Vibration and Shock Handbook 16

Vibration and Shock Handbook 16 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

16 Signal Conditioning and Modification

Clarence W de Silva

The University of British Columbia

16.1 Introduction 16-216.2 Amplifiers 16-2

Operational Amplifier † Use of Feedback in Opamp † Voltage, Current, and Power Amplifiers † Instrumentation Amplifiers † Amplifier Performance Ratings †

Component Interconnection

16.3 Analog Filters 16-15

Passive Filters and Active Filters † Low-Pass Filters † High-Pass Filters † Band-Pass Filters † Band-Reject Filters

16.4 Modulators and Demodulators 16-29

Amplitude Modulation † Application of Amplitude Modulation † Demodulation

16.5 Analog–Digital Conversion 16-37

Digital-to-Analog Conversion † Analog-to-Digital Conversion † Analog-to-Digital Converter Performance Characteristics † Sample-and-Hold Circuitry † Digital Filters

16.8 Miscellaneous Signal Modification Circuitry 16-56

Phase Shifters † Voltage-to-Frequency Converter † Frequency-to-Voltage Converter † Voltage-to-Current Converters † Peak-Hold Circuits

16.9 Signal Analyzers and Display Devices 16-62

Signal Analyzers † Oscilloscopes

Summary

This chapter concerns the conditioning of signals in a vibrating system and the conversion of signals in one form

to another as needed Amplification, filtering, modulation, demodulation, analog/digital conversion, voltage/frequency conversion, voltage/current conversion, linearization, bridge circuits, and signal analysis and displaydevices are presented Hardware, software, and techniques are considered Issues of impedance and loadingassociated with the interconnection of components are addressed

16-1

16.1 Introduction

Signal modification is an important function in many applications of vibration The tasks of signalmodification may include: signal conditioning (e.g., amplification, and analog and digital filtering); signalconversion (e.g., analog-to-digital conversion, digital-to-analog conversion, voltage-to-frequencyconversion, and frequency-to-voltage conversion); modulation (e.g., amplitude modulation, frequencymodulation, phase modulation, pulse-width modulation, pulse-frequency modulation, and pulse-codemodulation); and demodulation (the reverse process of modulation) In addition, many other types ofuseful signal modification operations can be identified For example, sample and hold circuits are used indigital data acquisition systems Devices such as analog and digital multiplexers and comparators areneeded in many applications of data acquisition and processing Phase shifting, curve shaping, offsetting,and linearization can also be classified as signal modification This chapter describes signal conditioningand modification operations that are useful in vibration applications Signal modification plays a crucialrole in component interfacing When two devices are interfaced, it is essential to ensure that a signalleaving one device and entering the other will do so at proper signal levels (voltage, current, power), inthe proper form (analog, digital), and without distortion (loading and impedance considerations) Asignal should be properly modified for transmission by amplification, modulation, digitizing, and so on,

so that the signal/noise ratio of the transmitted signal is sufficiently large at the receiver The significance

of signal modification is clear from these observations

16.2 Amplifiers

The level of an electrical signal can be represented by variables such as voltage, current, and power.Analogous across variables, through variables, and power variables can be defined for other types ofsignals (e.g., mechanical) as well Signal levels at various interface locations of components in a vibratorysystem have to be properly adjusted for proper performance of these components and of the overallsystem For example, input to an actuator should possess adequate power to drive the actuator A signalshould maintain its signal level above some threshold during transmission so that errors due to signalweakening will not be excessive Signals applied to digital devices must remain within the specified, logiclevels Many types of sensors produce weak signals that have to be upgraded before they can be fed into amonitoring system, data processor, controller, or data logger

Signal amplification concerns the proper adjustment of a signal level for performing a specific task.Amplifiers are used to accomplish signal amplification An amplifier is an active device that needs anexternal power source to operate Even though active circuits, amplifiers in particular, can be developed

in the monolithic form using an original integrated-circuit (IC) layout so as to accomplish a particularamplification task, it is convenient to study their performance using the operational amplifier (opamp) asthe basic element Of course, operational amplifiers are widely used not only for modeling and analyzingother types of amplifier but also as basic elements in building other kinds of amplifier For these reasons,our discussion on amplifiers will revolve around the operational amplifier

16.2.1 Operational Amplifier

The origin of the operational amplifier dates to the 1940s when the vacuum tube operational amplifierwas introduced The operational amplifier, or opamp, got its name due to the fact that originally it wasused almost exclusively to perform mathematical operations; for example, it was used in analogcomputers Subsequently, in the 1950s, the transistorized opamp was developed It used discrete elementssuch as bipolar junction transistors and resistors The opamp was still too large in size, consumed too muchpower, and was too expensive for widespread use in general applications This situation changed in thelate 1960s when IC opamp was developed in the monolithic form as a single IC chip Today, the ICopamp, which consists of a large number of circuit elements on a substrate, typically of a single siliconcrystal (the monolithic form), is a valuable component in almost any signal modification device

An opamp could be manufactured in the

discrete-element form using perhaps ten bipolar

junction transistors and as many discrete resistors;

alternatively (and preferably), it may be

manufac-tured in the modern monolithic form as an IC chip

that may be equivalent to over 100 discrete

elements In any form, the device has an input

impedance, Zi; an output impedance, Zo; and a gain,

K: Hence, a schematic model for an opamp can be

given as in Figure 16.1(a) The conventional

symbol of an opamp is shown in Figure 16.1(b)

Typically, there are about six terminals (lead

connections) to an opamp For example, there

may be two input leads (a positive lead with voltage

vipand a negative load with voltage vin), an output

lead (voltage vo), two bipolar power supply leads

đợvsand 2vsỡ; and a ground lead

Note from Figure 16.1(a) that, under open-loop

(no feedback) conditions

From Equation 16.1 and Equation 16.2, it is clear that if the negative input lead is grounded(i.e., vinỬ 0), then

1 5mV at the positive lead and 2 mV at the negative lead

2 25mV at the positive lead and 2 mV at the negative lead

3 5mV at the positive lead and 22 mV at the negative lead

4 25mV at the positive lead and 22 mV at the negative lead

5 1 V at the positive lead and negative lead grounded

6 1 V at the negative lead and positive lead grounded

(b)

vin

−+FIGURE 16.1 Operational amplifier: (a) a schematic model; (b) conventional symbol.

