19EEPC402- NETWORKS AND SYNTHESIS

Presented by

S.Arun

UNIT-I: NETWORK TOPOLOGY

• Basic definitions of a network graph – Oriented graph – Sub graph – Planar graph – Path and Circuit – Tree and its properties – Cut sets – Incidence matrix – Circuit matrix – Cut set matrix – Fundamental circuit of Tie set matrix – Fundamental cut set matrix. Network analysis using graph theory: Formation of network equations – Network equilibrium equations on the basis of loop analysis – Network equilibrium equations on the basis of node analysis – Application to DC networks.

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 Graph theory.

• Network topology is a graphical representation of electric circuits. It is useful for analyzing complex electric circuits by converting them into network graphs. Network topology is also called as Graph theory.

Basic Terminology of Network Topology

• Graph

Network graph is simply called as graph. It consists of a set of nodes connected by branches. In graphs, a node is a common point of two or more branches. Sometimes, only a single branch may connect to the node. A branch is a line segment that connects two nodes.

• Any electric circuit or network can be converted into its equivalent graph by replacing the passive elements and voltage sources with short circuits and the current sources with open circuits. That means, the line segments in the graph represent the branches corresponding to either passive elements or voltage sources of electric circuit.

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Example : Let us consider the following electric circuit.

In the above circuit, there are four principal nodes and, There are seven branches in the above circuit

An equivalent graph corresponding to the above electric circuit is shown in the following figure.

• In the above graph, there are four nodes and those are labelled with 1, 2, 3 & 4 respectively. These are same as that of principal nodes in the electric circuit. There are six branches in the above graph and those are labelled with a, b, c, d, e & f respectively.
• In this case, we got one branch less in the graph because the 4 A current source is made as open circuit, while converting the electric circuit into its equivalent graph.
• From this Example,

The number of nodes present in a graph will be equal to the number of principal nodes present in an electric circuit.

The number of branches present in a graph will be less than or equal to the number of branches present in an electric circuit.

Types of Graphs

• Connected Graph
• Unconnected Graph
• Directed Graph
• Undirected Graph

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Directed Graph

• If all the branches of a graph are represented with arrows, then that graph is called as a directed graph. These arrows indicate the direction of current flow in each branch. Hence, this graph is also called as oriented graph.

Subgraph and its Types

• A part of the graph is called as a subgraph. We get subgraphs by removing some nodes and/or branches of a given graph.
• Following are the two types of subgraphs.
• Tree
• Co-Tree

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Tree

• Tree is a connected subgraph of a given graph, which contains all the nodes of a graph. But, there should not be any loop in that subgraph. The branches of a tree are called as twigs.
• Consider the following connected subgraph of the graph, which is shown in the Example of the beginning of this chapter.
• This connected subgraph contains all the four nodes of the given graph and there is no loop. Hence, it is a Tree.
• From the above Tree, we can conclude that the number of branches that are present in a Tree should be equal to n - 1 where ‘n’ is the number of nodes of the given graph.

Co-Tree

• Co-Tree is a subgraph, which is formed with the branches that are removed while forming a Tree. Hence, it is called as Complement of a Tree. For every Tree, there will be a corresponding Co-Tree and its branches are called as links or chords.

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The number of branches that are present in a co-tree will be equal to the difference between the number of branches of a given graph and the number of twigs. 

Where,

l is the number of links.

b is the number of branches present in a given graph.

n is the number of nodes present in a given graph.

Matrices Associated with Network Graphs

• Incidence Matrix
• Fundamental Loop Matrix
• Fundamental Cut set Matrix

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Incidence Matrix

• An Incidence Matrix represents the graph of a given electric circuit or network. Hence, it is possible to draw the graph of that same electric circuit or network from the incidence matrix
• The elements of incidence matrix will be having one of these three values, +1, -1 and 0.
• If the branch current is leaving from a selected node, then the value of the element will be +1.
• If the branch current is entering towards a selected node, then the value of the element will be -1.
• If the branch current neither enters at a selected node nor leaves from a selected node, then the value of element will be 0.

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Example

• Consider the following directed graph

we can conclude that the summation of column elements of incidence matrix is equal to zero. That means, a branch current leaves from one node and enters at another single node only.

Fundamental Loop Matrix

• Fundamental loop or f-loop is a loop, which contains only one link and one or more twigs. So, the number of f-loops will be equal to the number of links. Fundamental loop matrix is represented with letter B. It is also called as fundamental circuit matrix and Tie-set matrix. This matrix gives the relation between branch currents and link currents.

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• the fundamental loop matrix will have ‘b-n+1’ rows and ‘b’ columns. Here, rows and columns are corresponding to the links of co-tree and branches of given graph. Hence, the order of fundamental loop matrix will be (b - n + 1) × b.

Procedure to find Fundamental Loop Matrix

• Select a tree of given directed graph.
• By including one link at a time, we will get one f-loop. Fill the values of elements corresponding to this f-loop in a row of fundamental loop matrix.
• Repeat the above step for all links.

