# Introduction to Complex Numbers in Physics/Engineering

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The real numbers (denoted R) are incomplete (not closed) in the sense that standard operations applied to some real numbers do not yield a real number result (e.g., square root: 1). It is surprisingly easy to enlarge the set of real numbers producing a set of numbers that is closed under standard operations: one simply needs to include 1 (and linear combinations of it). This enlarged field of numbers, called the complex numbers (denoted C), consists of numbers of the form: z = a + b 1 where a and b are real numbers. There are lots of notations for theses numbers. In mathematics, 1 is called i (so z = a + bi), whereas in electrical engineering i is frequently used for current, so 1 is called j (so z = a + bj). In Mathematica complex numbers are constructed using I for i. Since complex numbers require two real numbers to specify them they can also be represented as an ordered pair: z = (a,b). In any case a is called the real part of z: a = (z) and b is called the imaginary part of z: b = Im(z). Note that the imaginary part of any complex number is real and the imaginary part of any real number is zero. Finally there is a polar notation which reports the radius (a.k.a. absolute value or magnitude) and angle (a.k.a. phase or argument) of the complex number in the form: r θ. The polar notation can be converted to an algebraic expression because of a surprising relationship between the exponential function and the trigonometric functions: e jθ = cos θ + j sin θ (1) Thus there is a simple formula for the complex number in terms of its magnitude and angle: a 2 + b 2 = r (2) a = r cos θ = cos θ (3) b = r sinθ = sin θ (4) = a + bj = (cos θ + j sinθ) = e jθ (5) For example, we have the following notations for the complex number 1 + i: 1 + i = 1 + j = 1 +I = (1,1) = 2 45 = 2e jπ/4 (6) Note that Equation 1 can be used to express the usual trigonometric functions in terms of complex exponentials: cos θ = (e jθ ) = ejθ + e jθ 2 sin θ = Im(e jθ ) = ejθ e jθ Since complex numbers are closed under the standard operations, we can define things which previously made no sense: log( 1), arccos(2), ( 1) π, sin(i),... The complex numbers are large enough to define every function value you might want. Note that addition, subtraction, 2j (7)

2 Im Im a b a 2 +b 2 = r r θ a b Figure 1: Complex numbers can be displayed on the complex plane. A complex number = a + bi may be displayed as an ordered pair: (a,b), with the real axis the usual x- axis and the imaginary axis the usual y-axis. Complex numbers are also often displayed as vectors pointing from the origin to (a,b). The angle θ can be found from the usual trigonometric functions; = r is the length of the vector. multiplication, and division of complex numbers proceeds as usual, just using the symbol for 1 (let s use j): = a + bj z 2 = c + dj (8) + z 2 = (a + bj) + (c + dj) = (a + c) + (b + d)j z 2 = (a + bj) (c + dj) = (a c) + (b d)j z 2 = (a + bj) (c + dj) = ac + adj + bcj + bdj 2 = (ac bd) + (ad + bc)j 1 1 = a + bj = 1 a + bj a bj a bj = a bj a 2 + b 2 = a a 2 + b 2 + b a 2 + b 2 j Note in calculating 1/ we made use of the complex number a bj; a bj is called the complex conjugate of and it is denoted by z 1 or sometimes. See that zz = z 2. Note that, in terms of the ordered pair representation of C, complex number addition and subtraction looks just like component-by-component vector addition: (a,b) + (c,d) = (a + b,c + d) (9) Im Im +z 2 z 2 * Figure 2: The complex conjugate is obtained by reflecting the vector in the real axis. Complex number addition works just like vector addition.

