L L, R, C. Kirchhoff s rules applied to AC circuits. C Examples: Resonant circuits: series and parallel LRC. Filters: narrowband,

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "L L, R, C. Kirchhoff s rules applied to AC circuits. C Examples: Resonant circuits: series and parallel LRC. Filters: narrowband,"

Transcription

1 Today in Physics 1: A circuits Solving circuit problems one frequency at a time. omplex impedance of,,. Kirchhoff s rules applied to A circuits. it V in Examples: esonant circuits: i series and parallel. Filters: narrowband, lowpass, highpass V out November 1 Physics 1, Fall 1 1

2 By the book, not. Note that we are not following the techniques introduced in the book, in A circuits. it In the book, A circuit problems are solved by the phasor approach, in which the currents in circuit components are determined by a form of vector addition. In the judgment of your professors, this is a poor choice for introducing an important topic. It is much easier to solve A circuit problems by using the algebra of complex numbers. All but a few have seen complex numbers before, and we can bring the others up to speed quickly. eal engineers and scientists solve A circuit problems with complex numbers, not phasors. November 1 Physics 1, Fall 1

3 One frequency at a time The different time-dependent behavior of,, and makes for circuit problems that are superficially difficult. For example: What is the current, I(t), in the circuit at right, for a voltage V(t) which is a general function of time? Vt We can use the second Kirchhoff rule to generate an equation to solve for I = dq/dt: Q di Vt I dt I t dq dq 1 Q V t dt dt November 1 Physics 1, Fall 1 3

4 One frequency at a time (continued) That looks hard to solve, and indeed it is. an t be integrated directly, like all the other differential equations we ve met. But it turns out, as you will find in MTH 81 or ME 1, that any function V(t) can be represented td as a sum of sines and cosines; for example, 1 V t V costd Fourier s theorem where is an angular frequency. Suppose we deal with one frequency at a time. Vt I t November 1 Physics 1, Fall 1 4

5 One frequency at a time (continued) For one frequency component, dq dq 1 Q V cost dt dt and we solve for I dq dt. Q() and I() will still depend on t.? V cost Virtue: Q, and I, will also vary sinusoidally how could they not? apart from a delay resulting from a compromise between the delays between V and I for and : I Q Q cos t sin I I sin t November 1 Physics 1, Fall 1 5

6 One frequency at a time (continued) Even easier, in the algebra and calculus sense, is to use imaginary numbers: it V V e V cos t iv sin t under the agreement that we always take the real part at the end of the problem, as there are no imaginary voltages and currents in nature. In these terms, 1 i t Fourier s V t V e d theorem (again) and we use i only to simplify the math. i t? eve e I November 1 Physics 1, Fall 1 6

7 eminders about imaginary and complex numbers i i 1 i 1 e i e Im A i i i A A e A cos ia sin A e A Im A arctan Im A e 1 Phasor Im A representations of A e A i 1 i complex numbers e A A Electrical engineers usually refer to i as j, for perverse reasons best known to themselves. November 1 Physics 1, Fall 1 7

8 One frequency at a time (continued) et s prove that Q Q e is the solution to the differential equation above: i t dq dq Q V e dt dt Q e i 1 it iq e i t i t 1 Q e i t i t V e i t? eve Qe 1 i V e I November 1 Physics 1, Fall 1 8

9 One frequency at a time (continued) The term in brackets can be written as u1 i u e u euim u 1 i arctan 1 i V i so Qe e, whence u 1 Q V 1, and 1 i t? eve e I November 1 Physics 1, Fall 1 9

10 One frequency at a time (continued) arctan 1 arctan, 1 q.e.d. Furthermore, i t? eve dq it it I iq e Qe dt e I Q cos t Q sin t e I with Q and as given above. November 1 Physics 1, Fall 1 1

11 eactance and impedance eturn for a moment to the differential equation we solved: dq dq Q V e dt dt 1 i t and insert the solution. In the first and third terms we get i t? eve d i t d i t Q e i Q e i I dt dt 1 it 1 d i t Qe Qe 1 I i dt i e I November 1 Physics 1, Fall 1 11

12 eactance and impedance (continued) Thus we can rewrite the differential equation for Q a loop equation, remember as i I I 1 I i t V e i Apparently, at a given frequency, and have a property rather like resistance which relates their current and voltage. We call it reactance, X: Im X 1 e X i X i e V = IX: thus V lags I Phasor X 1 i e g by / for, and leads by / for. representation of the reactances November 1 Physics 1, Fall 1 1

13 eactance and impedance (continued) And thus the series A circuit looks just like a series of resistors in D circuits, the only change being a complex version of Ohm s law: it 1 V V e Ii IZ i Here Z is the impedance of the series combination. So what have we done? 1. By taking one frequency at a time, we have swapped a complicated calculus problem for a simple algebra problem. That s always a good trade. Im X 1 Z e X In phasor-speak, speak Z is the vector sum of the resistances and reactances. November 1 Physics 1, Fall 1 13

14 eactance and impedance (continued). To find currents and voltages in A circuits, inductors and capacitors can be treated with the Kirchhoff rules precisely as resistors are, using the reactances X and X. For example, the current in the series circuit again: 1 1 i V it Z i i Z e I e i Z 1 where Z Z 1 and arctan Im Z e Z arctan Three lines, instead of three pages. November 1 Physics 1, Fall 1 14

