Network Analysis

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NETWORK ANALYSIS AND SYNTHESIS (EEE-402)

SYLLABUS Unit – I: Graph Theory: Graph of a Network, definitions, tree, co tree , link, basic loop and basic cut set, Incidence matrix, cut set matrix, Tie set matrix Duality, Loop and Nodal methods of analysis. (7) Unit – II: Network Theorems (Applications to ac networks): Super-position theorem, Thevenin’s theorem, Norton’s theorem, maximum power transfer theorem, Reciprocity theorem. Millman’s theorem, compensation theorem, Tellegen’s theorem. (8) Unit – III: Network Functions: Concept of Complex frequency , Transform Impedances Network functions of one port and two port networks, concept of poles and zeros, properties of driving point and transfer functions, time response and stability from pole zero plot. (8) Unit – IV: Two Port Networks: Characterization of LTI two port networks ZY, ABCD and h parameters, reciprocity and symmetry. Inter-relationships between the parameters, inter-connections of two port networks, Ladder and Lattice networks. T & Π Representation. (8) Unit – V: (a) Network Synthesis: Positive real function; definition and properties; properties of LC, RC and RL driving point functions, synthesis of LC, RC and RL driving point immittance functions using Foster and Cauer first and second forms. (b) Filters: Image parameters and characteristics impedance, passive and active filter fundamentals, low pass, high pass, (constant K type) filters, and introduction to active filters. (9)

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UNIT- I

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GRAPH THEORY Graph theory is the branch of mathematics dealing with graphs. In network analysis, graphs are used extensively to represent a network being analyzed. The graph of a network captures only certain aspects of a network; those aspects related to its connectivity, or, in other words, its topology. This can be a useful representation and generalization of a network because many network equations are invariant across networks with the same topology. This includes equations derived from Kirchhoff's laws and Tellegen's theorem. Graph theory has been used in the network analysis of linear, passive networks almost from the moment that Kirchhoff's laws were formulated. This approach was later generalized to RLC circuits, replacing resistances with impedances. In 1892 Maxwell provided the dual of this analysis with node analysis. Maxwell is also responsible for the topological theorem that the determinant of the nodeadmittance matrix is equal to the sum of all the tree admittance products. In 1900 Henri Poincare introduced the idea of representing a graph by its incidence matrix, hence founding the field of algebraic topology.

Comprehensive cataloguing of network graphs as they apply to electrical circuits began with Percy McMahon in 1891 (with an engineer friendly article in The Electrician in 1892) who limited his survey to series and parallel combinations. McMahon called these graphs yoke-chains. Ronald Foster in 1932 categorized graphs by their nullity or rank and provided charts of all those with a small number of nodes. This work grew out of an earlier survey by Foster while collaborating with George Campbell in 1920 on 4-port telephone repeaters and produced 83,539 distinct graphs. For a long time topology in electrical circuit theory remained concerned only with linear passive networks. The more recent developments of semiconductor devices and circuits have required new tools in topology to deal with them. Enormous increases in circuit complexity have led to the use of combinatory in graph theory to improve efficiency of computer calculation.

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GRAPHS AND CIRCUIT DIAGRAM

Networks are commonly classified by the kind of electrical elements making them up. in a circuit diagram these element-kinds are specifically drawn, each with its own unique symbol.

Resistive networks are one-element-kind networks, consisting only of R elements. Likewise capacitive or inductive networks are one-element-kind. The RC, RL and LC circuits are simple two-element-kind network.

The RLC circuit is the simplest three-element-kind network. The LC ladder network commonly used for low-pass filters can have many elements but is another example of a two-element-kind network. Conversely, topology is concerned only with the geometric relationship between the elements of a network, not with the kind of elements themselves. The heart of a topological representation of a network is the graph of the network. Elements are represented as the edges of the graph. An edge is drawn as a line, terminating on dots or small circles from which other edges (elements) may emanate. In circuit analysis, the edges of the graph are called branches. The dots are called the vertices of the graph and represent the nodes of the network. Node and vertex are terms that can be used interchangeably when discussing graphs of networks. Graphs used in network analysis are usually, in addition, both directed graphs, to capture the direction of current flow and voltage, and labeled graphs, to capture the uniqueness of the branches and nodes. For instance, a graph consisting of a square of branches would still be the same topological graph if two branches were interchanged unless the branches were uniquely labeled. In directed graphs, the two nodes that a branch connects to be designated the source and target nodes. Typically, these will be indicated by an arrow drawn on the branch.

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Equivalence Graphs are equivalent if one can be transformed into the other by deformation. Deformation can include the operations of translation, rotation and reflection; bending and stretching the branches; and crossing or knotting the branches. Two graphs which are equivalent through deformation are said to be congruent. In the field of electrical networks, there are two additional transforms that are considered to result in equivalent graphs which do not produce congruent graphs. The first of these is the interchange of series connected branches. This is the dual of interchange of parallel connected branches which can be achieved by deformation without the need for a special rule. The second is concerned with graphs divided into two or more separate parts, that is, a graph with two sets of nodes which have no branches incident to a node in each set. Two such separate parts are considered an equivalent graph to one where the parts are joined by combining a node from each into a single node. Likewise, a graph that can be split into two separate parts by splitting a node in two is also considered equivalent.

Trees and links A tree is a graph in which all the nodes are connected, either directly or indirectly, by branches, but without forming any closed loops. Since there are no closed loops, there are no currents in a tree.

In network analysis, we are interested in spanning trees, that is, trees that connect every node present in the graph of the network. In this article, spanning tree is meant by an unqualified tree unless otherwise stated. A given network graph can contain a number of different trees. The branches removed from a graph in order to form a tree are called links. The branches remaining in the tree are called twigs. For a graph with n nodes, the number of branches in each tree, t, must be

An important relationship for circuit analysis is;

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Where b is the number of branches in the graph and l is the number of links removed to form the tree.

Tie sets and cut sets The goal of circuit analysis is to determine all the branch currents and voltages in the network. These network variables are not all independent. The branch voltages are related to the branch currents by the transfer function of the elements of which they are composed. A complete solution of the network can therefore be either in terms of branch currents or branch voltages only. Nor are all the branch currents independent from each other. The minimum number of branch currents required for a complete solution is l. This is a consequence of the fact that a tree has l links removed and there can be no currents in a tree. Since the remaining branches of the tree have zero current they cannot be independent of the link currents. The branch currents chosen as a set of independent variables must be a set associated with the links of a tree: one cannot choose any l branches arbitrarily. In terms of branch voltages, a complete solution of the network can be obtained with t branch voltages. This is a consequence the fact that short-circuiting all the branches of a tree results in the voltage being zero everywhere. The link voltages cannot, therefore, be independent of the tree branch voltages A common analysis approach is to solve for loop currents rather than branch currents. The branch currents are then found in terms of the loop currents. Again, the set of loop currents cannot be chosen arbitrarily.

To guarantee a set of independent variables the loop currents must be those associated with a certain set of loops. This set of loops consists of those loops formed by replacing a single link of a given tree of the graph of the circuit to be analyzed.

