《 Principles of Circuit Analysis》 rsistance Originality Chapter 4: Network Analysis Chapter 4: Network Analysis Lecture 1 Simple and normal 2009-10-29 ways to get responses For given structure and element parameters in from stimulations network which has b branches, the voltage and current of each branch are unique and certain In Circuit Analysis, the solving of the voltage and E current for each branch from given structure and element parameter is called Network Analysis. Chapter 4: Fundamentals of Linear Network Analysis Linear network analysis 54-1 Fundamentals of network topology analysis 口Fom9amsC0)(z( 54-2 Methods for Linear Network Analysis Static state(DC) analysis, complex method, Laplace transform malysis Four I Loop current method lectures (Ch1)(Ch1)(Ch2) 2: Nodal point voltage method KCL(node), kvl(loop), vCr(branch)law 84-3 Large network' analysis method (Nodal analysis e just for knowledge) (Ch1) Fundamental Law (KVL, KCL, VCr) Focus in this lecture Attention: 1.reference directior 1. KCL=3 equations (branch direction) a Besides VCR, KCL, KCL, is there any 2.The direction voltage decreasing I2-l]1so simple methods for linear network If for one group of parameters for solving(numbernumber of ①2 +V1+V=0 branches), they satisfy the following conditions V+V,+V=0 The set is complete, and the variables are independent. 4 -They are Nearly 3. VCR =equations ndependent What we can get: the voltages and currents of all t hranches Num of branches b=6 白3+3+6=12 equations Loop current? Mesh current? Nodal voltage? unknown quantity =2b=12 suppositive
北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 第 ?讲: 复习 北京大学 wwhu 北京大学 《Principles of Circuit Analysis》 Chapter 4: Network Analysis Lecture 1 2009-10-29 Simple and normal ways to get responses from stimulations Interest Focus Persistence Originality 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 第 ?讲: 复习 北京大学 wwhu 北京大学 《Principles of Circuit Analysis》 Chapter 4: Network Analysis Lecture 1 wwhu@2008-10-27 http://course.pku.edu.cn For given structure and element parameters in a network which has b branches, the voltage and current of each branch are unique and certain. In Circuit Analysis, the solving of the voltage and current for each branch from given structure and element parameter is called Network Analysis. Interest Focus Persistence Originality 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 §4-1 Fundamentals of network topology analysis §4-2 Methods for Linear Network Analysis 1. Loop current method 2. Nodal point voltage method §5-1 Network theorem Replacement theorem, Superposition theorem, Reciprocity theorem §4-3 Large network analysis method (Nodal analysis Æ just for knowledge) Chapter 4: Fundamentals of Linear Network Analysis Chapter 4: Fundamentals of Linear Network Analysis Four lectures 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Linear network analysis From state analysis: From structure analysis KCL(node), KVL(loop), VCR(branch) law (Ch.1) Static state (DC) analysis, complex method, Laplace transform analysis (Ch.1) (Ch.1) (Ch.2) Review Z(jw) Z(s) Z=R 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 北京大学 ① 1 2 3 4 5 6 ② ③ (4) Attention: 1.reference direction (branch direction) 2.The direction voltage decreasing (loop direction) -I1-I2-I4=0 I2-I3-I5=0 I1+I3-I6=0 -V4+V1+V6=0 -V4+V2+V5=0 -V5+V3+V6=0 2.KVL 1.KCL 3.VCR Vi =Ri Ii Num of branches b=6 unknown quantity =2b=12 =3 equations = 3 equations = 6 equations 3+3+6=12 equations Fundamental Law (KVL, KCL, VCR) Review 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Focus in this lecture Besides VCR,KCL,KCL, is there any simple methods for linear network analysis? If for one group of parameters for solving (number<number of branches), they satisfy the following conditions: The set is complete, and the variables are independent. Loop current? Mesh current? Nodal voltage? suppositive suppositive They are linearly independent . What we can get: the voltages and currents of all the branches Including virtual Including virtual voltages and currents