15 Parallel circuit Problem 3-93 in the book, except for a 9 phase shift: Determine the current through each component in this parallel circuit, and the total current leaving the voltage source. V cost November 1 Physics 1, Fall 1 15

16 Parallel circuit (continued) V cost I Individual currents, I I I it I V e X: V i t V i t i t I e I e I i V e i i V V e i t e or, as they would be measured, V V I sin t I cos t I V sin t t November 1 Physics 1, Fall 1 16

17 Parallel circuit (continued) V cost I I I In the second part of the problem we could just add the currents, but it will be useful to compare to the series if we work out the impedance of the parallel combination: i i Z i I Z November 1 Physics 1, Fall 1 17

18 Parallel circuit (continued) V cost I I I 1 arctan Im 1 Z e 1 Z arctan i So the impedance is Z Z e, with Z and as given above, and 1 V V 1 it it I 1 e I e Z I of which the real part gets measured: November 1 Physics 1, Fall 1 18 e I I cos t.

19 Parallel circuit (continued) V cost I I I I 11 4 The circuit draws a lot of current except for angular I frequencies near 1 ; 1 V the resonance in this case is a maximum of impedance. (See 1 below.) November 1 Physics 1, Fall 1 19

20 Parallel circuit (continued) ompare the impedance of the series and parallel circuits at the resonance frequency, 1 : 1 1 Z,series Z,parallel 1 For the series the impedance reaches its minimum at this frequency; for the parallel one, its maximum. November 1 Physics 1, Fall 1

21 Narrowband filter (continued) V in V cos t Vout This suggests the use of circuits as narrowband or notch th filters. For example, series as narrowband: V it V out I e Z Amplitude: V Z,series 1 V 1 V 1 Zseries,series out November 1 Physics 1, Fall 1 1

22 Narrowband filter (continued) g Vout V From this we define the voltage gain, and plot: g ) 1.5 For this plot, component values are chosen such that / = 3., This is a useful building block for signal-processing circuits: only voltages with frequencies near the resonance frequency are transmitted from the input to the output. November 1 Physics 1, Fall 1

23 ow-pass filter Many other useful signal-processing gizmos make good A circuit examples, like this one. Problem 3-13 in the book: Find the voltage gain, V out V in, in this circuit, and discuss the behavior of the gain when f and f. V in V cos t Vout Note: f = / is the frequency of oscillation, in cycles per unit time. November 1 Physics 1, Fall 1 3

24 ow-pass filter (continued) V in V cos t Vout As before, we compute the impedance and current: 1 i Z X X i i t V in V e i V i e i t I Z i i 1 V V IX it 1 i e out November 1 Physics 1, Fall 1 4 1

25 ow-pass filter (continued) V in V cos t Vout i Express in the form Ae : it Ve V out 1 e 1 V 1 e iarctan i tarctan Again, recall that t only the real part is measured. November 1 Physics 1, Fall 1 5

26 ow-pass filter (continued) V in V cos t Vout The voltage gain is the ratio of the input and output voltages: V g e out 1 cos arctan i t Ve 1 If tan ab, then cos b a b. November 1 Physics 1, Fall 1 6

27 ow-pass filter (continued) Note the limits of low and high frequency: 1 g 1 g g 1 g 1 1 lim g The circuit efficiently transmits angular frequencies below 1/, and attenuates those above this value; hence its name November 1 Physics 1, Fall 1 7

28 Single-pole lowpass and highpass filters owpass Highpass V in V out V in V out Vin V out V in Vout November 1 Physics 1, Fall 1 8

Course Updates. Reminders: 1) Assignment #10 due Today. 2) Quiz # 5 Friday (Chap 29, 30) 3) Start AC Circuits

Course Updates. Reminders: 1) Assignment #10 due Today. 2) Quiz # 5 Friday (Chap 29, 30) 3) Start AC Circuits ourse Updates http://www.phys.hawaii.edu/~varner/phys272-spr10/physics272.html eminders: 1) Assignment #10 due Today 2) Quiz # 5 Friday (hap 29, 30) 3) Start A ircuits Alternating urrents (hap 31) In this

More information

I. Impedance of an R-L circuit.

I. Impedance of an R-L circuit. I. Impedance of an R-L circuit. [For inductor in an AC Circuit, see Chapter 31, pg. 1024] Consider the R-L circuit shown in Figure: 1. A current i(t) = I cos(ωt) is driven across the circuit using an AC

More information

1 Phasors and Alternating Currents

1 Phasors and Alternating Currents Physics 4 Chapter : Alternating Current 0/5 Phasors and Alternating Currents alternating current: current that varies sinusoidally with time ac source: any device that supplies a sinusoidally varying potential

More information

12 Chapter Driven RLC Circuits

12 Chapter Driven RLC Circuits hapter Driven ircuits. A Sources... -. A ircuits with a Source and One ircuit Element... -3.. Purely esistive oad... -3.. Purely Inductive oad... -6..3 Purely apacitive oad... -8.3 The Series ircuit...

