Chapter 9 and Chapter 1 from reader OUTLINE Phasors as notation for Sinusoids Arithmetic with Complex Numbers Complex impedances Circuit analysis using complex impdenaces Dervative Integration as multiplication division Phasor Relationship for Circuit Elements Frequency Response and Bode plots Reading Chapter 9 from your book Chapter 1 from your reader EE100 Summer 2008 Slide 1 Bharathwaj Muthuswamy Types of Circuit Excitation Linear TimeInvariant Circuit Linear TimeInvariant Circuit Steady State Excitation DC Steady State Linear TimeInvariant Circuit OR Digital Pulse Source Sinusoidal SingleFrequency Excitation AC Steady State EE100 Summer 2008 Linear TimeInvariant Circuit Transient Excitation Slide 2 Bharathwaj Muthuswamy Why is Single Frequency Excitation Important Some circuits are driven by a single frequency sinusoidal source Some circuits are driven by sinusoidal sources whose frequency changes slowly over time You can express any periodic electrical signal as a sum of single frequency sinusoids so you can analyze the response of the linear timeinvariant circuit to each individual frequency component and then sum the responses to get the total response This is known as Fourier Transform and is tremendously important to all kinds of engineering disciplines EE100 Summer 2008 Slide 3 Bharathwaj Muthuswamy Representing a Square Wave as a Sum of Sinusoids b signal V Signal signal V Signal a T i me ms d Signal V Relative Amplitude c Frequency Hz a Square wave with 1 second period b Fundamental component dotted with 1 second period third harmonic solid black with1 3 second period and their sum blue c Sum of first ten components d Spectrum with 20 terms EE100 Summer 2008 Slide 4 Bharathwaj Muthuswamy Steady State Sinusoidal Analysis Also known as AC steady state Any steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoid This is a consequence of the nature of particular solutions for sinusoidal forcing functions All AC steady state voltages and currents have the same frequency as the source In order to find a steady state voltage or current all we need to know is its magnitude and its phase relative to the source We already know its frequency Usually an AC steady state voltage or current is given by the particular solution to a differential equation EE100 Summer 2008 Slide 5 Bharathwaj Muthuswamy Example 1 2nd Order RLC Circuit t 0 Vs EE100 Summer 2008 R C L Slide 6 Bharathwaj Muthuswamy Example 2 2nd Order RLC Circuit t 0 Vs EE100 Summer 2008 R C L Slide 7 Bharathwaj Muthuswamy Sinusoidal Sources Create Too Much Algebra xP t dxP t FA sin wt FB cos wt dt Two terms to be general Dervatives t A sin wt B cos wt Addition d A sin wt B cos wt Guess a solution xP A sin wt B cos wt dt FA sin wt FB cos wt A B FA sin wt B A FB cos wt 0 Equation holds for all time A B FA 0 and time variations are B A FB 0 independent and thus each time variation coefficient is individually zero A FA FB 2 1 B FA FB 2 1 Phasors vectors that rotate in the complex plane are a clever alternative EE100 Summer 2008 Slide 8 Bharathwaj Muthuswamy Complex Numbers 1 imaginary axis y z j 1 real axis x Rectangular Coordinates Z x jy Polar Coordinates Z z Exponential Form j Z Z e ze EE100 Summer 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 9 j 2 1 90 Bharathwaj Muthuswamy 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 EE100 Summer 2008 Slide 10 Bharathwaj Muthuswamy Arithmetic With Complex Numbers To compute phasor voltages and currents we need to be able to perform computation with complex numbers Addition Subtraction Multiplication Division And later use multiplication by j to replace Diffrentiation Integration EE100 Summer 2008 Slide 11 Bharathwaj Muthuswamy Addition Addition is most easily performed in rectangular coordinates A x jy B z jw A B x z j y w EE100 Summer 2008 Slide 12 Bharathwaj Muthuswamy Addition Imaginary Axis A B B EE100 Summer 2008 A Slide 13 Real Axis Bharathwaj Muthuswamy Subtraction Subtraction is most easily performed in rectangular coordinates A x jy B z jw A B x z j y w EE100 Summer 2008 Slide 14 Bharathwaj Muthuswamy Subtraction Imaginary Axis B A A B EE100 Summer 2008 Slide 15 Real Axis Bharathwaj Muthuswamy Multiplication Multiplication is most easily performed in polar coordinates A AM B BM A B AM BM EE100 Summer 2008 Slide 16 Bharathwaj Muthuswamy Multiplication A B Imaginary Axis B A EE100 Summer 2008 Slide 17 Real Axis Bharathwaj Muthuswamy Division Division is most easily performed in polar coordinates A AM B BM A B AM BM EE100 Summer 2008 Slide 18 Bharathwaj Muthuswamy Division Imaginary Axis B A Real Axis A B EE100 Summer 2008 Slide 19 Bharathwaj Muthuswamy 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 EE100 Summer 2008 Slide 20 Bharathwaj Muthuswamy Phasors Assuming a source voltage is a sinusoid 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 2 EE100 Summer 2008 Slide 21 Bharathwaj Muthuswamy Phasor 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 EE100 Summer 2008 Slide 22 Bharathwaj Muthuswamy 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 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 relationships for inductors …
View Full Document
Unlocking...