This problem can be solved using Equation 16.1 and Equation 16.2 The results are given in Table 16.1.Note that, in the last two cases, the output will saturate and Equation 16.1 will no longer hold.Field effect transistors (FET), for example, metal oxide semiconductor field effect transistors(MOSFET), could be used in the IC form of an opamp The MOSFET type has advantages over manyother types; for example, such opamps have higher input impedance and more stable output (almostequal to the power supply voltage) at saturation This makes the MOSFET opamps preferable overbipolar junction transistor opamps in many applications

In analyzing operational amplifier circuits under unsaturated conditions, we use the following twocharacteristics of an opamp:

1 Voltages of the two input leads should be (almost) equal

2 Currents through each of the two input leads should be (almost) zero

As explained earlier, the first property is credited to high open-loop gain and the second property tohigh input impedance in an operational amplifier We shall repeatedly use these two properties to obtaininput–output equations for amplifier systems

16.2.2 Use of Feedback in Opamp

The operation amplifier is a very versatile device, primarily due to its very high input impedance, lowoutput impedance, and very high gain However, it cannot be used without modification as an amplifierbecause it is not very stable, as shown inFigure 16.1 Two factors that contribute to this problem are:

1 Frequency response

2 Drift

Stated in another way, opamp gain, K; does not remain constant; it can vary with the frequency of theinput signal (i.e., frequency-response function is not flat in the operating range); also, it can vary withtime (i.e., drift) The frequency-response problem arises due to circuit dynamics of an operationalamplifier This problem is usually not severe unless the device is operated at very high frequencies Thedrift problem arises due to the sensitivity of gain, K; to environmental factors such as temperature, light,humidity, and vibration, and as a result of variation of K due to aging Drift in an opamp can besignificant and steps should be taken to remove that problem

It is virtually impossible to avoid drift in gain and frequency-response error in an operationalamplifier However, an ingenious way has been found to remove the effect of these two problems atthe amplifier output Since gain K is very large, by using feedback we can virtually eliminate its effect atthe amplifier output This closed loop form of an opamp is preferred in almost every application

In particular, the voltage follower and charge amplifier are devices that use the properties of high Zi;low Zo; and high K of an opamp, along with feedback through a precision resistor, to eliminateerrors due to nonconstant K: In summary, the operational amplifier is not very useful in its open-loopform, particularly because gain, K; is not steady However, since K is very large, the problem can beremoved by using feedback It is this closed-loop form that is commonly used in the practical applications

In addition to the nonsteady nature of gain, there are other sources of error that contribute to the lessthan ideal performance of an operational amplifier circuit Noteworthy are:

1 offset current present at input leads due to bias currents that are needed to operate the solid-statecircuitry

2 offset voltage that might be present at the output even when the input leads are open

3 unequal gains corresponding to the two input leads (i.e., the inverting gain not equal to thenoninverting gain)

Such problems can produce nonlinear behavior in opamp circuits, and they can be reduced by propercircuit design and through the use of compensating circuit elements

16.2.3 Voltage, Current, and Power Amplifiers

Any type of amplifier can be constructed from scratch in the monolithic form as an IC chip, or in thediscrete form as a circuit containing several discrete elements such as discrete bipolar junction transistors

or discrete FETs, discrete diodes, and discrete resistors However, almost all types of amplifiers can also bebuilt using operational amplifier as the basic element Since we are already familiar with opamps andsince opamps are extensively used in general amplifier circuitry, we prefer to use the latter approach,which uses discrete opamps for the modeling of general amplifiers

If an electronic amplifier performs a voltage amplification function, it is termed a voltage amplifier.These amplifiers are so common that, the term “amplifier” is often used to denote a voltage amplifier Avoltage amplifier can be modeled as

Voltage amplifiers are used to achieve voltage compatibility (or level shifting) in circuits

Current amplifiers are used to achieve current compatibility in electronic circuits A current amplifiermay be modeled by

Note that voltage follower has Kv¼ 1 and, hence, it may be considered to be a current amplifier Also,

it provides impedance compatibility and acts as a buffer between a low-current (high-impedance) outputdevice (the device that provides the signal) and a high-current (low-impedance) input device (the devicethat receives the signal) that are interconnected Hence, the name buffer amplifier or impedancetransformer is sometimes used for a current amplifier with unity voltage gain

If the objective of signal amplification is to upgrade the associated power level, then a power amplifiershould be used for that purpose A simple model for a power amplifier is

It is easy to see from Equation 16.5 to Equation 16.7 that

Note that all three types of amplification could be achieved simultaneously from the same amplifier.Furthermore, a current amplifier with unity voltage gain (for example, a voltage follower) is a poweramplifier as well Usually, voltage amplifiers and current amplifiers are used in the first stages of a signalpath (e.g., sensing, data acquisition, and signal generation) where signal levels and power levels arerelatively low Power amplifiers are typically used in the final stages (e.g., actuation, recording, anddisplay) where high signal levels and power levels are usually required

Figure 16.2(a) shows an opamp-based voltage amplifier Note the feedback resistor, Rf; that serves thepurposes of stabilizing the opamp and providing an accurate voltage gain The negative lead is groundedthrough an accurately known resistor, R: To determine the voltage gain, recall that the voltages at the twoinput leads of an opamp should be virtually equal The input voltage, vi, is applied to the positive lead of

SensorCharge

+

FIGURE 16.2 (a) A voltage amplifier; (b) a current amplifier; (c) a charge amplifier.