More information

Chapter 33. Alternating Current Circuits

Chapter 33. Alternating Current Circuits Chapter 33 Alternating Current Circuits 1 Capacitor Resistor + Q = C V = I R R I + + Inductance d I Vab = L dt AC power source The AC power source provides an alternative voltage, Notation - Lower case

More information

Physics 2112 Unit 20. Outline: Driven AC Circuits Phase of V and I Conceputally Mathematically With phasors

Physics 2112 Unit 20. Outline: Driven AC Circuits Phase of V and I Conceputally Mathematically With phasors Physics 2112 Unit 20 Outline: Driven A ircuits Phase of V and I onceputally Mathematically With phasors Electricity & Magnetism ecture 20, Slide 1 Your omments it just got real this stuff is confusing

More information

AC Circuits Homework Set

AC Circuits Homework Set Problem 1. In an oscillating LC circuit in which C=4.0 μf, the maximum potential difference across the capacitor during the oscillations is 1.50 V and the maximum current through the inductor is 50.0 ma.

More information

Learnabout Electronics - AC Theory

Learnabout Electronics - AC Theory Learnabout Electronics - AC Theory Facts & Formulae for AC Theory www.learnabout-electronics.org Contents AC Wave Values... 2 Capacitance... 2 Charge on a Capacitor... 2 Total Capacitance... 2 Inductance...

More information

Chapter 32A AC Circuits. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 32A AC Circuits. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University Chapter 32A AC Circuits A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Objectives: After completing this module, you should be able to: Describe

More information

Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits. Nov. 7 & 9, 2011

Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits. Nov. 7 & 9, 2011 Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits Nov. 7 & 9, 2011 Material from Textbook by Alexander & Sadiku and Electrical Engineering: Principles & Applications,

More information

Lecture 24. Impedance of AC Circuits.

Lecture 24. Impedance of AC Circuits. Lecture 4. Impedance of AC Circuits. Don t forget to complete course evaluations: https://sakai.rutgers.edu/portal/site/sirs Post-test. You are required to attend one of the lectures on Thursday, Dec.

More information

Part 4: Electromagnetism. 4.1: Induction. A. Faraday's Law. The magnetic flux through a loop of wire is

Part 4: Electromagnetism. 4.1: Induction. A. Faraday's Law. The magnetic flux through a loop of wire is 1 Part 4: Electromagnetism 4.1: Induction A. Faraday's Law The magnetic flux through a loop of wire is Φ = BA cos θ B A B = magnetic field penetrating loop [T] A = area of loop [m 2 ] = angle between field

More information

z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1)

z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1) 11 Complex numbers Read: Boas Ch. Represent an arb. complex number z C in one of two ways: z = x + iy ; x, y R rectangular or Cartesian form z = re iθ ; r, θ R polar form. (1) Here i is 1, engineers call

More information

RLC Circuit (3) We can then write the differential equation for charge on the capacitor. The solution of this differential equation is

RLC Circuit (3) We can then write the differential equation for charge on the capacitor. The solution of this differential equation is RLC Circuit (3) We can then write the differential equation for charge on the capacitor The solution of this differential equation is (damped harmonic oscillation!), where 25 RLC Circuit (4) If we charge

More information

Lecture 21. Resonance and power in AC circuits. Physics 212 Lecture 21, Slide 1

Lecture 21. Resonance and power in AC circuits. Physics 212 Lecture 21, Slide 1 Physics 1 ecture 1 esonance and power in A circuits Physics 1 ecture 1, Slide 1 I max X X = w I max X w e max I max X X = 1/w I max I max I max X e max = I max Z I max I max (X -X ) f X -X Physics 1 ecture

More information

8.1 Alternating Voltage and Alternating Current ( A. C. )

8.1 Alternating Voltage and Alternating Current ( A. C. ) 8 - ALTENATING UENT Page 8. Alternating Voltage and Alternating urrent ( A.. ) The following figure shows N turns of a coil of conducting wire PQS rotating with a uniform angular speed ω with respect to

More information

RLC Circuits. 1 Introduction. 1.1 Undriven Systems. 1.2 Driven Systems

RLC Circuits. 1 Introduction. 1.1 Undriven Systems. 1.2 Driven Systems RLC Circuits Equipment: Capstone, 850 interface, RLC circuit board, 4 leads (91 cm), 3 voltage sensors, Fluke mulitmeter, and BNC connector on one end and banana plugs on the other Reading: Review AC circuits

More information

Physics 4B Chapter 31: Electromagnetic Oscillations and Alternating Current

Physics 4B Chapter 31: Electromagnetic Oscillations and Alternating Current Physics 4B Chapter 31: Electromagnetic Oscillations and Alternating Current People of mediocre ability sometimes achieve outstanding success because they don't know when to quit. Most men succeed because

More information

Physics-272 Lecture 20. AC Power Resonant Circuits Phasors (2-dim vectors, amplitude and phase)

Physics-272 Lecture 20. AC Power Resonant Circuits Phasors (2-dim vectors, amplitude and phase) Physics-7 ecture 0 A Power esonant ircuits Phasors (-dim vectors, amplitude and phase) What is reactance? You can think of it as a frequency-dependent resistance. 1 ω For high ω, χ ~0 - apacitor looks

More information

12. Introduction and Chapter Objectives

12. Introduction and Chapter Objectives Real Analog - Circuits 1 Chapter 1: Steady-State Sinusoidal Power 1. Introduction and Chapter Objectives In this chapter we will address the issue of power transmission via sinusoidal or AC) signals. This

More information

Kirchhoff's Laws and Circuit Analysis (EC 2)

Kirchhoff's Laws and Circuit Analysis (EC 2) Kirchhoff's Laws and Circuit Analysis (EC ) Circuit analysis: solving for I and V at each element Linear circuits: involve resistors, capacitors, inductors Initial analysis uses only resistors Power sources,

