Network Theory. EC / EE / IN. for

Transcription

1 Network Theory for / / IN By

2 Syllabus Syllabus for Networks Network Graphs: Matrces Assocated Wth Graphs: Incdence, Fundamental ut Set and Fundamental rcut Matrces. Soluton Methods: Nodal and Mesh Analyss. Network Theorems: Superposton, Thevenn and Norton s, Maxmum Power Transfer, Wye-Delta Transformaton. Steady State Snusodal Analyss Usng Phasors. Lnear onstant oeffcent Dfferental quatons, Tme Doman Analyss of Smple RL rcuts, Soluton of Network quatons Usng Laplace Transform, Frequency Doman Analyss of RL rcuts. 2-Port Network Parameters, Drvng Pont And Transfer Functons, State quatons For Networks. Analyss of GAT Papers Year IN Over All Percentage 10.93% 11.87% 10.50% : , opyrght reserved. Web:

3 ontents ontents hapters Page No. #1. Network Soluton Methodology 1 29 Introducton 1 2 Seres and Parallel onnecton of rcut lements 2 4 Mesh Analyss 4 5 Nodal Analyss 5 6 Voltage and urrent Source 6 Dependent Sources 6 7 Power and nergy 8 Network Theorems 8 11 Solved xamples Assgnment Assgnment Answer Keys & xplanatons #2. Transent/Steady State Analyss of RL rcuts Transent Response Analyss of Network lements Solved xamples Assgnment Assgnment Answer Keys & xplanatons #3. Snusodal Steady State Analyss Snusodal xctaton Average Power Suppled by A.. Source 58 Resonance Sngle Phase rcut Analyss Polyphase rcut Analyss Magnetcally oupled rcuts Solved xamples Assgnment Assgnment Answer Keys & xplanatons : , nfo@thegateacademy.com opyrght reserved. Web:

4 ontents #4. Frequency Response Analyss Defnton 87 Laplace Transform of Standard Functons Intal Value Theorem 88 Fnal Value Theorem Transfer Functon of LTI System rcut Analyss at a Generalzed Frequency 90 Standard Structures Drvng Pont Functon Solved xamples Assgnment Assgnment Answer Keys & xplanatons #5. Two Port Networks Defnton 110 Types of Parameters Some Standard Networks Inter-onnecton of Two Port Networks Solved xamples Assgnment Assgnment Answer Keys & xplanatons #6. Network Topology Defntons Nodal Incdence Matrx 130 Reduced Incdence Matrx 130 Loop Incdence Matrx Fundamental ut-set Matrx Solved xamples Assgnment Assgnment Answer Keys & xplanatons Module Test Test Questons Answer Keys & xplanatons Reference Books 160 : , nfo@thegateacademy.com opyrght reserved. Web:

5 HAPTR 1 Network Soluton Methodology Success conssts of gong from falure to falure wthout loss of enthusasm.. Wnston hurchll Learnng Objectves After readng ths chapter, you wll know: 1. Seres and Parallel onnecton of rcut lements 2. Voltage and urrent Relaton of Network 3. KL, KVL 4. Mesh Analyss 5. Nodal Analyss 6. Voltage and urrent Sources 7. Dependent Sources 8. Power and nergy 9. Network Theorems 10. Delta to Star Transformaton Introducton The passve crcut elements resstance R, nductance L and capactance are defned by the manner by whch voltages and currents are related for the ndvdual element. The table below summarzes the voltage-current(v-i) relaton, nstantaneous power (P) consumpton and energy stored n the perod [t,t ] for each of above elements. SL No. rcut lement Symbol n lectrc rcut Unts Voltage urrent Relaton Instantaneous Power, P nergy Stored / Dsspated n [ ] 1 Resstance, R V Ohm ( Ω ) V= R (Ohm s law) V= t t ) 2 Inductance, L V Henry ( H ) L ) 3 apactance, V Farad ( F ) = v v v ) Table 1.1 Voltage urrent Relaton of Network lements : , nfo@thegateacademy.com opyrght reserved. Web: 1

6 Network Soluton Methodology In the above table, f s the current at nstant m and s the voltage at nstant m, total energy dsspated n a resstor (R) n t t v dt dt Seres and Parallel onnecton of rcut lements Fgure below summarzes equvalent resstance /nductance /capactance for dfferent combnatons of network elements. R =. R. =.... Seres and Parallel onnecton of rcut lements Voltage / urrent Relaton n Seres / Parallel onnecton of Resstor Fgures below summarze voltage/current relatons n seres and parallel connecton of resstors. Seres onnecton of Resstor For seres connecton of resstors, = = [ ] n : , nfo@thegateacademy.com opyrght reserved. Web: 2