the opamp Then the voltage at point A should also be equal to vi Next, recall that the current throughthe input lead of an opamp is virtually zero Hence, by writing the current balance equation for the nodepoint A, we have

vo2 vi

Rf ¼

viRThis gives the amplifier equation

A current amplifier is shown inFigure 16.2(b).The input current, ii; is applied to the negative lead ofthe opamp as shown and the positive lead is grounded There is a feedback resistor Rfconnected to thenegative lead through the load RL: The resistor Rfprovides a path for the input current since the opamptakes in virtually zero current There is a second resistor R through which the output is grounded Thisresistor is needed for current amplification To analyze the amplifier, note that the voltage at point A (i.e.,

at the negative lead) should be zero because the positive lead of the opamp is grounded (zero voltage).Furthermore, the entire input current, ii; passes through resistor, Rf; as shown Hence, the voltage atpoint B is Rfii: Consequently, current through resistor R is Rfii=R; which is positive in the directionshown It follows that the output current, io; is given by

io¼ iiþ RRfiior

If the feedback capacitance is large in comparison with the cable capacitance, the latter can be neglected.This is desirable in practice In any event, for large values of gain, K; we have the approximaterelationship

vo¼ 2 q

Note that the output voltage is proportional to the charge generated at the sensor and depends only onthe feedback parameter, Cf: This parameter can be appropriately chosen in order to obtain the requiredoutput impedance characteristics Actual charge amplifiers also have a feedback resistor, Rf, in parallelwith the feedback capacitor, Cf: Then, the relationship corresponding to Equation 16.12a becomes a first-order ordinary differential equation, which in turn determines the time constant of the charge amplifier.This time constant should be high If it is low, the charge generated by the piezoelectric sensor will leakout quickly, giving erroneous results at low frequencies

16.2.4 Instrumentation Amplifiers

An instrumentation amplifier is typically a special-purpose voltage amplifier dedicated to a particularinstrumentation application Examples include amplifiers used for producing the output from a bridgecircuit (bridge amplifier) and amplifiers used with various sensors and transducers An importantcharacteristic of an instrumentation amplifier is the adjustable gain capability The gain value can beadjusted manually in most instrumentation amplifiers In more sophisticated instrumentationamplifiers, gain is programmable and can be set by means of digital logic Instrumentation amplifiersare normally used with low-voltage signals

16.2.4.1 Differential Amplifier

Usually, an instrumentation amplifier is also a differential amplifier (sometimes termed differenceamplifier) Note that in a differential amplifier both input leads are used for signal input, whereas in asingle-ended amplifier one of the leads is grounded and only one lead is used for signal input Ground-loop noise can be a serious problem in single-ended amplifiers Ground-loop noise can be effectivelyeliminated by using a differential amplifier, because noise loops are formed with both inputs of theamplifier using a differential amplifier allows that these noise signals are subtracted at the amplifieroutput Since the noise level is almost the same for both inputs, it is canceled out Note that any othernoise (e.g., 60 Hz line noise) that might enter both inputs with the same intensity will also be canceledout in the output of a differential amplifier

A basic differential amplifier that uses a single opamp is shown inFigure 16.3(a).The input–outputequation for this amplifier can be obtained in the usual manner For instance, since current through theopamp is negligible, current balance at point B gives

ðvoR=Rfþ vi1Þð1 þ R=RfÞ

16.2.4.2 Common Mode

The voltage that is “common” to both input leads of a differential amplifier is known as the mode voltage This is equal to the smaller of the two input voltages If the two inputs are equal, then thecommon-mode voltage is obviously equal to each one of the two inputs When vi1¼ vi2; ideally, theoutput voltage vo should be zero In other words, ideally, common-mode signals are rejected by a

differential amplifier However, since the operational amplifiers are not ideal and since they usually donot have exactly identical gains with respect to the two input leads, the output voltage vowill not be zerowhen the two inputs are identical This common-mode error can be compensated for by providing avariable resistor with fine resolution at one of the two input leads of the differential amplifier As shown

inFigure 16.3(b),to compensate for the common-mode error (i.e., to achieve a satisfactory level ofcommon-mode rejection), first the two inputs are made equal and then dR4is varied carefully until theoutput voltage level is sufficiently small (minimum) Usually, the dR4that is required to achieve thiscompensation is small compared with the nominal feedback resistance R4:

Since ideally dR4¼ 0; we shall neglect dR4in the derivation of the instrumentation amplifier equation.Now, note from the basic characteristics of an opamp with no saturation (voltages at the two input leadshave to be almost identical) that, in Figure 16.3(b), the voltage at point 2 should be vi2and the voltage atpoint 1 should be vi1: Furthermore, current through each input lead of an opamp is negligible Hence,current through the circuit path B ! 2 ! 1 ! A has to be the same This gives the current continuityequations

16.2.5 Amplifier Performance Ratings

Main factors that affect the performance of an amplifier are:

1 Stability

2 Speed of response (bandwidth, slew rate)

3 Unmodeled signals

We have already discussed the significance of some of these factors

The level of stability of an amplifier, in the conventional sense, is governed by the dynamics of theamplifier circuitry and may be represented by a time constant However, a more important considerationfor an amplifier is the “parameter variation” due to aging, temperature, and other environmental factors.Parameter variation is also classified as a stability issue in the context of devices such as amplifiers,because it pertains to the steadiness of the response when the input is maintained steady Of particularimportance is temperature drift This may be specified as drift in the output signal per unit change intemperature (e.g.,mV/8C)

The speed of response of an amplifier dictates the ability of the amplifier to faithfully respond totransient inputs Conventional time-domain parameters such as rise time may be used to represent this.Alternatively, in the frequency domain, speed of response may be represented by a bandwidth parameter.For example, the frequency range over which the frequency-response function is considered constant(flat) may be taken as a measure of bandwidth Since there is some nonlinearity in any amplifier,bandwidth can depend on the signal level itself Specifically, small-signal bandwidth refers to thebandwidth that is determined using small input signal amplitudes.