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• The above Tree contains three branches d, e & f. Hence, the branches a, b & c will be the links of the Co-Tree corresponding to the above Tree By including one link at a time to the above Tree, we will get one f-loop. So, there will be three f-loops, since there are three links. These three f-loops

Fundamental Cut-set Matrix

• Fundamental cut set or f-cut set is the minimum number of branches that are removed from a graph in such a way that the original graph will become two isolated subgraphs. The f-cut set contains only one twig and one or more links. So, the number of f-cut sets will be equal to the number of twigs.
• Fundamental cut set matrix is represented with letter C. This matrix gives the relation between branch voltages and twig voltages.

• The elements of fundamental cut set matrix will be having one of these three values, +1, -1 and 0.
• The value of element will be +1 for the twig of selected f-cutset.
• The value of elements will be 0 for the remaining twigs and links, which are not part of the selected f-cutset.
• If the direction of link current of selected f-cut set is same as that of f-cutset twig current, then the value of element will be +1.
• If the direction of link current of selected f-cut set is opposite to that of f-cutset twig current, then the value of element will be -1.

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Procedure to find Fundamental Cut-set Matrix

• Select a Tree of given directed graph and represent the links with the dotted lines.
• By removing one twig and necessary links at a time, we will get one f-cut set. Fill the values of elements corresponding to this f-cut set in a row of fundamental cut set matrix.
• Repeat the above step for all twigs

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Example

• Select the branches d, e & f of this directed graph as twigs. So, the remaining branches a, b & c of this directed graph will be the links.
• The twigs d, e & f are represented with solid lines and links a, b & c are represented with dotted lines

• By removing one twig and necessary links at a time, we will get one f-cut set. So, there will be three f-cut sets, since there are three twigs. These three f-cut sets are shown

the ideal current source is open circuited and ideal voltage source is short circuited. The oriented graph is drawn for which d is the reference

Graph

Tree and Co-tree

Loop

To Find Possible trees in graph

To draw the graph,

1. replace all resistors, inductors and capacitors by line segments,
2. replace voltage source by short circuit and current source by an open circuit,
3. assume directions of branch currents arbitrarily, and
4. number all the nodes and branches.
5. First write The complete Incidence Matrix (Aa) is written as

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• The reduced incidence matrix A is obtained by eliminating the last row from matrix Aa

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1. The reduced incidence matrix of an oriented graph is given below. Draw the graph.

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• Solution : First, writing the complete incidence matrix from the matrix A such that the sum of all entries in each column of Aa will be zero, we have

• The reduced incidence matrix of an oriented graph is shown in below , (a) Draw the graph. (b) How many trees are possible for this graph? (c) Write the tieset and cutset matrices

The number of possible trees = 8.

NETWORK EQUILIBRIUM EQUATION

• KVL Equation
• If there is a voltage source vsk in the branch k having impedance zk and carrying current ik,

2. If there is a voltage source in series with an impedance and a current source in parallel with the combination

where Z B Zb BT is the loop impedance matrix. This is the generalised KVL equation.

KCL Equation

1. If the branch k contains an input current source isk and an admittance yk as shown

2. If there is a voltage source in series with an impedance and a current source in parallel with the combination

• For the given network, write down the tieset matrix and obtain the network equilibrium equation in matrix form using KVL. Calculate loop currents.

Unit - 2

NETWORK FUNCTIONS AND TWO PORT NETWORKS

Concept of complex frequency – Network functions – Driving point and transfer functions and their properties – Poles and Zeros and their significance – Time domain behaviour from pole-zero plot – Two port networks – Z, Y, ABCD and h parameters – Condition for reciprocity and symmetry – Parameter conversion – Interconnection of two port networks – Analysis of typical two port networks – Input and Output impedances of terminated two port networks – Image impedances.

Concept of complex frequency

Physical significance of complex frequency

Case 1:

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Purely exponential function it behaves according to the values of

Number of poles = 3, and they are P1=-3, P2 = -1+j, P3=-1-j

Number of zeros=1,and they are Z=0;

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Tale inverse laplace =

To obtain A , Consider Pole P1

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To obtain B , Consider Pole P2

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six types of parameters

• Open circuit impedance parameters(Z)
• Short circuit admittance parameters(Y)
• Transmission (or) Chain Parameters(ABCD (or) T)
• Inverse Transmission Parameters(T^-1)
• Hybrid parameters(h- parameters)
• Inverse Hybrid parameters(h^-1- parameters)

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Open circuit impedance parameters(Z)

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= 0, 2-2’open circuited secondary terminal

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= 0, 1-1’open circuited first terminal

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Short circuit admittance parameters(Y)

Transmission parameters(ABCD)

Hybrid parameters(h)

UNIT-III: STATE VARIABLE ANALYSIS

• State, State variables and State space – State space models – Continuous time models – State space models applicable for electric circuits – Classification of circuits in state variable analysis – State variable analysis of circuits with controlled sources – Formation of state equations using network graph theory – Zero state response of the state vector – Complete response of state vector.