Since replacing a single link in a tree forms exactly one unique loop, the number of loop currents so defined is equal to l. The term loop in this context is not the same as the usual meaning of loop in graph theory. The set of branches forming a given loop is called a tie set. The set of network equations are formed by equating the loop currents to the algebraic sum of the tie set branch currents.[ It is possible to choose a set of independent loop currents without reference to the trees and tie sets. A sufficient, but not necessary, condition for choosing a set of independent loops is to ensure that each chosen loop includes at least one branch that was not previously included by loops already chosen. 7


A particularly straightforward choice is that used in mesh analysis in which the loops are all chosen to be meshes. Mesh analysis can only be applied if it is possible to map the graph on to a plane or a sphere without any of the branches crossing over. Such graphs are called planar graphs. Ability to map onto a plane or a sphere is equivalent conditions. Any finite graph mapped onto a plane can be shrunk until it will map onto a small region of a sphere. Conversely, a mesh of any graph mapped onto a sphere can be stretched until the space inside it occupies nearly the entire sphere. The entire graph then occupies only a small region of the sphere. This is the same as the first case, hence the graph will also map onto a plane. There is an approach to choosing network variables with voltages which is analogous and dual to the loop current method. Here the voltage associated with pairs of nodes is the primary variables and the branch voltages are found in terms of them. In this method also, a particular tree of the graph must be chosen in order to ensure that all the variables are independent. The dual of the tie set is the cut set. A tie set is formed by allowing all but one of the graph links to be open circuit. A cut set is formed by allowing all but one of the tree branches to be short circuit. The cut set consists of the tree branch which was not short-circuited and any of the links which are not shortcircuited by the other tree branches. A cut set of a graph produces two disjoint sub graphs, that is, it cuts the graph into two parts, and is the minimum set of branches needed to do so. The set of network equations are formed by equating the node pair voltages to the algebraic sum of the cut set branch voltages. The dual of the special case of mesh analysis is nodal analysis.

Nullity and rank The nullity, N, of a graph with s separate parts is defined by;

The nullity of a graph represents the number of degrees of freedom of its set of network equations. For a planar graph, the nullity is equal to the number of meshes in the graph. The rank, R of a graph is defined by;

Rank plays the same role in nodal analysis as nullity plays in mesh analysis. That is, it gives the number of node voltage equations required. Rank and nullity are dual concepts and are related by;

Solving the network variables Once a set of geometrically independent variables have been chosen the state of the network is expressed in terms of these. The result is a set of independent linear equations which need to be solved simultaneously in order to find the values of the network variables. 8


This set of equations can be expressed in a matrix format which leads to a characteristic parameter matrix for the network. Parameter matrices take the form of an impedance matrix if the equations have been formed on a loop-analysis basis, or as an admittance matrix if the equations have been formed on a node-analysis basis. These equations can be solved in a number of well-known ways. One method is the systematic elimination of variables. Another method involves the use of determinants. This is known as Cramer's rule and provides a direct expression for the unknown variable in terms of determinants. This is useful in that it provides a compact expression for the solution. However, for anything more than the most trivial networks, a greater calculation effort is required for this method when working manually.

Duality Two graphs are dual when the relationship between branches and node pairs in one is the same as the relationship between branches and loops in the other. The dual of a graph can be found entirely by a graphical method. The dual of a graph is another graph. For a given tree in a graph, the complementary set of branches (i.e., the branches not in the tree) form a tree in the dual graph. The set of current loop equations associated with the tie sets of the original graph and tree are identical to the set of voltage node-pair equations associated with the cut sets of the dual graph links in the same way that the tree consists of nodes connected by tree branches.Duals cannot be formed for every graph. Duality requires that every tie set has a dual cut set in the dual graph. This condition is met if and only if the graph is map able on to a sphere with no branches crossing. The dual of a tree is sometimes called a maze. It consists of spaces connected by

To see this, note that a tie set is required to "tie off" a graph into two portions and its dual, the cut set, is required to cut a graph into two portions. The graph of a finite network which will not map on to a sphere will require an n-fold torus. A tie set that passes through a hole in a torus will fail to tie the graph into two parts. Consequently, the dual graph will not be cut into two parts and will not contain the required cut set. Consequently, only planar graphs have duals.

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Duals also cannot be formed for networks containing mutual inductances since there is no corresponding capacitive element. Equivalent circuits can be developed which do have duals, but the dual cannot be formed of a mutual inductance directly.

Node and mesh elimination Operations on a set of network equations have a topological meaning which can aid visualization of what is happening. Elimination of a node voltage from a set of network equations corresponds topologically to the elimination of that node from the graph. For a node connected to three other nodes, this corresponds to the well known Y-Δ transform. The transform can be extended to greater numbers of connected nodes and is then known as the star-mesh transform. The inverse of this transform is the Δ-Y transform which analytically corresponds to the elimination of a mesh current and topologically corresponds to the elimination of a mesh. However, elimination of a mesh current whose mesh has branches in common with an arbitrary number of other meshes will not, in general, result in a realizable graph. This is because the graph of the transform of the general star is a graph which will not map on to a sphere (it contains star polygons and hence multiple crossovers). The dual of such a graph cannot exist, but is the graph required to represents a generalized mesh elimination

Mesh analysis circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not. Mesh analysis (or the mesh current method) is a method that is used to solve planar Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.

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Mesh currents and essential meshes Mesh analysis works by arbitrarily assigning mesh currents in the essential meshes (also referred to as independent meshes). An essential mesh is a loop in the circuit that does not contain any other loop. Figure labels the essential meshes with one, two, and three.

A mesh current is a current that loops around the essential mesh and the equations are set solved in terms of them. A mesh current may not correspond to any physically flowing current, but the physical currents are easily found from them. It is usual practice to have all the mesh currents loop in the same direction. This helps prevent errors when writing out the equations. The convention is to have all the mesh currents looping in a clockwise direction. Figure shows the same circuit from Figure 1 with the mesh currents labeled. Solving for mesh currents instead of directly applying law and Kirchhoff's voltage law can greatly reduce the amount of calculation required. This is because there are fewer mesh currents than there are physical branch currents. In figure 2 for example, there are six branch currents but only three mesh currents.

Setting up the equations Each mesh produces one equation. These equations are the sum of the voltage drops in a complete loop of the mesh current. For problems more general than those including current and voltage sources, the voltage drops will be the impedance of the electronic component multiplied by the mesh current in that loop. If a voltage source is present within the mesh loop, the voltage at the source is either added or subtracted depending on if it is a voltage drop or a voltage rise in the direction of the mesh current. For a current source that is not contained between two meshes, the mesh current will take the positive or negative value of the current source depending on if the mesh current is in the same or opposite direction of the current source. The following is the same circuit from above with the equations needed to solve for all the currents in the circuit.

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Once the equations are found, the system of linear equations can be solved by using any technique to solve linear equations.

Special cases There are two special cases in mesh current: currents containing a supermesh and currents containing dependent sources.

Supermesh The circuit is first treated as if the current source is not there. This leads to one equation that incorporates two mesh currents. Once this equation is formed, an equation is needed that relates the two mesh currents with the current source. A supermesh occurs when a current source is contained between two essential meshes. This will be an equation where the current source is equal to one of the mesh currents minus the other. The following is a simple example of dealing with a supermesh.

Dependent sources A dependent source is a current source or voltage source that depends on the voltage or current of another element in the circuit. When a dependent source is contained within an essential mesh, the dependent source should be treated like an independent source.

After the mesh equation is formed, a dependent source equation is needed. This equation is generally called a constraint equation. This is an equation that relates the dependent source’s variable to the voltage or current that the source depends on in the circuit. The following is a simple example of a dependent source.

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Nodal analysis

In electric circuit analysis, nodal analysis, node-voltage analysis, or the branch current method is a method of determining the voltage (potential difference) between "nodes" (points where elements or branches connect) in an electrical circuit in terms of the branch currents.

In analyzing a circuit using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysis using Kirchhoff's voltage law (KVL). Nodal analysis writes an equation at each electrical node, requiring that the branch currents incident at a node must sum to zero. The branch currents are written in terms of the circuit node voltages. As a consequence, each branch constitutive relation must give current as a function of voltage; an admittance representation. For instance, for a resistor, I branch = V branch * G, where G (=1/R) is the admittance (conductance) of the resistor. Nodal analysis is possible when all the circuit elements' branch constitutive relations have an admittance representation. Nodal analysis produces a compact set of equations for the network, which can be solved by hand if small, or can be quickly solved using linear algebra by computer. Because of the compact system of equations, many circuit simulation programs (e.g. SPICE) use nodal analysis as a basis. When elements do not have admittance representations, a more general extension of nodal analysis, modified, can be used.