Another measure of the speed of response is the slew rate Slew rate is defined as the largest possible rate

of change in the amplifier output for a particular frequency of operation Since, for a given inputamplitude, the output amplitude depends on the amplifier gain, slew rate is usually defined for unity gain.Ideally, for a linear device, the frequency-response function (transfer function) does not depend on theoutput amplitude (i.e., the product of the DC gain and the input amplitude) However, for a device thathas a limited slew rate, the bandwidth, or the maximum operating frequency at which output distortionsmay be neglected, will depend on the output amplitude The larger the output amplitude, the smaller thebandwidth for a given slew rate limit

We have noted that stability problems and frequency-response errors are prevalent in the open-loopform of an operational amplifier These problems can be eliminated using feedback because the effect ofthe open-loop transfer function on the closed loop transfer function is negligible if the open-loop gain isvery large, which is the case for an operational amplifier

Unmodeled signals can be a major source of amplifier error Unmodeled signals include:

16.2.5.1 Common-Mode Rejection Ratio

Common-mode error in a differential amplifier was discussed earlier We noted that ideally the mode input voltage (the voltage common to both input leads) should have no effect on the outputvoltage of a differential amplifier However, since a practical amplifier has imbalances in the internalcircuitry (for example, gain with respect to one input lead is not equal to the gain with respect to theother input lead and, furthermore, bias signals are needed for operation of the internal circuitry), therewill be an error voltage at the output that depends on the common-mode input The common-moderejection ratio (CMRR) of a differential amplifier is defined as

common-CMRR ¼ Kvcm

in which

K ¼ gain of the differential amplifier (i.e., differential gain)

vcm¼ common-mode voltage (i.e., voltage common to both input leads)

vocm¼ common-mode output voltage (i.e., output voltage due to common-mode input voltage)Note that, ideally, vocm¼ 0 and CMRR should be infinity It follows that the larger the CMRR, thebetter the differential amplifier performance

The three types of unmodeled signals mentioned above can be considered as noise In addition, thereare other types of noise signals that degrade the performance of an amplifier For example, ground-loop

noise can enter the output signal Furthermore, stray capacitances and other types of unmodeled circuiteffects can generate internal noise Usually in amplifier analysis, unmodeled signals (including noise) can

be represented by a noise voltage source at one of the input leads Effects of unmodeled signals can bereduced by using suitably connected compensating circuitry, including variable resistors that can beadjusted to eliminate the effect of unmodeled signals at the amplifier output (e.g., see dR4 inFigure16.3(b)).Some useful information about operational amplifiers is summarized in Box 16.1

Box 16.1

Ideal Opamp Properties:

* Infinite open-loop differential gain

* Infinite input impedance

* Zero output impedance

* Infinite bandwidth

* Zero output for zero differential input

Ideal Analysis Assumptions:

* Voltages at the two input leads are equal

* Current through either input lead is zero

Definitions:

* Open-loop gain ¼ Output voltage

Voltage difference at input leads with no feedback

* Input impedance ¼ Voltage between an input lead and ground

Current through that lead with other input leadgrounded and the output in open circuit

* Output impedance ¼ Voltage between output lead and ground in open circuit

Current through that leadwith normal input conditions

* Bandwidth ¼ frequency range in which the frequency response is flat (gain is constant)

* Input bias current ¼ average (DC) current through one input lead

* Input offset current ¼ difference in the two input bias currents

* Differential input voltage ¼ voltage at one input lead with the other grounded when theoutput voltage is zero

* Common-mode gain ¼ Output voltage when input leads are at the same voltage

Common input voltage

* Common-mode rejection ratio ðCMRRÞ ¼ Open loop differential gainCommon-mode gain

* Slew rate ¼ speed at which steady output is reached for a step input

16.2.6 Component Interconnection

When two or more components are interconnected, the behavior of the individual components in theoverall system can deviate significantly from their behavior of each component when they operateindependently The matching of components in a multicomponent system should be done carefully inorder to improve system performance and accuracy, particularly with respect to their impedancecharacteristics This is particularly true in vibration instrumentation

16.2.6.1 Impedance Characteristics

When components such as measuring instruments, digital processing boards, process (plant) hardware,and signal-conditioning equipment are interconnected, it is necessary to match impedances properly ateach interface in order to realize the devices’ rated performance level One adverse effect of improperimpedance matching is the loading effect For example, in a measuring system, the measuring instrumentcan distort the signal that is being measured The resulting error can far exceed other types ofmeasurement error Loading errors will result from connecting a measuring device with low inputimpedance to a signal source

Impedance can be interpreted either in the traditional electrical sense or in the mechanical sense,depending on the signal that is being measured For example, a heavy accelerometer can introduce anadditional dynamic load that will modify the actual acceleration at the monitoring location Similarly, avoltmeter can modify the currents (and voltages) in a circuit In mechanical and electrical systems,loading errors can appear as phase distortions as well Digital hardware also can produce loading errors.For example, an ADC board can load the amplifier output from a strain-gage bridge circuit, therebysignificantly affecting digitized data

Another adverse effect of improper impedance consideration is inadequate output signal levels, whichcan make signal processing and transmission very difficult Many types of transducers (e.g., piezoelectricaccelerometers, impedance heads, and microphones) have high output impedances in the order of athousand megohms These devices generate low output signals, and they require conditioning to step upthe signal level Impedance-matching amplifiers, which have high input impedances (megohms) and lowoutput impedances (a few ohms), are used for this purpose (e.g., charge amplifiers are used inconjunction with piezoelectric sensors) A device with a high input impedance has the further advantagethat it usually consumes less power (v2=R is low) for a given input voltage The fact that a low inputimpedance device extracts a high level of power from the preceding output device may transpire to be thereason for a loading error

16.2.6.2 Cascade Connection of Devices

Consider a standard two-port electrical device The output impedance, Zo; of such a device is defined asthe ratio of the open-circuit (i.e., no-load) voltage at the output port to the short-circuit current at theoutput port

Open-circuit voltage at the output is the output voltage present when there is no current flowing at theoutput port This is the case if the output port is not connected to a load (impedance) As soon as a load

is connected at the output of the device, a current will flow through it and the output voltage will drop to

a value less than that of the open-circuit voltage To measure open-circuit voltage, the rated input voltage

is applied at the input port and maintained constant, and the output voltage is measured using

a voltmeter that has a very high (input) impedance To measure short-circuit current, a verylow-impedance ammeter is connected at the output port

The input impedance, Zi; is defined as the ratio of the rated input voltage to the sponding current through the input terminals while the output terminals are maintained as an opencircuit

corre-Note that these definitions are associated with electrical devices A generalization is possible thatincludes both electrical and mechanical devices; one must interpret voltage and velocity as acrossvariables, and current and force as through variables Then, mechanical mobility can be used in place ofelectrical impedance in the associated analysis

