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 16.1 Introduction 16-2 16.2 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 16.6 Bridge Circuits 16-43 Wheatstone Bridge † Constant-Current Bridge Amplifiers † Impedance Bridges † Bridge 16.7 Linearizing Devices 16-49 Linearization by Software † Linearization by Hardware Logic † Analog Linearizing Circuitry † Offsetting Circuitry † Proportional-Output Circuitry 16.8 Miscellaneous Signal Modification Circuitry 16-56 Clarence W de Silva The University of British Columbia 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 display devices are presented Hardware, software, and techniques are considered Issues of impedance and loading associated with the interconnection of components are addressed 16-1 © 2005 by Taylor & Francis Group, LLC 16-2 16.1 Vibration and Shock Handbook Introduction Signal modification is an important function in many applications of vibration The tasks of signal modification may include: signal conditioning (e.g., amplification, and analog and digital filtering); signal conversion (e.g., analog-to-digital conversion, digital-to-analog conversion, voltage-to-frequency conversion, and frequency-to-voltage conversion); modulation (e.g., amplitude modulation, frequency modulation, phase modulation, pulse-width modulation, pulse-frequency modulation, and pulse-code modulation); and demodulation (the reverse process of modulation) In addition, many other types of useful signal modification operations can be identified For example, sample and hold circuits are used in digital data acquisition systems Devices such as analog and digital multiplexers and comparators are needed 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 conditioning and modification operations that are useful in vibration applications Signal modification plays a crucial role in component interfacing When two devices are interfaced, it is essential to ensure that a signal leaving one device and entering the other will so at proper signal levels (voltage, current, power), in the proper form (analog, digital), and without distortion (loading and impedance considerations) A signal 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 of signals (e.g., mechanical) as well Signal levels at various interface locations of components in a vibratory system have to be properly adjusted for proper performance of these components and of the overall system For example, input to an actuator should possess adequate power to drive the actuator A signal should maintain its signal level above some threshold during transmission so that errors due to signal weakening will not be excessive Signals applied to digital devices must remain within the specified, logic levels Many types of sensors produce weak signals that have to be upgraded before they can be fed into a monitoring 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 an external 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 particular amplification task, it is convenient to study their performance using the operational amplifier (opamp) as the basic element Of course, operational amplifiers are widely used not only for modeling and analyzing other 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 amplifier was introduced The operational amplifier, or opamp, got its name due to the fact that originally it was used almost exclusively to perform mathematical operations; for example, it was used in analog computers Subsequently, in the 1950s, the transistorized opamp was developed It used discrete elements such as bipolar junction transistors and resistors The opamp was still too large in size, consumed too much power, and was too expensive for widespread use in general applications This situation changed in the late 1960s when IC opamp was developed in the monolithic form as a single IC chip Today, the IC opamp, which consists of a large number of circuit elements on a substrate, typically of a single silicon crystal (the monolithic form), is a valuable component in almost any signal modification device © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-3 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 manufactured 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 vip and a negative load with voltage vin ), an output lead (voltage vo ), two bipolar power supply leads ỵvs and 2vs ị; and a ground lead Note from Figure 16.1(a) that, under open-loop (no feedback) conditions vs (Power Supply) + vip Inputs vi vin Zo Zi Output vo= K vi + Kv i − − (a) vin (b) vip − + vo FIGURE 16.1 Operational amplifier: (a) a schematic model; (b) conventional symbol vo ẳ Kvi 16:1ị in which the input voltage, vi ; is the differential input voltage defined as the algebraic difference between the voltages at the positive and negative lead; thus vi ¼ vip vin ð16:2Þ The open loop voltage gain K is very high (10 to 