2. Equivalent Capacitance and Inductance [20 points] An engineer has a DAQ with a 14-bit A/D converter and a voltage range of 0 to 10 V. They are measuring vibrations around

20 Hz, so they sample at f = 100 Hz to avoid aliasing. They add a DC offset of 5 V to the signal so that the periodic components due to vibration are centered within the range of the DAQ. a) Calculate the quantization error of this DAQ in mV (express your answer in terms of +/- mV). b) If the input amplitude is 0.025 V, calculate the quantization error as a percentage of the amplitude (express your answer in terms of +/-%). Repeat for an input amplitude of 1.50 V and compare the percentage errors. Discuss. c) Suppose the input voltage has electronic noise at 6500 Hz with an amplitude of 0.0035 V. Calculate the maximum required cutoff frequency of a first-order low-pass filter such that the amplitude of the noise is less than the quantization error of the A/D converter and therefore not resolvable by the DAQ. d) Calculate the minimum required cutoff frequency of a first-order low-pass filter such that the amplitude of the desired vibration signal is attenuated by no more than 10% (i.e., such that |G(jw)| ≥ 0.90). Note: If you do your calculations correctly, the minimum cutoff frequency of part (d) should be less than the maximum cutoff frequency of part (c). e) Pick a cutoff frequency halfway between the minimum and maximum required values. If the engineer has a 25 kn resistor on hand, calculate the capacitance (in units of μF) of the capacitor they should use to construct a passive first-order low-pass filter. f) Sketch the Bode magnitude plot for the fist-order filter, with gain |G(jw) | with a log scale on the vertical axis and frequency in Hz with a log scale on the horizontal axis. g) Sketch the Bode phase plot for the first-order filter, with phase in degrees on the vertical axis and frequency in Hz with a log scale on the horizontal axis. h) Suppose the signal frequency at 20 Hz is sinusoidal. Calculate the phase lag caused by the low-pass filter, in degrees.