Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 EE 1105 : Introduction to EE Freshman Seminar Lecture 3: Circuit Analysis Ohm’s Law, Kirkhoff’s Laws

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Voltage and Current Voltage is the energy per unit of charge. Current is the rate of flow of charge. Derivative: slope, Integral: area under curve

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Power and Energy Power associated with a circuit element is consumed by that circuit element when the value of power is positive. Conversely, power is generated, or produced by the element if the value consumed is negative.

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Independent Voltage Source Voltage may be constant or time- dependent Delivers nominal terminal voltage under all conditions

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Independent Current Source Current may be constant or time- dependent Delivers nominal terminal current under all conditions

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Passive Sign Convention Assign current flow direction according to the voltage polarities Calculate p=vi, p 0 is power delivered to device

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Ohm’s Law Electrical resistance is the ratio of voltage drop across a resistor to current flow through the resistor. Polarities are governed by the passive sign convention.

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Power and Energy Power associated with a circuit element is consumed by that circuit element when the value of power is positive. Conversely, power is generated, or produced by the element if the value consumed is negative.

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Power Consumed by Resistors Resistors consume power. v and i are both positive or both negative.

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Conductance Defined Conductance is the reciprocal of resistance. The units of conductance are called siemens (S) The circuit symbol is G

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Calculating Resistance When conductor has uniform cross-section

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Temperature Coefficient of Resistance Metallic conductors have a linear increase of resistance with increased temperature. T o is the reference temperature (usually 20 o C) and R o is the resistance at the reference temperature.  is the temperature coefficient of resistance for the material. At 20 o C, some values for  are: 20 o C Aluminum Copper

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Power and Energy Power associated with a circuit element is consumed by that circuit element when the value of power is positive. Conversely, power is generated, or produced by the element if the value consumed is negative.

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Power Consumed by Resistors Resistors consume power. v and i are both positive or both negative.

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Conductance Defined Conductance is the reciprocal of resistance. The units of conductance are called siemens (S) The circuit symbol is G

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, Edges(Branches) –Voltage drop computed between nodes –Currents flowing along edges –Paths (collection of edges with no node appearing twice), –Loops (closed paths), meshes (loop containing no other loop). –Series connection (elements sharing the same current) –Parallel connection (elements sharing the same voltage)

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Example of a Circuit Model w/ Series Connection

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 Apply KCL to Example

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Combine KVL & Ohm’s Law

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Lamp Voltage & Battery Voltage

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Battery Power and Lamp Power Loss: Efficiency:

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Resistors in Series By KCL: I s = I 1 = I 2 By Ohm’s Law: V 1 = R 1 ·I 1 and V 2 = R 2 ·I 2 Combine: V s = R 1 I 1 + R 2 I 2 = (R 1 + R 2 ) I s = R eq I s In General: R eq = R 1 + R 2 +···+ R n

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Resistors in Parallel (1/2) By KVL: V s = V 1 = V 2 By KCL: I s = I 1 + I 2 By Ohm’s Law: and Combine:

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Resistors in Parallel (2/2) For two resistors: For many resistors: In terms of conductance:

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Voltage Divider Circuit

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Loaded Voltage Divider

Dan O. Popa, Intro to EE, Freshman Seminar Spring 2015 Voltage Divider Equations Unloaded: Loaded: If R L >> R 2 :

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 Homework 3 due next class!! Available online at course website Acknowledgements: Dr. Bill Dillon Questions? 40