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While simple examples of nodal analysis focus on linear elements, more complex nonlinear networks can also be solved with nodal analysis by using Newton's method to turn the nonlinear problem into a sequence of linear problems.

Method 1. Note all connected wire segments in the circuit. These are the nodes of nodal analysis. 2. Select one node as the ground reference. The choice does not affect the result and is just a matter of convention. Choosing the node with the most connections can simplify the analysis. 3. Assign a variable for each node whose voltage is unknown. If the voltage is already known, it is not necessary to assign a variable. 4. For each unknown voltage, form an equation based on Kirchhoff's current law. Basically, add together all currents leaving from the node and mark the sum equal to zero. 5. Finding the current between two nodes is nothing more than "the node you're on, minus the node you're going to, divided by the resistance between the two nodes." 6. If there are voltage sources between two unknown voltages, join the two nodes as a supernode. The currents of the two nodes are combined in a single equation, and a new equation for the voltages is formed. 7. Solve the system of simultaneous equations for each unknown voltage.

Example

Basic case The only unknown voltage in this circuit is V1. There are three connections to this node and consequently three currents to consider. The direction of the currents in calculations is chosen to be away from the node. 1. Current through resistor R1: (V1 - VS) / R1 2. Current through resistor R2: V1 / R2 3. Current through current source IS: -IS With Kirchhoff's current law, we get:

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This equation can be solved in respect to V1:

Finally, the unknown voltage can be solved by substituting numerical values for the symbols. Any unknown currents are easy to calculate after all the voltages in the circuit are known.

Supernodes In this circuit, we initially have two unknown voltages, V1 and V2. The voltage at V3 is already known to be VB because the other terminal of the voltage source is at ground potential.

The current going through voltage source VA cannot be directly calculated. Therefore we cannot write the current equations for either V1 or V2. However, we know that the same current leaving node V2 must enter node V1. Even though the nodes cannot be individually solved, we know that the combined current of these two nodes is zero. This combining of the two nodes is called the supernode technique, and it requires one additional equation: V1 = V2 + VA. The complete set of equations for this circuit is:

By substituting V1 to the first equation and solving in respect to V2, we get :

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UNIT II

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NETWORK THEOREMS 1. Superposition theorem The superposition theorem for electrical circuits states that for a linear system the response (Voltage or Current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal impedances. To ascertain the contribution of each individual source, all of the other sources first must be "turned off" (set to zero) by: 1. Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential. i.e. V=0, internal impedance of ideal voltage source is ZERO (short circuit)). 2. Replacing all other independent current sources with an open circuit (thereby eliminating current. i.e. I=0, internal impedance of ideal current source is infinite (open circuit). This procedure is followed for each source in turn, and then the resultant responses are added to determine the true operation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources. The superposition theorem is very important in circuit analysis. It is used in converting any circuit into its Norton equivalent or Thevenin equivalent. Applicable to linear networks (time varying or time invariant) consisting of independent sources, linear dependent sources, linear passive elements Resistors, Inductors, Capacitors and linear transformers. Another point that should be considered is that superposition only works for voltage and current but not power. In other words the sum of the powers is not the real consumed power. To calculate power we should first use superposition to find both current and voltage of that linear element and then calculate sum of the multiplied voltages and currents respectively.

3. Thevenin's theorem In circuit theory, Thevenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem can also be applied to general impedances, not just resistors. 17


This theorem states that a circuit of voltage sources and resistors can be converted into a Thevenin equivalent, which is a simplification technique used in circuit analysis. The Thevenin equivalent can be used as a good model for a power supply or battery (with the resistor representing the internal impedance and the source representing the electromotive force). The circuit consists of an ideal voltage source in series with an ideal resistor.

Calculating the Thevenin equivalent To calculate the equivalent circuit, the resistance and voltage are needed, so two equations are required. These two equations are usually obtained by using the following steps, but any conditions placed on the terminals of the circuit should also work:

1. Calculate the output voltage, VAB, when in open circuit condition (no load resistor— meaning infinite resistance). This is VTh.

2. Calculate the output current, IAB, when the output terminals are short circuited (load resistance is 0). RTh equals VTh divided by this IAB. The equivalent circuit is a voltage source with voltage VTh in series with a resistance RTh. Step 2 could also be thought of as: 2a. Replace voltage sources with short circuits, and current sources with open circuits. 2b. Calculate the resistance between terminals A and B. This is RTh. The Thevenin-equivalent voltage is the voltage at the output terminals of the original circuit. When calculating a Thevenin-equivalent voltage, the voltage divider principle is often useful, by declaring one terminal to be Vout and the other terminal to be at the ground point. The Thevenin-equivalent resistance is the resistance measured across points A and B "looking back" into the circuit. It is important to first replace all voltage- and current-sources with their internal resistances. For an ideal voltage source, this means replace the voltage source with a short circuit. For an ideal current source, this means replace the current source with an open circuit. Resistance can then be calculated across the terminals using the formulae for series and parallel circuits. 18


This method is valid only for circuits with independent sources. If there are dependent sources in the circuit, another method must be used such as connecting a test source across A and B and calculating the voltage across or current through the test source.

Example

In the example, calculating the equivalent voltage:

(notice that R1 is not taken into consideration, as above calculations are done in an open circuit condition between A and B, therefore no current flows through this part, which means there is no current through R1 and therefore no voltage drop along this part) Calculating equivalent resistance:

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Practical limitations Many, if not most circuits are only linear over a certain range of values, thus the Thevenin equivalent is valid only within this linear range and may not be valid outside the range. The Thevenin equivalent has an equivalent I–V characteristic only from the point of view of the load. The power dissipation of the Thevenin equivalent is not necessarily identical to the power dissipation of the real system. However, the power dissipated by an external resistor between the two output terminals is the same regardless of how the internal circuit is represented

Proof of the theorem The proof involves two steps. First use superposition theorem to construct a solution, and then use uniqueness theorem to show the solution is unique. The second step is usually implied. Firstly, using the superposition theorem, in general for any linear "black box" circuit which contains voltage sources and resistors, one can always write down its voltage as a linear function of the corresponding current as follows

Where the first term reflects the linear summation of contributions from each voltage source, while the second term measures the contribution from all the resistors. The above argument is due to the fact that the voltage of the black box for a given current identical to the linear superposition of the solutions of the following problems:

is

(1) to leave the black box open circuited but activate individual voltage source one at a time and, (2) to short circuit all the voltage sources but feed the circuit with a certain ideal voltage source so that the resulting current exactly reads I (or an ideal current source of current I).

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Once the above expression is established, it is straightforward to show that single voltage source and the single series resistor in question

and

is the

3. Norton's theorem Norton's theorem for linear electrical networks, known in Europe as the Mayer–Norton theorem, states that any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R. For single-frequency AC systems the theorem can also be applied to general impedances, not just resistors. The Norton equivalent is used to represent any network of linear sources and impedances, at a given frequency. The circuit consists of an ideal current source in parallel with an ideal impedance (or resistor for non-reactive circuits).

Norton's theorem is an extension of Thevenin's theorem and was introduced in 1926 separately by two people: Siemens & Halske researcher Hans Ferdinand Mayer (1895–1980) and Bell Labs engineer Edward Lawry Norton (1898–1983). The Norton equivalent circuit is a current source with current INo in parallel with a resistance RNo. To find the equivalent, 1. Find the Norton current INo. Calculate the output current, IAB, with a short circuit as the load (meaning 0 resistances between A and B). This is I No. 2. Find the Norton resistance RNo. When there are no dependent sources (all current and voltage sources are independent), there are two methods of determining the Norton impedance RNo. 

