EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

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1 EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN

2 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage and Current Sources, Ohm s Law, Krchhoff s Laws, Resstors n parallel and n seres, Voltage and Current Dvson 3. Technques of Crcut Analyss (Chapter 4 of Nlsson - 12 Hrs.) Node Analyss, Node-Voltage Method and Dependent Sources, Mesh Analyss, Mesh-Current Method and Dependent Sources, Source Transformatons, Thevenn and Norton Equvalents, Maxmum Power Transfer, Superposton Theorem 4. Operatonal Amplfer (Chapter 5 of Nlsson - 6 Hrs.) Op-Amp Termnals & Ideal Op-Amp, Basc Op-Amp Crcuts, Buffer crcut, Invertng and Non-nvertng Amplfers, Summng Inverter, Dfference Amplfer, Cascade OpAmp Crcuts 5. Capactors and Inductors (Chapter 6 of Nlsson - 3 Hrs.) Inductors, Capactors, Seres and Parallel Combnatons of them, Mutual Inductance 6. Frst Order Crcuts (Chapter 7 of Nlsson - 9 Hrs.) The Natural Response of an RL & RC Crcuts, The Step Response of RL and RC Crcuts, A General Soluton for Step and Natural Responses, Integratng Amplfer Crcut 7. Second Order Crcuts (Chapter 8 of Nlsson - 6 Hrs.) The Natural Response of a Parallel RLC Crcut, The Forms of Natural Response of a Parallel RLC Crcut, The Step Response of a Parallel RLC Crcut, Natural and Step Responses of a Seres RLC Crcut

3 Sources Voltage/Current AC/DC Dependent/Independent Ohm s Law Resstors Nodes, Branches, Loops Krchhoff s Laws Krchhoff s Votage Law (KVL) Krchhoff s Current Law (KCL) v =. R Seres & Parallel Connectons of Resstors Delta-to-Wye Transform Current & Voltage Dvson N n= 1 = 0 = 0 n M m= 1 v m

4 Havng understood the fundamental laws of crcut theory, Ohm s law Krchhoff s laws (KVL & KCL) Apply these laws to develop two powerful technques for crcut analyss. Nodal analyss, whch s based on a systematc applcaton of Krchhoff s current law (KCL) Mesh analyss, whch s based on a systematc applcaton of Krchhoff s voltage law (KVL).

5 Source Transformatons Maxmum Power Transfer

6 Nodal Analyss (Node-Voltage Method) Nodal analyss provdes a general procedure for analyzng crcuts usng node voltages as the crcut varables. Choosng node voltages nstead of element voltages as crcut varables s convenent and reduces the number of equatons one must solve smultaneously. To smplfy matters, we shall assume n ths secton that crcuts do not contan voltage sources. Crcuts that contan voltage sources wll be analyzed later. In nodal analyss, we are nterested n fndng the node voltages. Gven a crcut wth N nodes wthout voltage sources, the nodal analyss of the crcut nvolves takng the followng three steps.

7 Nodal Analyss (Node-Voltage Method) Steps to Determne Node Voltages: 1. Select a node as the reference node. Assgn voltages v1, v2,..., v N 1 to the remanng N 1 nodes. The voltages are referenced wth respect to the reference node. 2. Apply KCL to each of the N 1 nonreference nodes. Use Ohm s law to express the branch currents n terms of node voltages. 3. Solve the resultng smultaneous equatons to obtan the unknown node voltages.

8 Nodal Analyss (Node-Voltage Method) Steps to Determne Node Voltages: 1. Select a node as the reference node. Assgn voltages v1, v2,..., v N 1 to the remanng N 1 nodes. The voltages are referenced wth respect to the reference node. The reference node s commonly called the ground snce t s assumed to have zero potental.

9 Nodal Analyss (Node-Voltage Method) Steps to Determne Node Voltages: 2. Apply KCL to each of the N 1 nonreference nodes. Use Ohm s law to express the branch currents n terms of node voltages. Current flows from a hgher potental to a lower potental n a resstor.

10 Nodal Analyss (Node-Voltage Method) Steps to Determne Node Voltages: 3. Solve the resultng smultaneous equatons to obtan the unknown node voltages. Number of nonreference nodes s equal to the number of ndependent equatons that we wll derve.

11 Nodal Analyss (Node-Voltage Method) Example: Calculate the node voltages n the crcut gven? Answer : 20V, 40/3 V Steps: 1. Choose a reference node, assgn voltages to other nodes w.r.t. reference one. 2. Apply KCL to each node. (Arbtrary but Consstent). Apply Ohm s law to fnd node voltages. 3. Solve all obtaned equatons together. 1. Substtuton method 2. Elmnaton method 3. Cramer s rule 4. Matrx nverson 5...

12 Nodal Analyss (Node-Voltage Method) Example: Use the node-voltage method to fnd shown? (Assesment problem 4.2 from textbook) v n the crut Answer : 15 V

13 Nodal Analyss (Node-Voltage Method) wth Dependent Sources The node-voltage equatons must be supplemented wth the constrant equatons mposed by the presence of the dependent sources. v = 15 φ Constrant equaton : = v 20Ω φ 20

14 Modfed Nodal Analyss ( Modfed Node-Voltage Method) When a voltage source s drectly connected n between a reference node and a non-reference node. The voltage of that non-reference node = voltage of that source. Hence, the number of requred equatons s decreased by one. v 3 = v s

15 Modfed Nodal Analyss ( Modfed Node-Voltage Method) When a voltage source s the only element between two essental nodes (or non-reference nodes), the node-voltage method s smplfed. For ths case, those nodes can be combned to form a SUPERNODE.