Example 16.2

Input impedance, Zi, and output impedance, Zo;

can be represented schematically as in Figure

16.4(a) Note that vo is the open-circuit output

voltage When a load is connected at the output

port, the voltage across the load will be different

from vo: This is caused by the presence of a current

through Zo: In the frequency domain, viand voare

represented by their respective Fourier spectra The

corresponding transfer relation can be expressed in

terms of the complex frequency-response

(trans-fer) function G (jv) under open-circuit (no-load)

conditions:

vo¼ Gvi ð16:15ÞNow, consider two devices connected in cascade,

as shown in Figure 16.4(b) It can be easily verified

that the following relations apply:

Note that cascading has “distorted” the frequency-response characteristics of the two devices If

Zo1=Zi2p 1; this deviation becomes insignificant From this observation, it can be concluded that, whenfrequency-response characteristics (i.e., dynamic characteristics) are important in a cascaded device,cascading should be done such that the output impedance of the first device is much smaller than theinput impedance of the second device

16.2.6.3 AC-Coupled Amplifiers

The DC component of a signal can be blocked off by connecting that signal through a capacitor (Notethat the impedance of a capacitor is 1=ðjvCÞ and, hence, at zero frequency there will be an infiniteimpedance.) If the input lead of a device has a series capacitor, we say that the input is AC coupled and, ifthe output lead has a series capacitor, then the output is AC coupled Typically, an AC-coupled amplifierhas a series capacitor both at the input lead and the output lead Hence, its frequency-response functionwill have a high-pass characteristic; in particular, the DC components will be filtered out Errors due

to bias currents and offset signals are negligible for an AC-coupled amplifier Furthermore, in anAC-coupled amplifier, stability problems are not very serious

16.3 Analog Filters

Unwanted signals can seriously degrade the

performance of a vibration monitoring and

analysis system External disturbances, error

com-ponents in excitations, and noise generated

internally within system components and

instru-mentation are such spurious signals A filter is a

device that allows only the desirable part of a signal

to pass through, rejecting the unwanted part

In typical applications of acquisition and

processing of a vibration signal, the filtering task

requires allowing certain frequency components

through and filtering out certain other frequency

components in the signal In this context, we can

identify four broad categories of filters:

1 Low-pass filters

2 High-pass filters

3 Band-pass filters

4 Band-reject (or notch) filters

The ideal frequency-response characteristic of

each of these four types of filter is shown in Figure

16.5 Note that only the magnitude of the

frequency-response function is shown

It is understood, however, that the phase

distortion of the input signal also should be

small, within the pass band (the allowed frequency

range) Practical filters are less than ideal Their

frequency-response functions do not exhibit sharp

cutoffs as in Figure 16.5 and, furthermore, some

phase distortion will be unavoidable

A special type of band-pass filter that is

widely used in acquisition and monitoring of

vibration signals (e.g., in vibration testing) is the

tracking filter This is simply a band-pass filter

with a narrow pass band that is frequency

tunable The center frequency (the mid-value) of

the pass band is variable, usually by coupling it

to the frequency of a carrier signal In this

manner, signals whose frequency varies with

some basic variable in the system (e.g., rotor

speed, frequency of a harmonic excitation signal,

frequency of a sweep oscillator) can be

accu-rately tracked in the presence of noise The

inputs to a tracking filter are the signal that is being tracked and the variable tracking frequency(carrier input) A typical tracking filter that can simultaneously track two signals is schematicallyshown in Figure 16.6

Filtering can be achieved using digital filters as well as analog filters Before digital signalprocessing became efficient and economical, analog filters were exclusively used for signal filteringand they are still widely used In an analog filter, the signal is passed through an analog circuit

TrackingFilter

Input Channel 1 Output Channel 1

Output Channel 2Input Channel 2

Carrier Input(Tracking Frequency)FIGURE 16.6 Schematic representation of a two- channel tracking filter.

The dynamics of the circuit will be such that the desired signal components will be passed throughand the unwanted signal components will be rejected Earlier versions of analog filters employeddiscrete circuit elements such as discrete transistors, capacitors, resistors, and even discreteinductors Since inductors have several shortcomings, including susceptibility to electromagneticnoise, unknown resistance effects, and large size These days, they are rarely used in filter circuits.Furthermore, owing to well-known advantages of IC devices, analog filters in the form ofmonolithic IC chips are today extensively used in modem applications and are preferred overdiscrete-element filters Digital filters that employ digital signal processing to achieve filtering arealso widely used nowadays.

16.3.1 Passive Filters and Active Filters

Passive analog filters employ analog circuits containing only passive elements, such as resistors andcapacitors (and sometimes inductors) An external power supply is not needed in a passive filter.Active analog filters employ active elements and components, such as transistors and operationalamplifiers in addition to passive elements Since external power is needed for the operation of theactive elements and components, an active filter is characterized by the need of an external powersupply Active filters are widely available in a monolithic IC form and are usually preferred overpassive filters

Advantages of active filters include the following:

1 Loading effects are negligible because active filters can provide a very high input impedance andvery low output impedance

2 They can be used with low-level signals because signal amplification and filtering can be provided

by the same active circuit

3 They are widely available in a low cost and compact IC form

4 They can be easily integrated with digital devices

5 They are less susceptible to noise from electromagnetic interference than passive filters

Commonly mentioned disadvantages of active filters are the following:

1 They need an external power supply

2 They are susceptible to “saturation”-type nonlinearity at high signal levels

3 They can introduce many types of internal noise and unmodeled signal errors (offset, biassignals, etc.)

Note that advantages and disadvantages of passive filters can be directly inferred from the advantages and advantages of active filters as given above

dis-16.3.1.1 Number of Poles

Analog filters are dynamic systems and they can be represented by transfer functions, assuminglinear dynamics The number of poles of a filter is the number of poles in the associated transferfunction This is also equal to the order of the characteristic polynomial of the filter transferfunction (i.e., order of the filter) Note that poles (or eigenvalues) are the roots of the characteristicequation

In our discussion, we will show simplified versions of filters, typically consisting of a single filter stage.The performance of such a basic filter can be improved at the expense of circuit complexity (and anincreased pole count) Only simple discrete-element circuits are shown for passive filters Simpleoperational-amplifier circuits are given for active filters Even here, much more complex devices arecommercially available, but our purpose is to illustrate underlying principles rather than to providedescriptions and data sheets for commercial filters