3 Thus there is a tendency to denote complex numbers as vectors rather than points in the complex plane. Superposition of Oscillation While the closure property of the complex numbers is dear to the hearts of mathematicians, the main use of complex numbers in science is to represent sinusoidally varying quantities in a simple way allowing them to be combined with relative ease. (member that the superposition of sinusoidal quantities is itself sinusoidal, but with a new amplitude and phase.) For example, in a series RC circuit the voltage across the resistor might be given by V R (t) = Acos ωt whereas the voltage across the capacitor might be given by V C (t) = B sinωt, and the voltage across the combination (according to Kirchhoff) is the sum: V R (t) + V C (t) = Acos ωt + B sin ωt where: A,B R = ( ) A 2 + B 2 A B cos ωt + A 2 + B2 A 2 + B sin ωt 2 = A 2 + B 2 (cos δ cos ωt + sin δ sin ωt) where: cos δ = = A 2 + B 2 cos(ωt δ) A A 2 + B 2 Yuck! That s a lot of work just to add two sinusoidal functions; we seek a simpler method (which might not seem overly simple at first glance). Note that V R can be written as (Ae jωt ) and V C can be written as ( jbe jωt ) so: V R (t) + V C (t) = ( (A jb)e jωt) (10) Now using the polar form of the complex number A jb: we have: A jb = A 2 + B 2 e jφ where: tan φ = B/A (11) V R (t) + V C (t) = ( (A jb)e jωt) = ( A 2 + B 2 e jφ e jωt) = A 2 + B 2 (e j(ωt φ)) = A 2 + B 2 cos(ωt φ) If we have more sinusoidal waves to add up, the problem is not much more difficult. Consider the case of adding four waves, which for ease of notation we assume have unit amplitude and constant phase difference (δ). [I m also going to switch to 1 = i; you should get used to both notations.] f(t) = cos(ωt) + cos(ωt + δ) + cos(ωt + 2δ) + cos(ωt + 3δ) (12) We can write this as the real part of a complex expression: [( f(t) = 1 + e iδ + e i2δ + e i3δ) e iωt] (13)

4 Figure 3: Consider the problem of adding four sinusoidal functions with the same frequency and amplitude, but with different offsets (see left). The result (right) is a sinusoidal function with the same frequency we seek to easily determine the resulting amplitude and offset. We ll use some tricks below to add these four complex numbers, but for now the main point is that they add to some complex number which can be expressed in polar form: ( 1 + e iδ + e i2δ + e i3δ) = A e iφ (14) so [ f(t) = A e i(ωt+φ)] = Acos(ωt + φ) (15) Thus adding sinusoidal waves is as simple as adding the corresponding (complex) amplitudes. Now for the trick that applies to this particular problem. call the formula for the sum of the geometric series: 1 + r + r r N 1 = 1 rn (16) 1 r (The proof of this result is easy: just multiply both sides by (1 r), and notice that the rhs terms telescope down to 1 r N.) Here: So the sum is: A e iφ = 1 ei4δ ei2δ = 1 eiδ e iδ/2 r = e iδ (17) e i2δ e i2δ sin(2δ) e iδ/2 = ei3δ/2 e iδ/2 sin(δ/2) (18) Im 3δ 2δ δ Figure 4: Finding the sum of four cosines amounts to adding four complex amplitudes: ( 1 + e iδ + e i2δ + e i3δ)

5 Figure 5: A 2 is plotted as a function of δ. Note that if δ = 0,2π,4π,..., the four waves are in phase so the amplitude is 4 (so A 2 = 16). The first zero of A 2 occurs when the four amplitude vectors form a (closed) square for δ = π/2. and hence: A = sin(2δ) sin(δ/2) φ = 3δ/2 (19) In physics we are usually most interested in the value of A 2 which is plotted in Figure 5 as function of δ. In a more general case we might need to add N cosines with possibly different real amplitudes a k and offsets δ k : h(t) = a 1 cos(ωt + δ 1 ) + a 2 cos(ωt + δ 2 ) + + a N cos(ωt + δ N ) (20) N = a k cos(ωt + δ k ) (21) k=1 [( ) = a 1 e iδ 1 + a 2 e iδ a N e iδ N e iωt] (22) [( N ) ] = a k e iδ k e iωt (23) k=1 Once again all that is required is to express the sum of the complex amplitudes in polar fashion: then ( N ) a k e iδ k = Ae iφ (24) k=1 h(t) = Acos(ωt + φ) (25) Differential Equations and e iωt Complex exponentials provide a fast and easy solution for many differential equations. Consider the damped harmonic oscillator: F net = kx bv = ma (26)