More information

Alternating Current Circuits. Home Work Solutions

Alternating Current Circuits. Home Work Solutions Chapter 21 Alternating Current Circuits. Home Work s 21.1 Problem 21.11 What is the time constant of the circuit in Figure (21.19). 10 Ω 10 Ω 5.0 Ω 2.0µF 2.0µF 2.0µF 3.0µF Figure 21.19: Given: The circuit

More information

RLC Series Circuit. We can define effective resistances for capacitors and inductors: 1 = Capacitive reactance:

RLC Series Circuit. We can define effective resistances for capacitors and inductors: 1 = Capacitive reactance: RLC Series Circuit In this exercise you will investigate the effects of changing inductance, capacitance, resistance, and frequency on an RLC series AC circuit. We can define effective resistances for

More information

Alternating Current. Chapter 31. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman

Alternating Current. Chapter 31. PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Chapter 31 Alternating Current PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Modified by P. Lam 8_8_2008 Topics for Chapter 31

More information

LCR Series Circuits. AC Theory. Introduction to LCR Series Circuits. Module. What you'll learn in Module 9. Module 9 Introduction

LCR Series Circuits. AC Theory. Introduction to LCR Series Circuits. Module. What you'll learn in Module 9. Module 9 Introduction Module 9 AC Theory LCR Series Circuits Introduction to LCR Series Circuits What you'll learn in Module 9. Module 9 Introduction Introduction to LCR Series Circuits. Section 9.1 LCR Series Circuits. Amazing

More information

PHYS 3900 Homework Set #02

PHYS 3900 Homework Set #02 PHYS 3900 Homework Set #02 Part = HWP 2.0, 2.02, 2.03. Due: Mon. Jan. 22, 208, 4:00pm Part 2 = HWP 2.04, 2.05, 2.06. Due: Fri. Jan. 26, 208, 4:00pm All textbook problems assigned, unless otherwise stated,

More information

EIT Review 1. FE/EIT Review. Circuits. John A. Camara, Electrical Engineering Reference Manual, 6 th edition, Professional Publications, Inc, 2002.

EIT Review 1. FE/EIT Review. Circuits. John A. Camara, Electrical Engineering Reference Manual, 6 th edition, Professional Publications, Inc, 2002. FE/EIT eview Circuits Instructor: uss Tatro eferences John A. Camara, Electrical Engineering eference Manual, 6 th edition, Professional Publications, Inc, 00. John A. Camara, Practice Problems for the

More information

Solutions For the problem setup, please refer to the student worksheet that follows this section.

Solutions For the problem setup, please refer to the student worksheet that follows this section. 1 TASK 3.3.3: SHOCKING NUMBERS Solutions For the problem setup, please refer to the student worksheet that follows this section. 1. The impedance in part 1 of an AC circuit is Z 1 = 2 + j8 ohms, and the

More information

Chapter 28: Alternating Current

Chapter 28: Alternating Current hapter 8: Alternating urrent Phasors and Alternating urrents Alternating current (A current) urrent which varies sinusoidally in tie is called alternating current (A) as opposed to direct current (D).

More information

AC Circuit Analysis and Measurement Lab Assignment 8

AC Circuit Analysis and Measurement Lab Assignment 8 Electric Circuit Lab Assignments elcirc_lab87.fm - 1 AC Circuit Analysis and Measurement Lab Assignment 8 Introduction When analyzing an electric circuit that contains reactive components, inductors and

More information

AC Circuits. The Capacitor

AC Circuits. The Capacitor The Capacitor Two conductors in close proximity (and electrically isolated from one another) form a capacitor. An electric field is produced by charge differences between the conductors. The capacitance

More information

R-L-C Circuits and Resonant Circuits

R-L-C Circuits and Resonant Circuits P517/617 Lec4, P1 R-L-C Circuits and Resonant Circuits Consider the following RLC series circuit What's R? Simplest way to solve for is to use voltage divider equation in complex notation. X L X C in 0

More information

General Physics (PHY 2140)

General Physics (PHY 2140) General Physics (PHY 40) eminder: Exam this Wednesday 6/3 ecture 0-4 4 questions. Electricity and Magnetism nduced voltages and induction Self-nductance Circuits Energy in magnetic fields AC circuits and

More information

Review of Linear Time-Invariant Network Analysis

Review of Linear Time-Invariant Network Analysis D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x

More information

Sinusoidal Steady-State Analysis

Sinusoidal Steady-State Analysis Chapter 4 Sinusoidal Steady-State Analysis In this unit, we consider circuits in which the sources are sinusoidal in nature. The review section of this unit covers most of section 9.1 9.9 of the text.

