BASICS OF NETWORK ANALYSIS AND THEOREMS] Basic Electrical © Lecture teshimaryokan.info (teshimaryokan.info) Page 1 Basics of Network Analysis and. NETWORK ANALYSIS. Notes Credits to Prof Shivananda (CITECh, Blr) ENGINEERING EM. Notes Credits to Navya S M. Networks Theory (Hand Written Notes) - Ebook download as PDF File .pdf) or read book online. for GATE Prep.
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Engineering Class handwritten notes, exam notes, previous year questions, PDF free download. CH: 2 NETWORKS THEOREM: click here to download. CH: 3 INITIAL CONDITIONS: click here to download. CH: 4 FIRST ORDER DIFF. MODERN NETWORK ANALYSIS I. LECTURE NOTES. Aleksandar I. Zecevic. Dept. of Electrical Engineering. Santa Clara University.
Get New Updates Email Alerts Enter your email address to subscribe to this blog and receive notifications of new posts by email. Hence the series A. Two of them are expressed in. Millmans Theorem says that if a number of voltage sources with internal impedances are connected in parallel across two terminals, then the entire combination can be replaced by a single voltage source in series with single impedance. In Other words the evaluation of integral in equation 1 requires the use of contour integration In the complex plane, which is very difficult. Example 3: EasyEngineering Team Publication team comprises of highly renowned and experienced authors, Civil and Engineering Services qualified rankers and professors from top institutions who are well known all over India for their contribution in books and international journals.
Cut set Analysis A cut set of a graph is a set of branches whose removal , cuts the connected graph in to two parts such that the replacement of any one branch of the cut set renders the two parts connected. Note that FCS - 1 yields node A and the set of nodes B, C, D The Orientation of the fundamental cut set is usually assumed to be the same as the orientation of the tree branch in it, Which is shown by an arrow.
It should be noted that for each tree branch there will be a fundamental cut set. For a graph having n number of nodes the number of twigs is n Therefore there will be n-1 n-1 fundamental cut-sets. Once the fundamental cut sets are identified and their orientations are fixed, it is possible to write a schedule, known as cut set schedule which gives the relation between tree branch voltages and all other branch voltages of the graph. Let the element of a cut set schedule be denoted by Qik then,.
O If branch is k is not in cut set i. Each of the branches has a representation as shown in figure VK YK. The elements of the source current matrix are positive if the directions of the branch current and the source connect attached to that branch are same otherwise negative. Example 2: The tree marked by thick lines and the link marked by doffed lines are as shown.
The voltage source is Transformed in to an equivalent current source. It should be noted that all the circuit Passive elements must be admittances and the net work should contain only current sources.
The graph for the network is shown. A possible tree shown with thick lines and co tree shown by dotted lines are shown. Table of dual Quantities 1. Voltage Source 2. Loop currents 3. Iductances 4. Resistances 5. Capacitances 6. On KVL basis 7. Close of switch. Only planar networks have duals. Procedure for drawing dual network The duals of planar networks could be obtained by a graphical technique known as the dot method.
The dot method has the following procedure. Put a dot in each independent loop of the network. These dots correspond to independent nodes in the dual network.
Comparing the equations 1 and 2 ,we get the similarity between the networks of fig 1 and fig 2. The solution of equation 1 will be identical to the solution of equation 2 when the following exchanges are made R G, L C, CL and V i Hence networks of figure 1 and 2 are dual to each other.
Put a dot outside the network. This dot corresponds to the reference node in the dual network. Connect all internal dots in the neighboring loops by dotted lines cutting the common branches. These branches that are cut by dashed lines will form the branches connecting the corresponding independent nodes in the dual network. Join all internal dots to the external dot by dashed lines cutting all external branches.
Duals of these branches will form the branches connecting the independent nodes and the reference node. Join node 1 and reference node through a dotted line passing through 3 ohms resistor. This element appears as 3mho conductance between node1 and reference node in the dual. Join node 2 and reference node through a dotted line passing through the capacitor of 4 Farads. This element will appear as 4 Henry inductor between node 2 and reference node in the dual Join node 2 and reference node through a dotted line passing through the resistor of 4 ohms.
This element will appear as 4 mho conductance between node 2 and reference node. The Dual network drawn using these procedural steps is shown. Mark two independent nodes 1 and 2 and a reference node 0as shown in the figure.
Join node 1 and 2 by a dotted line passing through the inductance of 6H. Join node 1 and reference node through a dotted line passing the voltage source of 2sin6t volts.
This will appear as a current source of 2sin6t amperes between node1 and reference node. Give suitable example. Calculate all branch current. For the graph shown in Fig. Write in intergo differential form i mesh equations for the given network ii node equations for the dual. Draw the dual of the circuit shown in fig.