6 or: 0 = d2 x dt 2 + b dx m dt + k m x (27) Seeking a more compact notation, we redefine the constants in this expression: b m 2β k m ω2 0 (28) So: d 2 x dx + 2β dt2 dt + ω2 0 x = 0 (29) If we guess a solution of the form: x = Ae rt, we find: [ r 2 + 2β r + ω0 2 ] Ae rt = 0 (30) Since e rt is never zero, r must be a root of the quadratic equation in square brackets. r = 2β ± 4β 2 4ω0 2 = β ± i ω0 2 2 β2 (31) where we have assumed ω 0 > β. Defining the free oscillation frequency ω = ω0 2 β2, we have a solution: [ ] x = A e βt e iω t = A e βt cos(ω t + φ) (32) In the driven, damped harmonic oscillator, we have a driving force: F 0 cos ωt in addition to the other forces: F net = F 0 cos ωt kx bv = ma (33) Defining A 0 = F 0 /m yields the differential equation: d 2 x dx + 2β dt2 dt + ω2 0 x = A 0 cos(ωt) (34) If we seek a solution that oscillates at the driving frequency: x = [ Ae iωt], our differential equation becomes: [ ω 2 + 2βiω + ω0 2 ] Ae iωt = A 0 e iωt (35) So: A 0 A = (ω0 2 ω2 ) + 2βiω From which we can extract the amplitude and phase of the oscillation, for example: (36) A = A 0 (ω 2 0 ω 2 ) 2 + 4β 2 ω 2 (37) One can show that the amplitude is largest for ω = ω0 2 2β2 (38) Finally it is customary to describe driven oscillators in terms of a dimensionless quality factor, Q, Q = ω 0 (39) 2β If we then let x be the dimensionless frequency ratio ω/ω 0, we can write the oscillation amplitude in a particularly simple form: A = A 0 /ω 2 0 (1 x 2 ) 2 + x 2 /Q 2 (40) Notice that the case of small damping (small β) corresponds to large Q.

7 Figure 6: sonance: the amplitude factor: is plotted as a function of the (1 x 2 ) 2 +x 2 /Q2 dimensionless frequency ratio: x = ω/ω 0 for the case Q = 20. Clearly the largest amplitude occurs when x 1, i.e., ω ω 0 Homework 1. Prove that when you multiply complex numbers and z 2, the magnitude of the result is the product of the magnitudes of and z 2, and the phase of the product is the sum of the phases of and z 2. Im r 1 z 2 r 2 θ 2 θ 1 z 3 = z 2 θ 3 = θ 1 + θ 2 z 3 r 3 = r 1 r 2 2. Express the following in the r θ format (I bet your calculator can do this automatically): 1 (a) (b) 3 + i (c) 25e 2i (d) (1/(1 + i)) (e) i 1 + 3i (1 + i) 3. Find the following in (a,b) format (I bet your calculator can do this automatically): (a) 3i 7 (b) ( i) 4 (c) 3 + 4i (d) 25e 2i (e) ln( 1) i Three cosine functions with amplitudes and offsets: a 1 = 1.32, δ 1 =.253 rad; a 2 = 3.21, δ 2 =.532 rad; and a 3 = 2.13, δ 3 =.325 rad are to be added together. Find the result. 5. Find the formula for the amplitude A that results from adding 5 unit-amplitude cosine waves with constant phase difference: g(t) = cos(ωt) + cos(ωt + δ) + cos(ωt + 2δ) + cos(ωt + 3δ) + cos(ωt + 4δ) = Acos(ωt + φ) Use your favorite graphing program to make a hardcopy plot of A 2 vs. δ for δ (0,4π).