More information

Driven RLC Circuits Challenge Problem Solutions

Driven RLC Circuits Challenge Problem Solutions Driven LC Circuits Challenge Problem Solutions Problem : Using the same circuit as in problem 6, only this time leaving the function generator on and driving below resonance, which in the following pairs

More information

Phasors: Impedance and Circuit Anlysis. Phasors

Phasors: Impedance and Circuit Anlysis. Phasors Phasors: Impedance and Circuit Anlysis Lecture 6, 0/07/05 OUTLINE Phasor ReCap Capacitor/Inductor Example Arithmetic with Complex Numbers Complex Impedance Circuit Analysis with Complex Impedance Phasor

More information

The RC Circuit: An Approach with Fourier Transforms

The RC Circuit: An Approach with Fourier Transforms The RC Circuit: An Approach with Fourier Transforms In this article we shall mathematically analyse the Resistor- Capacitor RC) circuit with the help of Fourier transforms FT). This very general technique

More information

11. The Series RLC Resonance Circuit

11. The Series RLC Resonance Circuit Electronicsab.nb. The Series RC Resonance Circuit Introduction Thus far we have studied a circuit involving a () series resistor R and capacitor C circuit as well as a () series resistor R and inductor

More information

Chapter 31: RLC Circuits. PHY2049: Chapter 31 1

Chapter 31: RLC Circuits. PHY2049: Chapter 31 1 hapter 31: RL ircuits PHY049: hapter 31 1 L Oscillations onservation of energy Topics Damped oscillations in RL circuits Energy loss A current RMS quantities Forced oscillations Resistance, reactance,

More information

Chapter 21: RLC Circuits. PHY2054: Chapter 21 1

Chapter 21: RLC Circuits. PHY2054: Chapter 21 1 Chapter 21: RC Circuits PHY2054: Chapter 21 1 Voltage and Current in RC Circuits AC emf source: driving frequency f ε = ε sinωt ω = 2π f m If circuit contains only R + emf source, current is simple ε ε

More information

Inductance, RL Circuits, LC Circuits, RLC Circuits

Inductance, RL Circuits, LC Circuits, RLC Circuits Inductance, R Circuits, C Circuits, RC Circuits Inductance What happens when we close the switch? The current flows What does the current look like as a function of time? Does it look like this? I t Inductance

More information

CIRCUIT ANALYSIS II. (AC Circuits)

CIRCUIT ANALYSIS II. (AC Circuits) Will Moore MT & MT CIRCUIT ANALYSIS II (AC Circuits) Syllabus Complex impedance, power factor, frequency response of AC networks including Bode diagrams, second-order and resonant circuits, damping and

More information

Yell if you have any questions

Yell if you have any questions Class 31: Outline Hour 1: Concept Review / Overview PRS Questions possible exam questions Hour : Sample Exam Yell if you have any questions P31 1 Exam 3 Topics Faraday s Law Self Inductance Energy Stored

More information

EXP. NO. 3 Power on (resistive inductive & capacitive) load Series connection

EXP. NO. 3 Power on (resistive inductive & capacitive) load Series connection OBJECT: To examine the power distribution on (R, L, C) series circuit. APPARATUS 1-signal function generator 2- Oscilloscope, A.V.O meter 3- Resisters & inductor &capacitor THEORY the following form for

More information

Lecture 05 Power in AC circuit

Lecture 05 Power in AC circuit CA2627 Building Science Lecture 05 Power in AC circuit Instructor: Jiayu Chen Ph.D. Announcement 1. Makeup Midterm 2. Midterm grade Grade 25 20 15 10 5 0 10 15 20 25 30 35 40 Grade Jiayu Chen, Ph.D. 2

More information

Network Graphs and Tellegen s Theorem

Network Graphs and Tellegen s Theorem Networ Graphs and Tellegen s Theorem The concepts of a graph Cut sets and Kirchhoff s current laws Loops and Kirchhoff s voltage laws Tellegen s Theorem The concepts of a graph The analysis of a complex

More information

NETWORK ANALYSIS WITH APPLICATIONS

NETWORK ANALYSIS WITH APPLICATIONS NETWORK ANALYSIS WITH APPLICATIONS Third Edition William D. Stanley Old Dominion University Prentice Hall Upper Saddle River, New Jersey I Columbus, Ohio CONTENTS 1 BASIC CIRCUIT LAWS 1 1-1 General Plan

More information

Electricity & Magnetism Lecture 20

Electricity & Magnetism Lecture 20 Electricity & Magnetism Lecture 20 Today s Concept: AC Circuits Maximum currents & voltages Phasors: A Simple Tool Electricity & Magne?sm Lecture 20, Slide 1 Other videos: Prof. W. Lewin, MIT Open Courseware

More information

Module 4. Single-phase AC Circuits. Version 2 EE IIT, Kharagpur 1

Module 4. Single-phase AC Circuits. Version 2 EE IIT, Kharagpur 1 Module 4 Single-phase A ircuits ersion EE IIT, Kharagpur esson 4 Solution of urrent in -- Series ircuits ersion EE IIT, Kharagpur In the last lesson, two points were described:. How to represent a sinusoidal

More information

ECE145A/218A Course Notes

ECE145A/218A Course Notes ECE145A/218A Course Notes Last note set: Introduction to transmission lines 1. Transmission lines are a linear system - superposition can be used 2. Wave equation permits forward and reverse wave propagation

More information

In Chapter 15, you learned how to analyze a few simple ac circuits in the time

In Chapter 15, you learned how to analyze a few simple ac circuits in the time In Chapter 15, you learned how to analyze a few simple ac circuits in the time domain using voltages and currents expressed as functions of time. However, this is not a very practical approach. A more

More information

1 2 U CV. K dq I dt J nqv d J V IR P VI

1 2 U CV. K dq I dt J nqv d J V IR P VI o 5 o T C T F 3 9 T K T o C 73.5 L L T V VT Q mct nct Q F V ml F V dq A H k TH TC L pv nrt 3 Ktr nrt 3 CV R ideal monatomic gas 5 CV R ideal diatomic gas w/o vibration V W pdv V U Q W W Q e Q Q e Carnot