16.3.2 Low-Pass Filters

The purpose of a low-pass filter is to allow through all signal components below a certain (cutoff)frequency and block all signal components above that cutoff Analog low-pass filters are widely used asantialiasing filters in digital signal processing An error known as aliasing will enter the digitally processedresults of a signal if the original signal has frequency components above half the sampling frequency (Halfthe sampling frequency is called the Nyquist frequency.) Hence, aliasing distortion can be eliminated if,prior to sampling and digital processing, the signal is filtered using a low-pass filter with its cutoff set atNyquist frequency This is one of numerous applications of analog low-pass filters Another typicalapplication would be to eliminate high-frequency noise in a measured vibration response

A single-pole, passive low-pass filter circuit is shown inFigure 16.7(a).An active filter corresponding

to the same low-pass filter is shown in Figure 16.7(b) It can be shown that the two circuits have identicaltransfer functions Hence, it might seem that the opamp in Figure 16.7(b) is redundant This is not true,however If two passive filter stages, each similar to Figure 16.7(a), are connected together, the overalltransfer function is not equal to the product of the transfer functions of the individual stages The reasonfor this apparent ambiguity is the circuit loading that arises due to the fact that the input impedance ofthe second stage is not sufficiently larger than the output impedance of the first stage However, if twoactive filter stages, similar to those in Figure 16.7(b), are connected together, such loading errors will benegligible because the opamp with feedback (i.e., a voltage follower) introduces a very high inputimpedance and very low output impedance, while maintaining the voltage gain at unity

To obtain the filter equation for the scenario depicted in Figure 16.7(a), note that, since the output isopen circuit (zero load current), the current through capacitor C is equal to the current through resistorR: Hence,

Cdvo

dt ¼

vi2 voRor

The frequency-response function corresponding to Equation 16.19 is obtained by setting s ¼ jv; thus

This gives the response of the filter when a sinusoidal signal of frequency, v; is applied Themagnitude lGðjvÞl of the frequency-transfer function gives the signal amplification and phase angle/GðjvÞ gives the phase lead of the output signal with respect to the input The magnitude curve(Bode magnitude curve) is shown in Figure 16.7(c) Note from Equation 16.20 that, for smallfrequencies (i.e., v p 1=t), the magnitude is approximately unity Hence, 1=t can be considered the

FIGURE 16.7 A single-pole low-pass filter: (a) a passive filter stage; (b) an active filter stage; (c) the response characteristic; (d) a two-pole, low-pass Butterworth filter.

R C

Rf

+A

(c)

Magnitude(Log)

C1

A

R2

(d)

t2v2þ 1 ¼

12or

t2v2þ 1 ¼ 2or

t2v2¼ 1Hence, the half-power bandwidth is

This is identical to the cutoff frequency given by Equation 16.11

Now, forv q 1=t (i.e., tv q 1) Equation 16.20 can be approximated by

GðjvÞ ¼ tjv1This has the magnitude

lGðjvÞl ¼ tv1

In the log scale

log10lGðjvÞl ¼ 2log10v 2 log10t

It follows that the log10(magnitude) vs log10(frequency) curve is a straight line with slope 21 In otherwords, when frequency increases by a factor of ten (i.e., a decade), the log10magnitude decreases byunity (i.e., by 20 dB) Hence, the roll-off rate is 220 dB/decade These observations are shown in

Figure 16.7(c).Note that an amplitude change by a factor ofpffiffi2(or power by a factor of 2) corresponds to

3 dB Hence, when the DC (zero-frequency) magnitude is unity (0 dB), the half power magnitude

is 23 dB

Cutoff frequency and the roll-off rate are the two main design specifications for a low-pass filter.Ideally, we would like a low-pass filter magnitude curve to be flat until the required pass-band limit(cutoff frequency) and then roll off very rapidly The low-pass filter shown in Figure 16.7 only

approximately meets these requirements In particular, the roll-off rate is not as large as is desirable Wewould like a roll-off rate of at least 240 dB/decade and, preferably, 260 dB/decade in practical filters.This can be realized by using a higher order filter (i.e., a filter having many poles) The low-passButterworth filter is a widely used filter of this type.

16.3.2.1 Low-Pass Butterworth Filter

A low-pass Butterworth filter having two poles can provide a roll-off rate of 240 dB/decade, and onehaving three poles can provide a roll-off rate of 260 dB/decade Furthermore, the steeper the slope of theroll-off, the flatter is the filter magnitude curve within the pass band

A two-pole, low-pass Butterworth filter is shown inFigure 16.7(d) We could construct a two-polefilter simply by connecting two single-pole stages of the type shown in Figure 16.7(b) Then, we wouldrequire two opamps, whereas the circuit shown in Figure 16.7(d) achieves the same objective by usingonly one opamp (i.e., at a lower cost)

the damping ratio is

z ¼ t2ffiffiffiffiffiffiffiþt34t1t2

The filter equation is obtained by considering current balance in Figure 16.8(a), noting that the output

is in open circuit (zero load current) Accordingly,

C d

dtðv12 voÞ ¼

voR

frequency-above is not perfect, as observed from the frequency-response characteristic shown inFigure 16.8(c).

It can be easily verified that the half-power bandwidth of the basic high-pass filter is equal to the cutofffrequency given by Equation 16.37, as in the case of the basic low-pass filter The roll-up slope of thesingle-pole high-pass filter is 20 dB/decade Steeper slopes are desirable Multiple-pole, high-passButterworth filters can be constructed to give steeper roll-up slopes and reasonably flat pass-bandmagnitude characteristics

16.3.4 Band-Pass Filters

An ideal band-pass filter passes all signal components within a finite frequency band and blocks offall signal components outside that band The lower frequency limit of the pass band is called thelower cutoff frequency ðvc1Þ; and the upper frequency limit of the band is called the upper cutofffrequency ðvc2Þ:

The most straightforward way to form a band-pass filter is to cascade a high-pass filter of cutofffrequencyvc1with a low-pass filter of cutoff frequencyvc2: Such an arrangement is shown in Figure 16.9.The passive circuit shown in Figure 16.9(a) is obtained by connecting the circuits shown inFigure 16.7(a)

and Figure 16.8(a) The passive circuit shown in Figure 16.9(b) is obtained by connecting a voltagefollower opamp circuit to the original passive circuit Passive and active filters have the same transferfunction, assuming that loading problems are not present in the passive filter Since loading errors can beserious in practice, however, the active version is preferred

To obtain the filter equation, first consider the high-pass portion of the circuit shown inFigure 16.9(a).