8 V o cos(ωt) V o cos(ωt) 6. Consider a driven RC circuit (see below left). According to Kirchhoff s law the voltage drop across the resistor (IR, for current I) plus the voltage drop across the capacitor (Q/C, for charge Q) must equal the voltage applied by the a.c. generator (V 0 cos(ωt)) (where the generator is operating at an angular frequency ω). Thus: Q C + R I = V 0 cos(ωt) (41) Since the the current flowing must accumulate as charge on the capacitor we have: Thus we have the differential equation: I = dq dt (42) Q C + R dq dt = V 0 cos(ωt) (43) Using complex variable methods and guessing a solution of the form: Q = Q 0 e iωt (44) Determine the amplitude and phase of Q 0 so you can express your final answer in real form: Q = Acos(ωt + φ) (45) I I Q 7. Consider a driven RL circuit (see above right). According to Kirchhoff s law the voltage drop across the resistor (IR, for current I) plus the voltage drop across the inductor (L di/dt) must equal the voltage applied by the a.c. generator (V 0 cos(ωt)) (where the generator is operating at an angular frequency ω). Thus: L di dt + R I = V 0 cos(ωt) (46) Using complex variable methods and guessing a solution of the form: I = I 0 e iωt (47) Determine the amplitude and phase of I 0 so you can express your final answer in real form: I = Acos(ωt + φ) (48)

### How to add sine functions of different amplitude and phase

Physics 5B Winter 2009 How to add sine functions of different amplitude and phase In these notes I will show you how to add two sinusoidal waves each of different amplitude and phase to get a third sinusoidal

### 2.2 Magic with complex exponentials

2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or

### A few words about imaginary numbers (and electronics) Mark Cohen mscohen@g.ucla.edu

A few words about imaginary numbers (and electronics) Mark Cohen mscohen@guclaedu While most of us have seen imaginary numbers in high school algebra, the topic is ordinarily taught in abstraction without

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### Unit2: Resistor/Capacitor-Filters

Unit2: Resistor/Capacitor-Filters Physics335 Student October 3, 27 Physics 335-Section Professor J. Hobbs Partner: Physics335 Student2 Abstract Basic RC-filters were constructed and properties such as

### Phasors. Phasors. by Prof. Dr. Osman SEVAİOĞLU Electrical and Electronics Engineering Department. ^ V cos (wt + θ) ^ V sin (wt + θ)

V cos (wt θ) V sin (wt θ) by Prof. Dr. Osman SEVAİOĞLU Electrical and Electronics Engineering Department EE 209 Fundamentals of Electrical and Electronics Engineering, Prof. Dr. O. SEVAİOĞLU, Page 1 Vector

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### CIRCUITS LABORATORY EXPERIMENT 3. AC Circuit Analysis

CIRCUITS LABORATORY EXPERIMENT 3 AC Circuit Analysis 3.1 Introduction The steady-state behavior of circuits energized by sinusoidal sources is an important area of study for several reasons. First, the

### EE 221 AC Circuit Power Analysis. Instantaneous and average power RMS value Apparent power and power factor Complex power

EE 1 AC Circuit Power Analysis Instantaneous and average power RMS value Apparent power and power factor Complex power Instantaneous Power Product of time-domain voltage and time-domain current p(t) =

### COMPLEX NUMBERS AND PHASORS

COMPLEX NUMBERS AND PHASORS 1 Professor Andrew E. Yagle, EECS 206 Instructor, Fall 2005 Dept. of EECS, The University of Michigan, Ann Arbor, MI 48109-2122 I. Abstract The purpose of this document is to

### COMPLEX NUMBERS. a bi c di a c b d i. a bi c di a c b d i For instance, 1 i 4 7i 1 4 1 7 i 5 6i

COMPLEX NUMBERS _4+i _-i FIGURE Complex numbers as points in the Arg plane i _i +i -i A complex number can be represented by an expression of the form a bi, where a b are real numbers i is a symbol with

### Oscillations. Vern Lindberg. June 10, 2010

Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT 4 Understand single-phase alternating current (ac) theory Single phase AC

### 7.1 POWER IN AC CIRCUITS

C H A P T E R 7 AC POWER he aim of this chapter is to introduce the student to simple AC power calculations and to the generation and distribution of electric power. The chapter builds on the material

### Alternating-Current Circuits

hapter 1 Alternating-urrent ircuits 1.1 A Sources... 1-1. Simple A circuits... 1-3 1..1 Purely esistive load... 1-3 1.. Purely Inductive oad... 1-5 1..3 Purely apacitive oad... 1-7 1.3 The Series ircuit...