More information

EXPERIMENT 07 TO STUDY DC RC CIRCUIT AND TRANSIENT PHENOMENA

EXPERIMENT 07 TO STUDY DC RC CIRCUIT AND TRANSIENT PHENOMENA EXPERIMENT 07 TO STUDY DC RC CIRCUIT AND TRANSIENT PHENOMENA DISCUSSION The capacitor is a element which stores electric energy by charging the charge on it. Bear in mind that the charge on a capacitor

More information

Physics 294H. lectures will be posted frequently, mostly! every day if I can remember to do so

Physics 294H. lectures will be posted frequently, mostly! every day if I can remember to do so Physics 294H l Professor: Joey Huston l email:huston@msu.edu l office: BPS3230 l Homework will be with Mastering Physics (and an average of 1 hand-written problem per week) Help-room hours: 12:40-2:40

More information

The simplest type of alternating current is one which varies with time simple harmonically. It is represented by

The simplest type of alternating current is one which varies with time simple harmonically. It is represented by ALTERNATING CURRENTS. Alternating Current and Alternating EMF An alternating current is one whose magnitude changes continuously with time between zero and a maximum value and whose direction reverses

More information

20. Alternating Currents

20. Alternating Currents University of hode sland DigitalCommons@U PHY 204: Elementary Physics Physics Course Materials 2015 20. lternating Currents Gerhard Müller University of hode sland, gmuller@uri.edu Creative Commons License

More information

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the timevarying

More information

Exam 3 Topics. Displacement Current Poynting Vector. Faraday s Law Self Inductance. Circuits. Energy Stored in Inductor/Magnetic Field

Exam 3 Topics. Displacement Current Poynting Vector. Faraday s Law Self Inductance. Circuits. Energy Stored in Inductor/Magnetic Field Exam 3 Topics Faraday s Law Self Inductance Energy Stored in Inductor/Magnetic Field Circuits LR Circuits Undriven (R)LC Circuits Driven RLC Circuits Displacement Current Poynting Vector NO: B Materials,

More information

AC Source and RLC Circuits

AC Source and RLC Circuits X X L C = 2π fl = 1/2π fc 2 AC Source and RLC Circuits ( ) 2 Inductive reactance Capacitive reactance Z = R + X X Total impedance L C εmax Imax = Z XL XC tanφ = R Maximum current Phase angle PHY2054: Chapter

More information

The Basic Elements and Phasors

The Basic Elements and Phasors 4 The Basic Elements and Phasors 4. INTRODUCTION The response of the basic R, L, and C elements to a sinusoidal voltage and current will be examined in this chapter, with special note of how frequency

More information

CM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers

CM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers CM2202: Scientific Computing and Multimedia Applications General Maths: 3. Complex Numbers Prof. David Marshall School of Computer Science & Informatics A problem when solving some equations There are

More information

Low Pass Filters, Sinusoidal Input, and Steady State Output

Low Pass Filters, Sinusoidal Input, and Steady State Output Low Pass Filters, Sinusoidal Input, and Steady State Output Jaimie Stephens and Michael Bruce 5-2-4 Abstract Discussion of applications and behaviors of low pass filters when presented with a steady-state

More information

Notes on Electric Circuits (Dr. Ramakant Srivastava)

Notes on Electric Circuits (Dr. Ramakant Srivastava) Notes on Electric ircuits (Dr. Ramakant Srivastava) Passive Sign onvention (PS) Passive sign convention deals with the designation of the polarity of the voltage and the direction of the current arrow

More information

RC, RL, and LCR Circuits

RC, RL, and LCR Circuits RC, RL, and LCR Circuits EK307 Lab Note: This is a two week lab. Most students complete part A in week one and part B in week two. Introduction: Inductors and capacitors are energy storage devices. They

More information

Revision of Basic A.C. Theory

Revision of Basic A.C. Theory Revision of Basic A.C. Theory 1 Revision of Basic AC Theory (Much of this material has come from Electrical & Electronic Principles & Technology by John Bird) Electricity is generated in power stations

More information

Sinusoidal Steady-State Analysis

Sinusoidal Steady-State Analysis Sinusoidal Steady-State Analysis Almost all electrical systems, whether signal or power, operate with alternating currents and voltages. We have seen that when any circuit is disturbed (switched on or

More information

PHYSICS NOTES ALTERNATING CURRENT

PHYSICS NOTES ALTERNATING CURRENT LESSON 7 ALENAING CUEN Alternating current As we have seen earlier a rotating coil in a magnetic field, induces an alternating emf and hence an alternating current. Since the emf induced in the coil varies

More information

Sinusoidal Steady- State Circuit Analysis

Sinusoidal Steady- State Circuit Analysis Sinusoidal Steady- State Circuit Analysis 9. INTRODUCTION This chapter will concentrate on the steady-state response of circuits driven by sinusoidal sources. The response will also be sinusoidal. For

More information

Lecture 9 Time Domain vs. Frequency Domain

Lecture 9 Time Domain vs. Frequency Domain . Topics covered Lecture 9 Time Domain vs. Frequency Domain (a) AC power in the time domain (b) AC power in the frequency domain (c) Reactive power (d) Maximum power transfer in AC circuits (e) Frequency

More information

Dynamic circuits: Frequency domain analysis

Dynamic circuits: Frequency domain analysis Electronic Circuits 1 Dynamic circuits: Contents Free oscillation and natural frequency Transfer functions Frequency response Bode plots 1 System behaviour: overview 2 System behaviour : review solution

More information

0-2 Operations with Complex Numbers

0-2 Operations with Complex Numbers Simplify. 1. i 10 2. i 2 + i 8 3. i 3 + i 20 4. i 100 5. i 77 esolutions Manual - Powered by Cognero Page 1 6. i 4 + i 12 7. i 5 + i 9 8. i 18 Simplify. 9. (3 + 2i) + ( 4 + 6i) 10. (7 4i) + (2 3i) 11.

