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Berkeley ELENG 105 - Lecture 2: Frequency domain analysis, Phasors

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1Department of EECS University of California, BerkeleyEECS 105 SPRING 2004, Lecture 1Lecture 2: Frequency domain analysis, PhasorsProf. J. Stephen SmithDepartment of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithAnnouncementsz The course web site is http://inst.eecs.berkeley.edu/~ee105z Today’s discussion section will meetz The Wednesday discussion section will move to Tuesday, 5:00-6:00, 293 Coryz You can go to any or all discussions you like.z Labs will start Feb 3.z Reading assignments from the text will start next weekz There is a homework set due Wed. 1/282Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithContextIn 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, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithLinear Circuit model↔ set of linear differential equationsrtitvrr⋅= )()(dttdvCtiCc)()( =dttdiLtvLL)()( =The wires convey the variables (voltages and currents) between the equations (components) , by applying Kirchoff’s laws.For the low pass example:outcrincrincoutiiiivvvtvtv+==+== )()(resistors capacitorsinductors3Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithDifferential equation for low pass:dttdvRCvvvvtvtvdttdvRCRiRivoutoutoutrincoutCcrr)()()()(+=+=====We are going to take the output current equal to zero, for simplicity, so:Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen Smith→Equation∫∫∫∫∫∫+++++++==LL)()()()()()()}({)}({3212221tvctvctvctvdtdbtvdtdbtavtvtvinininininininout 21LL{}{},21LLHere 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:)}({)}({)}()({ tgbtfatgbtfa LLL⋅+⋅=⋅+⋅For any linear circuit, you will be able to write:4Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithIt’s now just mathematics, and therefore easy!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, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithFourier TransformOne important linear analysis technique we will use is the Fourier Transform:The Fourier transform states: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: ∫+∞∞−−= dtetfFtjωω)()(∫+∞∞−=ωωπωdeFtftj)(21)()(ωF5Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithLinearity Let’s use functions of frequency (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.We just substitute this form into each of our equations)(ωF∫+∞∞−=ωωπωdeFtftj)(21)(Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen Smithrtitvrr⋅= )()(Resistors:∫∫+∞∞−+∞∞−⋅=ωωπωωπωωdreidevtjrtjr)(21)(216Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithdttdvCtiCc)()( =Capacitors:∫∫+∞∞−+∞∞−=ωωπωωπωωdevdtdCdeitjctjc)(21)(21∫∫+∞∞−+∞∞−=ωωωπωωπωωdevjCdeitjctjc)()(21)(21Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithdttdiLtvLL)()( =Inductors:∫∫+∞∞−+∞∞−=ωωπωωπωωdeidtdLdevtjLtjL)(21)(21∫∫+∞∞−+∞∞−=ωωωπωωπωωdeijLdevtjLtjL)()(21)(217Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen Smithz In each of these, we can eliminate the integration over frequency, and the constant, to get:rivrr⋅= )()(ωω)()()(ωωωjCvicc⋅=)()()(ωωωjLivLL⋅=)()(1)(ωωωccijCv =Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithConversion of linear circuits to algebraic equationsz 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)8Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen Smithz 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 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.)(tvinDepartment of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen SmithComplex numbersz 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: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. ABBABAABACABCBA+=+=+=+ )(9Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 2 Prof. J. Stephen Smithz If you have a calculator


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Berkeley ELENG 105 - Lecture 2: Frequency domain analysis, Phasors

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