Lecture 2: Measurement and Instrumentation

Time vs. Frequency Domain Different ways of looking at a problem –Interchangeable: no information is lost in changing from one domain to another –Benefits from changing perspective: the solution to difficult problems can often become quite clear in the other domain

Time Domain Analysis The traditional way of observing signals is to view them in the time domain. Record in the time domain typically describes the variation of system output or system parameter over time.

Record from Seismograph This record (of displacement, also called seismogram) displays 24 hours of data, beginning approximately 8 hours before the mainshock, recorded by the BDSN station BKS at a distance of 525 km.

Frequency Domain Analysis It was shown over one hundred years ago by Jean B. Fourier that any waveform that exists in the real world can be generated by adding up sine waves. where the Fourier coefficients of f(x) can be calculated using the so-called Euler formulas

Frequency Component of a Signal This frequency domain representation of our signal is called the spectrum of the signal, in which every sine wave from the signal appears as a vertical line. Its height represents its amplitude and its position represents its frequency. Each sine wave line of the spectrum is called a component of the total signal.

Spectrum of Various Signal Types Figures on the right show a few common signals in both the time and frequency domains.

Laplace Transformation

Measurement System Components –Transducer: convert a physical quantity into a time- varying electrical signal, i.e. analog signal –Signal conditioner: modify/enhance the analog signal (profiltering, amplification, etc) –Analog-to-digital converter: convert analog signal into digital format –Digital signal processing –Recorder: display or data storage Actual Input Measurement System Recorder output

Transfer Function Block diagram and transfer function provide an efficient way to describe a dynamic system Transfer Function –Used to describe linear time-invariant systems –Can be expressed using the Laplace transform of the ratio of the output and input variables of the system –Can be experimentally determined by curve-fitting Frequency response (sinusoidal input) Impulse response (short-duration pulse input) Input, r(t)Output, y(t) System

Frequency Response of a System

Characteristics of a Good Measurement System Amplitude Linearity Adequate Bandwidth Phase Linearity A: Amplitude f: frequency   phase

Amplitude Linearity Factors impact linearity Limited range of input amplitude Bandwidth  : proportion constant

Frequency Bandwidth Requirement: a measurement system should replica all frequency components Unit-Decibel scale: Bode plot: frequency response curve of a system, i.e. a plot of amplitude ratio A out /A in, vs. the input frequency

Bode Plot fLfL fcfc fHfH Frequency Amplitude (dB)

Bandwidth Bandwidth: the range of frequencies where the input of the system is not attenuated by morn than -3dB Low & high cutoff frequency: Bandwidth: Questions: –what happen if a square wave is measured with a limited BW system –How to determine the BW of a measurement system