More information

Gabriel Kron's biography here.

Gabriel Kron's biography here. Gabriel Kron, Electric Circuit Model of the Schrödinger Equation, 1945 - Component of :... Page 1 of 12 {This website: Please note: The following article is complete; it has been put into ASCII due to

More information

1 of 7 8/2/ :10 PM

1 of 7 8/2/ :10 PM {This website: Please note: The following article is complete; it has been put into ASCII due to a) space requirement reduction and b) for having thus the possibility to enlarge the fairly small figures.

More information

Inductance, RL and RLC Circuits

Inductance, RL and RLC Circuits Inductance, RL and RLC Circuits Inductance Temporarily storage of energy by the magnetic field When the switch is closed, the current does not immediately reach its maximum value. Faraday s law of electromagnetic

More information

Chapter 10. Experiment 8: Electromagnetic Resonance Introduction V L V C V R

Chapter 10. Experiment 8: Electromagnetic Resonance Introduction V L V C V R Chapter 10 Experiment 8: Electromagnetic esonance 10.1 Introduction If a pendulum whose angular frequency is ω 0 for swinging freely is subjected to a small applied force F(t) = F 0 cos ωt oscillating

More information

Electrical Engineering Fundamentals for Non-Electrical Engineers

Electrical Engineering Fundamentals for Non-Electrical Engineers Electrical Engineering Fundamentals for Non-Electrical Engineers by Brad Meyer, PE Contents Introduction... 3 Definitions... 3 Power Sources... 4 Series vs. Parallel... 9 Current Behavior at a Node...

More information

PES 1120 Spring 2014, Spendier Lecture 36/Page 1

PES 1120 Spring 2014, Spendier Lecture 36/Page 1 PES 0 Spring 04, Spenier ecture 36/Page Toay: chapter 3 - R circuits: Dampe Oscillation - Driven series R circuit - HW 9 ue Wenesay - FQs Wenesay ast time you stuie the circuit (no resistance) The total

More information

INDUCTORS AND AC CIRCUITS

INDUCTORS AND AC CIRCUITS INDUCTOS AND AC CICUITS We begin by introducing the new circuit element provided by Faraday s aw the inductor. Consider a current loop carrying a current I. The current will produce a magnetic flux in

More information

PHASOR DIAGRAMS HANDS-ON RELAY SCHOOL WSU PULLMAN, WA.

PHASOR DIAGRAMS HANDS-ON RELAY SCHOOL WSU PULLMAN, WA. PHASOR DIAGRAMS HANDS-ON RELAY SCHOOL WSU PULLMAN, WA. RON ALEXANDER - BPA What are phasors??? In normal practice, the phasor represents the rms maximum value of the positive half cycle of the sinusoid

More information

Electricity & Magnetism Lecture 21

Electricity & Magnetism Lecture 21 Electricity & Magnetism Lecture 21 Uncertainties DC Meters! Digital error of at least ±1 digit in the rightmost digit.! Analog 'calibra:on' error is specified by a percentage of the reading, o?en about

More information

CORRESPONDENCE BETWEEN PHASOR TRANSFORMS AND FREQUENCY RESPONSE FUNCTION IN RLC CIRCUITS

CORRESPONDENCE BETWEEN PHASOR TRANSFORMS AND FREQUENCY RESPONSE FUNCTION IN RLC CIRCUITS CORRESPONDENCE BETWEEN PHASOR TRANSFORMS AND FREQUENCY RESPONSE FUNCTION IN RLC CIRCUITS Hassan Mohamed Abdelalim Abdalla Polytechnic Department of Engineering and Architecture, University of Studies of

More information

ELEC 202 Electric Circuit Analysis II Lecture 10(a) Complex Arithmetic and Rectangular/Polar Forms

ELEC 202 Electric Circuit Analysis II Lecture 10(a) Complex Arithmetic and Rectangular/Polar Forms Dr. Gregory J. Mazzaro Spring 2016 ELEC 202 Electric Circuit Analysis II Lecture 10(a) Complex Arithmetic and Rectangular/Polar Forms THE CITADEL, THE MILITARY COLLEGE OF SOUTH CAROLINA 171 Moultrie Street,

More information

Lecture #3. Review: Power

Lecture #3. Review: Power Lecture #3 OUTLINE Power calculations Circuit elements Voltage and current sources Electrical resistance (Ohm s law) Kirchhoff s laws Reading Chapter 2 Lecture 3, Slide 1 Review: Power If an element is

More information

Review of DC Electric Circuit. DC Electric Circuits Examples (source:

Review of DC Electric Circuit. DC Electric Circuits Examples (source: Review of DC Electric Circuit DC Electric Circuits Examples (source: http://hyperphysics.phyastr.gsu.edu/hbase/electric/dcex.html) 1 Review - DC Electric Circuit Multisim Circuit Simulation DC Circuit

More information

Capacitor and Inductor

Capacitor and Inductor Capacitor and Inductor Copyright (c) 2015 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or

More information

Electrical measurements:

Electrical measurements: Electrical measurements: Last time we saw that we could define circuits though: current, voltage and impedance. Where the impedance of an element related the voltage to the current: This is Ohm s law.