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state vector

The state vector is a (nx1) column matrix (or vector) whose elements are state variables of the system. (Where n is the order of the system).It is denoted by X(t).

input and output space

The set of all possible values which the input vector U(t) can have (or assume) at time t forms the input space of the system. The set of all possible values which the output vector Y(t) can have (or assume) at time t forms the output space of the system.

advantages of state space analysis

• The state space analysis is applicable to any type of systems. They can be used for modeling and analysis of linear & non linear systems, time invariant & time variant systems and multiple input & multiple output systems.

• The state space analysis can be performed with initial conditions.

• The variables used to represent the system can be any variables in the system.

• Using this analysis the internal states of the system at any time instant can be predicted

state and state variable

• The state is the condition of a system at any time instant.
• A set of variable which describes the state of the system at any time instant are called state variables

solution of state equations

• Homogeneous state equations.
• Non-homogeneous state equations

state diagram

• The pictorial representation of the state model of the system is called state diagram. The state diagram of the system can be either in block diagram or in signal flow graph form

state model of nth order electric circuits

• The state model of a system consists of state equation and output equation.The state model of a nth order system with m-inputs and p-outputs are
• =AX (t) +BU (t) …..State Equation.
• Y (t) =CX (t) +DU (t) …..Output Equation.
• Where X (t) =state vector of order (nx1), U (t) =Input vector of order (mx1)
• A= system matrix of order (nxn), B=Input matrix of order (nxm)
• Y (t) =output vector of order (px1), C=output matrix of order (pxn),
• D= Transmission matrix of order (pxm).

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Obtain the state model of the electrical network by choosing V₁(t) and V₂(t) as state variable

UNIT-IV: ELEMENTS OF REALIZABILITY AND SYNTHESIS OF ONE – PORT NETWORKS

• Hurwitz polynomials – Positive real functions – Frequency response of reactive one ports – Synthesis of reactive one ports by Foster method and Cauer method – Synthesis of RL and RC networks by Foster method and Cauer method.

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• Synthesis of Reactive One-Ports by Foster’s method.

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• There are two forms of Foster networks for reactive one-ports
• first Foster form or Impedance form.

One is a series combination of parallel LC circuits with capacitance C0 and inductance L∞

• second Foster form or admittance form

parallel combination of series LC circuits with inductance L0 and capacitance C∞

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first Foster form

second Foster form

Synthesis of Reactive One-Ports by the Cauer Method

• In the Cauer method, there are two types of ladder networks to realise the one-port network. In one type of network,
• the series arms are inductors and the shunt arms are capacitors
• In the other network, the series arms are capacitors and the shunt arms are inductors

Cauer Form I

Cauer Form II

Foster-I form of RL network

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Foster-II form of RL network

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• Cauer –I form for RL network structure

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Cauer –II form for RL network structure

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• Foster-I form of RC network

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Foster-II form of RC network

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Cauer –I form for RC network structure

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Cauer –II form for RC network structure

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UNIT – 5�FILTERS AND ATTENUATORS

• Classification of filters – Filter networks – Equations of filter networks – Classification of pass band and stop band – Characteristic impedance in pass and stop bands – Constant K low pass, high pass, band pass and band elimination filters – Limitations of constant K filters – M derived filters – Composite filter. Attenuators: T type, type, Lattice, Bridged T and L type attenuators.

Filter

• A filter is a reactive network that freely passes the desired bands of frequencies while almost totally suppressing all other bands.

Classification of filters

• low pass, high pass, band pass and band elimination.

Lowpass and high pass filters

• A low pass (LP) filter is one which passes without attenuation all frequencies up to the cut-off frequency fc, and attenuates all other frequencies greater than fc.
• A high pass (HP) filter attenuates all frequencies below a designated cut-off frequency, fc, and passes all frequencies above fc.

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Band pass filter and band elimination filter.

• A band pass filter passes frequencies between two designated cut-off frequencies and attenuates all other frequencies It is abbreviated as BP filter. a BP filter has two cut-off frequencies and will have the pass band f2 – f1; f1 is called the lower cut-off frequency, while f2 is called the upper cut-off frequency.

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• A band elimination filter passes all frequencies lying outside a certain range, while it attenuates all frequencies between the two designated frequencies. It is also referred as band stop filter., All frequencies between f1 and f2 will be attenuated while frequencies below f1 and above f2 will be passed.

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Characteristic impedance of symmetrical T-Section.

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Structure of constant K low pass filter

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Structure of constant K high pass filter

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M - derived low pass filter T section and pi Section

M derived High pass filter T section and pi section

Bandpass filter T section and pi section

Band elimination filter T section and Pi section

Attenuator

• T-type attenuator
• Pi -type attenuators
• Bridged T-attenuator.

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• volume controls in radio broadcasting sections, used in laboratory to obtain small value of voltage or current for testing circuits

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