Fundamentals of network topology analysis Fundamentals of network topology analysis *lek Directed graph(digraph): number each node and branch in the network, and each numbered branch(edge) is an directed edge. Network planning: it is a set of nodes and branches. The two ends of each branch combine the different nodes. Connected graph: In an undirected graph G, two vertices u Number of branches: be and y are called connected if G contains a branch from u to v Number of nodes: na4 Fundamentals of network topology analysis *ik Fundamentals of network topology analy a Tree: a subgraph of the Connected graph which satisfies Conclusion( theorem): the foLlowing three conditions: "Aa. The number of independent nodes in a connected network equals to the number of twigs n including all the nodes e which satisfies: nan-I In this example we can sel Using the induction for proof 《 a Twig: the edges of tree(red lines) n=2 3 n'=n+1 Number of twigs=number of tree nodes-l n=1 nt nt n+1 Fundamentals of network topology analysis pe Fundamentals of network topology analysis link: the branch(edge) which is not a twig(blue Cut-set: A set of branches in a graph which satisfy ()The network will be unconnected if w e the cut-set and it will 零分 divided into two pa Method to get: draw closed surface in the network .So, the set of links is also called cotree. Closed surfaceEl-0-supernode The number of links satisfy: L=b-n=b-n+I Each link and some twigs should make up a unique(single-link) loop. The number of independent loops in a network equals the number of links-b-n+I
北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Connected graph: In an undirected graph G, two vertices u and v are called connected if G contains a branch from u to v. Connected graph: In an undirected graph G, two vertices u and v are called connected if G contains a branch from u to v. Network planning: it is a set of nodes and branches. The two ends of each branch combine the different nodes. Network planning: it is a set of nodes and branches. The two ends of each branch combine the different nodes. Fundamentals of network topology analysis 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Directed graph (digraph): number each node and branch in the network, and each numbered branch (edge) is an directed edge. Number of branches: b=6 Number of nodes: n=4 Real network Real network + - ① ④ ② ③ 1 2 3 4 5 6 Fundamentals of network topology analysis *** 1 2 3 4 5 6 ① ④ ② ③ digraph digraph 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Tree: a subgraph of the Connected graph which satisfies the following three conditions: (1)Connected; (2)No cycle; (3)Including all the nodes Twig: the edges of tree (red lines) In this example, we can select: **** Conclusion: Number of twigs = number of tree nodes -1 nt =n-1 Fundamentals of network topology analysis 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Conclusion (theorem): The number of independent nodes in a connected network equals to the number of twigs nt Æ which satisfies: nt =n-1 Conclusion (theorem): The number of independent nodes in a connected network equals to the number of twigs nt Æ which satisfies: nt =n-1 Using the induction for proof: n = 2 nt = 1 n = 3 nt = 2 n’ = n+1 nt’ = nt+1 Fundamentals of network topology analysis 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 link:the branch (edge) which is not a twig (blue line) Conclusion (theorem): The number of independent loops in a network equals the number of links=b-n+1. Conclusion (theorem): The number of independent loops in a network equals the number of links=b-n+1. So, the set of links is also called cotree. The number of links satisfy: L=b- nt =b-n+1 So, the set of links is also called cotree. The number of links satisfy: L=b- nt =b-n+1 Each link and some twigs should make up a unique (single-link) loop. Each link and some twigs should make up a unique (single-link) loop. Fundamentals of network topology analysis *** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Cut-set: A set of branches in a graph which satisfy: (1)The network will be unconnected if we remove the cut-set and it will be divided into two parts. (2)Keep any branch in the cut-set will make the network connected. Method to get: draw closed surface in the network 例: Closed surfaceÆΣI=0Æsupernode e.g.: Fundamentals of network topology analysis
