Linear Circuits. Concept Map 9/10/ Resistive Background Circuits. 5 Power. 3 4 Reactive Circuits. Frequency Analysis

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1 Linear Circuits Dr. Bonnie Ferri Professor School of Electrical and Computer Engineering An introduction to linear electric components and a study of circuits containing such devices. School of Electrical and Computer Engineering Concept Map 1 2 Resistive Background Circuits 5 Power 3 4 Reactive Circuits Frequency Analysis 2 1

2 Resistive vs Reactive Circuit Voltage Time 3 Concept Map Background Resistive Circuits Current, voltage, sources, resistance Reactive Circuits Capacitors RC Circuits Inductors RL Circuits Differential Reactive 2-order Diff Eqn Equations Circuits RLC Circuits Applications Methods to obtain circuit equations (KCL, KVL, mesh, node, Thévenin) Frequency Analysis Power 4 2

3 Capacitance Nathan V. Parrish PhD Candidate & Graduate Research Assistant School of Electrical and Computer Engineering Describe the behavior of capacitors by calculating: the charge stored on the capacitor plates the current flowing through the capacitor the voltage across the capacitor the capacitance of the capacitor School of Electrical and Computer Engineering Lesson Objectives Describe the construction of a capacitor Find charge stored on a capacitor Find the current through a capacitor Find the voltage across a capacitor Calculate the capacitance of a capacitor Explain how current flows through a capacitor 5 1

4 Capacitors 6 Capacitors and Charge 7 2

5 Current and Voltage 8 Calculating Capacitance 9 3

6 Permittivity of Common Materials Material Approximate ε r (or k) Air 1 Teflon 2.1 Paper 3.9 Glass 4.7 Rubber 7.0 Silicon 11.7 Water 78.5 (varies by T) 10 Current Through A Capacitor 11 4

7 Summary Identified how capacitors work Calculated charge stored on a capacitor Identified the relationship between current and voltage on a capacitor Calculated capacitance Explained how current flows through a capacitor 12 5

8 Capacitors Nathan V. Parrish PhD Candidate & Graduate Research Assistant School of Electrical and Computer Engineering Present how capacitors work in a system Identify behavior in DC circuits Graphically represent the relationships between current, voltage, power, and energy School of Electrical and Computer Engineering Lesson Objectives Analyzing capacitors in series/parallel Analyze DC circuits with capacitors Calculate energy in a capacitor Sketch current/voltage/power/energy curves 5 1

9 Capacitors in Parallel 6 Capacitors in Series 7 2

10 Behavior in DC Circuits 8 Stored Energy 9 3

11 Graphs 10 Summary Calculated capacitance for capacitors in parallel/series configurations Identified how capacitors in DC circuits behave like open circuits Derived an equation for the energy stored by a capacitor as an electric field Showed graphically the relationships between voltage/current/power/energy in capacitors 11 4

12 Inductance Nathan V. Parrish PhD Candidate & Graduate Research Assistant School of Electrical and Computer Engineering Introduce inductors and describe how they work Calculate current and voltage for inductors School of Electrical and Computer Engineering Lesson Objectives Describe the construction and behavior of an inductor Find current through an inductor Find voltage across an inductor Explain how a voltage is created across an inductor 5 1

13 Inductors 6 Current and Voltage 7 2

14 Ampère s Law 8 How Inductors Work 9 3

15 Voltages Across a Wire 10 Summary Presented the equations for current and voltage in inductors Introduced Ampère s Law and showed how inductors work in context of this law Showed how a voltage is created across an inductor as currents change in a system 11 4

16 Inductors Nathan V. Parrish PhD Candidate & Graduate Research Assistant School of Electrical and Computer Engineering Present how inductors work in a system Identify behavior in DC circuits Graphically represent the relationships between current, voltage, power, and energy School of Electrical and Computer Engineering Learning Objectives Analyze inductors in series/parallel Analyze DC circuits with inductors Calculate the energy in an inductor Sketch current/voltage/power/energy curves 5 1

