DOC PREVIEW
BU EECE 301 - Discussion 01 - Complex Numbers and Sinusoids

This preview shows page 1-2-3-4-5 out of 16 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

EECE 301 Signals & Systems Prof. Mark FowlerEECE 301 Signals & SystemsProf. Mark FowlerDiscussion #1• Complex Numbers and Complex-Valued Functions • Reading Assignment: Appendix A of Kamen and HeckComplex NumbersComplex numbers arise as roots of polynomials.1))((1))((112−=−−⇒=−⇒−=⇒−=jjjjjjRectangular form of a complex number:}Im{}Re{zbzajbaz==+=real numbersRecall that the solution of differential equations involves finding roots of the “characteristic polynomial”So…Definition of imaginary # j and some resulting properties:)()())((:)()()()(:bcadjbdacjdcjbaMultiplydbjcajdcjbaAdd++−=+++++=+++The rules of addition and multiplication are straight-forward:Polar Formθjrez =Polar form… an alternate way to express a complex number…Polar Form…good for multiplication and divisionr > 0Note: you may have learned polar form as r∠θ… we will NOT use that here!!The advantage of the rejθis that when it is manipulated using rules of exponentials and it behaves properly according to the rules of complex #s:yxyxyxyxaaaaaa−+== /))((Dividing Using Polar Form()())(21212121θθθθ−=jjjerrerer22222111θθjjererz−==Multiplying Using Polar Form()( ))(21212121θθθθ+=jjjerrerer()njnnjnnnjnerzerrez//1/1θθθ===baz = a + jbImRerθGeometry of Complex NumbersWe need to be able convert between Rectangular and Polar Forms… this is made easy and obvious by looking at the geometry (and trigonometry) of complex #s:θrabθθcossinrarb==⎟⎠⎞⎜⎝⎛=+=−abbar122tanθConversion Formulas⇒ Since cosθ + jsinθ has the same expansion as e jθwe can say that:θθθjej =+ sincosFrom Calc II:...!6!4!21cos642+−+−=θθθθ...!7!5!3sin753+−+−=θθθθθjjjjj...!4!3!21sincos432++−−+=+θθθθθθjjj...!4!3!21432+++++=xxxxexAlso From Calc II:...!4!3!21432++−−+=θθθθθjjejQ: Why the form rejθfor polar form?? Start with trigonometry:[]θθθθsincos)sin(cos jrrjrjbaz+=+=+=Complex Exponentials vs. Sines and CosinesEuler’s Equations:(A)(B)(C)(D)Summary of Rectangular & Polar FormsWarning: Your calculator will give you the wrong answerwhenever you have a < 0. In other words, for z values that lie in the II and III quadrants. You can always fix this by either adding or subtracting π. Use common sense… looking at the signs of a and b will tell you what quadrant z is in… make sure your angle agrees with that!!!Summary of Rectangular & Polar Forms⎟⎠⎞⎜⎝⎛==∠+==−∈≥=−abzbarzrrezj122tan],(0θππθθPolar Form:θθsin}Im{cos}Re{rbzrazjbaz====+=Rect Form:Conjugate of Zθθjjrezrezjbazjbaz−=⇒=−=⇒+=**Denoted aszz or*Properties of z*222**))((.2}Re{2.1zbajbajbazzzzz=+=−+=×=+Unit CircleImRe1θUnit Circle: A set of complex numbers with magnitude of 1 (|z| =1)All z on the unit circle look like: θje1ReImFour special points on the unit circle:2πjej =01je=2πjej−=−πje=−1⎪⎪⎩⎪⎪⎨⎧=−==−===⎩⎨⎧−=±±,...10,6,2,1,...8,4,0,1...,11,7,3,...,9,5,1,integers allfor 1integereven 1,integer odd ,122nnnjnjeennejnjnjnπππKnow these!!!A sinusoid is completely defined by its three parameters:-Amplitude A (for EE’s typically in volts or amps or other physical unit)-Frequency ωin radians per second-Phase shift φin radiansT is the period of the sinusoid and is related to the frequencyPhase shift (often just shortened to phase) shows up explicitly in the equation but shows up in the plot as a time shift (because the plot is a function of time). Q: What is the relationship between the plot-observed time shift and the equation-specified phase shift?A: We can write the time shift of a function by replacing t by t + to(more on this later, but you should be able to verify that this is true!) Then we get:)sin())(sin()(ooottAttAttfωωω+=+=+= φSo we get that: φ= ωto (unit-wise this makes sense!!!)Frequency can be expressed in two common units:-Cyclic frequency: f = 1/T in Hz (1 Hz = 1 cycle/second)-Radian Frequency: ω= 2π/T (in radians/second) From this we can see that these two frequency units have a simple conversion factor relationship (like all other unit conversions – e.g. feet and meters): fπω2=In circuits you used “phasors” (we’ll call them “static” phasors here)… the point ofusing them is to make it EASY to analyze circuits that are driven by a singlesinusoid.Here is an example to refresh your memory!!Find output voltage of the following circuit:R = 1ΩL = 2mH()41000cos5)(π+= ttx?)(=tyUse phasor and impedance ideas:45ˆ :Input ofPhasor 2 :Inductor of ImpedanceπωjLexjLjZ===Use voltage divider to find output:[]46.0489.05212ˆˆjjeejjxyπ=⎥⎦⎤⎢⎣⎡+=Output phasor:25.145.4ˆjey =Output signal:)25.11000cos(45.4)(+=ttyNote that in using “static” phasors there was no need to “carry around the frequency” … it gets suppressed in the static phasorBUT… if you have multiple driving sinusoids (each at its own unique frequency) then you’ll need to keep that frequency in the phasor representation… that leads to:{})cos(Re)sin()cos(oootjjoootjtAeeAtjteooφωωωωφω+=+=system)cos(oootAφω+input rotatingphasorModifiedsystemtjjoooeeAωφRotating phasoroutput rotatingphasorRotating PhasorsKeeping the frequency “part”“static” phasor parttjjotjooooooooeeAeAtAωφφωφω=→++ )()cos([]()[]{})cos(2)(~Re22)(*~)(~)(*~ :isWhat )cos()()(~ IfootjjtjootjtAtxtxtxeAeAetxtAtxAetxooooooφωφωωφφωφω+==+==+==−−+−++Because rotating phasors take the value of a complex number at each Instant of time they must follow all the rules of complex numbers…Especially: EULER’S EQUATIONS!!Rotating PhasorsEuler’s EquationsViewing rotating phasor on the complex


View Full Document

BU EECE 301 - Discussion 01 - Complex Numbers and Sinusoids

Download Discussion 01 - Complex Numbers and Sinusoids
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Discussion 01 - Complex Numbers and Sinusoids and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Discussion 01 - Complex Numbers and Sinusoids 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?