Syllabus of unit: Superposition, Reciprocit y and Millmans theorems Recommended readings: A number of simultaneous equations are to be set up. Solving these equations, the response in all the branches of the network may be attained. But in many cases, we require the response in one branch or in a small part of the network. In such cases, we can use network theorems, which are the aides to simplify the analysis. To reduce the amount of work involved by considerable amount, as compared to mesh or nodal analysis.
Let us discuss some of them. The response of a linear network with a number of excitations applied simultaneously is equal to the sum of the responses of the network when each excitation is applied individually replacing all other excitations by their internal impedances.
Here the excitation means an independent source. Initial voltage across a capacitor and the initial current in an inductor are also treated as independent sources. This theorem is applicable only to linear responses and therefore power is not subject to superposition. During replacing of sources, dependent sources are not to be replaced.
Replacing an ideal voltage source is by short circuit and replacing an ideal current source is by open circuit. In any linear network containing a number of sources, the response current in or voltage across an element may be calculated by superposing all the individual responses caused by each independent source acting alone, with all other independent voltage sources replaced by short circuits and all other independent current sources replaced by open circuits.
Initial capacitor voltages and initial inductor currents, if any, are to be treated as independent sources. To prove this theorem consider the network shown in fig. We consider only one-voltage sources and only one current sources for simplicity. In an initially relaxed linear network containing one independent source only.
The ratio of the response to the excitation is invariant to an interchange of the position of the excitation and the response. Similarly if the single current source Ix between nodes X and X produces the voltage response Vy between nodes Y and Y then the removal of the current source from X and X and its insertion between Y and Y will produce the voltage response Vy between the nodes X and X. Between the excitation and the response, one is voltage and other is current.
It should be noted that after the source and response are interchanged, the current and the voltages in other parts of the network will not remain the same. Consider a network as shown in which the excitation is E and the response is I in Z4.
The reading of the ammeter is. The transfer impedance between any two pairs of terminals of a linear passive network is the ratio of the voltage applied at one pair of terminals to the resulting current at the other pair of terminals.
With this definition the reciprocity theorem can be stated as: Only one value of transfer impedance is associated with two pairs of terminals of a linear passive network. Sum of the product of individual voltage sources and their series admittances Sum of all series admittances and the single series impedance is the reciprocal of sum of all series admittances. Certain simple combinations of potential and current source equivalents are of use because they offer simplification in solutions of more extensive networks in which combinations occur.
Millmans Theorem says that if a number of voltage sources with internal impedances are connected in parallel across two terminals, then the entire combination can be replaced by a single voltage source in series with single impedance.
Let E1, E2. En be the voltage sources and Z1, Z2Zn are their respective impedances.
Transform each voltage into its equivalent current source. Then the circuit is as in Fig. The theorem can be stated as If a number of current sources with their parallel admittances are connected in series between terminals A and B, then they can be replaced by a single current source in parallel with a single admittance.
The single current source is the ratio Sum of products of individual current sources and their impedances Sum of all shunt impedances. Let I1, I2,.. In be the n number of current sources and Y1,Y Also find maximum power. Using superposition theorem, obtain the response I for the network shown in Fig. Thevinins and Nortons Theorems: If we are interested in the solution of the current or voltage of a small part of the network, it is convenient from the computational point of view to simplify the network, except that small part in question, by a simple equivalent.
This is achieved by Thevinins Theorem or Nortons theorem. If two linear networks one M with passive elements and sources and the other N with passive elements only and there is no magnetic coupling between M and N, are connected together at terminals A and B, then with respect to terminals A and B, the network M can be replaced by an equivalent network comprising a single voltage source in series with a single impedance.
The single voltage source is the open circuit voltage across the terminals A and B and single series impedance is the impedance of the network M as viewed from A and B with independent voltage sources short circuited and independent current sources open circuited. Dependent sources if any are to be retained. Suppose the required response is the current IL in ZL. Connected between A and B.
According to Thevinins theorem the following steps are involved to calculate IL Step 1: Remove ZL and measure the open circuit voltage across AB. To obtain the single impedance as viewed from A and B, replace the network in Fig.
The Thevinins equivalent consists of a voltage source and a series impedance. If the circuit is transformed to its equivalent current source, we get Nortons equivalent. Thus Nortons theorem is the dual of the Thevinins theorem. The proof of the Nortons theorem is simple Consider the same network that is considered for the Thevinins Theorem and for the same response. Step 1: Short the terminals A and B and measure the short circuit current in AB, this is Nortons current source.
Z1 Z2 Zs. If two linear networks, one M with passive elements and sources and the other N with passive elements only and with no magnetic coupling between M and N, are connected together at terminals A and B, Then with respect to terminals A and B, the network M can be replaced by a single current source in parallel with a single impedance.