More information

Designing Information Devices and Systems II Spring 2016 Anant Sahai and Michel Maharbiz Midterm 2

Designing Information Devices and Systems II Spring 2016 Anant Sahai and Michel Maharbiz Midterm 2 EECS 16B Designing Information Devices and Systems II Spring 2016 Anant Sahai and Michel Maharbiz Midterm 2 Exam location: 145 Dwinelle (SIDs ending in 1 and 5) PRINT your student ID: PRINT AND SIGN your

More information

Calculus/Physics Schedule, Second Semester. Wednesday Calculus

Calculus/Physics Schedule, Second Semester. Wednesday Calculus / Schedule, Second Semester Note to instructors: There are two key places where the calculus and physics are very intertwined and the scheduling is difficult: GaussÕ Law and the differential equation for

More information

Fourier series. XE31EO2 - Pavel Máša. Electrical Circuits 2 Lecture1. XE31EO2 - Pavel Máša - Fourier Series

Fourier series. XE31EO2 - Pavel Máša. Electrical Circuits 2 Lecture1. XE31EO2 - Pavel Máša - Fourier Series Fourier series Electrical Circuits Lecture - Fourier Series Filtr RLC defibrillator MOTIVATION WHAT WE CAN'T EXPLAIN YET Source voltage rectangular waveform Resistor voltage sinusoidal waveform - Fourier

More information

Get Discount Coupons for your Coaching institute and FREE Study Material at ELECTROMAGNETIC INDUCTION

Get Discount Coupons for your Coaching institute and FREE Study Material at  ELECTROMAGNETIC INDUCTION ELECTROMAGNETIC INDUCTION 1. Magnetic Flux 2. Faraday s Experiments 3. Faraday s Laws of Electromagnetic Induction 4. Lenz s Law and Law of Conservation of Energy 5. Expression for Induced emf based on

More information

Phasor Diagram. Figure 1: Phasor Diagram. A φ. Leading Direction. θ B. Lagging Direction. Imag. Axis Complex Plane. Real Axis

Phasor Diagram. Figure 1: Phasor Diagram. A φ. Leading Direction. θ B. Lagging Direction. Imag. Axis Complex Plane. Real Axis 1 16.202: PHASORS Consider sinusoidal source i(t) = Acos(ωt + φ) Using Eulers Notation: Acos(ωt + φ) = Re[Ae j(ωt+φ) ] Phasor Representation of i(t): = Ae jφ = A φ f v(t) = Bsin(ωt + ψ) First convert the

More information

AC Circuits and Electromagnetic Waves

AC Circuits and Electromagnetic Waves AC Circuits and Electromagnetic Waves Physics 102 Lecture 5 7 March 2002 MIDTERM Wednesday, March 13, 7:30-9:00 pm, this room Material: through next week AC circuits Next week: no lecture, no labs, no

More information

Phasor mathematics. Resources and methods for learning about these subjects (list a few here, in preparation for your research):

Phasor mathematics. Resources and methods for learning about these subjects (list a few here, in preparation for your research): Phasor mathematics This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

More information

Physics 2112 Unit 11

Physics 2112 Unit 11 Physics 2112 Unit 11 Today s oncept: ircuits Unit 11, Slide 1 Stuff you asked about.. what happens when one resistor is in parallel and one is in series with the capacitor Differential equations are tough

More information

Differential Equations and Linear Algebra Supplementary Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University

Differential Equations and Linear Algebra Supplementary Notes. Simon J.A. Malham. Department of Mathematics, Heriot-Watt University Differential Equations and Linear Algebra Supplementary Notes Simon J.A. Malham Department of Mathematics, Heriot-Watt University Contents Chapter 1. Linear algebraic equations 5 1.1. Gaussian elimination

More information

Algebraic substitution for electric circuits

Algebraic substitution for electric circuits Algebraic substitution for electric circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

More information

Capacitance, Resistance, DC Circuits

Capacitance, Resistance, DC Circuits This test covers capacitance, electrical current, resistance, emf, electrical power, Ohm s Law, Kirchhoff s Rules, and RC Circuits, with some problems requiring a knowledge of basic calculus. Part I. Multiple

More information

Review of Basic Electrical and Magnetic Circuit Concepts EE

Review of Basic Electrical and Magnetic Circuit Concepts EE Review of Basic Electrical and Magnetic Circuit Concepts EE 442-642 Sinusoidal Linear Circuits: Instantaneous voltage, current and power, rms values Average (real) power, reactive power, apparent power,

More information

Exam 3 Solutions. The induced EMF (magnitude) is given by Faraday s Law d dt dt The current is given by

Exam 3 Solutions. The induced EMF (magnitude) is given by Faraday s Law d dt dt The current is given by PHY049 Spring 008 Prof. Darin Acosta Prof. Selman Hershfield April 9, 008. A metal rod is forced to move with constant velocity of 60 cm/s [or 90 cm/s] along two parallel metal rails, which are connected

More information