### A complex number W consists of real and imaginary parts a and b respectively, and the imaginary constant j which is the square root of negative one.

eactance and Impedance A Voltage and urrent In a D circuit, we learned that the relationship between voltage and current was V=I, also known as Ohm's law. We need to find a similar law for A circuits,

### Applications of Second-Order Differential Equations

Applications of Second-Order Differential Equations Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration

### 7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )

34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using

### Chapter 12 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...

### Mathematics. (www.tiwariacademy.com : Focus on free Education) (Chapter 5) (Complex Numbers and Quadratic Equations) (Class XI)

( : Focus on free Education) Miscellaneous Exercise on chapter 5 Question 1: Evaluate: Answer 1: 1 ( : Focus on free Education) Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) =

### Lecture 21. AC Circuits, Reactance.

Lecture 1. AC Circuits, Reactance. Outline: Power in AC circuits, Amplitude and RMS values. Phasors / Complex numbers. Resistors, Capacitors, and Inductors in the AC circuits. Reactance and Impedance.

### 11.7 Polar Form of Complex Numbers

11.7 Polar Form of Complex Numbers 989 11.7 Polar Form of Complex Numbers In this section, we return to our study of complex numbers which were first introduced in Section.. Recall that a complex number

### MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:

MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex

### 2 Session Two - Complex Numbers and Vectors

PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

### Using the Impedance Method

Using the Impedance Method The impedance method allows us to completely eliminate the differential equation approach for the determination of the response of circuits. In fact the impedance method even

### COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS

COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS BORIS HASSELBLATT CONTENTS. Introduction. Why complex numbers were introduced 3. Complex numbers, Euler s formula 3 4. Homogeneous differential equations 8 5.

### THE COMPLEX EXPONENTIAL FUNCTION

Math 307 THE COMPLEX EXPONENTIAL FUNCTION (These notes assume you are already familiar with the basic properties of complex numbers.) We make the following definition e iθ = cos θ + i sin θ. (1) This formula

### Module P5.4 AC circuits and electrical oscillations

F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P5.4 Opening items. Module introduction.2 Fast track questions.3 Ready to study? 2 AC circuits 2. Describing alternating currents

### Complex Numbers and the Complex Exponential

Complex Numbers and the Complex Exponential Frank R. Kschischang The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto September 5, 2005 Numbers and Equations

### Bharathwaj Muthuswamy EE100 Active Filters

Bharathwaj Muthuswamy EE100 mbharat@cory.eecs.berkeley.edu 1. Introduction Active Filters In this chapter, we will deal with active filter circuits. Why even bother with active filters? Answer: Audio.

### 1 Introduction. 2 Complex Exponential Notation. J.L. Kirtley Jr.

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.06 Introduction to Power Systems Class Notes Chapter AC Power Flow in Linear Networks J.L. Kirtley Jr.

### Understanding Poles and Zeros

MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

### Chapter 5: Analysis of Time-Domain Circuits

Chapter 5: Analysis of Time-Domain Circuits This chapter begins the analysis of circuits containing elements with the ability to store energy: capacitors and inductors. We have already defined each of

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1

19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point

### Mechanical Vibrations

Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement,

### 2.6 The driven oscillator

2.6. THE DRIVEN OSCILLATOR 131 2.6 The driven oscillator We would like to understand what happens when we apply forces to the harmonic oscillator. That is, we want to solve the equation M d2 x(t) 2 + γ

### April 8. Physics 272. Spring Prof. Philip von Doetinchem

Physics 272 April 8 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem philipvd@hawaii.edu Phys272 - Spring 14 - von Doetinchem - 218 L-C in parallel

### Basic AC Reactive Components IMPEDANCE

Basic AC Reactive Components Whenever inductive and capacitive components are used in an AC circuit, the calculation of their effects on the flow of current is important. EO 1.9 EO 1.10 EO 1.11 EO 1.12

### Simple Harmonic Motion

Simple Harmonic Motion -Theory Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is eecuted by any quantity obeying the Differential

Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

### Fourier Analysis. A cosine wave is just a sine wave shifted in phase by 90 o (φ=90 o ). degrees

Fourier Analysis Fourier analysis follows from Fourier s theorem, which states that every function can be completely expressed as a sum of sines and cosines of various amplitudes and frequencies. This

### 12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*

+ v C C R L - v i L FIGURE 12.24 The parallel second-order RLC circuit shown in Figure 2.14a. 12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* We will now analyze the undriven parallel RLC circuit shown in Figure

