EE40 Lecture 12 Josh Hug 7 21 2010 EE40 Summer 2010 Hug 1 Logistical Things HW6 due Friday at 5PM also short Midterm next Wednesday 7 28 Focus is heavily on HW4 5 6 and Labs P1 4 5 Will reuse concepts from HW 1 2 3 EE40 Summer 2010 Hug 2 Filtering For the past couple of lectures we ve discussed using phasors and impedances to solve circuits Usually we ve assumed we have some single frequency source and found the resulting output Last time in lecture we showed that we could apply two different frequencies at one time using superposition Each was scaled and shifted by different amounts EE40 Summer 2010 Hug 3 Transfer Functions e g Maps system input signal to system output signal Plug an input voltage into Get an output voltage Output is scaled and shifted in time Scaling and shifting depend on frequency Frequency is unchanged linear system Tells you how system will respond to any frequency a k a frequency response EE40 Summer 2010 Hug 4 Using a Transfer Function Suppose is 10 1 2 10 1 1 Output phasor is just 4 EE40 Summer 2010 Hug 5 Using a Transfer Function Suppose is 50 1 26 5 0 5 1 Output phasor is just 1 37 EE40 Summer 2010 Hug 6 Transfer Function For each frequency different Scaling magnitude Delay phase shift It is useful to graphically depict the magnitude and phase of the transfer function EE40 Summer 2010 Hug 7 Bode Magnitude Plot 1 1 0 01 2 Magnitude plot is just a plot of as a function of EE40 Summer 2010 Linear Scale Log Scale Hug 8 Bode Magnitude Plot in Context of Circuit 0 1 1 0 1 1 1 0 01 2 All frequencies below get through pretty well Above that increasingly attenuated EE40 Summer 2010 Hug 9 Bode Phase Plot 0 1 1 Phase plot is just a plot of as a function of EE40 Summer 2010 Linear Scale Semilog Scale Hug 10 Bode Phase Plot in Context of Circuit 0 1 1 0 1 0 1 1 All frequencies below move in time with the source above that gets out of phase EE40 Summer 2010 Hug 11 Multiple Frequencies Real signals are often a combination of a continuum of many frequencies Radio antenna input Microphone input Intuitively Thunder contains a bunch of low frequency sounds Boiling kettles contains a bunch of high frequency sounds There is a mathematically well defined idea of what it means for a signal to contain many frequencies EE40 Summer 2010 Hug 12 Time vs Frequency Domain The Fourier Transform takes a function and converts it into a function Every signal can be made out of a sum of an infinite number of sinusoids Fourier transform tells you how much of each one to include sin 3000 EE40 Summer 2010 Hug 13 Multiple Frequencies The 1 button on a phone is a combination of a 697 Hz tone and a 1209 Hz tone sin 2 697 sin 2 1209 EE40 Summer 2010 Hug 14 Multiple Frequencies Bill and Ted saying the word bogus is a more complex set of frequencies EE40 Summer 2010 Hug 15 Filtering Example If we apply a filter with the frequency response on the right to the signal on the left Then we ll get EE40 Summer 2010 Hug 16 More complex filtering EE40 Summer 2010 Each frequency individually scaled Hug 17 Phase Effects If we shift the phase of the larger sine we get Original 1 button tone EE40 Summer 2010 Hug 18 Magnitude and Phase Demo Let s try the ever risky live demo EE40 Summer 2010 Hug 19 Bode Plots Hopefully I ve convinced you that magnitude and phase plots are useful Now the goal will be to draw them straight from the transfer function First some reminders on loglog plots EE40 Summer 2010 Hug 20 Loglog Plots On a loglog plot looks like a straight line w slope and y offset because Y offset intercept at 0 1 2 EE40 Summer 2010 10 1 Hug 21 Loglog Plots On a loglog plot will Be flat for values of Be a straight line of slope n for values of Have a y offset of 2 200 EE40 Summer 2010 2 2 20 2 Hug 22 Loglog Plots On a loglog plot will Be flat for values of Be a straight line of slope n for values of Have an y offset of 2 20 2 EE40 Summer 2010 10 2 20 2 Hug 23 Manual Bode Plots On board using handout EE40 Summer 2010 Hug 24 2nd Order Filter Example Also on board EE40 Summer 2010 Hug 25 2nd order Bode Plots Also on board This is where we stopped in class EE40 Summer 2010 Hug 26 Active filter example On board EE40 Summer 2010 Hug 27 Magnitude Plot Units So far we ve been plotting our Bode plots on where represents a signal getting through perfectly EE40 Summer 2010 Hug 28 Bel and Decibel dB A bel symbol B is a unit of measure of ratios of power levels i e relative power levels B log10 P1 P2 where P1 and P2 are power levels The bel is a logarithmic measure Zero bels corresponds to a ratio of 1 1 One bel corresponds to a ratio of 10 1 Three bels corresponds to a ratio of 1000 1 The bel is too large for everyday use so the decibel dB equal to 0 1B is more commonly used 1dB 10 log10 P1 P2 0 dB corresponds to a ratio of 1 1 10 dB corresponds to a ratio of 10 1 10 dB corresponds to a ratio of 1 10 dB are used to measure Electric power filter magnitude EE40 Summer 2010 Hug 29 Logarithmic Measure for Power To express a power in terms of decibels one starts by choosing a reference power Preference and writing Power P in decibels 10 log10 P Preference Exercise Express a power of 50 mW in decibels relative to 1 watt P dB 10 log10 50 x 10 3 13 dB Use logarithmic scale to express power ratios varying over a large range dB P 10 log 1 P2 EE40 Summer 2010 dB Note dB is not a unit for a physical quantity since power ratio is unitless It is just a notation to remind us we are in the log scale Hug 30 Decibels for measuring transfer function magnitude Decibels provide a measure of relative power However we said they also can be used for 2 2 P V I transfer functions The key is in realizing that Thus if a voltage gets reduced by a factor of 100 we d say the power ratio between the output and in the input would be EE40 Summer 2010 Hug 31 Transfer Function in dB Given a transfer function magnitude We can convert this transfer function into dB by There s not that much to this but it s common in datasheets EE40 Summer 2010 Hug 32 Example EE40 Summer 2010 20 log 10 Hug 33
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