EECE 301 Signals & Systems Prof. Mark Fowler1/16EECE 301 Signals & SystemsProf. Mark FowlerComplex Sinusoids•Complex-Valued Sinusoidal Functions2/16A 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 frequency3/16Phase 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=4/16In 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)(+=tty5/16Note 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(rotating phasor6/16[]()[]{})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!!7/16Rotating Phasors… Complex SinusoidsEuler’s Equations8/16Viewing rotating phasor on the complex
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