### G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 31. Alternating Current Circuits Assignment is due at 2:00am on Wednesday, March 21, 2007 Credit for problems submitted late will decrease to 0% after the

### Lecture L5 - Other Coordinate Systems

S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates

### Lecture 3 The Laplace transform

S. Boyd EE12 Lecture 3 The Laplace transform definition & examples properties & formulas linearity the inverse Laplace transform time scaling exponential scaling time delay derivative integral multiplication

### 19.7. Applications of Differential Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Applications of Differential Equations 19.7 Introduction Blocks 19.2 to 19.6 have introduced several techniques for solving commonly-occurring firstorder and second-order ordinary differential equations.

### Physics 9 Fall 2009 Homework 8 - Solutions

1. Chapter 34 - Exercise 9. Physics 9 Fall 2009 Homework 8 - s The current in the solenoid in the figure is increasing. The solenoid is surrounded by a conducting loop. Is there a current in the loop?

### Trigonometry Lesson Objectives

Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### LABORATORY 10 TIME AVERAGES, RMS VALUES AND THE BRIDGE RECTIFIER. Bridge Rectifier

LABORATORY 10 TIME AVERAGES, RMS VALUES AND THE BRIDGE RECTIFIER Full-wave Rectification: Bridge Rectifier For many electronic circuits, DC supply voltages are required but only AC voltages are available.

### Trigonometric Functions and Equations

Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending

### Positive Feedback and Oscillators

Physics 3330 Experiment #6 Fall 1999 Positive Feedback and Oscillators Purpose In this experiment we will study how spontaneous oscillations may be caused by positive feedback. You will construct an active

### Transmission Lines. Smith Chart

Smith Chart The Smith chart is one of the most useful graphical tools for high frequency circuit applications. The chart provides a clever way to visualize complex functions and it continues to endure

### Fourier Transforms The Fourier Transform Properties of the Fourier Transform Some Special Fourier Transform Pairs 27

24 Contents Fourier Transforms 24.1 The Fourier Transform 2 24.2 Properties of the Fourier Transform 14 24.3 Some Special Fourier Transform Pairs 27 Learning outcomes In this Workbook you will learn about

### Reactance and Impedance

Reactance and Impedance Capacitance in AC Circuits Professor Andrew H. Andersen 1 Objectives Describe capacitive ac circuits Analyze inductive ac circuits Describe the relationship between current and

### Electrical Resonance

Electrical Resonance (R-L-C series circuit) APPARATUS 1. R-L-C Circuit board 2. Signal generator 3. Oscilloscope Tektronix TDS1002 with two sets of leads (see Introduction to the Oscilloscope ) INTRODUCTION

### Complex Algebra. What is the identity, the number such that it times any number leaves that number alone?

Complex Algebra When the idea of negative numbers was broached a couple of thousand years ago, they were considered suspect, in some sense not real. Later, when probably one of the students of Pythagoras

### L and C connected together. To be able: To analyse some basic circuits.

circuits: Sinusoidal Voltages and urrents Aims: To appreciate: Similarities between oscillation in circuit and mechanical pendulum. Role of energy loss mechanisms in damping. Why we study sinusoidal signals

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Chapter 1 Quadratic Equations in One Unknown (I)

Tin Ka Ping Secondary School 015-016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real

### Trigonometry for AC circuits

Trigonometry for AC 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/,

### Trigonometry for AC circuits

Trigonometry for AC 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/,

### Chapter 22: Alternating current. What will we learn in this chapter?

Chapter 22: Alternating current What will we learn in this chapter? Contents: Phasors and alternating currents Resistance and reactance Series R L C circuit Power in ac-circuits Series resonance Parallel

### Physics 53. Oscillations. You've got to be very careful if you don't know where you're going, because you might not get there.