17 Inductors in Series 6 Inductors in Parallel 7 2

18 Behavior in DC Circuits 8 Stored Energy 9 3

19 Graphs 10 Summary Calculated inductance for inductors in parallel/series configurations Identified how inductors in DC circuits behave like short circuits Derived an equation for the energy stored by an inductor as a magnetic field Showed graphically the relationships between voltage/current/power/energy in inductors 11 4

20 Dr. Bonnie H. Ferri Professor and Associate Chair School of Electrical and Computer Engineering First-Order Differential Equations Solve and graph solutions to first-order differential equations School of Electrical and Computer Engineering Lesson Objectives Examine first-order differential equations with a constant input Write the solution Sketch the solution 5 1

21 Ordinary Differential Equations ODE: Include functions of variables and their derivatives. dy dt + 2y= 4 dy dt d 2 y dy dv dt y= f (t) dt 2y = 4sin(ωt) dt + 2v= i(t) 6 Models of Physical Systems Inputs, f(t) System Model Outputs, y(t) dy dt + ay= f (t) Heat source Thermal System Temp Production Biological System Population Voltage or current source Linear Circuit Voltage or current 7 2

22 Solution to First-Order Differential Equation dy dt + ay= K, y(0) Has solution: y(t) = K a (1 e at )+ y(0)e at, t 0 If a 0, e at 0 y(t) K a = steady-state 8 Graph of Response y(t) = K a (1 e at )+ y(0)e at, t 0 K a y(0) K e at, t 0 a 9 3

23 Time Constant y(t) Time (sec) y(t) Time (sec) time,τ, for exponential transient to decay to e of its initial value (or 63% to its final value) 10 Sample Problems Pause 11 4

24 Summary Discussed how various physical phenomena are modeled by differential equations Showed the solution to a generic first-order differential equation with a constant input and initial condition Introduced the transient and steady-state responses Showed how to sketch the response and plot the time constant 12 5

25 RC Circuits Nathan V. Parrish PhD Candidate & Graduate Research Assistant School of Electrical and Computer Engineering Generate a differential equation that describes the behavior of a circuit with resistors and capacitors Solve the differential equation for step inputs (or switching constant inputs) Graph the behavior School of Electrical and Computer Engineering Lesson Objectives Generate a differential equation from a circuit Identify initial and final conditions Solve the differential equation Graph the result 5 1

26 Behavior of RC circuits 6 Example 1: Initial and Final Conditions 7 2

27 Example 1: Differential Equation 8 Example 1: Graph 9 3

28 Example 2: Initial and Final Conditions 10 Example 2: Differential Equation 11 4

29 Example 2: Graph 12 Summary Got some intuition about how RC circuits behave Identified initial and final conditions Found differential equations for the circuit and solved them Graphed the results 13 5

30 Dr. Bonnie Ferri Professor and Associate Chair School of Electrical and Computer Engineering Module 3 Lab Demo: RC Circuits Summary of Reactive Circuits Module School of Electrical and Computer Engineering RC Circuit + - v - + R + C v c - ~ C v c v s R

31 RC Circuit + - v - + R + C v c - ~ C v c v s R RC Circuit + - v - + R + C v c - ~ C v c v s R

32 RC Circuit + - v - + R + C v c - ~ C v c v s R RC Circuit + - v - + R + C v c - ~ C v c v s R

33 Lab Demo: RC Circuits 8 Summary An oscilloscope is used to measure and record voltage signals versus time. A function generator allows you to input voltage signals into a circuit Inputting a square wave into a circuit allows you to capture RC circuit transient behavior and to measure the time constant 9 4

34 RL Circuits Nathan V. Parrish PhD Candidate & Graduate Research Assistant School of Electrical and Computer Engineering Use differential equations to show the behavior of an RL circuit as the system changes. School of Electrical and Computer Engineering Lesson Objectives Generate a differential equation from a circuit Identify initial and final conditions Solve the differential equation Graph the result 5 1

