Supplementary Reader I EECS 40 Introduction to Microelectronic Circuits Prof C Chang Hasnain Fall 2006 EE 40 University of California Berkeley Chang Hasnain Professor 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 i EE 40 University of California Berkeley Prof Chang Hasnain 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 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 B 2 when B or when is much less than the break frequency 3 when B or when is much greater than the break frequency The break frequency B 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 when B ends and the next when B 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 function H 1 1 j RC We find the break frequency by examining the denominator because it consists of two terms 1 and j RC The first term is real and has a 0 dependence The second term j RC has a dependence The break frequency B occurs when the MAGNITUDE two components are equal Thus the break frequency for a filter with this transfer function 1 is at B 1 RC After finding the break frequency we can examine the trends on either side of B When B the real term in the denominator of Exp 1 dominates and we ignore the imaginary component When B 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 1 EE 40 University of California Berkeley Professor Chang Hasnain e g For a b a c and c d a c d ad b b 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 H in decibels unit dB This is because the transfer function describes the output to input voltage ratio Since 2 power is proportional to V 2 we plot H which in dB is H dB 10 log H 2 1 20 log H Both of these expressions can be useful depending on the form of the transfer function We plot H dB on the y axis against a logarithmic scale of on the x axis One thing to keep in mind about logarithmic functions is the ability to pull multiplicative factors out for instance 20 log A B 20 log A 20 log B C 10 log 10 log C 10 log D D 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 B we simply plug B 2 When 0 B in H dB and get the actual value Examine SEPERATELY in the numerator and denominator all terms containing 1 0 e g The one with the lowest power e g 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 B Again we examine the numerator and denominator separately Leave only the term with the highest power 1 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 construct the magnitude Bode plot We start at the break frequency B and then plot the trends for B and B Example 2 Find the Transfer Function H and plot the magnitude Bode Plot Given the following circuit construct the magnitude Bode plot R2 A 100 R1 200 VT Vin R1 AVT C 2 Vout R2 1k C 10 F EE 40 University of California Berkeley We can see VT Vin where Z c Professor Chang Hasnain By voltage divider we have Vout 1 j C Zc AVin Z c R2 Vout Zc A Vin Z c R2 1 j C A A 1 j C R 2 1 j R 2 C Thus the transfer function is given by H The magnitude plot will be the transfer function in dB or H dB 20 log H 20 log A 1 j R2C Plotting this function would yield the exact behavior of our filter but we only need the asymptotic behavior for the Bode plot Now we begin a 3 part analysis Step 1 Break Frequency Setting equal the real and imaginary components in the denominator we find the break frequency 1 1 R2C 1 k 10 F 1 rad 2 100 rad s 10 s B Step 2 Asymptotes 1 At B 2 H B dB 2 A2 A A 10 log 10 log 10 log 1 j 2 2 20 log A 10 log 2 20 log A 3dB With A 100 H B dB 37 dB The first line of the above steps shows that at B the output is at half the maximum power 3dB is half in linear scale the transfer function voltage has a value of A 2 and power is proportional to the square of voltage The break frequency for a first order circuit is also referred to as the half …
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