10 ) for a typical opamp Furthermore, the input impedance, Zi ; could be as high as MV and the output impedance is low, of the order of 10 V Since vo is typically to 10 V, from Equation 16.1 it follows that vi ø since K is very large Hence, from Equation 16.2, we have vip ø vin : In other words, the voltages at the two input leads are nearly equal Now, if we apply a large voltage differential vi (say, V) at the input then, according to Equation 16.1, the output voltage should be extremely high This never happens in practice, however, since the device saturates quickly beyond moderate output voltages (of the order of 15 V) From Equation 16.1 and Equation 16.2, it is clear that if the negative input lead is grounded (i.e., vin ¼ 0), then vo ẳ Kvip 16:3ị and, if the positive input lead is grounded (i.e., vip ¼ 0) vo ¼ 2Kvin ð16:4Þ Accordingly, vip is termed noninverting input and vin is termed inverting input Example 16.1 Consider an opamp having an open-loop gain of £ 105 If the saturation voltage is 15 V, determine the output voltage in the following cases: mV at the positive lead and mV at the negative lead mV at the positive lead and mV at the negative lead mV at the positive lead and 22 mV at the negative lead mV at the positive lead and 2 mV at the negative lead V at the positive lead and negative lead grounded V at the negative lead and positive lead grounded © 2005 by Taylor & Francis Group, LLC 16-4 Vibration and Shock Handbook TABLE 16.1 vip mV 25 mV mV 25 mV 1V Solution to Example 16.1 vin vi vo mV mV 22 mV 22 mV 1V mV 27 mV mV 23 mV 1V 21 V 0.3 V 20.7 V 0.7 V 20.3 V 15 V 215 V Solution 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 many other types; for example, such opamps have higher input impedance and more stable output (almost equal to the power supply voltage) at saturation This makes the MOSFET opamps preferable over bipolar junction transistor opamps in many applications In analyzing operational amplifier circuits under unsaturated conditions, we use the following two characteristics of an opamp: Voltages of the two input leads should be (almost) equal 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 to high input impedance in an operational amplifier We shall repeatedly use these two properties to obtain input –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, low output impedance, and very high gain However, it cannot be used without modification as an amplifier because it is not very stable, as shown in Figure 16.1 Two factors that contribute to this problem are: Frequency response Drift Stated in another way, opamp gain, K; does not remain constant; it can vary with the frequency of the input signal (i.e., frequency-response function is not flat in the operating range); also, it can vary with time (i.e., drift) The frequency-response problem arises due to circuit dynamics of an operational amplifier This problem is usually not severe unless the device is operated at very high frequencies The drift 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 be significant and steps should be taken to remove that problem It is virtually impossible to avoid drift in gain and frequency-response error in an operational amplifier However, an ingenious way has been found to remove the effect of these two problems at the amplifier output Since gain K is very large, by using feedback we can virtually eliminate its effect at the 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 eliminate errors due to nonconstant K: In summary, the operational amplifier is not very useful in its open-loop form, particularly because gain, K; is not steady However, since K is very large, the problem can be removed by using feedback It is this closed-loop form that is commonly used in the practical applications of an opamp © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-5 In addition to the nonsteady nature of gain, there are other sources of error that contribute to the less than ideal performance of an operational amplifier circuit Noteworthy are: offset current present at input leads due to bias currents that are needed to operate the solid-state circuitry offset voltage that might be present at the output even when the input leads are open unequal gains corresponding to the two input leads (i.e., the inverting gain not equal to the noninverting gain) Such problems can produce nonlinear behavior in opamp circuits, and they can be reduced by proper circuit 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 the discrete 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 be built using operational amplifier as the basic element Since we are already familiar with opamps and since 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 A voltage amplifier can be modeled as in which vo ẳ K v vi 16:5ị vo ẳ output voltage vi ¼ input voltage Kv ¼ voltage gain 