a co-tree. So, cotree is the complement of a tree. These are shown by dotted lines in Fig.

Note: The branches of a co-tree may or may not be connected,

whereas the branches of a tree are always connected.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

To summarize,

• Number of nodes in a graph = n
• Number of independent voltages = n-1
• Number of tree-branches = n-1
• Number of links = L = (Total number of branches)-(Number of tree-branches) = b-(n-1) = b-n+1
• Total number of branches = b = L + (n-1)

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Properties of a Tree

1. In a tree, there exists one and only one path between any pairs of nodes.

2. Every connected graph has at least one tree.

3. A tree contains all the nodes of the graph.

4. There is no closed path in a tree and hence, tree is circuitless.

5. The rank of a tree is (n 1).

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TIE-SET MATRIX

A tie-set is a set of branches contained in a loop such that each loop contains one link or chord and the remainder are tree branches. Consider the graph and the tree as shown. This selected tree will result in three fundamental loops as we connect each link, in turn to the tree.

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Fundamental Loop 1 (FL1): Connecting link 1 to the tree. Fundamental Loop 2 (FL2): Connecting link 5 to the tree. Fundamental Loop 3 (FL3): Connecting link 6 to the tree. These sets of branches (1, 2, 3), (2, 4, 5) and (3, 4, 6) form three tie-sets.

Department of Computer Science and Engineering, BVCOE New Delhi

For a given graph having n nodes and b branches, tie-set matrix is a rectangular matrix with b columns and as many rows as there are loops. Its elements have the following values:

Bij = 1, if branch j is in loop i and their orientations coincide (i.e., loop current and branch current flow in the same direction);

= -1, if branch j is in loop i and their orientations do not coincide; = 0, if branch j is not in loop i.

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Example For the graph shown in Fig. (a) and tree selected in Fig. (b), the tie-set matrix is written as follows. The entries in the Tie-set schedule are given as +1 or 1 if the branch current is in the same direction as the link current or not. If the branch current does not depend on the link current, then entry is zero.

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CUT-SET MATRIX

A cut-set is a minimum set of elements that when cut, or removed, separates the graph into two groups of nodes. A cut-set is a minimum set of branches of a connected graph, such that the removal of these branches from the graph reduces the rank of the graph by one. In other words, for a given connected graph (G), a set of branches (C) is defined as a cut-set if and only if:

(i) the removal of all the branches of C results in an unconnected graph. (ii) the removal of all but one of the branches of C leaves the graph still connected.

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

The rank of the graph of Fig.(a) = (4-1) = 3

The rank of the graph of Fig.(b) = Addition of the ranks of the subgraphs = (1 + 1) = 2

So, branches [1, 3] may be a cut-set.

Also, removal of the branches 1, 3 and 5 reduces the graph into two connected subgraphs as shown in Fig.(c) and the rank becomes 2. So, [1, 3, 5] may also be a cut-set.

As cut-set is the minimum set of branches and [1, 3] is a subset of [1, 3, 5], so [1, 3] is the cut-set, [1, 3, 5] is not a cut-set.t

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Example Consider the graph shown in Fig.(a). The rank of the graph is 3. The removal of branches 1 and 3 reduces the graph into two connected subgraphs as shown in Fig. (b).

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Procedure for Finding the Fundamental Cut-sets

1. First, select a tree of the given graph.
2. Focus on a tree branch (bk).
3. Check whether removing this tree branch (bk) from the

tree disconnects the tree into two separate parts.

4. All the links which go from one part of this disconnected

tree to the other, together with the tree branch (bk) forms a

fundamental cut-set.

Following this procedure, the fundamental cut-sets for

the graph of Fig. will be

f-cut-set 1: [1, 2, 6];

f-cut-set 2: [2, 3, 5, 6];

f-cut-set 3: [4, 5, 6]

Fundamental Cut-Set A fundamental cut-set (FCS) is a cut-set that cuts or contains one and only one tree branch. Therefore, for a given tree, the number of fundamental cut-sets will be equal to the number of twigs.

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Properties of Cut-Set

1. A cut-set divides the set of nodes into two subsets.

2. Each fundamental cut-set contains one tree-branch, the remaining elements being links.

3. Each branch of the cut-set has one of its terminals incident at a node in one subset and its other terminal at a node in the other subset.

4. A cut-set is oriented by selecting an orientation from one of the two parts to the other. Generally, the direction of cut-set is chosen same as the direction of the tree branch.

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Cut-Set Matrix (Qc)

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For a given graph, a cut-set matrix (QC) is defined as a rectangular matrix whose rows correspond to cut-sets and columns correspond to the branches of the graph. Its elements have the following values:

Qij = 1, if branch j is in the cut-set i and the orientations coincide.

= -1, if branch j is in the cut-set i and the orientations do not coincide.

= 0, if branch j is not in the cut-set i.