Calculate the output voltage, VAB, when in open circuit condition (i.e., no load resistor — meaning infinite load resistance). RNo equals this VAB divided by INo.

Or 21

NETWORK ANALYSIS AND SYNTHESIS (EEE-402) 

Replace independent voltage sources with short circuits and independent current sources with open circuits. The total resistance across the output port is the Norton impedance RNo.

This is equivalent to calculating the Thevenin resistance. However, when there are dependent sources, the more general method must be used. This method is not shown below in the diagrams.   

Connect a constant current source at the output terminals of the circuit with a value of 1 Ampere and calculate the voltage at its terminals. This voltage divided by the 1A current is the Norton impedance RNo. This method must be used if the circuit contains dependent sources, but it can be used in all cases even when there are no dependent sources.

Example

In the example, the total current Itotal is given by:

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The current through the load is then, using the current divider rule:

And the equivalent resistance looking back into the circuit is:

So the equivalent circuit is a 3.75 mA current source in parallel with a 2 kΩ resistor

Relation between Thevenin and Norton equivalent circuit

4. Maximum power transfer theorem

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obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law". In electrical engineering, the maximum power transfer theorem states that, to The theorem results in maximum power transfer, and not maximum efficiency. If the resistance of the load is made larger than the resistance of the source, then efficiency is higher, since a higher percentage of the source power is transferred to the load, but the magnitude of the load power is lower since the total circuit resistance goes up. If the load resistance is smaller than the source resistance, then most of the power ends up being dissipated in the source, and although the total power dissipated is higher, due to a lower total resistance, it turns out that the amount dissipated in the load is reduced. The theorem states how to choose (so as to maximize power transfer) the load resistance, once the source resistance is given. It is a common misconception to apply the theorem in the opposite scenario. It does not say how to choose the source resistance for a given load resistance. In fact, the source resistance that maximizes power transfer is always zero, regardless of the value of the load resistance. The theorem can be extended to AC circuits that include reactance, and states that maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance. 2 2 ET h I N RN Pmax   4 RT h 4

Maximizing power transfer versus power efficiency The theorem was originally misunderstood (notably by Joule) to imply that a system consisting of an electric motor driven by a battery could not be more than 50% efficient since, when the impedances were matched, the power lost as heat in the battery would always be equal to the power delivered to the motor. In 1880 this assumption was shown to be false by either Edison or his colleague Francis Robbins Upton, who realized that maximum efficiency was not the same as maximum power transfer. 24


To achieve maximum efficiency, the resistance of the source (whether a battery or a dynamo) could be made close to zero. Using this new understanding, they obtained an efficiency of about 90%, and proved that the electric motor was a practical alternative to the heat engine

The condition of maximum power transfer does not result in maximum efficiency. If we define the efficiency as the ratio of power dissipated by the load to power developed by the source, then it is straightforward to calculate from the above circuit diagram that

Consider three particular cases: 

If



If

or



If

, then

, then then

The efficiency is only 50% when maximum power transfer is achieved, but approaches 100% as the load resistance approaches infinity, though the total power level tends towards zero. Efficiency also approaches 100% if the source resistance can be made close to zero. When the load resistance is zero, all the power is consumed inside the source (the power dissipated in a short circuit is zero) so the efficiency is zero.

Impedance matching A related concept is reflection less impedance matching. In radio, transmission lines, and other electronics, there is often a requirement to match the source impedance (such as a transmitter) to the load impedance (such as an antenna) to avoid reflections in the transmission

Calculus-based proof for purely resistive circuits

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In the diagram opposite, power is being transferred from the source, with voltage and fixed source resistance , to a load with resistance , resulting in a current . By Ohm's law, is simply the source voltage divided by the total circuit resistance:

The power

dissipated in the load is the square of the current multiplied by the resistance :

The value of for which this expression is a maximum could be calculated by differentiating it, but it is easier to calculate the value of for which the denominator

is a minimum. The result will be the same in either case. Differentiating the denominator with respect to :

For a maximum or minimum, the first derivative is zero, so

or

In practical resistive circuits, and are both positive, so the positive sign in the above is the correct solution. To find out whether this solution is a minimum or a maximum, the denominator expression is differentiated again:

This is always positive for positive values of and minimum, and the power is therefore a maximum, when In reactive circuits

26

, showing that the denominator is a


However, this equation only applies if the source resistance cannot be adjusted, e.g., with antennas (see the first line in the proof stating "fixed source resistance"). For any given load resistance a source resistance of zero is the way to transfer maximum power to the load. As an example, a 100 volt source with an internal resistance of 10 ohms connected to a 10 ohm load will deliver 250 watts to that load. Make the source resistance zero ohms and the load power jumps to 1000 watts.

5. Tellegen's theorem Tellegen's theorem is one of the most powerful theorems in network theory. Most of the energy distribution theorems and extremism principles in network theory can be derived from it. It was published in 1952 by Bernard Tellegen’s. Fundamentally, Tellegen's theorem gives a simple relation between magnitudes that satisfy Kirchhoff's laws of electrical circuit theory. The Tellegen’s theorem is applicable to a multitude of network systems. The basic assumptions for the systems are the conservation of flow of extensive quantities (Kirchhoff's current law, KCL) and the uniqueness of the potentials at the network nodes (Kirchhoff's voltage law, KVL). The Tellegen’s theorem provides a useful tool to analyze complex network systems among the electrical circuits, biological and metabolic networks, pipeline flow networks, and chemical process networks.

The theorem Consider an arbitrary lumped network whose graph has branches and nodes. In an electrical network, the branches are two-terminal components and the nodes are points of interconnection. Suppose that to each branch of the graph we assign arbitrarily a branch potential difference and a branch current for , and suppose that they are measured with respect to arbitrarily picked associated reference directions. If the branch potential differences satisfy all the constraints imposed by KVL and if the branch currents

satisfy all the constraints imposed by KCL, then

Tellegen's theorem is extremely general; it is valid for any lumped network that contains any elements, linear or nonlinear, passive or active, time-varying or time-invariant.The generality is extended when and are linear operations on the set of potential differences and on the set of branch currents (respectively) since linear operations don't affect KVL and KCL. For instance, the linear operation may be the average or the Laplace transform. The set of currents can also be sampled at a different time from the set of potential differences since KVL and KCL are true at all instants of time. Another extension is when the set of potential differences is from one network and the set of currents is from an entirely different network, so long as the 27


two networks have the same topology (same incidence matrix) Tellegen's theorem remains true. This extension of Tellegen's Theorem leads to many theorems relating to two-port networks.

Definitions We need to introduce a few necessary network definitions to provide a compact proof. Incidence matrix: The matrix elements being

A reference or datum node

matrix

is called node-to-branch incidence matrix for the

is introduced to represent the environment and connected to all

dynamic nodes and terminals. The elements of the reference node

matrix , where the row that contains the is eliminated, is called reduced incidence matrix.

The conservation laws (KCL) in vector-matrix form:

The uniqueness condition for the potentials (KVL) in vector-matrix form:

Where

are the absolute potentials at the nodes to the reference node

.

Proof Using KVL:

Because

by KCL. So:

Applications 



Network analogs have been constructed for a wide variety of physical systems, and have proven extremely useful in analyzing their dynamic behavior. The classical application area for network theory and Tellegen's theorem is electrical circuit theory. It is mainly in use to design filters in signal processing applications. A more recent application of Tellegen's theorem is in the area of chemical and biological processes. The assumptions for electrical circuits (Kirchhoff laws) are generalized for dynamic systems obeying the laws of irreversible thermodynamics. Topology and 28




structure of reaction networks (reaction mechanisms, metabolic networks) can be analyzed using the Tellegen’s theorem. Another application of Tellegen's theorem is to determine stability and optimality of complex process systems such as chemical plants or oil production systems. The Tellegen’s theorem can be formulated for process systems using process nodes, terminals, flow connections and allowing sinks and sources for production or destruction of extensive quantities.