The single current source is the short circuit current in AB and the single impedance is the impedance of the network M as viewed from A and B with independent sources being replaced by their internal impedances. This is the same as in the case of thevnins theorem Step 3: When network contains both dependent and independent sources. It is convenient to determine ZTH by finding both the open circuit voltage and short circuit current If the network contains only dependent sources both VTH and IN are zero in the absence of independent sources.
Then apply a constant voltage source or resultant source and the ratio of voltage to current gives the ZTH. Maximum Transfer Theorem: When a linear network containing sources and passive elements is connected at terminals A and B to a passive linear network, maximum power is transferred to the passive network when its impedance becomes the complex conjugate of the Thevinins impedance of the source containing network as viewed form the terminals A and B.
Efficiency of Power Transfer: Such a low efficiency cannot be permitted in power systems involving large blocks of power where RL is very large compared to Rs. Therefore constant voltage power systems are not designed to operate on the basis of maximum power transfer. However in communication systems the power to be handled is small as these systems are low current circuits. Thus impedance matching is considerable factor in communication networks. Case iii: Case i: Series and parallel resonance, frequency response of series and Parallel circuits, Q factor, Bandwidth.
Recommended readings:. Resonant Circuits Resonance is an important phenomenon which may occur in circuits containing both inductors and capacitors. In a two terminal electrical network containing at least one inductor and one capacitor, we define resonance as the condition, which exists when the input impedance of the network is purely resistive.
In other words a network is in resonance when the voltage and current at the network in put terminals are in phase. Resonance condition is achieved either by keeping inductor and capacitor same and varying frequency or by keeping the frequency same and varying inductor and capacitor. Study of resonance is very useful in the area of communication.
The ability of a radio receiver to select the correct frequency transmitted by a broad casting station and to eliminate frequencies from other stations is based on the principle of resonance. The resonance circuits can be classified in to two categories Series Resonance Circuits. Parallel Resonance Circuits. Series Resonance Circuit.
A series resonance circuit is one in which a coil and a capacitance are connected in series across an alternating I voltage of varying frequency as shown in figure. Hence the series A.
Thus the resonance curve will be as shown in figure. Another feature of a resonant circuit is the Q rise of voltage across the resonating elements. The output response during limited band of frequencies only will be in the useful range. If the out put power is equal to.
Selectivity is a useful characteristic of the resonant circuit. Hence larger the value of Q Smaller will be the selectivity. The Selectivity of a resonant circuit depends on how sharp the out put is contained with in limited band of frequencies.
The circuit is said to be highly selective if the resonance curve falls very sharply at off resonant frequencies.
In the figure it is seen that there are two frequencies where the out put power is half of the maximum power. These frequencies are called as half power points f1 and f2 A frequency f1 which is below fr where power is half of maximum power is called as lower half power frequency or lower cut off frequency.
Similarly frequency f2 which is above fr is called upper half power frequency or upper cut-off frequency The band of frequencies between f2 and f1 are said to be useful band of frequencies since during these frequencies of operation the out put power in the circuit is more than half of the maximum power. Thus their band of frequencies is called as Bandwidth. Relation between Resonant frequency and cut-off frequencies Let fr be the resonant frequency of a series resonant circuit consisting of R,L and C elements.
From the Characteristic it is seen that at both half frequencies f2 and f1 the out put current is 0.
At lower cut-off frequency f1 f1 Ir 0. A parallel resonant circuit is one in which a coil and a capacitance are connected in parallel across a variable frequency A. The response of a parallel resonant circuit is somewhat different from that of a series resonant circuit. Impedance at resonance We know that at resonance the susceptive part of the admittance is zero. Resonance by varying Inductance Resonance in RLC series circuit can also be obtained by varying resonating circuit elements.
Let us consider a circuit where in inductance is varied as shown in figure. Hence the current at resonance is minimum. Hence this type of circuit is called rejector circuit. Frequency response characterisitics The frequency response curve of a parallel resonant circuit is as shown in the figure.
We find that current is minimum at resonance. The half power points are given by the points at which the current is vf2 Ir.
From the above characteristic it is clear that the characteristic is exactly opposite to that of series resonant. Quality factor Q-factor The quality factor of a parallel resonant circuit is defined as the current magnification.
An Impedance coil of 25 ohms Resistance and 25 mH inductance is connected in parallel with a variable capacitor. For what value of Capacitor will the circuit resonate. If 90 volts, Hz source is used, what will be the line Current under these conditions Solution: Derive the condition for parallel resonance when RL connected parallel to RC.