Since the output is open circuit (zero current), we have from Equation 16.35:

16.3.4.1 Resonance-Type Band-Pass Filters

There are many applications where a filter with a very narrow pass band is required The tracking filtermentioned in the beginning of the section on analog filters is one such application A filter circuit with asharp resonance can serve as a narrow-band filter Note that the cascaded RC circuit shown in Figure 16.9does not provide an oscillatory response (the filter poles are all real) and, hence, it does not form aresonance-type filter A slight modification to this circuit using an additional resistor, R1; as shown in

Figure 16.10(a),will produce the desired effect

To obtain the filter equation, note that, for the voltage follower unit

Next, since current through an opamp lead is zero, for the high-pass circuit unit (see Equation 16.35), wehave

It can be shown that, unlike Equation 16.41, the present characteristic equation

t1t2s2þ ðt1þt2þt3Þs þ 2 ¼ 0 ð16:44Þcan possess complex roots

or

kvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðv22v2Þ2þ 4z2v2v2

2pffiffi2zvn

v2c1þ 2zvnvc12v2n¼ 0

v2 c22 2zvnvc22v2¼ 0Accordingly, by solving these two quadratic equations and selecting the appropriate sign, we obtain

vc1¼ 2zvnþqffiffiffiffiffiffiffiffiffiffiffiffiffiv2þz2v2

ð16:49Þ

vc2¼zvnþqffiffiffiffiffiffiffiffiffiffiffiffiffiv2þz2v2

ð16:50ÞThe half-power bandwidth is

16.3.5 Band-Reject Filters

Band-reject filters, or notch filters, are commonly used to filter out a narrow band of noise componentsfrom a signal For example, 60 Hz line noise in signals can be eliminated by using a notch filter with anotch frequency of 60 Hz

An active circuit that could serve as a notch filter is shown inFigure 16.11(a).This is known as theTwin T circuit because its geometric configuration resembles two T-shaped circuits connected together

To obtain the filter equation, note that the voltage at point P is vo because of unity gain of the voltagefollower Now, we write the current balance at nodes A and B; thus

C d

dtðvi2 vAÞ ¼

vAR=2 þ C

d

dtðvA2 voÞNext, since the current through the positive lead of the opamp (voltage follower) is zero, we have thecurrent through point P as

vB2 vo

d

dtðvo2 vAÞThese three equations are written in the Laplace form as

vo¼ 1

This is known as the notch frequency The magnitude of the frequency-response function of the notchfilter is sketched in Figure 16.11(b) It is noticed that any signal component at frequencyvo will becompletely eliminated by the notch filter Sharp roll-down and roll-up are needed to allow the other(desirable) signal components through without too much attenuation

Whereas the previous three types of filters achieve their frequency-response characteristics through thepoles of the filter transfer function, a notch filter achieves its frequency-response characteristic throughits zeros (roots of the numerator polynomial equation) Some useful information about filters issummarized in Box 16.2

16.4 Modulators and Demodulators

Sometimes signals are deliberately modified to maintain the accuracy during signal transmission,conditioning, and processing In signal modulation, the data signal, known as the modulating signal, isused to vary a property (such as amplitude or frequency) of a carrier signal We say that the carrier signal

is modulated by the data signal After transmitting or conditioning the modulated signal, the data signal

is usually recovered by removing the carrier signal This is known as demodulation or discrimination.Many modulation techniques exist, and several other types of signal modification (e.g., digitizing)could be classified as signal modulation even though they might not be commonly termed as such Fourtypes of modulation are illustrated inFigure 16.12 In amplitude modulation (AM), the amplitude of aperiodic carrier signal is varied according to the amplitude of the data signal (modulating signal),frequency of the carrier signal (carrier frequency) being kept constant Suppose that the transient signal

Box 16.2

Active Filters (Need External Power)

Advantages:

* Smaller loading errors (have high input impedance and low output impedance, and hence

do not affect the input circuit conditions and output signals)

* Lower cost

* Better accuracy

Passive Filters (No External Power, Use Passive Elements)

Advantages:

* Useable at very high frequencies (e.g., radio frequency)

* No need for a power supply

* Band pass: Allows frequency components within an interval and rejects the rest

* Notch (or band reject): Rejects frequency components within an interval (usually narrow)and allows the rest

Definitions

* Filter order: Number of poles in the filter circuit or transfer function

* Antialiasing filter: Low-pass filter with cutoff at less than half the sampling rate (i.e., Nyquistfrequency), for digital processing

* Butterworth filter: A high-order filter with a very flat pass band

* Chebyshev filter: An optimal filter with uniform ripples in the pass band

* Sallen-Key filter: An active filter whose output is in phase with input

(b)

(c)

t Time t

frequency-shown inFigure 16.12(a)is used as the modulating signal A high-frequency sinusoidal signal is used asthe carrier signal The resulting amplitude-modulated signal is shown in Figure 16.12(b) Amplitudemodulation is used in telecommunications, radio and TV signal transmission, instrumentation, andsignal conditioning The underlying principle is useful in other applications such as fault detection anddiagnosis in rotating machinery.