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 amplifier may be modeled by io ẳ Ki ii 16:6ị in which io ẳ output current ii ¼ input current Ki ¼ current gain Note that voltage follower has Kv ¼ 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) output device (the device that provides the signal) and a high-current (low-impedance) input device (the device that receives the signal) that are interconnected Hence, the name buffer amplifier or impedance transformer 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 amplifier should be used for that purpose A simple model for a power amplifier is po ¼ Kp Pi in which po ¼ output power pi ¼ input power Kp ẳ power gain â 2005 by Taylor & Francis Group, LLC ð16:7Þ 16-6 Vibration and Shock Handbook It is easy to see from Equation 16.5 to Equation 16.7 that Kp ẳ Kv Ki 16:8ị 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 power amplifier as well Usually, voltage amplifiers and current amplifiers are used in the first stages of a signal path (e.g., sensing, data acquisition, and signal generation) where signal levels and power levels are relatively low Power amplifiers are typically used in the final stages (e.g., actuation, recording, and display) 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 the purposes of stabilizing the opamp and providing an accurate voltage gain The negative lead is grounded through an accurately known resistor, R: To determine the voltage gain, recall that the voltages at the two input leads of an opamp should be virtually equal The input voltage, vi , is applied to the positive lead of Input vi Output vo + A − R Rf (a) R B ii A Input Rf RL Load − ii io (Output) + (b) Cf Feedback Capacitor −vo /k − − Sensor Charge q Zi + vo k + Ce + − Zo vo Voltage Drop Across Zo = (c) FIGURE 16.2 K + Output vo − (a) A voltage amplifier; (b) a current amplifier; (c) a charge amplifier © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-7 the opamp Then the voltage at point A should also be equal to vi Next, recall that the current through the input lead of an opamp is virtually zero Hence, by writing the current balance equation for the node point A, we have vo vi v ¼ i Rf R This gives the amplier equation vo ẳ ỵ Rf v R i ð16:9aÞ Hence, the voltage gain is given by Kv ẳ ỵ Rf R 16:9bị Note the Kv depends on R and Rf and not on the opamp gain Hence, the voltage gain can be accurately determined by selecting the two resistors, R and Rf ; precisely Also note that the output voltage has the same sign as the input voltage Hence, this is a noninverting amplifier If the voltages are of the opposite sign, we will have an inverting amplifier A current amplifier is shown in Figure 16.2(b) The input current, ii ; is applied to the negative lead of the opamp as shown and the positive lead is grounded There is a feedback resistor Rf connected to the negative lead through the load RL : The resistor Rf provides a path for the input current since the opamp takes in virtually zero current There is a second resistor R through which the output is grounded This resistor 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 at point B is Rf ii : Consequently, current through resistor R is Rf ii =R; which is positive in the direction shown It follows that the output current, io ; is given by io ¼ ii ỵ Rf i R i io ẳ ỵ Rf i R i or ð16:10aÞ The current gain of the amplier is Ki ẳ ỵ Rf R 16:10bị This gain can be accurately set using the high-precision resistors, R and Rf 16.2.3.1 Charge Amplifiers The principle of capacitance feedback is utilized in charge amplifiers These amplifiers are commonly used for conditioning the output signals from piezoelectric transducers A schematic diagram for the charge amplifier is shown in Figure 16.2(c) The feedback capacitance is denoted by Cf and the connecting cable capacitance by Cc : The charge amplifier views the sensor as a charge source (q), even though there is an associated voltage Using the fact that charge ¼ voltage £ capacitance, a charge balance equation can be written: qỵ vo v C ỵ vo ỵ o C f ẳ K c K 16:11ị From this, we obtain vo ẳ â 2005 by Taylor & Francis Group, LLC K q K ỵ 1ịCf ỵ Cc 16:12aị 16-8 Vibration and Shock Handbook 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 approximate relationship vo ẳ q Cf 16:12bị Note that the output voltage is proportional to the charge generated at the sensor and depends only on the feedback parameter, Cf : This parameter can be appropriately chosen in order to obtain the required output impedance characteristics Actual charge amplifiers also have a feedback resistor, Rf , in parallel with the feedback capacitor, Cf : Then, the relationship corresponding to Equation 16.12a becomes a firstorder 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 leak out 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 particular instrumentation application Examples include amplifiers used