35 Behavior of RL circuits 6 Example 1: Initial and Final Conditions 7 2

36 Example 1: Differential Equation 8 Example 1: Graph v L 5V 1.84V (63.2%) 0.68V (86.5%) t 9 3

37 Example 2: Initial and Final Conditions 10 Example 2: Differential Equation 11 4

38 Example 2: Graph i L 4mA 1.16mA (63.2%) 0.11mA (86.5%) 1 2 ma t 12 Summary Got some intuition about how RL circuits behave Identified initial and final conditions Found differential equations for the circuit and solved them Graphed the results 13 5

39 Second-Order Differential Equations Dr. Bonnie H. Ferri Professor and Associate Chair School of Electrical and Computer Engineering Recognize types of second-order responses School of Electrical and Computer Engineering Lesson Objectives Examine second-order differential equations with a constant input: Determine the steady-state solution Determine the type of transient response Recognize the characteristics of the plot of the solution 5 1

40 Ordinary Differential Equations ODE: Include functions of variables and their derivatives d 2 y dy dt y= f (t) dt 6 Models of Physical Systems Inputs, f(t) System Model Outputs, y(t) d 2 y dy dt y= f (t) dt Force Vibratory System Position Voltage or current source RLC Circuit Voltage or current 7 2

41 Solutions to Second-Order Differential Equation d 2 y dt 2+ a dy 1 dt + a 2y=K, y(0)and dy dt t=0 Solution: y(t) = steady-state + transient y( t) K a 2 Three possible forms depending on roots of (s 2 +a 1 s+a 2 )=0. 8 Transient Response (s 2 +a 1 s+a 2 )=0. ((real and distinct roots, r 1 and r 2 ) r t r t K e 1 K e ((real and equal roots, r and r) rt K1 e + K2 ((complex roots, a±jb) Ke at te rt sin( bt+ ϕ) 9 3

42 Sample Problems Overdamped 2 d y dy y= 1, 2 dt dt y(0) = 0, dy dt t= 0 = y(t) Time (sec) Sample Problems Underdamped 2 d y dy y= 1, 2 dt dt y(0) = 0, dy dt t= 0 = y(t) Time (sec) 11 4

43 Summary Examined generic 2 nd order differential equation Vibratory systems, RLC circuits Showed steady-state solution Showed generic transient solutions to underdamped and overdamped responses Showed characteristic plots of under damped and overdamped responses to a constant input applied at t=0 12 5

44 RLC Circuits Part 1 Nathan V. Parrish PhD Candidate & Graduate Research Assistant School of Electrical and Computer Engineering Use differential equations to show the behavior of an RLC circuit as the system changes. School of Electrical and Computer Engineering Lesson Objectives Generate a second-order differential equation from a RLC circuit Identify initial and final conditions Solve the differential equation Recognize if a system is underdamped/overdamped 5 1

45 Example 1: Initial and Final Conditions 6 Example 1: Differential Equation 7 2

46 Example 1: Transient Characteristic Equation: 8 Example 1: Steady State 9 3

47 Example 1: Solving for Constants 10 Example 1: Final Solution 11 4

48 Summary Looked at an overdamped case Identified initial and final conditions Found and solved representative differential equation Plotted the results 12 5

49 RLC Circuits Part 2 Nathan V. Parrish PhD Candidate & Graduate Research Assistant School of Electrical and Computer Engineering Use differential equations to show the behavior of an RLC circuit as the system changes. School of Electrical and Computer Engineering Lesson Objectives Generate a second-order differential equation from an underdamped RLC circuit Identify initial and final conditions Solve the differential equation Recognize if a system is underdamped/overdamped Identify the effect of damping on a second-order system 5 1

50 Example 2: Initial and Final Conditions 6 Example 2: Differential Equation 7 2

51 Example 2: Final Solution 8 Damping Ratio 9 3

52 Summary Got some intuition about how RLC circuits behave and contrasted overdamped and underdamped cases Identified initial and final conditions Found and solved representative differential equations Plotted the results Animated response as the resistance changes to show the effect of damping on the system 10 4

53 Lab Demo: RLC Circuit Dr. Bonnie Ferri Professor and Associate Chair School of Electrical and Computer Engineering Transient response of an RLC circuit School of Electrical and Computer Engineering RLC Circuit Measurements + +15v Buffer (Op Amp) - -15v + 3.3mH v s 20kΩ 0.01µf v c - 4 1