Fundamentals of network topology analysis Fundamentals of network topology analysis *lek Cut-set: A set of branches in a graph which satisfy Planar network: a network which may be drawn on a plane surface in such a way that no branch passes over or under any other branch 2)Keep any branch in the cut-set will make the network co Nonplanar network: any network which is not planar Method to get: draw closed surface in the network Closed surface21-0-supernode Each twig and some links can make up a unique cut-set: yes Single-twig cut-set (basic cut-set) Planar network: Like a net in forn→→mesh Number of hasic cut-sets numher of twigs a n=n-I Conclusion (theorem): for the planar network Number of mesh =number of independent loop = number of links= b-n+l Fundamentals of network topology analysis Fundamentals of network topology analysis-tree, twig, and link Principles of establishing the network equation u Tree: a subgraph of the C graph which satisfies the following three independent and complete COnnected; (2)No cycle: )Including all the nod No lack move the root, n-l independent nodes are left The number of independent nodes in a connected network equals to the number of twigs n, s which satisfies: n=B- I The number of independent loops in a network equals the number of linksmb-n+ Links single twig, n-l independent cut-sets >n-lindependent KCL equations Conclusion (theorem): ?cut-set analysis Number of basic cut-sets m number of twigs -n, n-l wigs+single links, b-n+I independent loop Conclusion(theorem): for the planar network >b-n+l n-lindependent KVL equations Number of mesh- number of independent loop- number of nks- b-n+I Fundamentals of network topology analysis Tea break/ n networks: analysis Establish KCL equations using Cut-set all the independent cut-set Establish KVL equations using Homework all the independent loop analys →4-1,241 Sth ind endequanoes using D Nodal analysIs Establish KvL equations using all the independent mesh Planar network
北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 例:e.g.: Each twig and some links can make up a unique cut-set: Single-twig cut-set (basic cut-set) Conclusion (theorem): Number of basic cut-sets = number of twigs = nt =n-1 Conclusion (theorem): Number of basic cut-sets = number of twigs = nt =n-1 Fundamentals of network topology analysis Cut-set: A set of branches in a graph which satisfy: (1)The network will be unconnected if we remove the cut-set and it will be divided into two parts. (2)Keep any branch in the cut-set will make the network connected. Method to get: draw closed surface in the network Closed surfaceÆΣI=0Æsupernode 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Planar network: a network which may be drawn on a plane surface in such a way that no branch passes over or under any other branch. Nonplanar network: any network which is not planar e.g.: Planar network: Like a net in form →→ mesh yes yes no Conclusion (theorem): for the planar network Number of mesh = number of independent loop = number of links = b-n+1 Conclusion (theorem): for the planar network Number of mesh = number of independent loop = number of links = b-n+1 Fundamentals of network topology analysis *** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Conclusion (theorem): The number of independent nodes in a connected network equals to the number of twigs nt Æ which satisfies: nt =n-1 Conclusion (theorem): The number of independent nodes in a connected network equals to the number of twigs nt Æ which satisfies: nt =n-1 Principles of establishing the network equation: independent and complete Fundamentals of network topology analysis No repeat No lack Conclusion (theorem): The number of independent loops in a network equals the number of links=b-n+1 Conclusion (theorem): The number of independent loops in a network equals the number of links=b-n+1 Conclusion (theorem): Number of basic cut-sets = number of twigs = nt =n-1 Conclusion (theorem): Number of basic cut-sets = number of twigs = nt =n-1 