16 Modfed Nodal Analyss ( Modfed Node-Voltage Method) Note the followng propertes of a supernode: 1. The voltage source nsde the supernode provdes a constrant equaton needed to solve for the node voltages. 2. A supernode has no voltage of ts own. 3. A supernode requres the applcaton of both KCL and KVL. Steps for Supernode case: 1. Assgn a current for that branches. 2. Use these currents as the addtonal varables. 3. For supernode, use KCL to obtan supernode equaton. To obtan Node-voltage equatons, nodes can be used separately or used as a supernode (f avalable). Both gves the same result but usng supernode s decreased the number of equatons.

17 Modfed Nodal Analyss ( Modfed Node-Voltage Method) Example: Use the node-voltage method to fnd (Assesment Problem 4.6 from textbook) v 1 n the crcut shown? Answer : 48V

18 Mesh Analyss (Mesh-Current Method) Mesh: A loop does not enclose any other loops. Nodal analyss apples KCL to fnd unknown voltages n a gven crcut, whle, mesh analyss apples KVL to fnd unknown currents. Nodal analyss can be appled all crcuts n general. On the other hand, Mesh analyss s not qute as general as nodal analyss because t s only applcable to a crcut that s planar. A planar crcut s one that can be drawn n a plane wth no branches crossng one another; otherwse t s nonplanar. A crcut may have crossng branches and stll be planar f t can be redrawn such that t has no crossng branches.

19 Mesh Analyss (Mesh-Current Method)

20 Mesh Analyss (Mesh-Current Method) The current through a mesh s known as mesh current. In mesh analyss, we are nterested n applyng KVL to fnd the mesh currents n a gven crcut. Steps to Determne Mesh Currents: 1. Assgn mesh currents 1, 2,..., N 1 to the N meshes. 2. Apply KVL to each of the N meshes. Use Ohm s law to express the voltages n terms of the mesh currents. 3. Solve the resultng N smultaneous equatons to get the unknown mesh currents.

21 Mesh Analyss (Mesh-Current Method) Steps to Determne Mesh Currents: 1. Assgn mesh currents 1, 2,..., N 1 to the N meshes. Although a mesh current may be assgned to each mesh n an arbtrary drecton, t s conventonal to assume that each mesh current flows clockwse.

22 Mesh Analyss (Mesh-Current Method) Steps to Determne Mesh Currents: 2. Apply KVL to each of the N meshes. Use Ohm s law to express the voltages n terms of the mesh currents = = + + R R R R v v v mesh b KVL v v v mesh a KVL 0 ) ( 0 ) ( = + + = + + a b b b a a R R v R R v Ohm s Law

23 Mesh Analyss (Mesh-Current Method) Steps to Determne Mesh Currents: 3. Solve the resultng N smultaneous equatons to get the unknown mesh currents. 0 ) ( 0 ) ( = + + = + + a b b b a a R R v R R v = v v R R R R R R b a a b & +1 = n b l

24 Mesh Analyss (Mesh-Current Method) Example : For the crcut gven below, fnd the branch currents usng mesh analyss. I 1, I 2 and I3 Answer: I = I = A, I =

25 Mesh Analyss (Mesh-Current Method) Exercse : Calculate the mesh currents 1 and 2 n the crcut below.? Answer: = / 3A, =

26 Mesh Analyss (Mesh-Current Method) wth dependent sources Only dfference s that we have an extra constrant equaton. Example: Use mesh analyss to fnd the current 0 n the crcut below? Answer: =1. 0 5A

27 Mesh Analyss (Mesh-Current Method) wth dependent sources Example: Use mesh analyss to fnd the current 0 n the crcut below? Answer: 0 = 5 A

28 Modfed Mesh Analyss (Modfed Mesh-Current Method) When a current source exst only n one mesh: The curent of that mesh = current of that source. Hence, the number of requred equatons s decreased by one. 2 = 5A

29 Modfed Mesh Analyss (Modfed Mesh-Current Method) When a current source exst between two meshes, the mesh-current method s smplfed. For ths case, those meshes can be combned to form a SUPERMESH. 1 2 = 3.2A = 2.8A

30 Modfed Mesh Analyss (Modfed Mesh-Current Method) Example: For the crcut gven below, fnd 1 to 4 usng mesh analyss? = 7.5A = 2.5A = A = A

31 Mesh Analyss (Mesh-Current Method) Practce Problem : Use mesh analyss to determne 1 2,, = 3.474A = A = A

32 Mesh Analyss (Mesh-Current Method) Practce Problem : Use mesh-current method to fnd the mesh current the crcut shown (Assesment problem 4.11 from textbok) a n a = 15A

33 Mesh Analyss (Mesh-Current Method) Practce Problem : Use mesh-current method to fnd the power dsspated n the 1Ω resstor n the crcut shown (Assesment problem 4.12 from textbok) P = 36W

34 Mesh and Nodal Analyss Example 4.6 & 4.7 from textbook are useful to understand when & why are these methods (Mesh and Node Analyss) requred. Try to solve and try to understand these examples. Also try assessment problem 4.13 from textbook.

35 END OF CHAPTER 3, Part 1 (Nodal & Mesh Analyss) Dr. Yılmaz KALKAN

+ + + - - This circuit than can be reduced to a planar circuit

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