7 Network Soluton Methodology For parallel connecton of resstors, Parallel ombnaton of Resstors ( ) n = V Krchoff s urrent Law (KL) The algebrac sum of current at a node n a electrcal crcut s equal to zero. At any pont n electrcal crcut the phasor sum of the current flowng towards a juncton s equal to the phasor sum of the currents flowng away from the juncton. Assumng that current approachng node O bears postve sgn, and vce versa, At any pont n an electrcal crcut the phasor sum of the currents flowng towards that juncton s equal to the phasor sum of the current flowng away from the juncton. Node = Juncton Krchoff s oltage aw (KVL) Demonstraton of KL In any closed loop electrcal crcut, the algebrac sum of voltage drops across all the crcut elements s equal to emf rse n the same. Fgure below demonstrates KVL and KL equaton can be wrtten as, In any closed loop n a networks the phasor sum of the voltage chops (.e., product of current and mpedance) taken around the loop s equal to the phasor sum of the emf s actng n that loop. Node O Demonstraton of KVL : , nfo@thegateacademy.com opyrght reserved. Web: 3

8 Network Soluton Methodology + From Krchoff s current law, Also from KVL, ) ) Demonstraton of KL & KVL Mesh Analyss In the mesh analyss, a current s assgned to each wndow of the network such that the currents complete a closed loop. They are also referred to as loop currents. ach element and branch therefore wll have an ndependent current. When a branch has two of the mesh currents, the actual current s gven by ther algebrac sum. Once the currents are assgned, Krchhoff s voltage law s wrtten for each of the loops to obtan the necessary smultaneous equatons. Use mesh analyss to fnd and, ( ) ) ) ) The above set of smultaneous equatons should be solved for and. Soluton of Smultaneous quatons Matrx Inverson Method The above set of equatons can be wrtten as below n matrx form: [ ] [ ] = * + Ths s of form, AX = B Where, Demonstraton of Mesh Analyss ramer s Rule To fnd ether of and use ramer s rule as below, : , nfo@thegateacademy.com opyrght reserved. Web: 4

9 Network Soluton Methodology Mesh Analyss (Usng Super Mesh) When two of the loops have a common element as a current source, mesh analyss s not appled to both loops separately. Instead both the loops are merged and a super mesh s formed. Now KVL s appled to super mesh. For the crcut n fgure shown below, = I Demonstraton of Mesh Analyss Usng Super Mesh Nodal Analyss Typcally, electrcal networks contan several nodes, where some are smple nodes and some are prncpal nodes. In the node voltage method, one of the prncpal nodes s selected as the reference and equatons based on KL are wrtten at the other prncpal nodes. At each of these other prncpal nodes, a voltage s assgned, where t s understood that ths voltage s wth respect to the reference node. These voltages are the unknowns and are determned by Nodal Analyss. ~ ~ Demonstraton of Nodal Analyss Use nodal analyss to fnd, When the node voltages to be found by nodal analyss are more than 1, the node voltages can be found from smultaneous equatons by matrx nverson method or ramer s rule. Nodal Analyss (Includng Super Node) When deal voltage source s connected between two non-reference Node, then t s easy to get soluton usng Super Node Technque..e., nstead of solvng Nodal equatons separately they were merged and treated as a sngle Node. : , nfo@thegateacademy.com opyrght reserved. Web: 5

10 Network Soluton Methodology By nodal analyss, Demonstraton of Nodal Analyss Usng Super Node Voltage and urrent Source Ideal vs Practcal Voltage Source Practcal Voltage Source Fgure above depcts the symbol of a practcal voltage source. Here s the emf of source and s the nternal resstance of the source. For an deal voltage source, s zero and for a practcal source, s fnte & small. Ideal vs Practcal urrent Source Practcal urrent Source Fgure above depcts the symbol of a practcal current source. Here,I s the current of source and s nternal resstance of source. For an deal voltage source, s nfnte and for a practcal source, s fnte and large. Dependent Sources A source s called dependent f voltage / current of the source depends on voltage / current n some other part of the network. Dependng upon the nature of the source, dependent sources can be classfed as below. Voltage ontrolled Voltage Source (VVS) Here the voltage of the voltage source depends on voltage across some other element n the network. : , nfo@thegateacademy.com opyrght reserved. Web: 6