A formulation for Tellegen's theorem of process systems:

Where are the production terms, are the terminal connections, and storage terms for the extensive variables.

are the dynamic

6. Millman’s Theorem In electrical engineering, Millman's theorem (or the parallel generator theorem) is a method to simplify the solution of a circuit. Specifically, Millman's theorem is used to compute the voltage at the ends of a circuit made up of only branches in parallel. It is named after Jacob Millman, who proved the theorem.

Explanation

Let ek be the voltage generators and am the current generators. Let Ri be the resistances on the branches with no generator. Let Rk be the resistances on the branches with voltage generators. Let Rm be the resistances on the branches with current generators. Then Millman states that the voltage at the ends of the circuit is given by:

29


It can be proved by considering the circuit as a single supernode. Then, according to Ohm and Kirchhoff, "the voltage between the ends of the circuit is equal to the total current entering the supernode divided by the total equivalent conductance of the supernode". The total current is the sum of the currents flowing in each branch. The total equivalent conductance of the supernode is the sum of the conductance of each branch, since all the branches are in parallel. When computing the equivalent conductance all the generators have to be switched off, so all voltage generators become short circuits and all current generators become open circuits. That's why the resistances on the branches with current generators do not appear in the expression of the total equivalent conductance.

7. Reciprocity Theorem The reciprocity theorem is applicable only to single-source networks and states the following: 1. The current I in any branch of a network, due to a single voltage source E anywhere in the network, will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current I was originally measured. 2. The location of the voltage source and the resulting current may be interchanged without a change in current

30


UNIT III

31


NETWORK FUNCTIONS

Driving point impedance

The driving point impedance is a mathematical representation of the input impedance of a filter in the frequency domain using one of a number of notations such as Laplace transform (sdomain) or Fourier transform ( jω-domain). Treating it as a one-port network, the expression is expanded using continued fraction or partial fraction expansions. The resulting expansion is transformed into a network (usually a ladder network) of electrical elements. Taking an output from the end of this network, so realized, will transform it into a two-port network filter with the desired transfer function. Not every possible mathematical function for driving point impedance can be realized using real electrical components. Wilhelm Cauer (following on from R. M. Foster) did much of the early work on what mathematical functions could be realized and in which filter topologies. The ubiquitous ladder topology of filter design is named after Cauer. There are a number of canonical forms of driving point impedance that can be used to express all (except the simplest) realizable impedances. The most well known ones are    

Cauer's first form of driving point impedance consists of a ladder of shunt capacitors and series inductors and is most useful for low-pass filters. Cauer's second form of driving point impedance consists of a ladder of series capacitors and shunt inductors and is most useful for high-pass filters. Foster's first form of driving point impedance consists of parallel connected LC resonators and is most useful for band-pass filters. Foster's second form of driving point impedance consists of series connected LC antiresonators and is most useful for band-stop filters.

32


Transfer Function A transfer function is defined as the ratio of the Laplace transform of the output to the input with all initial conditions equal to zero. Transfer functions are defined only for linear time invariant systems. Transfer functions can usually be expressed as the ratio of two polynomials in the complex variable, s. A transfer function can be factored into the following form.

G( s) 

K ( s  z1 )( s  z2 ) ... ( s  zm ) ( s  p1 )( s  p2 ) ... ( s  pn )

The roots of the numerator polynomial are called zeros. The roots of the denominator polynomial are called poles.

Visualizing Pole-Zero plot Pole-Zero plots is an important tool, which helps us to relate the Frequency domain and Zdomain representation of a system. Understanding this relation will help in interpreting results in either domain. It also helps in determining stability of a system; given its transfer function H (z). Since the z-transform is a function of a complex variable, it is convenient to describe and interpret it using the complex z-plane. In the z-plane, the contour corresponding to |z| = 1 is a circle of unit radius. This contour is referred to as the Unit Circle. Also, the z-transform is most useful when the infinite sum can be expressed as a simple mathematical formula. One important form of representation is to represent it as a rational function inside the Region of Convergence, i.e.

Where, the numerator and denominator are polynomials in z. The values of z for which H (z) = 0 are called the zeros of H(z), and the values of z for which H(z) is ∞ are referred to as the poles of H(z). In other words, the zeros are the roots of the numerator polynomial and the poles of H (z) for finite values of z are the roots of the denominator polynomial.

33


A plot of Pole and Zeros of a system on the z-plane is called a Pole-Zero plots. Usually, a Zero is represented by a 'o'(small-circle) and a pole by a 'x'(cross). Since H (z) evaluated on the unit-circle gives the frequency response of a system. The pole-zero plots gives us a convenient way of visualizing the relationship between the Frequency domain and Z-domain. The frequency response H (e jw) is obtained from the transfer function H(z), by evaluating the transfer function at specific values of z = e jw. Since, the frequency response is periodic with period 2p, we need to evaluate it over one period, such as -p < w < p. If we substitute these values of w in z= e jw, values of z lie on the unit circle and range from z = 1 all the way around and back to the point z = -1. This is shown in Figure 1 below. From this the periodicity of 2p in frequency domain corresponds to moving through an angle of 2p on the unit circle.

An Example: You are given the following transfer function. Show the poles and zeros in the s-plane.

G( s ) 

(s  8)(s  14) s(s  4)(s  10)

34


Bode Plots A Bode plot is a (semi log) plot of the transfer function magnitude and phase angle as a function of frequency. The gain magnitude is many times expressed in terms of decibels (dB). Bode plot is the representation of the magnitude and phase of G(j*w) (where the frequency vector w contains only positive frequencies). dB = 20 log10 A where A is the amplitude or gain a decade is defined as any 10-to-1 frequency range an octave is any 2-to-1 frequency range 20 dB/decade = 6 dB/octave A Bode plot is a standard format for plotting frequency response of LTI systems. Becoming familiar with this format is useful because:  It is a standard format, so using that format facilitates communication between engineers.  Many common system behaviors produce simple shapes (e.g. straight lines) on a Bode plot, so it is easy to either look at a plot or recognize the system behavior, or to sketch a plot from what you know about the system behavior.  The format is a log frequency scale on the horizontal axis and, on the vertical axis, phase in degrees and magnitude in decibels.  Straight-line approximations of the Bode plot may be drawn quickly from knowing the poles and zeros o response approaches a minimum near the zeros o response approaches a maximum near the poles

The overall effect of constant, zero and pole terms

Term

Magnitude Break

Asymptotic Magnitude Slope

Asymptotic Phase Shift

Constant (K)

N/A

0

0

Zero

upward

+20 dB/decade

+ 90

Pole

downward

–20 dB/decade

– 90

35


Method •

Express the transfer function in standard form





K  j  (1  j1 ) 1  2 2 ( j2 )  ( j2 ) 2  H( j )  (1  ja ) 1  2 b ( jb )  ( jb ) 2  N



•

•



There are four different factors: – Constant gain term, K – Poles or zeros at the origin, (j)±N – Poles or zeros of the form (1+ j) – Quadratic poles or zeros of the form 1+2(j)+(j)2 We can combine the constant gain term (K) and the N pole(s) or zero(s) at the origin such that the magnitude crosses 0 dB at

Pole :

K ( j ) N

Zero : K ( j ) N

0 dB  K 1/ N 0 dB  (1 / K )1/ N

Define the break frequency to be at ω=1/ with magnitude at ±3 dB and phase at ±45°