54 Lab Demo: RLC Circuits 5 Summary An underdamped RLC circuit has a small R value and results in large peaks An overdamped RLC circuit has a large R value 6 2

55 Lab Demo: Applications of Capacitance Dr. Bonnie H. Ferri Professor and Associate Chair School of Electrical and Computer Engineering Show common applications of capacitance School of Electrical and Computer Engineering Applications of Capacitance C = εwl d 1

56 Lab Demo: Applications of Capacitance 5 Summary Showed capacitive sensors such as touch pads and capacitive microphone, and antenna tuner 6 2

57 Credits Thanks to Allen Robinson, James Steinberg, Kevin Pham, and Al Ferri for help with demonstration ideas. Thanks to Marion Crowder for videotaping the demonstration. Capacitance drawings done by Nathan Parrish. 7 3

58 Dr. Bonnie H. Ferri Professor and Associate Chair School of Electrical and Computer Engineering Lab Demo: Applications of Inductance Show common applications of inductance School of Electrical and Computer Engineering Lab Demo: Applications of Inductance 4 1

59 Summary Discussed energy exchange in inductors mechanical to electrical and vice versa Moving conductor in magnetic field induces current Changing current in coiled wire causes a magnetic field Showed inductance applications Passive Sensing (guitar pick-up) Active Sensing (metal detector) Actuation (solenoid, speaker) 5 Credits Thanks to Allen Robinson, James Steinberg, Kevin Pham, and Al Ferri for help with demonstration ideas. Thanks to Ken Connor and Don Millard for the guitar string experiment. Thanks to Marion Crowder for videotaping the demonstration. Inductance drawings done by Nathan Parrish. 6 2

60 Dr. Bonnie Ferri Professor and Associate Chair School of Electrical and Computer Engineering Module 3 Reactive Circuit Wrap Up Summary of Reactive Circuits Module School of Electrical and Computer Engineering Concept Map Background Resistive Circuits Current, voltage, sources, resistance Reactive Circuits Capacitors RC Circuits Inductors RL Circuits Differential Reactive 2-order Diff Eqn Equations Circuits RLC Circuits Applications Methods to obtain circuit equations (KCL, KVL, mesh, node, Thévenin) Frequency Analysis Power 3 1

61 Important Concepts and Skills Understand the basic structure of a capacitor and its fundamental physical behavior Be able to use the i-v relationship to calculate current from voltage or vice versa Be able to reduce capacitor connections using using parallel and series connections Be able to calculate energy in a capacitor Be able to sketch current/voltage/power/energy curves 4 Important Concepts and Skills Be able to describe the construction and behavior of an inductor Be able to use the i-v relationship to find current through an inductor from the voltage across it, and vice versa Be able to explain how a voltage is created across an inductor Be able to analyze inductors in series/parallel Be able to calculate the energy in an inductor Be able to sketch current/voltage/power/energy curves 5 2

62 Important Concepts and Skills Given a constant input, be able to determine the steady-state value, time constant, and sketch the response Be able to write a differential equation governing the behavior of the circuit Be able to calculate the time constant, steadystate value, and sketch the response 6 Important Concepts and Skills Be able to identify the steady-state value Be able to predict the type of response from the roots (underdamped, critically damped, overdamped) Be able to write the differential equation that governs the behavior Be able to predict the type of response (underdamped, overdamped, critically damped) Be able to compute the damping factor and the resonant frequency Know that the smaller the damping factor, the larger the oscillations 7 3

63 Important Concepts and Skills Know the purposes of an oscilloscope and a function generator Know several applications of inductors and capacitors when they are used with non-electrical components 8 Concept Map 1 2 Resistive Background Circuits 5 Power 3 4 Reactive Circuits Frequency Analysis 9 4

64 Reminder Do all homework for this module Study for the quiz Continue to visit the forum 10 5

ENGR 2405 Chapter 8. Second Order Circuits

ENGR 2405 Chapter 8. Second Order Circuits ENGR 2405 Chapter 8 Second Order Circuits Overview The previous chapter introduced the concept of first order circuits. This chapter will expand on that with second order circuits: those that need a second