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Example

For the graph shown in Fig., fundamental cut-sets have

been identified as follows.

f-cut-set 1: [1, 2, 6];

f-cut-set 2: [2, 3, 5, 6];

f-cut-set 3: [4, 5, 6]

So, the cut-set matrix is written as,

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Two port Network

• Condition for Reciprocity
• A two port network is said to be reciprocal if the ratio of the excitation to response is invariant to an interchange of the positions of the excitation and response.

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• (i)

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• (ii)

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• (iii)

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• Condition for symmetry
• A two port network is said to be symmetrical if the ports can be interchanged without changing the port voltages and current.

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• (i)

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• (ii) In terms of Y-parameter

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• (iii)

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• For the network shown below find Z, Y, T parameters.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• Q2. For the T network shown below obtain z-parameters.

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• Solution:

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• Q3. For a π-network given below obtain Y-parameters.

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Department of Computer Science and Engineering, BVCOE New Delhi

• Q1.Obtain the z-parameter of the following network.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• Q2. Two identical sections of the network shown in fig. below are cascaded Calculate the transmission parameters of the resulting network.

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Department of Computer Science and Engineering, BVCOE New Delhi

• Q3. Determine the transmission parameters of a T network shown below. Considering three sections as shown below (assuming connected in cascade manner)

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Department of Computer Science and Engineering, BVCOE New Delhi

• Q4. Determine the transmission parameter of the network shown below where N₁ and N₂ are in cascade.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• Q. Determine the transmission parameters of the network shown below using the concept of interconnection of four two port networks N₁, N₂, N₃ and N₄ in cascade.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• Open circuit and short ckt impedences

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• Open ckt and short ckt impedences in terms of T-parameters.

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Image Impedence

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• Image Impedences in terms of Input and Output impedences

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• Image Impedences in terms of T-parameters.

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Department of Computer Science and Engineering, BVCOE New Delhi

NETWORK FUNCTIONS

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• The transform impedence and transform admittance must relate to the port 1-1´ or 2-2´ as shown in fig. a and b below

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• The impedence (or admittance ) found at a given port is called driving point impedence (or admittance) i.e transform impedence(or admittances) of port 1-1´ and 2-2´ are also called as input and output driving point impedences(or admittances ) respectively.
• Because of the similarity of impedence and admittance these two quantities are assigned one name which is called immittance(a combination of impedence and admittance).

Department of Computer Science and Engineering, BVCOE New Delhi

• An immittance is thus an impedence or an admittance.
• Table below shows the immittance functions of the ckt. Elements

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• It is conventional to define transfer functions as the ratio of an output quantity to an input quantity.
• In terms of two port network as shown in fig. below; the output quantities are V₂(s) and I₂(s) and the input quantities are V₁(s) and I₁(s).

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• Using this scheme there are only four basic transfer functions for the two port network and these are given as;

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Department of Computer Science and Engineering, BVCOE New Delhi

• Q. obtain the driving point impedence functions or transform impedences Z(s) for the networks shown below.
• Fig (a) fig(b)

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• Solution a

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• Solution b

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• Q. Determine the driving point admittance functions or Transform admittances Y(s) for the network shown below

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• Q. Find the voltage transfer functions of the network as shown below

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• Solution

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Ladder network and its solution�(Driving point impedence at terminal 1-1´)

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• Q. Find the open ckt driving point impedence at terminal 1-1´ of the ladder network shown below.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Poles and Zeros of a network function

• In linear RLC networks all network functions T(s) may be expressed as the ratio of two polynomials namely N(s) [Numerator polynomial] and D(s) the denominator polynomial

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• Necessary conditions for driving point Immittance functions;
• 1. The coefficients in the polynomials N(s) and D(s) must be real and positive

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• Q. Check wheather given functions are suitable in representing the driving point immittance functions or not.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• Necessary conditions for transfer functions:
• 1. The coefficients in the polynomial N(s) and D(s) (of T=N/D) must be real and those for D(s) must be positive.

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Department of Computer Science and Engineering, BVCOE New Delhi

• Q. check wheather given functions are suitable in representing the transfer functions or not.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Network Synthesis

• In this topic we will be dealing with the problem of synthesizing a network
• Causality and stability:
• The first step in a synthesis procedure is to determine wheather T(s) can be realized as a physical passive network.
• For this there are two important considerations are: Causality and stability

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• Causality: The response of the network must be zero for t<0.
• And for stability the following three conditions must be satisfied:
• (i) T(s) cannot have poles in the right hand side of s-plane.
• (ii) T(s) cannot have multiple poles in the imaginary (jω) axis.

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• (iii) The degree of the numerator of T(s) cannot exceed the degree of the denominator by more than unity.
• Hurwitz polynomial

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• Properties of Hurwitz polynomial:

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• Q.1

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Department of Computer Science and Engineering, BVCOE New Delhi

• Q2.

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Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

Department of Computer Science and Engineering, BVCOE New Delhi

• Q3 Check wheather the given polynomial is hurwitz or not.

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Department of Computer Science and Engineering, BVCOE New Delhi