for producing the output from a bridge circuit (bridge amplifier) and amplifiers used with various sensors and transducers An important characteristic of an instrumentation amplifier is the adjustable gain capability The gain value can be adjusted manually in most instrumentation amplifiers In more sophisticated instrumentation amplifiers, gain is programmable and can be set by means of digital logic Instrumentation amplifiers are normally used with low-voltage signals 16.2.4.1 Differential Amplifier Usually, an instrumentation amplifier is also a differential amplifier (sometimes termed difference amplifier) Note that in a differential amplifier both input leads are used for signal input, whereas in a single-ended amplifier one of the leads is grounded and only one lead is used for signal input Groundloop noise can be a serious problem in single-ended amplifiers Ground-loop noise can be effectively eliminated by using a differential amplifier, because noise loops are formed with both inputs of the amplifier using a differential amplifier allows that these noise signals are subtracted at the amplifier output Since the noise level is almost the same for both inputs, it is canceled out Note that any other noise (e.g., 60 Hz line noise) that might enter both inputs with the same intensity will also be canceled out in the output of a differential amplifier A basic differential amplifier that uses a single opamp is shown in Figure 16.3(a) The input–output equation for this amplifier can be obtained in the usual manner For instance, since current through the opamp is negligible, current balance at point B gives vi2 vB v ẳ B R Rf iị in which vB is the voltage at B Similarly, current balance at point A gives vo vA v vi1 ¼ A Rf R iiị vA ẳ vB iiiị Now, we use the property for an operational amplifier to eliminate vA and vB from Equation i and Equation ii This gives vi2 ðv R=Rf ỵ vi1 ị ẳ o ỵ R=Rf ị ỵ R=Rf ị â 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-9 Rf Inputs R vi1 vi2 R A − Output vo B + Rf (a) vi1 + A − R3 R3 R1 − vi2 FIGURE 16.3 − R2 Inputs (b) R4 R1 + + Output vo R4+δ R4 B (a) A basic differential amplifier; (b) a basic instrumentation amplifier or vo ¼ Rf ðv vi1 Þ R i2 ð16:13Þ Two things are clear from Equation 16.13 First, the amplifier output is proportional to the difference between, and not the absolute value of, the two inputs vi1 and vi2 : Second, voltage gain of the amplifier is Rf =R: This is known as the differential gain Note that the differential gain can be accurately set by using high-precision resistors R and Rf : The basic differential amplifier, shown in Figure 16.3(a) and discussed above, is an important component of an instrumentation amplifier In addition, an instrumentation amplifier should possess the adjustable gain capability Furthermore, it is desirable to have a very high input impedance and very low output impedance at each input lead An instrumentation amplifier that possesses these basic requirements is shown in Figure 16.3(b) The amplifier gain can be adjusted using the precisely variable resistor, R2 : Impedance requirements are provided by two voltage-follower-type amplifiers, one for each input, as shown The variable resistance, dR4 ; is necessary to compensate for errors due to unequal common-mode gain Let us first consider this aspect and then obtain an equation for the instrumentation amplifier 16.2.4.2 Common Mode The voltage that is “common” to both input leads of a differential amplifier is known as the commonmode voltage This is equal to the smaller of the two input voltages If the two inputs are equal, then the common-mode voltage is obviously equal to each one of the two inputs When vi1 ¼ vi2 ; ideally, the output voltage vo should be zero In other words, ideally, common-mode signals are rejected by a © 2005 by Taylor & Francis Group, LLC 16-10 Vibration and Shock Handbook differential amplifier However, since the operational amplifiers are not ideal and since they usually not have exactly identical gains with respect to the two input leads, the output voltage vo will not be zero when the two inputs are identical This common-mode error can be compensated for by providing a variable resistor with fine resolution at one of the two input leads of the differential amplifier As shown in Figure 16.3(b), to compensate for the common-mode error (i.e., to achieve a satisfactory level of common-mode rejection), first the two inputs are made equal and then dR4 is varied carefully until the output voltage level is sufficiently small (minimum) Usually, the dR4 that is required to achieve this compensation is small compared with the nominal feedback resistance R4 : Since ideally dR4 ¼ 0; we shall neglect dR4 in the derivation of the instrumentation amplifier equation Now, note from the basic characteristics of an opamp with no saturation (voltages at the two input