Conclusion (theorem): for the planar network Number of mesh = number of independent loop = number of links = b-n+1 Conclusion (theorem): for the planar network Number of mesh = number of independent loop = number of links = b-n+1 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Fundamentals of network topology analysis – tree, twig, and link Tree: a subgraph of the Connected graph which satisfies the following three conditions:(1)Connected; (2)No cycle; (3)Including all the nodes Twig: the edges of tree (red lines). Link: the branch (edge) which is not a twig (gray line) Remove the root, n-1 independent nodes are left Æn-1 independent KCL equations Ænodal voltage method Remove the root, n-1 independent nodes are left Æn-1 independent KCL equations Ænodal voltage method Links + single twig,n-1 independent cut-sets Æn-1independent KCL equations Æcut-set analysis Links + single twig,n-1 independent cut-sets Æn-1independent KCL equations Æcut-set analysis Twigs + single links, b-n+1 independent loop Æb-n+1 n-1independent KVL equations Æloop current method, mesh current method Twigs + single links, b-n+1 independent loop Æb-n+1 n-1independent KVL equations Æloop current method, mesh current method 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 analysis Establish KCL equations using all the independent cut-set Cut-set analysis Establish KVL equations using all the independent loop Loop analysis Establish KCL equations using all the independent nodes Nodal analysis Establish KVL equations using all the independent mesh Mesh analysis Planar network only In networks: today next next Fundamentals of network topology analysis 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Tea break! Tea break! Homework: Æ4-1,2,4,11 Homework: Æ4-1,2,4,11
Basic network analysis- Loop or mesh current methduPr u Loop current methed OStepl: set numbers and directions in the circuit ding to the strueture of tree; It can be used in nonplanar network. aStep2: select the loop current direction usually the clockwise This method is easier to learnt aStep3: write the kvl equations: It ca planar network. Basic network analysis- Introduction to loop current method Basic network analysis- Introduction to loop current method *k e…g Z4+z2+z4-z2 Iz11-vsx+z4(I1-I3)+vs2+z2(I1-I2)=0 -z2Z2+22+25-zs I2 IIz2(L2-1)-v2+z3(I2-I3)+Ms3+z3I2=0 -z4 z。z2+2+2八- IIz63+25(I3-I2)+z(I3-I2)=0 Step4: build up a matrix, and solve the aStep6: get each branch voltage from the current Basic network analysis-Introduction to loop current method Basic network analysis- Introduction to loop current method *alk z1+22 -z2z2+23+25-z5 Loop current voltage z5z4+2s+26 0 impedance L column vector Source the algebraic sum of all the voltage source matru vector (increasing voltage)along the loop i
北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Basic network analysis – Loop or mesh current method Mesh current method Loop current method Loop current method: Equation establishing according to the structure of tree; It can be used in nonplanar network. Mesh current method: This method is easier to learn; It can only be used in planar network. Z2 Z3 Z5 Z1 Z4 Z6 Vs1 Vs3 Vs2 + - - + + - 6 3 2 4 5 1 6 3 2 4 5 1 6 3 2 4 5 1 *** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu Step3: write the KVL equations: 北京大学 wwhu 北京大学 Z2 Z3 Z5 Z1 Z4 Z6 Vs1 Vs3 Vs2 + - - + + - e.g.: 3 2 4 5 1 6 Step1: set numbers and directions in the circuit Step2: select the loop current direction usually the clockwise. I3 I2 I1 Basic network analysis – Introduction to loop current method 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 第三步:写出 北京大学KVLwwhu 方程: 北京大学 Z2 Z3 Z5 Z1 Z4 Z6 Vs1 Vs3 Vs2 + - - + + - e.g.: 3 2 4 5 1 6 第一步:给电路标号、方向 第二步:选定回路电流方向 一般选取顺时针方向 I3 I2 I1 第三步:写出KVL方程: I Z1I1 -VS1 +Z4(I1 -I3)+ VS2 +Z2(I1 -I2) = 0 II Z2(I2 -I1)-VS2 +Z5(I2 -I3)+ VS3 +Z3I2 = 0 III Z6I3 +Z5(I3 -I2)+Z4(I3 -I2) = 0 Step4: build up a matrix, and solve the loop current Basic network analysis – Introduction to loop current method 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 * 6 Z2 Z3 Z5 Z1 Z4 Z6 Vs1 Vs3 Vs2 + - - + + - e.g.: 3 2 4 5 1 I3 I2 I1 = Z1 +Z2 +Z4 Z2 - Z2 +Z3 +Z5 Z4 +Z5 +Z6 Z4 - Z5 - Z2 - Z4 - Z5 - I1 I2 I3 S1 VS2 V - S2 VS3 V - 0 Step5: get each branch current from the loop current. Step6: get each branch voltage from the current. Basic network analysis – Introduction to loop current method 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 = Z1 +Z2 +Z4 Z2 - Z2 +Z3 +Z5 Z4 +Z5 +Z6 Z4 - Z5 - Z2 - Z4 - Z5 - I1 I2 I3 S1 VS2 V - S2 VS3 V - 0 VS Z⋅I = Loop impedance matrix Loop current column vector Loop voltage source column vector Basic network analysis – Introduction to loop current method 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Z2 Z3 Z5 Z1 Z4 Z6 Vs1 Vs3 Vs2 + - - + + - e.g.: 3 2 4 5 1 6 I3 I2 I1 = Z1 +Z2 +Z4 Z2 - Z2 +Z3 +Z5 Z4 +Z5 +Z6 Z4 - Z5 - Z2 - Z4 - Z5 - I1 I2 I3 S1 VS2 V - S2 VS3 V - 0 VSi : the algebraic sum of all the voltage source (increasing voltage) along the loop i Basic network analysis – Introduction to loop current method ***
t method iek Mesh current method ? -z, I(V-V z+2+2-乙2v2v2 7z4+z+z6 z2z2+2+2-z5 I2|=v2v2 z 4+Z+zL/V,-V z4-2524+25+z6 sum of the branch impedance in loop i(self impedance) positive Loop current method--matrixes op Loop current method-examples Loop current method: Loop current matrix Write down the matrix intuitivly, swiftly and accurately ZI=Vs =(工1I2I3) Step2. write the Z matrix +R2 R2+R3+R Review matrixes operatio self-impedance Z 0 R, R,+RS symnetr R4+R5 al impedan Loop current method-examples 02G Step4. solve the loop current Step3.Zvs s1( Z8 0 R R4+R5人I3丿(-Vs4 Steps: get each branch current from the loop current. Step6: get each branch voltage from the current
北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Z2 Z3 Z5 Z1 Z4 Z6 Vs1 Vs3 Vs2 + - - + + - e.g.: 3 2 4 5 1 6 I3 I2 I1 = Z1 +Z2 +Z4 Z2 - Z2 +Z3 +Z5 Z4 +Z5 +Z6 Z4 - Z5 - Z2 - Z4 - Z5 - I1 I2 I3 S1 VS2 V - S2 VS3 V - 0 Zii : sum of the branch impedance in loop i (self impedance) positive Basic network analysis – Introduction to loop current method *** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 6 3 2 4 5 1 6 3 2 4 5 1 ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + + + + + + 0 s2 s3 s1 s2 3 2 1 4 5 4 5 6 2 2 3 5 5 1 2 4 2 4 V -V V -V I I I -Z -Z Z Z Z -Z Z Z Z -Z Z Z Z -Z -Z I III I II II III ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + + + + + + s1 s3 s2 s3 s1 s2 3 2 1 1 3 1 3 6 2 2 3 5 3 1 2 4 2 1 V -V V -V V -V I I I Z Z Z Z Z -Z Z Z Z Z Z Z Z -Z Z Mesh current method Loop current method Zij : the common branch impedance between loop I and j (mutual impedance) if the direction of Ii and Ij are the same, Zij is positive, otherwise it is negative. Basic network analysis – Introduction to loop current method *** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Loop current method — matrixes operation Loop current method: Loop current matrix ∑= Δ = L k 1 Sk ki i V Z I Z·I=Vs I = Z-1 · Vs Review matrixes operation Review matrixes operation *** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Loop current method — examples Write down the matrix intuitivly, swiftly and accurately. R1 R3 R2 R4 + - - +VS1 + R5 - Vs4 V2 I1 I2 I3 Step1. define the current direction of loop (mesh) T I = (I1 ,I2 ,I3) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + + + + 4 5 2 3 4 1 2 R R R R R R R Step2.write the Z matrix self-impedance ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + + + − + − 4 5 2 3 4 4 1 2 2 R R R R R R R R R 0 mutual impedance ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − + − + + − + − 4 4 5 2 2 3 4 4 1 2 2 0 R R R R R R R R R R R 0 symmetry *** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 R1 R3 R2 R4 + - - +VS1 + R5 - V2 I1 I2 I3 Step3.ZI=Vs ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − + − + + − + − S4 S4 S1 3 2 1 4 4 5 2 2 3 4 4 1 2 2 -V V V I I I 0 R R R R R R R R R R R 0 Vs4 Step4. solve the loop current Loop current method — examples *** Step5: get each branch current from the loop current. Step6: get each branch voltage from the current. 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 02G Z1 Z2 Z5 Z3 Z6 Z4 Z7 Z8 + Vs1 - Is1 aIZ1 IZ1