Physics 53 Oscillations You've got to be very careful if you don't know where you're going, because you might not get there. Yogi Berra Overview Many natural phenomena exhibit motion in which particles

### Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

### Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

### Department of Electrical Engineering and Computer Sciences University of California, Berkeley. First-Order RC and RL Transient Circuits

Electrical Engineering 42/100 Summer 2012 Department of Electrical Engineering and Computer Sciences University of California, Berkeley First-Order RC and RL Transient Circuits When we studied resistive

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Experiment V: The AC Circuit, Impedance, and Applications to High and Low Pass Filters

Experiment : The AC Circuit, Impedance, and Applications to High and Low Pass Filters I. eferences Halliday, esnick and Krane, Physics, ol. 2, 4th Ed., Chapters 33 Purcell, Electricity and Magnetism, Chapter

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### 580.439 Course Notes: Linear circuit theory and differential equations

58.439 ourse Notes: Linear circuit theory and differential equations eading: Koch, h. ; any text on linear signal and system theory can be consulted for more details. These notes will review the basics

### ε: Voltage output of Signal Generator (also called the Source voltage or Applied

Experiment #10: LR & RC Circuits Frequency Response EQUIPMENT NEEDED Science Workshop Interface Power Amplifier (2) Voltage Sensor graph paper (optional) (3) Patch Cords Decade resistor, capacitor, and

### P517/617 Lec3, P1 R-L-C AC Circuits

P517/617 Lec3, P1 R-L-C AC Circuits What does AC stand for? AC stands for "Alternating Current". Most of the time however, we are interested in the voltage at a point in the circuit, so we'll concentrate

### Equivalent Circuits and Transfer Functions

R eq isc Equialent Circuits and Transfer Functions Samantha R Summerson 14 September, 009 1 Equialent Circuits eq ± Figure 1: Théenin equialent circuit. i sc R eq oc Figure : Mayer-Norton equialent circuit.

### SIGNAL PROCESSING & SIMULATION NEWSLETTER

1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty

### DRAFT. Further mathematics. GCE AS and A level subject content

Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed

### Vectors and Phasors. A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles. Owen Bishop

Vectors and phasors Vectors and Phasors A supplement for students taking BTEC National, Unit 5, Electrical and Electronic Principles Owen Bishop Copyrught 2007, Owen Bishop 1 page 1 Electronics Circuits

### Spring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations

Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring

### Circuits with inductors and alternating currents. Chapter 20 #45, 46, 47, 49

Circuits with inductors and alternating currents Chapter 20 #45, 46, 47, 49 RL circuits Ch. 20 (last section) Symbol for inductor looks like a spring. An inductor is a circuit element that has a large

### Figure 1.1 Vector A and Vector F

CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

### Module M6.3 Solving second order differential equations

F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module M6. Solving second order differential equations Opening items. Module introduction.2 Fast track questions. Rea to stu? 2 Methods

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Print ] Eðlisfræði 2, vor 2007 30. Inductance Assignment is due at 2:00am on Wednesday, March 14, 2007 Credit for problems submitted late will decrease to 0% after the deadline has

### 5 Indefinite integral

5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse

### C. Complex Numbers. 1. Complex arithmetic.

C. Complex Numbers. Complex arithmetic. Most people think that complex numbers arose from attempts to solve quadratic equations, but actually it was in connection with cubic equations they first appeared.

### Frequency response of a general purpose single-sided OpAmp amplifier

Frequency response of a general purpose single-sided OpAmp amplifier One configuration for a general purpose amplifier using an operational amplifier is the following. The circuit is characterized by:

### Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure 2. R. Figure 1.

Examples of Transient and RL Circuits. The Series RLC Circuit Impulse response of Circuit. Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure.

### Physics 231 Lecture 15

Physics 31 ecture 15 Main points of today s lecture: Simple harmonic motion Mass and Spring Pendulum Circular motion T 1/f; f 1/ T; ω πf for mass and spring ω x Acos( ωt) v ωasin( ωt) x ax ω Acos( ωt)

### Frequency response: Resonance, Bandwidth, Q factor

Frequency response: esonance, Bandwidth, Q factor esonance. Let s continue the exploration of the frequency response of circuits by investigating the series circuit shown on Figure. C + V - Figure The

### A Resonant Circuit. I. Equipment

Physics 14 ab Manual A Resonant Circuit Page 1 A Resonant Circuit I. Equipment Oscilloscope Capacitor substitution box Resistor substitution box Inductor Signal generator Wires and alligator clips II.

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through