In frequency modulation (FM), the frequency of the carrier signal is varied in proportion to theamplitude of the data signal (modulating signal), while the amplitude of the carrier signal is keptconstant If the data signal shown in Figure 16.12(a) is used to frequency modulate a sinusoidal carriersignal, then the result will appear as shown in Figure 16.12(c) Since information is carried as frequencyrather than amplitude, any noise that might alter the signal amplitude will have virtually no effect on thetransmitted data Hence, FM is less susceptible to noise than AM Furthermore, since the carrieramplitude is kept constant in FM, signal weakening and noise effects that are unavoidable in long-distance data communication will have less effect than in the case of AM, particularly if the data signallevel is low in the beginning However, more sophisticated techniques and hardware are needed for signalrecovery (demodulation) in FM transmission, because FM demodulation involves frequencydiscrimination rather than amplitude detection Frequency modulation is also widely used in radiotransmission and in data recording and replay

In pulse-width modulation (PWM), the carrier signal is a pulse sequence The pulse width is changed inproportion to the amplitude of the data signal, while keeping the pulse spacing constant This isillustrated in Figure 16.12(d) Pulse-width modulated signals are extensively used in controlling electricmotors and other mechanical devices such as valves (hydraulic, pneumatic) and machine tools Notethat, in a given short time interval, the average value of the pulse-width modulated signal is an estimate ofthe average value of the data signal in that period Hence, PWM signals can be used directly in controlling

a process without one having to demodulate it Advantages of PWM include better energy efficiency (lessdissipation) and better performance with nonlinear devices For example, a device may stick at lowspeeds due to Coulomb friction This can be avoided by using a PWM signal that provides the signalamplitude that is necessary to overcome friction while maintaining the required average control signal,which might be very small

In pulse-frequency modulation (PFM) as well, the carrier signal is a pulse sequence In this method,the frequency of the pulses is changed in proportion to the data signal level, while the pulse width iskept constant PFM has the advantage of ordinary frequency modulation Additionaladvantages result due to the fact that electronic circuits (digital circuits, in particular) can handlepulses very efficiently Furthermore, pulse detection is not susceptible to noise because it involvesdistinguishing between the presence and absence of a pulse rather than accurate determination ofthe pulse amplitude (or width) PFM may be used in place of PWM in most applications withbetter results

Another type of modulation is phase modulation (PM) In this method, the phase angle of the carriersignal is varied in proportion to the amplitude of the data signal

Conversion of discrete (sampled) data into the digital (binary) form is also considered to bemodulation In fact, this is termed pulse-code modulation (PCM) In this case, each discrete data sample isrepresented by a binary number containing a fixed number of binary digits (bits) Since each digit in thebinary number can take only two values, 0 or 1, it can be represented by the absence or presence of avoltage pulse Hence, each data sample can be transmitted using a set of pulses This is known asencoding At the receiver, the pulses have to be interpreted (or decoded) in order to determine the datavalue As with any other pulse technique, PCM is quite immune to noise because decoding involvesdetection of the presence or absence of a pulse rather than determination of the exact magnitude of thepulse signal level Also, since pulse amplitude is constant, long-distance signal transmission (of thisdigital data) can be accomplished without the danger of signal weakening and associated distortion Ofcourse, there will be some error introduced by the digitization process itself, which is governed by thefinite word size (or dynamic range) of the binary data element This is known as quantization error and isunavoidable in signal digitization

In any type of signal modulation, it is essential to preserve the algebraic sign of the modulating signal(data) Different types of modulators handle this in different ways For example, in PCM an extra sign bit

is added to represent the sign of the transmitted data sample In AM and FM, a phase-sensitivedemodulator is used to extract the original (modulating) signal with the correct algebraic sign Note that,

in these two modulation techniques, a sign change in the modulating signal can be represented by a 1808phase change in the modulated signal This is not noticeable inFigure 16.12(b)and (c) In PWM andPFM, a sign change in the modulating signal can be represented by changing the sign of the pulses, asshown in Figure 16.12(d) and (e) In PM, a positive range of phase angles (say 0 top) can be assigned forthe positive values of the data signal and a negative range of phase angles (say 2p to 0) can be assignedfor the negative values of the signal

16.4.1 Amplitude Modulation

Amplitude modulation can naturally enter into many physical phenomena More important, perhaps, isthe deliberate (artificial) use of AM to facilitate data transmission and signal conditioning Let us firstexamine the related mathematics

Amplitude modulation is achieved by multiplying the data signal (modulating signal), xðtÞ; by a highfrequency (periodic) carrier signal, xcðtÞ: Hence, amplitude-modulated signal, xaðtÞ; is given by

Note that the carrier could be any periodic signal such as one which is harmonic (sinusoidal), squarewave, or triangular The main requirement is that the fundamental frequency of the carrier signal (carrierfrequency), fc; be significantly larger (say, by a factor of five or ten) than the highest frequency of interest(bandwidth) of the data signal Analysis can be simplified by assuming a sinusoidal carrier frequency;thus

16.4.1.1 Modulation Theorem

Modulation theorem is also known as the frequency-shifting theorem, and it relates the fact that if a signal

is multiplied by a sinusoidal signal, the Fourier spectrum of the product signal is simply the Fourierspectrum of the original signal shifted through the frequency of the sinusoidal signal In other words, theFourier spectrum, Xaðf Þ; of the amplitude-modulated signal, xaðtÞ; can be obtained from the Fourierspectrum, Xðf Þ; of the data signal, xðtÞ; simply by shifting through the carrier frequency, fc:

To mathematically explain the modulation theorem, we use the definition of the Fourier integraltransform to obtain

Xað f Þ ¼ acð1

21xðtÞ cos 2p fct expð2j2p ftÞdtHowever, since

inFigure 16.13 Consider a transient signal, xðtÞ; with a (continuous) Fourier spectrum, Xðf Þ; whosemagnitude, lXðf Þl; is as shown in Figure 16.13(a) If this signal is used to modulate the AM of ahigh-frequency sinusoidal signal, the resulting modulated signal, xaðtÞ; and the magnitude of its

Fourier spectrum are as shown in Figure 16.13(b) It should be kept in mind that the magnitude has beenmultiplied by ac=2: Note that the data signal is assumed to be band limited, with bandwidth fb: Of course,the theorem is not limited to band-limited signals but, for practical reasons, we need to have some upperlimit on the useful frequency of the data signal Also for practical reasons (not for the theorem itself), thecarrier frequency, fc; should be several times larger than fbso that there is a reasonably wide frequencyband from 0 to ð fc2 fbÞ, within which the magnitude of the modulated signal is virtually zero Thesignificance of this should be clear when we discuss applications of amplitude modulation.

Figure 16.13 shows only the magnitude of the frequency spectra It should be remembered, however,that every Fourier spectrum has a phase angle spectrum as well This is not shown for conciseness, butclearly the phase-angle spectrum is also similarly affected (frequency shifted) by AM

16.4.1.2 Side Frequencies and Side Bands

The modulation theorem, as described above, assumed transient data signals with associated continuousFourier spectra The same ideas are applicable to periodic signals (with discrete spectra) as well The case