CI 6 A series resonant circuit includes 1 11F capacitor and a resistance of 16? Resonance is to be achieved by variation of RL and RC. Calculate the resonance frequency for the following cases: Consider The R-L series circuit shown in the fig.
Electrical circuits are connected to supply by closing the switch and disconnected from the supply by opening the switch. This switching operation will change the current and voltage in the device. A purely resistive device will allow instantaneous change in current and voltage. An inductive device will not allow sudden change in current or delay the change in current.
A capacitive device will not allow sudden change in voltage or delay the change in voltage. Hence when switching operation is performed in inductive or capacitive device the current and voltage in the device will take a certain time to change from preswitching value to steady value after switching. This study of switching condition in network is called transient analysis.
The state or condition of the current from the instant of switching to attainment of steady state is called transient state or transient.
The current and voltage of circuit elements during transient period is called transient response. The transient may also occur due to variation in circuit elements. Transient analysis is an useful tool in electrical engineering for analysis of switching conditions in Circuit breakers, Relays, Generators etc.
It is also useful for the analysis of faulty conditions in electrical devices. Transient analysis is also useful for analyzing switching Conditions in analog and digital Electronic devices. The equation clearly indicates transient nature of current, which is also shown in figure. Hence Time constant for an R-L series current circuit is defined as the time taken by the circuit to reach Network Analysis Consider the RC circuit shown.
When the circuit is switched to position 2, this 1 Amp current constituted the stored energy in the coil. The steady state current having been previously established in R-L circuit.
Find the current i t after switching. A series R-C circuit is shown in figure. The capacitor has an initial charge of Coulombs on its plates, at the time the switch is closed. Find the resulting current transient. For the circuit shown in figure the relay coil is adjusted to operate at a current of 5 Amps. Find the value of inductance L of the relay. In figure the switch K is closed.
Find the time when the current in the circuitry reaches to mA Soln: Intial conditions: The reason for studying initial and final conditions in a network is to evaluate the arbitrary constants that appear in the general solution of the differential equations written for the network. In this chapter we concentrate on finding the change in selected variables in a circuit when a switch is thrown from open to closed or vice versa position.
Past history will show up in the form of capacitor voltages and inductor currents. Initial and final conditions in elements. The resistor: From this equation we find that the current through a resistor will change instantaneously, if the voltage changes instantaneously.
Similarly voltage will change instantaneously if current changes instantaneously. Write the equivalent from of the elements in terms of the initial condition of the element. Laplace transform is a very useful and powerful tool in circuit analysis. Integro-differential equations Can be transformed in to algebraic equations using the technique of Laplace transformation and complete solution involviong both natural response and forced response is obtained in one step Definition of Laplace Transform: Let f t be a function of time.
Important properties of Laplace transform Linearity Property: Network Analysis 5. Let us consider a Function x t that is periodic as shown in figure. The function x t can be represented as the sum of time-shifted functions as shown in figure. Where x1 t is the waveform described over the first period of x t. Initial value Theorem: The Initial -value theorem allows us to find the initial value x 0 directly from the Laplace Transform X S.
The function X S as defined by equation 2 is said to be rational function of S,since It is a ratio of two polynomials. The denominator Q S can be factored in to linear factors. A partial fraction expansion allows a strictly proper rational function P S to be expressed Q S As a factor of terms whose numerators are constants and whose denominator corresponds to Linear or a combination of linear and repeated factors.
This in turn allows us to relate such terms To their corresponding inverse Laplace transform. For performing partial fraction technique on X S the function X S has to meet the following conditions.
When X S is improper we can use long division. Partial Fraction method: In Other words the evaluation of integral in equation 1 requires the use of contour integration In the complex plane, which is very difficult.
Hence we will avoid using equation 1 to compute Inverse Laplace transform. We go for indirect methods to get the inverse Laplace transform of The given function,which are Partial Fraction method Convolution integral method. Network Analysis to reduce it to proper fraction.
Convolution-integral method: The step function can have a discontinuiy For example in sequential switching. They are called Singularity functions because they are either not finite or they do not posess finite derivative everywhere. The three important singularity functions employed in circuit analysis are the unit step function u t the delta function t the ramp function r t.
We use step function to represent an abrupt change in voltage or current , like the changes that oo Occur in the circuit of control engineering and digital systems. The derivative of the unit step function is the unit impulse function t t i. Since the area under the unit impulse is unity, it is practice to write 1 beside the arrow. When the impulse has a strength other than unity the area of the impulse function is equal to its strength.
In general a ramp is a function that changes at a Constant rate. A delayed ramp function is shown in figure Mathematically it is described as follows. Also find the Laplace transform of the following wave form shown in Fig. Assume zero initial condition. What are the limitations of each theorem? Definition of z, y, h and transmission parameters, modeling with these parameters, relationship between parameters sets.
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