leads have to be almost identical) that, in Figure 16.3(b), the voltage at point should be vi2 and the voltage at point should be vi1 : Furthermore, current through each input lead of an opamp is negligible Hence, current through the circuit path B ! ! ! A has to be the same This gives the current continuity equations vB vi2 v vi1 v vA ¼ i2 ¼ i1 R1 R2 R1 in which VA and VB are the voltages at points A and B, respectively Hence, we obtain the two equations vB ẳ vi2 ỵ R1 v vi1 ị R2 i2 vA ẳ vi1 R1 v vi1 Þ R2 i2 Now, by subtracting the second equation from the first, we have the equation for the first stage of the amplifier; thus 2R1 ðvi2 vi1 ị iị vB vA ẳ ỵ R2 From the previous result (see Equation 16.13) for a differential amplifier, we have (with dR4 ¼ 0) R vo ¼ ðvB vA Þ ðiiÞ R3 Note that only the resistor R2 is varied to adjust the gain (differential gain) of the amplifier In Figure 16.3(b), the two input opamps (the voltage-follower opamps) not have to be exactly identical as long as the resistors R1 and R2 are chosen so that they are accurate This is so because the opamp parameters such as open-loop gain and input impedance not enter the amplifier equations provided that their values are sufficiently high, as noted earlier 16.2.5 Amplifier Performance Ratings Main factors that affect the performance of an amplifier are: Stability Speed of response (bandwidth, slew rate) 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 the amplifier circuitry and may be represented by a time constant However, a more important consideration for 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 particular importance is temperature drift This may be specified as drift in the output signal per unit change in temperature (e.g., mV/8C) © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-53 of operation Now, the output of the curve shaper can be utilized for any task that requires the device output The advantage here is that linear assumptions are valid with the curve shaper, which is not the case for the actual device When the operating range changes, the curve shaper must be returned to the new range Comparison (calibration) of the curve shaper and the nonlinear device can be done off line and, once a set of gain values corresponding to a set of operating ranges is determined in this manner for the curve shaper, it is possible to completely replace the nonlinear device with the curve shaper Then the gain of the curve shaper can be adjusted depending on the actual operating range during system operation This is known as gain scheduling Note that we can replace a nonlinear device with a linear device (curve shaper) within a multicomponent system in this manner without greatly sacrificing the accuracy of the overall system 16.7.4 Offsetting Circuitry Common-mode outputs and offsets in amplifiers R and other analog devices can be minimized by Input including a compensating resistor that can provide R B − Output vi fine adjustments at one of the input leads vo vref A+ Furthermore, the larger is the feedback signal (DC Reference) Rc level in a feedback system, the smaller is the steadyRo state error Hence, steady-state offsets can be reduced by reducing the feedback resistance (thereby increasing the feedback signal) Furthermore, since a ballast (potentiometer) circuit FIGURE 16.21 An inverting amplifier circuit for offset provides an output of vo þ dvo and a bridge compensation circuit provides an output of dvo ; the use of a bridge circuit can be interpreted as an offset compensation method The most straightforward way of offsetting is by using a differential amplifier (or a summing amplifier) to subtract (or add) a DC voltage to the output of the nonlinear device The DC level has to be variable so that various levels of offset can be provided using the same circuit This is accomplished by using an adjustable resistance at the DC input lead of the amplifier An operational-amplifier circuit for offsetting is shown in Figure 16.21 Since the input, vi ; is connected to the negative lead of the opamp, we have an inverting amplifier, and the input signal will appear in the output, vo ; with its sign reversed This is also a summing amplifier because two signals can be added together by this circuit If the input, vi ; is connected to the positive lead of the opamp, we will have a noninverting amplifier The DC voltage, vref ; provides the offsetting voltage The resistor, Rc (compensating resistor), is variable so that different values of offset can be compensated using the same circuit To obtain the circuit equation, we write the current balance equation for node A, using the usual assumption that the current through an input lead is zero for an opamp because of very high input impedance; thus vref vA v ¼ A Rc Ro or vA ¼ Ro v Ro ỵ Rc ị ref iị Similarly, the current balance at node B gives vi vB v vB ỵ o ẳ0 R R or vo ẳ 