Supplementary Reader I EECS 40 Introduction to Microelectronic Circuits Prof. C. Chang-Hasnain Fall 2006EE 40, University of California Berkeley Professor Chang-Hasnain i Table of Contents Table of Contents ............................................................................................................................................ i Chapter 1. Bode Plots............................................................................................................................... 1 1.1 Introduction ................................................................................................................................. 1 1.2 First Order Circuits...................................................................................................................... 1 1.2.1 General Construction and Break Frequency ........................................................................... 1 1.2.2 Bode Magnitude Plot .............................................................................................................. 2 1.2.3 Bode Phase Plot...................................................................................................................... 5 1.3 Second Order Circuits ................................................................................................................. 7 1.3.1 General Construction and Resonant Frequency...................................................................... 7 1.3.2 Bode Magnitude Plot .............................................................................................................. 7 1.3.3 Bode Phase Plot.................................................................................................................... 10 1.3.4 Definitions ............................................................................................................................ 12 Acknowledgements We thank the writing assistance of Abhinav Gupta, Kevin Wang, Henry Wang, and Wendy Xiao-Xue Zhao, and detailed editing by Isaac Seetho, Yu Ben, Timothy C Loo, Jia Zou, and Michael Krishnan.EE 40, University of California Berkeley Prof. Chang-Hasnain 1 Chapter 1. Bode Plots 1.1 Introduction Bode plots are widely used in various fields of engineering because they characterize the magnitude and the phase response of a system. In this section, we will present step-by-step analysis to create the Bode plots of a given transfer function. We analyze the trend of the transfer function at different frequency regimes based on the value of the break or resonant frequency. This approach helps to understand the frequency behavior of the circuit, and also works for first order, second and higher order circuits. 1.2 First Order Circuits 1.2.1 General Construction and Break Frequency A Bode plot illustrates the behavior of a circuit by generalizing its response into trends and graphing it against a log scale of frequency. Given a transfer function, H(ω), we may produce a magnitude and phase Bode plot. In each plot, we break down the analysis of the transfer function into 3 regimes, depending on the frequency in question: 1. At Bωω=, 2. when Bωω<< , or when ω is much less than the break frequency 3. when Bωω>>, or when ω is much greater than the break frequency The break frequency,Bω, is a property of the filter that can be found by examining the transfer function. It describes the frequency where the trends on the Bode plot are broken, where one trend (whenBωω<< ) ends and the next (whenBωω>>) begins. For first-order circuits, the break frequency can be found by looking solving for the frequency at which the real and complex components are equal. Example 1 – Find the Break Frequency Given the transfer functionRCjHωω+=11)(. We find the break frequency by examining the denominator, because it consists of two terms, 1 and RCjω. The first term is real and has a 0ω dependence. The second termRCjω has a 1ωdependence. The break frequency,Bω, occurs when the MAGNITUDE two components are equal. Thus, the break frequency for a filter with this transfer function is at RCB1=ω. After finding the break frequency, we can examine the trends on either side of Bω. When Bωω<< , the real term (in the denominator of Exp. 1) dominates and we ignore the imaginary component. When Bωω>> , ω dominates real term, so we work with only the imaginary term, as will be shown next. Note: We can only disregard terms that are added and subtracted and not those that are multiplied or divided when making approximations.EE 40, University of California Berkeley Professor Chang-Hasnain 2 e.g For a<<b, a<c and c<<d, badbdca ≈+×)( Try to plug in any number you like and see if it is true. 1.2.2 Bode Magnitude Plot For the magnitude plot, we plot on the y-axis: the ‘square’ of the magnitude of the transfer function, )(ωH in decibels (unit dB). This is because the transfer function describes the output to input voltage ratio. Since power is proportional to V 2, we plot 2)(ωH, which in dB is: (1) Both of these expressions can be useful, depending on the form of the transfer function. We plot dBH )(ωon the y-axis, against a logarithmic scale of ω on the x-axis. One thing to keep in mind about logarithmic functions is the ability to pull multiplicative factors out, for instance: The key to our analysis is that we identify and examine the dominant terms of the transfer function in each of the 3 regimes in the following. (1) At Bωω=, we simply plugBω in dBH )(ω and get the actual value. (2) When Bωω<<. Examine SEPERATELY in the numerator and denominator all terms containing ω (e.g. ω0 , ω1). The one with the lowest power (e.g. ω0) dominates in this frequency range. Hence we keep only the dominating term, one each for the numerator and denominator, respectively. The resulting formula will be used to determine the asymptotic behavior , or “trend”. (3) When Bωω>>. Again, we examine the numerator and denominator separately. Leave only the term with the highest power (ω1 in the previous example), one each for numerator and dominator. The resulting formula will be used to determine the asymptotic behavior , or “trend”. By analyzing these three regimes, we can
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