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Berkeley PHYSICS 111 - Lab 2 Linear Circuits II

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Physics 111 ~ BSC Student Evaluation of Lab Write-UpUniversity of California at Berkeley Physics 111 Laboratory Basic Semiconductor Circuits (BSC) Lab 2 Linear Circuits II ©2009 by the Regents of the University of California at Berkeley. All rights reserved. References: Hayes & Horowitz Chapter 1, p. 32–60 Horowitz & Hill Chapters 1.06, 1.12-1.24, 5.01-5.02, 5.04-5.05, Appendix H Appendix B (skim) Web resources Wikipedia.org Legends: See the Main BSC Web pages for details on introductory Advice next to Glossary. In this week’s lab you will continue investigating linear circuits. Fourier analysis, scope probe fre-quency properties, inductors, resonant circuits, and computer simulations will be studied. Before coming to class complete this list of tasks: • Completely read the Lab Write-up • Answer the pre-lab questions utilizing the references and the write-up • Perform any circuit calculations or anything that can be done outside of lab. • Plan out how to perform Lab tasks. Pre-lab questions: 1. Calculate the Fourier coefficients for the ramp waveform shown below: tV(t) 1V T 2. How does the scope probe work? 3. Approximately what is the Q of the KFOG-FM transmitter? 4. The Q in the RLC tank circuit drawn in Sec 2.6 is proportional to R. By shifting the location of the resistor, draw a circuit in which the Q is proportional to 1/R. Last Revision: 2009 Page 1 of 12 ©2009 by the Regents of the University of California at Berkeley. All rights reserved.Physics 111 BSC Laboratory Lab 2 Linear Circuits II Time Dependent Circuits Circuit analysis is straightforward if all the signals are time independent, i.e. DC. The response of circuit to time dependent (AC) signals like sine waves is more complicated because the response to the signal may not be in phase with the signal, and may depend on frequency. For example, a circuit driven by a voltage source VV t=1cos( )ω might produce an output current phase-shifted by φ, namely II t=1cos( )+ωφ. We can incorporate such phase shifts into Ohm’s law by allowing the volt-ages, currents, and resistances to be complex. Thus, It1cos( )ωφ+ becomes1 Ijexp( )ωt, where II jexp( )φ=1. While we do all our algebra with complex quantities, we have to take the real part in the end (e.g. IIjt)]= Re[ exp(ω) because we can measure only real quantities in the lab. Since and IV are not necessarily in phase, the resistance can no longer be a pure real quantity. We use a new term for complex resistances: the impedance Z. The magnitude of the impedance has much the same function in Ohm’s law (now VZI=), as did the resistance R; it determines the rela-tion between the magnitudes of I and V. The phase angle of Z determines the phase shift between I and V. Note that resistance is redefined to be the real part of the impedance, and the reactance is defined to be the imaginary part of the impedance. Clearly, a resistor has pure real impedance ZRR= and induces no phase shifts. Capacitors have impedance ZjCC=1ω and inductors have impedance ZjLL=ω. Capacitor impedance decreases with frequency; inductor impedance increases with frequency. Both capacitors and inductors induce 90° phase shifts, but the phase shifts are in opposite directions. Any linear circuit2 can be analyzed using the impedance formulas. The familiar parallel and series resistor addition formulas carry over directly; just substitute the capacitative and inductive imped-ances for R. For example, the impedance of two capacitors in parallel is ()ZZZZZ jCCjC jCjC jC=+==++C1 C2C1 C2 1 2111121121ωωωωω. Analyze any circuit just as you would if all the components were resistors, but keep track of the complex parts, and you will get the right answer. Thévenin circuit reduction works as well, though the Thévenin resistance becomes a complex, frequency dependent impedance. And that’s all we need to know about complex impedances for this class. But as physicists we should understand the formal differential equations methods that underlie these simplifications, which can be found in most E&M texts. Background Fourier Analysis and Repetitive Waveforms In last week’s lab, we showed how the concept of complex impedances extends our ability to analyze circuits to those driven by pure sinusoidal waveforms. However, our circuits will be frequently driven by waveforms more complicated than pure sinusoids. Fortunately complex impedance analysis is easily extended to include circuits driven by any repetitive waveform by using the fact that any repetitive waveform can be decomposed into a (possibly) infinite series of harmonic sinusoids. The 1 To avoid confusion with the symbol for current, we use j instead of i for the −1 . Last Revision: 2009 Page 2 of 12 2 More specifically, any circuit that consists only of resistors, capacitors, inductors, voltage sources, and current sources. ©2009 by the Regents of the University of California. All rights reserved.Physics 111 BSC Laboratory Lab 2 Linear Circuits II mathematical technique, which performs the decomposition, is called Fourier Analysis – you should have already studied this technique in your mathematics classes. In summary, any waveform F(t) which is repetitive [has a period T defined such that F(t+T) = F(t) for all t] can be synthesized from the infinite sum () ()Ft A n t B n tnnn() sin cos=+=∞∑ωω0 where the Fourier coefficients are defined as ()()ATFt n tdtBTFt n tdtBTFtdtnTnTT===∫∫∫2210000()sin()cos() ,ωω Last Revision: 2009 Page 3 of 12 ©2009 by the Regents of the University of California. All rights reserved. where ωπ= 2 T . The lowest order sinusoid (n=1) in the infi-nite sum is called the fundamental; all the other waveforms are referred to as the nth harmonic. For example, the square wave shown to the right has Fourier coefficients Ann= 4π for n odd, with all other coefficients equal to zero. The beauty of using Fourier Analysis in circuit analysis is that the response of any linear circuit to a repetitive waveform is the sum of the responses of the circuit to each individual harmonic. Thus a circuit with a transfer function H(ω), driven by the repetitive waveform F(t), has an output 1 1 1ms () ()[]∑∞=+0cossinnononHtnBtnAωωτ=×−4105)(onω. For example, when a 1kHz square wave is passed through a s low pass filter [transfer function Hj111ms() ( )ωωτ=+11 ], the calculation is summarized in the table below and the output


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