Bode Plot Summary Magnitude Behavior Factor Constant

Low Freq

Break

Asymptotic

Phase Behavior Low Freq

Break

Asymptotic

20 log10(K) for all frequencies

0 for all frequencies

Poles or zeros at origin

±20N dB/decade for all frequencies with a crossover of 0 dB at ω=1

±90(N) for all frequencies

First order (simple) poles or zeros

0 dB

±3N dB at ω=1/

±20N dB/decade

0

±45(N) with slope ±45(N) per decade

±90(N)

Quadratic poles or zeros

0 dB

see ζ at ω=1/

±40N dB/decade

0

±90(N)

±180(N)

36


Single Pole and Zero Bode Plots

Gain Margin and Phase Margin

37




 

The gain margin is defined as the change in open loop gain required to make the system unstable. Systems with greater gain margins can withstand greater changes in system parameters before becoming unstable in closed loop. Keep in mind that unity gain in magnitude is equal to a gain of zero in dB. The phase margin is defined as the change in open loop phase shift required to make a closed loop system unstable. The phase margin is the difference in phase between the phase curve and -180 deg at the point corresponding to the frequency that gives us a gain of 0dB (the gain cross over frequency, Wgc).

 Likewise, the gain margin is the difference between the magnitude curve and 0dB at the point corresponding to the frequency that gives us a phase of -180 deg (the phase cross over frequency, Wpc).

38


UNIT IV

39


TWO PORT NETWORK A two-port network is an electrical network (circuit) or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the electric current entering one terminal must equal the current emerging from the other.

The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port and port 2 is considered the output port. The two-port network model is used in mathematical circuit analysis techniques to isolate portions of larger circuits. A two-port network is regarded as a "black box" with its properties specified by amatrix of numbers. This allows the response of the network to signals applied to the ports to be calculated easily, without solving for all the internal voltages and currents in the network. It also allows similar circuits or devices to be compared easily. For example, transistors are often regarded as two-ports, characterized by their h-parameters (see below) which are listed by the manufacturer. Any linear circuit with four terminals can be regarded as a two-port network provided that it does not contain an independent source and satisfies the port conditions. Examples of circuits analyzed as two-ports are filters, matching networks, transmission lines, transformers, and small-signal models for transistors (such as the hybrid-pi model). The analysis of passive two-port networks is an outgrowth of reciprocity theorems first derived by Lorentz. In two-port mathematical models, the network is described by a 2 by 2 square matrix of complex numbers. The common models that are used are referred to as z-parameters, y-parameters, hparameters, g-parameters, and ABCD-parameters, each described individually below. These are all limited to linear networks since an underlying assumption of their derivation is that any given circuit condition is a linear superposition of various short-circuit and open circuit conditions. They are usually expressed in matrix notation, and they establish relations between the variables

40


General Properties There are certain properties of two-ports that frequently occur in practical networks and can be used to greatly simplify the analysis. These include: Reciprocal Networks. A network is said to be reciprocal if the voltage appearing at port 2 due to a current applied at port 1 is the same as the voltage appearing at port 1 when the same current is applied to port 2. Exchanging voltage and current results in an equivalent definition of reciprocity. In general, a network will be reciprocal if it consists entirely of linear passive components (that is, resistors, capacitors and inductors). In general, it will not be reciprocal if it contains active components such as generators. Symmetrical Networks. A network is symmetrical if its input impedance is equal to its output impedance. Most often, but not necessarily, symmetrical networks are also physically symmetrical. Sometimes also anti symmetrical networks are of interest. These are networks where the input and output impedances are the duals of each other. Lossless Network. A lossless network is one which contains no resistors or other dissipative elements.

Impedance parameters (z-parameters)

Where z11 is the impedance seen looking into port 1 when port 2 is open. z12 is a transfer impedance. It is the ratio of the voltage at port 1 to the current at port 2 when port 1 is open. z21 is a transfer impedance. It is the ratio of the voltage at port 2 to the current at port 1 when port 2 is open. z22 is the impedance seen looking into port 2 when port 1 is open.

41


Notice that all the z-parameters have dimensions of ohms. For reciprocal networks lossless networks all the

. For symmetrical networks are purely imaginary.

. For reciprocal

Admittance parameters (y-parameters)

Where y11 is the admittance seen looking into port 1 when port 2 is shorted. y12 is a transfer admittance. It is the ratio of the current at port 1 to the voltage at port 2 when port 1 is shorted. y21 is a transfer impedance. It is the ratio of the current at port 2 to the voltage at port 1 when port 2 is shorted y22 is the admittance seen looking into port 2 when port 1 is shorted Notice that all the Y-parameters have dimensions of Mho. For reciprocal networks lossless networks all the

. For symmetrical networks are purely imaginary.

Hybrid parameters (h-parameters)

Where

42

. For reciprocal


This circuit is often selected when a current amplifier is wanted at the output. The resistors shown in the diagram can be general impedances instead. Notice that off-diagonal h-parameters are dimensionless, while diagonal members have dimensions the reciprocal of one another.

ABCD-parameters The ABCD-parameters are known variously as chain, cascade, or transmission line parameters. There are a number of definitions given for ABCD parameters, the most common is

For reciprocal networks . For symmetrical networks . For networks which are reciprocal and lossless, A and D are purely real while B and C are purely imaginary. This representation is preferred because when the parameters are used to represent a cascade of two-ports, the matrices are written in the same order that a network diagram would be drawn, that is, left to right. However, the examples given below are based on a variant definition.

Where

The negative signs in the definitions of parameters the opposite sense to , that is, .

and

arise because

is defined with

The reason for adopting this convention is so that the output current of one cascaded stage is equal to the input current of the next. Consequently, the input voltage/current matrix vector can be directly replaced with the matrix equation of the preceding cascaded stage to form a combined matrix. The terminology of representing the parameters designated a11 etc. as adopted by some authors and the inverse 43

as

a

matrix of elements parameters as a matrix


of elements designated b11 etc. is used here for both brevity and to avoid confusion with circuit elements.

There is a simple relationship between these two forms: one is the matrix inverse of the other, that is;

Two Port Parameter Conversions

44


Combinations of two-port networks When two or more two-port networks are connected, the two-port parameters of the combined network can be found by performing matrix algebra on the matrices of parameters for the component two-ports. The matrix operation can be made particularly simple with an appropriate choice of two-port parameters to match the form of connection of the two-ports. For instance, the z-parameters are best for series connected ports. The combination rules need to be applied with care. Some connections (when dissimilar potentials are joined) result in the port condition being invalidated and the combination rule will no longer apply. This difficulty can be overcome by placing 1:1 ideal transformers on the outputs of the problem two-ports. This does not change the parameters of the two-ports, but does ensure that they will continue to meet the port condition when interconnected. An example of this problem is shown for seriesseries connections in figures 11 and 12 below.

1. Series-series connection:

When two-ports are connected in a series-series configuration as shown in figure 10, the best choice of two-port parameter is the z-parameters. The z-parameters of the combined network are found by matrix addition of the two individual zparameter matrices

45


As mentioned above, there are some networks which will not yield directly to this analysis. [11] A simple example is a two-port consisting of a L-network of resistors R1 and R2. The z-parameters for this network are;

Figure 11 shows two identical such networks connected in series-series. The total z-parameters predicted by matrix addition are;

However, direct analysis of the combined circuit shows that ,

The discrepancy is explained by observing that R1 of the lower two-port has been by-passed by the short-circuit between two terminals of the output ports. This results in no current flowing through one terminal in each of the input ports of the two individual networks. Consequently, the port condition is broken for both the input ports of the original networks since current is still able to flow into the other terminal. This problem can be resolved by inserting an ideal transformer in the output port of at least one of the two-port networks. While this is a common text-book approach to presenting the theory of

46


two-ports, the practicality of using transformers is a matter to be decided for each individual design.

