Modeling, Computing, & Measurement: Measurement Systems # 4 Dr. Kevin Craig Professor of Mechanical Engineering Rensselaer Polytechnic Institute 1
Frequency Response and Filters When you hear music and see color, you are experiencing the frequency domain. It is all around you, just like the time domain. The frequency domain is a kind of hidden companion to our everyday world of time. We describe what happens in the time domain as temporal and in the frequency domain as spectral. Roughly speaking, in the time domain we measure how long something takes, whereas in the frequency domain we measure how fast or slow it is. These are two ways of viewing the same thing! 2
Most signals and processes involve both fast and slow components happening at the same time. Frequency domain analysis separates these components and helps to keep track of them. Mechanical Spectrum Second hand of a clock: 1 rpm Audio CDs: 200 to 500 rpm Dentist s drill: 400,000 rpm Two-foot diameter tire on a car traveling at 60 mph: 840 rpm Earth s rotation: 0.00069 rpm (1000 mph at the equator!) 1 Hz = 60 rpm = 2π rad/sec = 6.28 rad/sec. All have dimensions 1/time. 3
Mechanical Spectrum Electromagnetic Spectrum 4
Electromagnetic Spectrum Electromagnetic effects can be described in the frequency domain as well. Electromagnetic waves travel at the speed of light c, which depends on the medium (fastest in a vacuum, slower in other media). The frequency of vibration f depends on the wavelength λ of the electromagnetic phenomenon and the speed of propagation c of the medium according to f = c/λ. Long-wavelength electromagnetic waves are radio waves (see spectrum diagram). Frequencies range from a few khz to 300 GHz. Higher frequencies are emitted by thermal motion, which we call infrared radiation. 5
Frequencies of visible light range from 440 THz (red light) to 730 THz (violet light). Humans perceive different frequencies within the visible light spectrum as different colors. Unlike the ear, the eye has a nonlinear response to combinations of frequencies. Filters Filters react differently to signals at different frequencies, e.g., low-pass, high-pass, band-pass. A band-pass filter is useful for separating one radio station from another on a crowded dial. Each AM (amplitude modulated) station is allocated a frequency range of bandwidth 10 khz within the electromagnetic spectrum 535 to 1605 KHz. The amplitude of a carrier frequency at the center of the 10-KHz frequency range is modulated. 6
FM (frequency modulated) radio stations are allocated 150 khz of electromagnetic spectrum bandwidth, from 87.925 to 108.075 MHz. This is almost 20 times as much bandwidth as the AM radio spectrum. To be useful, a receiver must be sure to allow only the signal from one station to pass through the circuit, whether it is an AM or FM signal. A band pass filter is used to pass the frequencies of the desired station while attenuating the signals from all other stations, which may be transmitted at higher or lower frequencies than the signal of the desired station. For controls engineers, a feedback controller is essentially a filter whose gain and phase are chosen to modify the response of the controlled system. 7
Electromagnetic Spectrum 8
Radio Spectrum 9
What Will We Do Today? Introduce the Frequency Domain and Filters. Build a passive RC circuit (see below) and use as the input from the function generator a sine wave. Set the amplitude of the sine wave and vary the frequency from low to high. Use the oscilloscope and describe what you observe at the output? Sine Wave? 10
Take data as you vary the frequency from low to high. Choose 3 frequencies below the frequency 1/τ (rad/sec), where τ is the time constant RC, and 3 frequencies above the frequency 1/τ. Record the amplitude ratio between the output and input sine waves, as well as the phase angle between the two sine waves, at each frequency. Use a linear-log plot for your data. For the amplitude ratio use the unit decibels (x db = 20 log 10 x) and for the frequency use the unit Hz (cycles/sec). What happens at the frequency 1/τ? Sketch the input sine wave and the output sine wave on the same amplitude vs. time plot. Why is the RC circuit called a Low-Pass Filter? Now use the ELVIS Bode Analyzer to generate the same plot. Now reverse the R and the C. Follow the same steps as for the RC circuit. Why is this circuit called a High-Pass Filter? 11
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Connections for the ELVIS Bode Analyzer 15
RC Circuit Unit Step Response 0.63 Amplitude 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 R = 15 KΩ C = 0.01 µf 0 0 1 2 3 4 5 Time (sec) τ = RC = 0.15 msec 16
RC Circuit Amplitude Ratio = 0.707 = -3 db Phase Angle = -45 Input 1061 Hz Sine Wave 1 0.8 0.6 Response to Input 1061 Hz Sine Wave 0.4 0.2 R = 15 KΩ C = 0.01 µf amplitude 0-0.2-0.4-0.6 Output -0.8-1 0 0.5 1 1.5 2 2.5 3 3.5 4 time (sec) x 10-3 17
RC Circuit Frequency Response -3 db 0 Bode Diagram 1 1 rad = = 6666.7 τ RC sec = 1061 Hz Magnitude (db) -10-20 -30-40 0 Phase (deg) -45-90 10 1 10 2 10 3 10 4 10 5 Frequency (Hz) 18