1Department of EECS University of California, BerkeleyEECS 105 Fall 2004, Lecture 40Lecture 40: Review Phasor notation, Transfer FunctionsProf. J. S. SmithDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithContext–Converting a linear circuit into a set of differential equations, –How to convert the set of differential equations into the frequency domain, a set of algebraic equations.–Analyzing circuits with a sinusoidal input, (in the frequency domain, a single frequency at a time)–How to simplify our notation with Phasors–How to present information about the circuit directly in the frequency domain using diagrams of amplitude and phase at different frequencies (Bode plots)Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithLast Week of LecturezMonday:–Review of Frequency domain analysis of linear circuits, Bode plots.zWednesday:–Semiconductor materials, FET physics and modelszFriday:–Review of active linear circuits, amplifiers wrapupDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. 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)()(ωF2Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. 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 Spring 2004, Lecture 40 Prof. J. S. Smithrtitvrr⋅= )()(Resistors:∫∫+∞∞−+∞∞−⋅=ωωπωωπωωdreidevtjrtjr)(21)(21Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithdttdvCtiCc)()( =Capacitors:∫∫+∞∞−+∞∞−=ωωπωωπωωdevdtdCdeitjctjc)(21)(21∫∫+∞∞−+∞∞−=ωωωπωωπωωdevjCdeitjctjc)()(21)(21Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithdttdiLtvLL)()( =Inductors:∫∫+∞∞−+∞∞−=ωωπωωπωωdeidtdLdevtjLtjL)(21)(21∫∫+∞∞−+∞∞−=ωωωπωωπωωdeijLdevtjLtjL)()(21)(213Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithzIn 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 Spring 2004, Lecture 40 Prof. J. S. SmithConversion of linear circuits to algebraic equationszSomething 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.zOf 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, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithSingle frequency approachzAnother way to look at the situation is that since the circuit is linear, we can view any input as the sum of sin waves at various amplitudes, frequencies, and phases. (each piece of the Fourier transform)zIf we can under the circuit for an arbitrary sinusoidal input, we then can figure out what the circuit will do for an arbitrary input, or inputs. (Just break all the inputs up into sinusoids, put them through one at a time, and then add the results for each back up at the end, and that’s your answer!) Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithSinusoidal stimuluszWe are going to analyze circuits for a single sinusoid at a time which we are going to write:zBut we are going to use exponential notation)sin()(φω+=tVtviin..][21)(2/)()(2/)()sin()()()()()()()()()(CCeeVtveeVeeVtveeVtVtvtjjiintjjitjjiintjtjiiin+=−=−=+=−−+−+ωφωφωφφωφωφωComplex conjugate(same as first term, but with (j)→(-j) wherever it occurs)4Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithSin in→Sin at every node!It is especially interesting because any voltage or current in our circuit, if this is the only input, must also be sinusoidal with the same frequency, and so can also be written in this form.Because our equations will be linear, the same things will happen to the complex conjugate terms as happen to the first terms, so they will just tag alongCCeeItiCCeeVtvtjjanyanytjjanyany.][21)(.][21)()()()()(+=+=ωφωφDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithPhasorszEach of the voltages between nodes, and each of the currents, can then be represented by a single complex number (remember, this is for a single frequency input of a particular phase and amplitude)anytjjanyanyanytjjanyanyICCeeItiVCCeeVtvˆ.][21)(ˆ.][21)()()()()(⇒+=⇒+=ωφωφanyVˆanyIˆ{{Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithTricky Bits:zPhasor notation is very convenient, but there are some tricky parts to look out for:zYou can notuse phasor notation (without added precautions) if you need to multiply voltages and currents (such as in a power calculation), because that is not linear!zAnother way to look at phasor notation is that instead of adding the CC ( and dividing by 2), you take the real part, which gives the same result.zHowever, you must nottake the real part (or add the complex conjugate) before you put back in the time dependence e-jωtDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 40 Prof. J. S. SmithComplex Transfer
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