2. Parallel-parallel connection: When two-ports are connected in a parallelparallel configuration as shown in figure 13, the best choice of two-port parameter is the y-parameter

The y-parameters of the combined network are found by matrix addition of the two individual yparameter matrices.

3. Series-parallel connection: When two-ports are connected in a seriesparallel configuration as shown in figure 14, the best choice of two-port parameter is the h-parameters. The h-parameters of the combined network are found by matrix addition of the two individual h-parameter matrices.

4. Parallel-series connection:

When two-ports are connected in a parallelseries configuration as shown in figure 15, the best choice of two-port parameter is the g-parameters. The g-parameters of the 47


combined network are found by matrix addition of the two individual g-parameter

matrices.

5. Cascade connection: When two-ports are connected with the output port of the first connected to the input port of the second (a cascade connection) as shown in figure 16, the best choice of twoport parameter is the ABCD-parameters. The a-parameters of the combined network are found by matrix multiplication of the two individual a-parameter matrices.

A chain of n two-ports may be combined by matrix multiplication of the n matrices. To combine a cascade of b-parameter matrices, they are again multiplied, but the multiplication must be carried out in reverse order, so that;

48


UNIT V

49


NETWORK SYNTHESIS Hurwitz polynomial Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative. One sometimes uses the term Hurwitz polynomial simply as a (real or complex) polynomial with all zeros in the left-half plane (i.e., a Hurwitz stable polynomial). A polynomial is said to be Hurwitz if the following conditions are satisfied: 1. P(s) is real when s is real. 2. The roots of P(s) have real parts which are zero or negative. Note: Here P(s) is any polynomial in s. A simple example of a Hurwitz polynomial is the following:

The only real solution is −1, as it factors to

Properties For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. For all of a polynomial's roots to lie in the left half-plane, it is necessary and sufficient that the polynomial in question pass the Routh-Hurwitz stability criterion. A given polynomial can be tested to be Hurwitz or not by using the continued fraction expansion technique. 1. All the poles and zeros of a function are in the left half plane or on its boundary the imaginary axis. 2. Any poles and zeros on the imaginary axis are simple (have a multiplicity of one). 3. Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeros on the imaginary axis, the function has a real strictly positive derivative. 4. Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle). 5. there have no any missing term of 's' but it possible after the testing the prf stability

50


Positive-real functions, often abbreviated to PR function, are a kind of mathematical function that first arose in electrical network analysis. They are complex functions, Z(s), of a complex variable, s. A rational function is defined to have the PR property if it has a positive real part and is analytic in the right half plane of the complex plane and takes on real values on the real axis. In symbols the definition is,

In electrical network analysis, Z(s) represents an impedance expression and s is the complex frequency variable, often expressed as its real and imaginary parts; In which terms the PR condition can be stated;

The importance to network analysis of the PR condition lies in the reliability condition. Z(s) is realizable as one-port rational impedance if and only if it meets the PR condition. Realizable in this sense means that the impedance can be constructed from a finite (hence rational) number of discrete ideal passive linear elements (resistors, inductors and capacitors in electrical terminology).

Definition The term positive-real function was originally defined as any function Z(s) which   

is rational (the quotient of two polynomials), is real when s is real has positive real part when s has a positive real part

Many authors strictly adhere to this definition by explicitly requiring rationality, or by restricting attention to rational functions, at least in the first instance. However, a similar more general condition, not restricted to rational functions had earlier been considered by Cauer, and some authors ascribe the term positive-real to this type of condition, while other consider it to be a generalization of the basic definition.

Positive real Function Positive-real functions, often abbreviated to PR function, are a kind of mathematical function that first arose in electrical network analysis. They are complex functions, Z(s), of a complex 51


variable, s. A rational function is defined to have the PR property if it has a positive real part and is analytic in the right half plane of the complex plane and takes on real values on the real axis. In symbols the definition is,

In electrical network analysis, Z(s) represents an impedance expression and s is the complex frequency variable, often expressed as its real and imaginary parts;

in which terms the PR condition can be stated;

The importance to network analysis of the PR condition lies in the realizability condition. Z(s) is realizable as a one-port rational impedance if and only if it meets the PR condition. Realizable in this sense means that the impedance can be constructed from a finite (hence rational) number of discrete ideal passive linear elements (resistors, inductors and capacitors in electrical terminology)

Properties     



The sum of two PR functions is PR. The composition of two PR functions is PR. In particular, if Z(s) is PR, then so are 1/Z(s) and Z (1/s). All the poles and zeros of a PR function are in the left half plane or on its boundary the imaginary axis. Any poles and zeroes on the imaginary axis are simple (have a multiplicity of one). Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative. Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle). 52

NETWORK ANALYSIS AND SYNTHESIS (EEE-402) 

For a rational PR function, the number of poles and number of zeroes differ by at most one.

FILTERS Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both. Electronic filters can be:      

passive or active analog or digital High-pass, low-pass, band pass, band-reject (band reject; notch), or all-pass. discrete-time (sampled) or continuous-time linear or non-linear infinite impulse response (IIR type) or finite impulse response (FIR type)

The most common types of electronic filters are linear filters, regardless of other aspects of their design. See the article on linear filters for details on their design and analysis

L filter An L filter consists of two reactive elements, one in series and one in parallel.

T and π filters

Three-element filters can have a 'T' or 'π' topology and in geometries, a low-pass, high-pass, band-pass, or band-stop characteristic is possible. The components can be chosen symmetric or not, depending on the required frequency characteristics. The high-pass T filter in the illustration has very low impedance at high frequencies, and a very high impedance at low frequencies. That means that it can be inserted in a transmission line, resulting in the high frequencies being passed and low frequencies being reflected Likewise, for the illustrated low-pass π filter, the circuit can be connected to a transmission line, transmitting low frequencies and reflecting high frequencies. Using m-derived filter sections with correct termination impedances, the input impedance can be reasonably constant in the pass band. 53


Low-pass filter A low-pass filter is an electronic filter that passes low-frequency signals and attenuates (reduces the amplitude of) signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in audio applications. A low-pass filter is the opposite of a high-pass filter. A band-pass filter is a combination of a low-pass and a high-pass. Low-pass filters exist in many different forms, including electronic circuits (such as a hiss filter used in audio), anti-aliasing filters for conditioning signals prior to analog-to-digital conversion, digital filters for smoothing sets of data, acoustic barriers, blurring of images, and so on. The moving average operation used in fields such as finance is a particular kind of low-pass filter, and can be analyzed with the same signal processing techniques as are used for other lowpass filters. Low-pass filters provide a smoother form of a signal, removing the short-term fluctuations, and leaving the longer-term trend. In an electronic low-pass RC filter for voltage signals, high frequencies contained in the input signal are attenuated but the filter has little attenuation below its cutoff frequency which is determined by its RC time constant. For current signals, a similar circuit using a resistor and capacitor in parallel works in a similar manner. See current divider discussed in more detail below. Electronic low-pass filters are used on input signals to subwoofers and other types of loudspeakers, to block high pitches that they can't efficiently reproduce. Radio transmitters use low-pass filters to block harmonic emissions which might cause interference with other communications. The tone knob found on many electric guitars is a low-pass filter used to reduce the amount of treble in the sound. An integrator is another example of a single time constant low-pass filter. Telephone lines fitted with DSL splitters use low-pass separate DSL and POTS signals sharing the same pair of wires.

and high-pass filters

to

Low-pass filters also play a significant role in the sculpting of sound for electronic music as created by analogue synthesizers.

