Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 EE 1105 : Introduction to EE Freshman Seminar Lecture 4: Circuit Analysis Node Analysis, Mesh Currents Superposition

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Circuits Abstraction describing how (the topology) electrical or electronic modules are interconnected. Closely related to a GRAPH. Nomenclature: –Nodes, Extraordinary nodes, Supernodes (adjacent nodes sharing a voltage source) –Edges(Branches) –Paths (collection of edges with no node appearing twice), –Loops (closed paths) –Meshes (loop containing no other loop), Supermeshes (adjacent meshes sharing a current source)

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Kirchhoff’s Voltage Law The sum of the voltage drops around a closed path is zero. Example: V 1 + V 2 + V 3 + V 4 = 0

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Kirchhoff’s Current Law A node is a point where two or more circuit elements are connected together. The sum of the currents leaving a node is zero.

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Using Loops to Write Equations a: Loop b: Loop c: Loop c equation same as a & b combined.

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Using Nodes to Write Equations Node x: Node y: Node z: Node w: <== Redundant

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Combining the Equations There are 5 circuit elements in the problem. v a and v b are known. R 1, R 2 and R 3 are known. v 1, v 2 and v 3 are unknowns. i a, i b, i 1, i 2 and i 3 are unknowns. There are 2 loop (KVL) equations. There are 3 node (KCL) equations. There are 3 Ohm’s Law equations. There are 8 unknowns and 8 equations.

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example 1 (1/3) By KCL: By Ohm’s Law:

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example 1 (2/3) By KVL: Power:

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example 1 (3/3)

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example 2 (1/4) Find Source Current, I, and Resistance, R.

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example 2 (2/4) Ohm’s Law: 36 VKVL: 48 VOhm’s Law: 6 A

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example 2 (3/4) KCL: 3 AOhm’s Law: 12 VKVL: 60 V

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example 2 (4/4) Ohm’s Law: 3 AKCL: 6 A Ohm’s Law: R=3  KCL: I=9 A KVL: 24 V

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Superposition Principle Fundamental Property of Linear Circuits Replace all but one source in the circuit with a short (voltage source) or an open (current sources). Apply analysis to find nodal voltages. Repeat for all sources Add all nodal voltages to find the total result.

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Combining Voltage Sources Voltage sources are added algebraically

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Combining Voltage Sources Voltage sources are added algebraically

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Combining Voltage Sources Don’t do this. Why is this illogical? Whose fundamental circuit law is violated by this?

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Combining Current Sources Current sources are added algebraically

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Combining Current Sources Current sources are added algebraically

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Combining Current Sources Don’t do this. Why is this illogical? Whose fundamental circuit law is violated by this?

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Nodal Analysis Identify all extraordinary nodes. Select one as ground reference and assign node voltages to the other ones. Write KCL at the non-zero voltage nodes in conjunction with Ohm’s law. Solve a system of simultaneous equations In the case of a supernode, apply KVL along the connection, and ignore any resistors in parallel to a voltage source.

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 More complex example Image Source: Textbook

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example Replace E G1 with a short (zero). Solve resulting circuit for Va1. Replace E G2 with a short. Solve resulting circuit for Va2. Total Va=Va1+Va2 Exercise in Lab – you should obtain the same result as in the previous case.

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Mesh Analysis Identify all meshes and assign them an unknown current, clockwise. Write KVL on each mesh Solve a system of simultaneous equations In the case of a supermesh, add an extra equation with the dependence between the currents

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Example Two meshes with currents I1 and I2. KVL: Resulting current through R 3 is I 1 -I 2.

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 More complex example Image Source: Textbook

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Mesh Analysis by Inspection Applies only if all sources be independent voltage sources Same procedure to assign mesh currents. Define R ij – resistances as follows: –Rii – sum of all resistances connected to mesh I –Rij=Rji – minus sum of all resistances shared between mesh I and J Define total voltages from voltage sources along mesh I as Vi. Write and solve matrix equation RI=V, in which R=(Rij), V=(Vi), I=(Ii).

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Nodal Analysis By Inspection Applies only if all sources be independent current sources Same procedure to assign node voltages. Define G ij – conductances as follows: –Gii – sum of all conductances connected to node I –Gij=Gji – minus sum of all conductances connected between node I and J Define currents from current sources entering node I as Ii. Write and solve matrix equation GV=I, in which G=(Gij), V=(Vi), I=(Ii).

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Solving Linear Systems of Equations AX=B a 11 x 1 + a 12 x 2 + · · · + a 1n x n =b 1 a 21 x 1 + a 22 x 2 + · · · + a 2n x n =b 2 a m1 x 1 + a m2 x 2 + · · · + a mn x n =b m Methods to solve: 1)Elimination 2)Substitution 3)Cramer’s rule 4)Matrix inverse

Dan O. Popa, Intro to EE, Freshman Seminar, Spring 2015 Homework 4 due next class!! Available online at course website Acknowledgements: Dr. Bill Dillon Questions? 32