03G cos(ot)=Re{l·e} Loop current method- dealing with the current source sin(o)=Re{-j·e! 2jw 29 (2)Virtual loop current method ()Assuming the branch voltage 2 29 29 Loop current method- dealing with the current source Loop current method- dealing with the current sourced* l) Source equivalence( Norton's→ Thevenin’s) (2)Virtual loop current method (when the current source branch Weakness: changing the structures at t o boundary of the network) 70 2z2+2+25-z5=Vx 2+z4 Norton'sX Thevenin?s Loop current method-examples Loop current method- dealing with the current source Write down the matrix intuitivly, swiftly and accurately. (2)Virtual loop current method Evaluate the current of When current source is not at the boundary branch I3 US Q ls↑ EVsl Z8 1=2-=2← Z7
北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 + cos(ωt) - sin(ωt) 2Ω L=2H 2Ω C=0.5F 2Ω 2Ω 2Ω 2Ω 03G 2jw 2/jw 1 -j sin( ) Re{ } cos( ) Re{1 } j t j t t j e t e ω ω ω ω = − • = • 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Loop current method — dealing with the current source (1) Source equivalence(Norton’s →Thevenin’s) (2) Virtual loop current method R1 R3 R2 R4 + - - +VS1 + R5 - V2 I1 I2 I3 Vs4 (3)Assuming the branch voltage 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Z2 Z3 Z5 Z1 Z4 Vs1 Vs3 Vs2 Z2 Z3 Z5 Z1 Z4 Vs1 Vs3 Vs2 Is6 I=? I= I6-Is6 ? (Norton’s X →Thevenin’s) ? Z2 Z3 Z5 Z1 Z4 Vs1 2I Vs2 ? Loop current method — dealing with the current source (1)Source equivalence(Norton’s →Thevenin’s) Weakness: changing the structure of circuits 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 (2) Virtual loop current method (when the current source branch is at the boundary of the network) Z2 Z3 Z5 Z1 Z4 Vs1 Vs3 Vs2 Is6 = Z1 +Z2 +Z4 Z2 - Z2 +Z3 +Z5 1 Z4 - Z5 - Z2 - 0 0 I1 I2 I3 S1 VS2 V - S2 VS3 V - Is 1 2 4 = Z +Z +Z Z2 - Z2 Z2 +Z3 +Z5 - I1 I2 S1 VS2 Z4IS V - + S2 VS3 Z5IS V - + I3=Is6 Loop current method — dealing with the current source*** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Evaluate the current of 40 ohms I=? ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − − 50 136 3 10 40 50 8 50 40 1 0 0 3 2 1 I I I ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ 6 8 3 3 2 1 I I I I = I2 −I3 =2A Loop current method — examples Write down the matrix intuitivly, swiftly and accurately. 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Z1 Z2 Z5 Z3 Z6 Z4 Z7 Z8 + Vs1 - Is1 aIZ1 IZ1 (2) Virtual loop current method When current source is not at the boundary branch Loop current method — dealing with the current source
interpret the mid-term exam -5 oop current method-dealing with the current sourcaoop Is 1 When current source is not at the boundary branch (3)Assuming the branch voltage -vx+ Z5 Z6 Z8 al 10150l2|=-x 25-10 -10 o-1 0A.e 6 Loop current method- dealing with the current source?lk 02G When current source is not at the boundary branch (3)Assuming the branch voltage Z3 Z az 1015 1016八2)(vs 乙t he next lecture
北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 interpret the mid-term exam - 5 Z1 Z2 Z5 Z3 Z6 Z4 Z7 Z8 + Vs1 - Is1 aIZ1 IZ1 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 (3) Assuming the branch voltage - Vx + Vs ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + − − = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − V Vx Vx V I I I s s 3 2 1 0 0 1 10 15 0 25 10 0 II + III ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛− = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − 0 1 1 6 10 15 1 25 10 0 3 2 1 s s V V I I I -I2 + I3 = 6 *** Vx Loop current method — dealing with the current source When current source is not at the boundary branch 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛− = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − 0 1 1 6 10 15 1 25 10 0 3 2 1 s s V V I I I *** 1 2 25 10 10 16 6 S S I V I V ⎛ ⎞ − ⎛ ⎞ ⎛ ⎞ − ⎜ ⎟⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ − ⎝ ⎠ ⎝ ⎠ − 1 3 25 10 60 10 16 90 S S I V I V ⎛ ⎞ − ⎛⎞⎛ ⎞ − − ⎜ ⎟⎜⎟⎜ ⎟ = ⎝ ⎠ − ⎝⎠⎝ ⎠ + Zij ji = Z Loop current method — dealing with the current source When current source is not at the boundary branch (3)Assuming the branch voltage 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 02G Z1 Z2 Z5 Z3 Z6 Z4 Z7 Z8 + Vs1 - Is1 aIZ1 IZ1 The next lecture…