EECS 105 SPRING 2004 Lecture 1 Lecture 2 Frequency domain analysis Phasors Prof J Stephen Smith Department of EECS University of California Berkeley EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Announcements z z z z z z z The course web site is http inst eecs berkeley edu ee105 Today s discussion section will meet The Wednesday discussion section will move to Tuesday 5 00 6 00 293 Cory You can go to any or all discussions you like Labs will start Feb 3 Reading assignments from the text will start next week There is a homework set due Wed 1 28 Department of EECS University of California Berkeley 1 EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Context In the last lecture we covered z how circuits can be modeled as linear circuits by design or approximation z How to convert a linear circuit into a set of differential equations In this lecture we will cover z How to use complex analysis to solve circuits by converting the differential equations in the time domain into algebraic equations in the frequency domain Department of EECS University of California Berkeley Linear Circuit model set of linear differential equations EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith resistors vr t ir t r inductors vL t L capacitors ic t C diL t dt dvC t dt The wires convey the variables voltages and currents between the equations components by applying Kirchoff s laws For the low pass example vout t vc t vin vr vc iin ir ic iout Department of EECS University of California Berkeley 2 EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Differential equation for low pass We are going to take the output current equal to zero for simplicity so vr ir R ic R RC dvC t dt vout t vc t vin vr vout vout RC dvout t dt Department of EECS University of California Berkeley EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Equation For any linear circuit you will be able to write L1 vout t L 2 vin t d d2 avin t b1 vin t b2 2 vin t L dt dt c1 vin t c2 vin t c3 vin t L L L 1 2 Here represent Linear operators that is if you apply it to a function you get a new function it maps functions to functions and linear operators also have the property that L a f t b g t a L f t b L g t Department of EECS University of California Berkeley 3 It s now just mathematics and therefore easy EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Once we establish a linear model for a circuit by design or approximation z We can directly use the powerful methods of linear analysis from mathematics z We can develop our intuition as to what will happen allowing us to design Department of EECS University of California Berkeley EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Fourier Transform One important linear analysis technique we will use is the Fourier Transform The Fourier transform states F f t e j t dt f t 1 2 F e j t d Notice that what this says is that information that is expressed as a function of time voltage or current for example can be completely expressed as a function of frequency F Department of EECS University of California Berkeley 4 EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Linearity Let s use functions of frequency F voltages and currents rather than functions of time as our new variables The Fourier relationship shows us that if we can find these functions of frequency we can then convert them into the voltages and currents as a function of time that we want We can do this in every equation for our linear components 1 f t 2 F e j t d We just substitute this form into each of our equations Department of EECS University of California Berkeley EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Resistors vr t ir t r 1 2 Department of EECS 1 v e d r 2 j t i re r j t d University of California Berkeley 5 EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Capacitors ic t C 1 2 1 2 dvC t dt d 1 i e d C c dt 2 j t 1 ic e d 2 j t v e j t c d C j v e c j t d Department of EECS University of California Berkeley EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Inductors vL t L 1 2 1 2 Department of EECS diL t dt d 1 v e d L L dt 2 j t 1 v e d L 2 j t i L e j t d L j i L e j t d University of California Berkeley 6 EECS 105 Fall 2004 Lecture 2 z Prof J Stephen Smith In each of these we can eliminate the integration over frequency and the constant to get vr ir r ic vc C j vc 1 ic C j vL iL L j Department of EECS University of California Berkeley Conversion of linear circuits to algebraic equations EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith z Something wonderful just happened each of our simultaneous linear differential equations were just converted to algebraic equations just multiplication by a constant for these examples and the same thing happens to every linear circuit z Of course the same thing happens to the relationships derived from Kirchoff s laws they are linear too adding voltages for example Department of EECS University of California Berkeley 7 EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith z The advantage of changing differential equations into algebraic equations comes at a small price the constants that we are multiplying by and the functions of frequency for both voltage and current are now complex numbers z No matter how complicated the circuit if we drive the circuit with a real function vin t when we find the output by using the inverse Fourier transform it will be real as well as well as any voltage or current at any node at all times Department of EECS University of California Berkeley EECS 105 Fall 2004 Lecture 2 Prof J Stephen Smith Complex numbers z It is important to think of complex numbers as just an expansion over the definition of real numbers For example if A B and C are complex A B C AB AC AB BA A B B A This seems trivial but there is only one other definition for numbers which obeys these properties The other one is quaternions finite but not countable Department of EECS University of California Berkeley 8 EECS 105 Fall 2004 Lecture 2 z Prof J Stephen Smith If you have a calculator which can handle complex numbers you can just plug them in Otherwise you can use these …
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