EE40 Lecture 14 Venkat Anantharam 2 27 08 Reading Chap 5 phasors EE40 Spring 2008 Slide 1 Venkat Anantharam Complex Numbers 1 j 1 z y imaginary axis real axis x Rectangular Coordinates Z x jy Polar Coordinates Z z Exponential Form j Z Z e ze EE40 Spring 2008 j x is the real part y is the imaginary part z is the magnitude 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 2 j 2 1 90 Venkat Anantharam 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 ze j z EE40 Spring 2008 Slide 3 Venkat Anantharam Arithmetic With Complex Numbers To compute phasor voltages and currents we need to be able to perform computations with complex numbers Addition Subtraction Multiplication Division EE40 Spring 2008 Slide 4 Venkat Anantharam Addition Addition is most easily performed in rectangular coordinates A x jy B z jw A B x z j y w EE40 Spring 2008 Slide 5 Venkat Anantharam Addition Imaginary Axis A B B EE40 Spring 2008 A Slide 6 Real Axis Venkat Anantharam Subtraction Subtraction is most easily performed in rectangular coordinates A x jy B z jw A B x z j y w EE40 Spring 2008 Slide 7 Venkat Anantharam Subtraction Imaginary Axis B A A B EE40 Spring 2008 Slide 8 Real Axis Venkat Anantharam Multiplication Multiplication is most easily performed in polar coordinates A AM B BM A B AM BM EE40 Spring 2008 Slide 9 Venkat Anantharam Multiplication A B Imaginary Axis B A EE40 Spring 2008 Slide 10 Real Axis Venkat Anantharam Division Division is most easily performed in polar coordinates A AM B BM A B AM BM EE40 Spring 2008 Slide 11 Venkat Anantharam Division Imaginary Axis B A A B EE40 Spring 2008 Slide 12 Real Axis Venkat Anantharam Phasors Assuming a source voltage is a sinusoidal timevarying function v t V cos t We can write v t V cos t V Re e j t Re Ve j t Define Phasor as Ve j V Similarly if the function is v t V sin t j t 2 v t V sin t V cos t Re Ve 2 Phasor V EE40 Spring 2008 2 Slide 13 Venkat Anantharam Phasor from rotating Complex Vector v t V cos t Re Ve j e jwt Re V e j t Imaginary Axis Rotates at uniform angular velocity t V cos t Real Axis The head start angle is EE40 Spring 2008 Slide 14 Venkat Anantharam Complex Exponentials We represent a real valued sinusoid as the real part of a complex exponential after multiplying j t by e Complex exponentials provide the link between time functions and phasors Allow derivatives 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 relationships for inductors and capacitors much like we express the current voltage relationship for a resistor EE40 Spring 2008 Slide 15 Venkat Anantharam 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 Spring 2008 Slide 16 Venkat Anantharam Capacitor Impedance 1 i t C v t v t V cos 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 2 dt 2 dt CV j t j t e CV sin t CV cos t e 2j 2 V 1 1 1 V V j Zc CV I 2 2 C C j C I 2 i t C EE40 Spring 2008 Slide 17 Venkat Anantharam Capacitor Impedance 2 i t C v 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 CVe dt dt V 1 V V Zc j CV j C I I EE40 Spring 2008 Slide 18 Venkat Anantharam Example v t 120V cos 377t 30 C 2 F What is V What is I What is i t EE40 Spring 2008 Slide 19 Venkat Anantharam Computing the Current Note The differentiation and integration operations become algebraic operations d j dt EE40 Spring 2008 Slide 20 1 dt j Venkat Anantharam Inductor Impedance i t v t L di t v t L dt V j L I EE40 Spring 2008 Slide 21 Venkat Anantharam Example i t 1 A cos 2 9 15 107t 30 L 1 H What is I What is V What is v t EE40 Spring 2008 Slide 22 Venkat Anantharam
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