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Lecture 30 Review: First order highpass and lowpass filters Cutoff frequency Checking frequency response results Time-to-frequency domain relations for first order filters Bode plots Related educational materials: –Chapter 11.3, 11.4

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Filters Circuits categorized by their amplitude response – Filters pass some frequencies and stop others – Lowpass filters pass low frequencies – Highpass filters pass high frequencies Filters are also categorized by their order – The filter order corresponds to the order of the circuits governing differential equation Circuits I: only first order low- and highpass filters

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First order filters Review: Low- and highpass filter response examples Cutoff frequency separates the pass- and stopbands – Where the amplitude response is times the max amplitude

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Annotate previous slide to show cutoff frequencies – Note that cutoff frequency is related to maximum gain of filter Change max gain on figure (write in) => gain at corner frequency changes

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Time vs. frequency domain characterization In the time domain, we characterized first order circuits by their step response – DC gain, time constant ( ) We can likewise characterize first order circuits by their frequency response – DC gain, cutoff frequency ( c )

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On previous slide, sketch step response, show DC gain Then show that cosine input becomes a step input as frequency -> zero

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Time-to-Frequency domain relations – contd Governing differential equation for first order system: Converting to frequency domain:

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On previous slide, show: – (1) time domain DC gain, tau – (2) conversion to frequency response => u(t)=Ue^(jwt), etc. – (3) freqency response -> magnitude response – (4) DC gain and cutoff frequency from mag. Resp.

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Checking amplitude responses We can (fairly) easily check our amplitude responses at very low and very high frequencies Capacitors, inductors replaced by open, short circuits – Results in purely resistive network – Analyze resistive network, and compare result to amplitude response – Provides physical insight into low, high frequency operation

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Inductors at low, high frequencies Inductor impedance: = 0 – Inductor behaves like short circuit at low frequencies – Inductor behaves like open circuit at high frequencies

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Capacitors at low, high frequencies Capacitor impedance: = 0 – Capacitor behaves like open circuit at low frequencies – Capacitor behaves like short circuit at high frequencies

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Example – checking amplitude response What is the amplitude response of the circuit below as 0 and ?

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Bode plots – introduction We have used linear scales to plot frequency responses Selective use of logarithmic scales has a number of advantages: – Amplitudes and frequencies tend to span large ranges – Logarithms convert multiplication and division to addition and subtraction – Human senses work logarithmically Bode plots use logarithmic scales to simplify plotting of frequency responses and interpretation of plots

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Do demos on previous slide – Note large ranges of frequencies; sensitivity to frequency ranges – Note human senses work logarithmically

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Nomenclature relative to log scales Logarithmic scales convert multiplicative factors to linear differences Some of these multiplicative factors have special names – A factor of 10 change in frequency is a decade difference on a log scale – A factor of 2 change in frequency is an octave difference on a log scale

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Bode Plots Bode plots are a specific format for plotting frequency responses – Frequencies are on logarithmic scales – Amplitudes are on a decibel (dB) scale – Phases are on a linear scale

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Decibels Named for Alexander Graham Bell Common gains and their decibel values: 10 20dB 1 0dB dB dB -3dB

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Bode plots of first order filters Frequency response for typical first order lowpass filter: Magnitude, phase responses:

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Bode plot – magnitude response

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Annotate previous slide to show asymptotic behavior, -3dB point at wc, intersection of asymptotes is at wc Note that I can muliply this by a gain, without affecting the shape of the curve – it simply moves up and down

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Bode plot – phase response

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Annotate previous slide to show asymptotic behavior, -45 degrees at wc

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Example Sketch a Bode plot for the circuit below.

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