EE40 Lec 10 Complex Numbers and Phasors Prof Nathan Cheung Prof 09 29 2009 Reading Hambley Chapter 5 and Appendix A EE40 Fall 2009 Slide 1 Prof Cheung OUTLINE Phasors as notation for Sinusoids Arithmetic with Complex Numbers Complex impedances Circuit analysis using complex impedances Dervative Integration as multiplication division Phasor Relationship for Circuit Elements EE40 Fall 2009 Slide 2 Prof Cheung Types of Circuit Excitation Linear TimeI Invariant i t Circuit Steady State Excitation DC Steady State Linear TimeInvariant Ci it Circuit Sinusoidal SingleFrequency Excitation AC Steady State EE40 Fall 2009 Linear TimeI Invariant i t Circuit Step Excitation OR Digital Pulse Source Linear Time TimeInvariant Circuit Transient Excitations Slide 3 Prof Cheung Sinusoids Amplitude VvM t VM cos t Angular frequency 2 f Radians sec Phase angle Frequency f 1 T Unit 1 sec or Hz Period T Time necessary to go through one cycle EE40 Fall 2009 Slide 4 Prof Cheung Sinusoids What is the amplitude period frequency and d radian di ffrequency off thi this sinusoid i id 8 6 4 2 0 2 20 4 6 8 EE40 Fall 2009 0 01 0 02 Slide 5 0 03 0 04 0 05 Prof Cheung Sinusoidal Sources Create Too Much Algebra xP t Guess a solution dxP t FA sin wt FB cos wt dt xP t A sin wt B cos wt A sin i wt B cos wt d A sin wt B cos wt FA sin i wt FB cos wt dt A B FA sin wt B A FB cos wt 0 A B FA 0 B A FB 0 A FA FB 2 1 B FA FB 2 1 Phasors so s vec vectors o s that rotate o e in thee co complex pe plane are a clever alternative EE40 Fall 2009 Slide 6 Prof Cheung Complex Numbers 1 y iimaginary i axis j 1 real axis x Rectangular Coordinates Z x jy Polar Coordinates Z z Exponential Form j Z Z e ze EE40 Fall 2009 j x is i th the reall partt y is the imaginary part z is i the h magnitude i d is the phase y z sin x z cos z x y 2 2 tan Z z cos j sin 1 y x 1 1e j 0 1 0 j 1e Slide 7 j 2 1 90 Prof Cheung Complex Numbers 2 Euler s Identities e j e j cos 2 e j e j sin 2j e j cos j sin e j cos 2 sin 2 1 Exponential Form of a complex number Z Z e j zee j z EE40 Fall 2009 Slide 8 Prof Cheung Arithmetic With Complex Numbers To compute phasor voltages and currents we need to be able to perform computation with complex l numbers b Addition Subtraction S bt ti Multiplication Division And later use multiplication by j to replace Differentiation Integration g EE40 Fall 2009 Slide 9 Prof Cheung Addition Addition is most easily performed in rectangular coordinates A x jy B z jw A B x z j y w Imaginary g y Axis A B B EE40 Fall 2009 A Slide 10 Real Axis Prof Cheung Subtraction Subtraction is most easily performed in rectangular coordinates A x jy B z jw A B x z j y w Imaginary A i Axis B A A B EE40 Fall 2009 Slide 11 Real Axis Prof Cheung Multiplication Multiplication is most easily performed in polar coordinates A AM B BM A B AM BM A B IImaginary i Axis B A EE40 Fall 2009 Slide 12 Real A i Axis Prof Cheung Division Division is most easily performed in polar coordinates A AM B BM A B AM BM Imaginary Axis B A A B EE40 Fall 2009 Slide 13 Real Axis Prof Cheung Arithmetic Operations of Complex Numbers Add and Subtract it is easiest to do this in rectangular format Add subtract the real and imaginary parts separately Multiply and Divide it is easiest to do this in exponential polar format Multiply divide the magnitudes Add subtract the phases Z1 z1e j 1 z1 1 z1 cos 1 jz1 sin 1 Z 2 z2 e j 2 z2 2 z2 cos 2 jz2 sin 2 Z1 Z 2 z1 cos 1 z2 cos 2 j z1 sin 1 z2 sin 2 Z1 Z 2 z1 cos 1 z2 cos 2 j z1 sin 1 z2 sin 2 Z1 Z 2 z1 z2 e j 1 2 z1 z2 1 2 Z1 Z 2 z1 z2 e j 1 2 z1 z2 1 2 EE40 Fall 2009 Slide 14 Prof Cheung Summary of Complex Numbers EE40 Fall 2009 Slide 15 Prof Cheung Phasor Rotating Complex Vector j v t V cos t Re Ve e Imaginary Axis jwt Re Ve j t Phasor Rotates at uniform angular velocity t V cos t Real Axis The head start angle is EE40 Fall 2009 Slide 16 Prof Cheung Phasors and Complex Exponentials A sinusoid can be described using a complex exponential ej t cos t j sin t So So v t VM cos t VM Re ej t The phasor of v t is given by V VM EE40 Fall 2009 Slide 17 Prof Cheung Sinusoids Complex Exponentials Phasors Sinusoid z cos t Complex exponential Aej t z ej t Phasor V z EE40 Fall 2009 Slide 18 Prof Cheung Sinusoids Phasors EE40 Fall 2009 Slide 19 Prof Cheung Complex Exponentials Complex Exponentials provide the link between time functions and phasors phasors Allow dervatives and integrals to be replaced by multiplying or dividing by j make solving for AC steady state simple algebra with complex numbers Phasors allow us to express current voltage g relationships for inductors and capacitors much like we express the current voltage relationship for a resistor resistor EE40 Fall 2009 Slide 20 Prof Cheung Complex Impedance AC steady state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm s law V IZ Z is called impedance impedance EE40 Fall 2009 Slide 21 Prof Cheung I V Relationship for a Capacitor i t v t C dv t i t C dt Suppose that v t is a sinusoid v t Re VM ej t Find i t EE40 Fall 2009 Slide 22 Prof Cheung Capacitor i t C v t t v t V cos t i t dv t i t C dt V j t j t e e 2 dv t CV d j t j t CV e e j e j t e j t dt 2 dt 2 CV j t j t e CV sin t CV cos t e 2j 2 V V V 1 1 1 Zc j CV I 2 2 C C j C I 2 i t C EE40 Fall 2009 Slide 23 Prof Cheung Capacitor Impedance i t C v t t dv t i t C dt Phasor definition v t V cos t Re Ve j t V V dv t de j t j t I I Re CV i t C Re j …
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