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CHAPTER 5
The Response of Measuring Systems
5.1 Introduction
Quite simply, response is a measure of a system’s fidelity to purpose. It may be defined as an evaluation of the system’s ability to
faithfully sense, transmit, md present all the pertinent information included in the measurand and to exclude ude all else. We would
like to know if the output information truly represents the uput. If the input information is in the form of a sine wave, a square wave,
or a sawtooth wave, does the output appear as a sine wave, a square wave, or a sawtooth wave, as the case may be? Is each of the
harmonic components in a complex wave treated equally, or are some attenuated, completely ignored, or erhaps shifted timewise
relative to the others? These questions are answered by the response characteristics of the particular system—that is, (1) amplitude
response, (2) frequency response, (3) phase response, and (4) slew rate.
5.2 Amplitude Response
Amplitude response is governed by the system’s ability to treat all input amplitudes uniformly. If an input of 5 units is fed into a
system and an output of 25 indicator divisions is obtained, we can generally expect that an input of 10 units will result in an output of
50 divisions. Although this is the most non case, other special nonlinear responses are also occasionally required. Whatever the
arrangement, whether it be linear, exponential, or some other amplitude function, discrepancy between design expectations in this
respect and actual performance results in poor amplitude response. Of course no system exists that is capable of responding faithfully
over an unlimited range of amplitudes.
All systems can be overdriven. Figure 5.1 shows the amplitude response of a voltage amplifier suitable for
connecting a strain-gage bridge to an oscilloscope. The usable range of the amplifier is restricted to the
horizontal portion of the curve. The plot shows that for inputs above about 0.01 V the amplifier becomes
overloaded and the amplification ceases to be linear.
5.3 Frequency Response
Good frequency response is obtained when a system reacts to all frequency components in the
same way. If a 100-Hz sine wave with an input amplitude of 5 units is fed into a system and a peak-topeak output of 2 in. results on an oscilloscope screen, we can expect that a 500-Hz sine-wave input
of the same amplitude would also result in a 2-in. peak-to-peak output. Changing the frequency of
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the input signal should not alter the system’s output magnitude so long as the input amplitude
remains unchanged. Yet here again there must be a limit to the range over which good frequency
response may be expected. This is true for any dynamic system, regardless of its quality. Figure 5.2
illustrates the frequency response relations for the same voltage amplifier used in Fig. 5.1. Frequencies
above about 10 kHz are attenuated. Only inputs below this frequency limit are amplified in the
correct relative proportion.
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signal that the filter transmits to the computer recording system excludes the noise component. We
discuss filters in more detail when we consider signal conditioning in Chapter 7 (Sections 7.20—7.22).
In contrast to the filter, a well-designed sensor or transducer should respond to all frequencies
equally. Unfortunately, most actual sensors and transducers do not. Instead, such devices are
characterized by an upper or lower frequency beyond which response is attenuated (much like a
filter) or by a high or low frequency at which the sensor resonates with the input, producing an output
that is ridiculously large. Determination of such limiting frequencies is extremely important if a
dynamic measurand is to be accurately recorded.
The frequency and phase response of a system element (or of the entire chain of elements) can be
determined several ways. For simple elements, we may be able to construct a physical model of the
device that accurately predicts its response. For more complex systems, we may wish to test the
response experimentally, for example, by using a sine-wave test. In other circumstances, we may rely
on response data provided by the manufacturer. In any case, once we have determined the range
of frequencies for which the system responds accurately, we will disregard frequencies outside this
range, if they can be identified, and we will take precautions to prevent such frequencies from
entering the measuring chain.
The remainder of this chapter is concerned with the identification of system response. Sections 5.8
through 5.19 address the modeling of frequency and phase response for simple physical systems.
Section 5.20 returns to the matter of experimental determination and calibration of system response.
5.8 Simplified Physical Systems
What basic physical factors govern response? In terms of practical systems, we are confronted with
two fundamental segments of construction: mechanical and electrical. The basic mechanical
elements are mass, damping, and some form of equilibrium-restoring element, such as a spring.
Corresponding electrical elements are inductance, resistance, and capacitance. Although many, if
not most, devices and systems involve both electrical and mechanical elements, for our immediate
purposes it is advantageous to consider the two separately. In the next several sections we will
discuss some of the mechanical aspects; and beginning with Section 5.17, we will consider the
electrical.
5.9 Mechanical Elements
A discussion of the dynamic characteristics of an elementary mechanical system necessitates a short
description of the elements composing such a system.
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