Ideal and real filters

54


frequency while unchanged.

passing

those

below

its frequency response is a rectangular function, and is abrick-wall filter. The transition region present in practical filters does not exist in an ideal filter. An ideal low-pass filter completely eliminates all frequencies above the cutoff An ideal low-pass filter can be realized mathematically (theoretically) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with its impulse response, a sinc function, in the time domain However, the ideal filter is impossible to realize without also having signals of infinite extent in time, and so generally needs to be approximated for real ongoing signals. Real filters for real-time applications approximate the ideal filter by truncating and windowing the infinite impulse response to make a finite impulse response; applying that filter requires delaying the signal for a moderate period of time, allowing the computation to "see" a little bit into the future.

There are many different types of filter circuits, with different responses to changing frequency. The frequency response of a filter is generally represented using a Bode plot, and the filter is 55


characterized by its cutoff frequency and rate of frequency roll off. In all cases, at the cutoff frequency, the filter attenuates the input power by half or 3 dB. So the order of the filter determines the amount of additional attenuation for frequencies higher than the cutoff frequency. 

A first-order filter, for example, will reduce the signal amplitude by half (so power reduces by a factor of 4), or 6 dB, every time the frequency doubles (goes up one octave); more precisely, the power roll off approaches 20 dB per decade in the limit of high frequency. The magnitude Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency, and a diagonal line above the cutoff frequency. There is also a "knee curve" at the boundary between the two, which smoothly transitions between the two straight line regions. If the transfer function of a first-order low-pass filter has a zero as well as a pole, the Bode plot will flatten out again, at some maximum attenuation of high frequencies; such an effect is caused for example by a little bit of the input leaking around the one-pole filter; this onepole–one-zero filter is still a first-order low-pass. See Pole–zero plot and RC circuit.



A second-order filter attenuates higher frequencies more steeply. The Bode plot for this type of filter resembles that of a first-order filter, except that it falls off more quickly. For example, a second-order Butterworth filter will reduce the signal amplitude to one fourth its original level every time the frequency doubles (so power decreases by 12 dB per octave, or 40 dB per decade). Other all-pole second-order filters may roll off at different rates initially depending on their Q factor, but approach the same final rate of 12 dB per octave; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote. See RLC circuit.

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Third- and higher-order filters are defined similarly.

Electronic Low Pass Filter One simple electrical circuit that will serve as a low-pass filter consists of a resistor in series with a load, and a capacitor in parallel with the load. The capacitor exhibits reactance, and blocks lowfrequency signals, causing them to go through the load instead. At higher frequencies the reactance drops, and the capacitor effectively functions as a short circuit. The combination of resistance and capacitance gives the time constant of the filter (represented by the Greek letter tau). The break frequency, also called the turnover frequency Or cutoff frequency (in hertz), is determined by the time constant: 56


Or equivalently (in radians per second):

One way to understand this circuit is to focus on the time the capacitor takes to charge. It takes time to charge or discharge the capacitor through that resistor:  

At low frequencies, there is plenty of time for the capacitor to charge up to practically the same voltage as the input voltage. At high frequencies, the capacitor only has time to charge up a small amount before the input switches direction. The output goes up and down only a small fraction of the amount the input goes up and down. At double the frequency, there's only time for it to charge up half the amount.

Another way to understand this circuit is with the idea of reactance at a particular frequency:  

Since DC cannot flow through the capacitor, DC input must "flow out" the path marked (analogous to removing the capacitor). Since AC flows very well through the capacitor — almost as well as it flows through solid wire — AC input "flows out" through the capacitor, effectively short circuiting to ground (analogous to replacing the capacitor with just a wire).

The capacitor is not an "on/off" object (like the block or pass fluidic explanation above). The capacitor will variably act between these two extremes. It is the Bode plot and frequency response that show this variability.

Active electronic realization

In the operational amplifier circuit shown in the figure, the cutoff frequency (in hertz) is defined as: 57


or equivalently (in radians per second):

The gain in the passband is −R2/R1, and the stopband drops off at −6 dB per octave (that is −20 dB per decade) as it is a first-order filter.

High-pass filter A high-pass filter (HPF) is an electronic filter that passes highfrequency signals but attenuates (reduces the amplitude of) signals with frequencies lower than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. A high-pass filter is usually modeled as a linear time-invariant system. It is sometimes called a low-cut filter or bass-cut filter. High-pass filters have many uses, such as blocking DC from circuitry sensitive to non-zero average voltages or RF devices. They can also be used in conjunction with a low-pass filter to make a bandpass filter.

First-order implementation The simple first-order electronic high-pass filter shown in Figure 1 is implemented by placing an input voltage across the series combination of a capacitor and a resistor and using the voltage across the resistor as an output. The product of the resistance and capacitance (R×C) is the time constant (τ); it is inversely proportional to the cutoff frequency fc, that is,

where fc is in hertz, τ is in seconds, R is in ohms, and C is in farads.

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Figure shows an active electronic implementation of a first-order highpass filter using an operational amplifier. In this case, the filter has a pass band gain of -R2/R1 and has a corner frequency of

Because this filter is active, it may have non-unity pass band gain. That is, high-frequency signals are inverted and amplified by R2/R1.

Band-pass filter and rejects (attenuates) frequencies outside that range. Optical band-pass filters are of common usage. An example of an analogue electronic band-pass filter is an RLC circuit (a resistor–inductor– capacitor circuit). These filters can also be created by combining a low-pass filter with a high-pass filter A band-pass filter is a device that passes frequencies within a certain range . Bandpass is an adjective that describes a type of filter or filtering process; it is to be distinguished from pass band, which refers to the actual portion of affected spectrum. Hence, one might say "A dual bandpass filter has two pass bands." A bandpass signal is a signal containing a band of frequencies away from zero frequency, such as a signal that comes out of a bandpass filter An ideal bandpass filter would have a completely flat pass band (e.g. with no gain/attenuation throughout) and would completely attenuate all frequencies outside the pass band. Additionally, the transition out of the pass band would be instantaneous in frequency. In practice, no bandpass filter is ideal. The filter does not attenuate all frequencies outside the desired frequency range completely; in particular, there is a region just outside the intended pass band where frequencies are attenuated, but not rejected. 59


This is known as the filter roll-off, and it is usually expressed in dB of attenuation per octave or decade of frequency. Generally, the design of a filter seeks to make the roll-off as narrow as possible, thus allowing the filter to perform as close as possible to its intended design. Often, this is achieved at the expense of pass-band or stop-band ripple. The bandwidth of the filter is simply the difference between the upper and lower cutoff frequencies. The shape factor is the ratio of bandwidths measured using two different attenuation values to determine the cutoff frequency, e.g., a shape factor of 2:1 at 30/3 dB means the bandwidth measured between frequencies at 30 dB attenuation is twice that measured between frequencies at 3 dB attenuation. Outside of electronics and signal processing, one example of the use of band-pass filters is in the atmospheric sciences. It is common to band-pass filter recent meteorological data with a period range of, for example, 3 to 10 days, so that only cyclones remain as fluctuations in the data fields.

Band-stop Filter or band-rejection Filter In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a bandpass filter. A notch filter is a band-stop filter with a narrow stopband (high Q factor).

Narrow notch filters (optical) are used in Raman spectroscopy, live sound reproduction (public address systems, or PA systems) and in instrument amplifiers (especially amplifiers or preamplifiers for acoustic instruments such as acoustic guitar, mandolin, bass instrument amplifier, etc.) to reduce or prevent audio feedback, while having little noticeable effect on the rest of the frequency spectrum (electronic or software filters). Other names include 'band limit filter', 'T-notch filter', 'band-elimination filter', and 'band-reject filter'. Typically, the width of the stopband is 1 to 2 decades (that is, the highest frequency attenuated is 10 to 100 times the lowest frequency attenuated). However, in the audio band, a notch filter has high and low frequencies that may be only semitones apart.

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