1 / 25

Lecture 30 Review: First Order Highpass and Lowpass Filters

This lecture reviews the concepts of first order highpass and lowpass filters, including cutoff frequency, frequency response results, and time-to-frequency domain relations. Bode plots are also discussed.

pittmang
Download Presentation

Lecture 30 Review: First Order Highpass and Lowpass Filters

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. 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 circuit’s governing differential equation • Circuits I: only first order low- and highpass filters

  3. 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

  4. 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

  5. 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)

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

  7. Time-to-Frequency domain relations – cont’d • Governing differential equation for first order system: • Converting to frequency domain:

  8. 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.

  9. 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

  10. Inductors at low, high frequencies • Inductor impedance: •  = 0  • Inductor behaves like short circuit at low frequencies •     • Inductor behaves like open circuit at high frequencies

  11. Capacitors at low, high frequencies • Capacitor impedance: •  = 0  • Capacitor behaves like open circuit at low frequencies •     • Capacitor behaves like short circuit at high frequencies

  12. Example – checking amplitude response • What is the amplitude response of the circuit below as 0 and  ?

  13. 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

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

  15. 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

  16. 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

  17. Decibels • Named for Alexander Graham Bell • Common gains and their decibel values: 10  20dB 1  0dB 0.1  -20dB 0.5  -6dB  -3dB

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

  19. Bode plot – magnitude response

  20. 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

  21. Bode plot – phase response

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

  23. Example • Sketch a Bode plot for the circuit below.

More Related