2vi ỵ 2vB â 2005 by Taylor & Francis Group, LLC iiị 16-54 Vibration and Shock Handbook Since vA ẳ vB for the opamp (because of very high open-loop gain), we can substitute Equation i into Equation ii Then, vo ẳ 2vi ỵ 2Ro v Ro ỵ Rc ị ref ð16:97Þ Note the sign of vi is reversed at the output (because this is an inverting amplifier) This is not a problem because polarity can be reversed at input or output in connecting this circuit to other circuitry, thereby recovering the original sign The important result here is the presence of a constant offset term on the RHS of Equation 16.97 This term can be adjusted by picking the proper value for Rc so as to compensate for a given offset in vi : 16.7.5 Proportional-Output Circuitry An operational-amplifier circuit may be employed Active Element to linearize the output of a capacitive transverseR1 displacement sensor In constant-voltage and constant-current resistance bridges and in a R3 constant-voltage half bridge, the relation between B − DC Supply Output vref the bridge output, dvo ; and the measurand (the vo A + change in resistance in the active element) is R4 RL Load nonlinear The nonlinearity is least for the R2 constant-current bridge and it is highest for the half bridge Since dR is small compared with R; however, the nonlinear relations can be linearized without introducing large errors However, the FIGURE 16.22 A proportional-output circuit for an linear relations are inexact and are not suitable if active resistance element (strain gage) dR cannot be neglected in comparison to R: Under these circumstances, the use of a linearizing circuit would be appropriate One way to obtain a proportional output from a Wheatstone bridge is to feed back a suitable factor of the bridge output into the bridge supply, vref : Another way is to use the opamp circuit shown in Figure 16.22 This should be compared with the Wheatstone bridge shown in Figure 16.17(a) Note that R represents the only active element (e.g., an active strain gage) First, let us show that the output equation for the circuit in Figure 16.22 is similar to Equation 16.68 Using the fact that the current through an input lead of an unsaturated opamp can be neglected, we have the following current balance equations for nodes A and B: vref vA v ¼ A R4 R2 vref vB vo vB ỵ ẳ0 R3 R1 Hence, vA ẳ R2 v R2 ỵ R4 ị ref vB ẳ R1 vref ỵ R3 vo R1 ỵ R3 Þ and Now, using the fact vA ¼ vB for an opamp, we obtain R1 vref ỵ R3 vo R2 ẳ v R1 ỵ R3 ị R2 ỵ R4 ị ref © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-55 Accordingly, we have the circuit output equation vo ẳ R2 R3 R1 R4 ị v R3 R2 ỵ R4 ị ref 16:98ị Note that this relation is quite similar to the Wheatstone bridge equation (Equation 16.68) The balance condition (i.e., vo ¼ 0) is again given by Equation 16.69 Suppose that R1 ¼ R2 ¼ R3 ¼ R4 ¼ R in the beginning (the circuit is balanced), so vo ¼ 0: Then suppose that the active resistance R1 is changed by dR (say, owing to a change in strain in the strain gage R1 ) Then, using Equation 16.98, we can write an expression for the charge in circuit output as dvo ẳ ẵR2 RR ỵ dRị vref RR ỵ Rị or dvo dR ẳ2 R vref 16:99ị By comparing this result with Equation 16.71, we observe that the circuit output, dvo , is proportional to the measurand, dR: Furthermore, note that the sensitivity of the circuit in Figure 16.22 (1/2) is double that of a Wheatstone bridge that has one active element (1/4), which is a further advantage of the proportional-output circuit The sign reversal is not a drawback because it can be accounted for by reversing the load polarity 16.7.5.1 Curve-Shaping Circuitry A curve shaper can be interpreted as an amplifier Resistance Switching Circuit whose gain is adjustable A typical arrangement for Rf a curve-shaping circuit is shown in Figure 16.23 The feedback resistance, Rf ; is adjustable by some Input R means For example, a switching circuit with a A − Output vi bank of resistors (say, connected in parallel vo through solid-state switches as in the case of + weighted-resistor DAC) can be used to switch the feedback resistance to the required value Automatic switching can be realized by using Zener diodes that will start conducting at certain voltage FIGURE 16.23 A curve-shaping circuit levels In both cases (external switching by switching pulses or automatic switching using Zener diodes), amplifier gain is variable in discrete steps Alternatively, a potentiometer can be used as Rf so that the gain can be continuously adjusted (manually or automatically) The output equation for the curve-shaping circuit shown in Figure 16.23 is obtained by writing the current balance at node A, noting that vA ẳ 0; thus vi v ỵ o ¼0 R Rf or vo ¼ Rf v R i It follows that the gain ðRf =RÞ of the amplifier can be adjusted by changing Rf : © 2005 by Taylor & Francis Group, LLC ð16:100Þ 16-56 16.8 Vibration and Shock Handbook Miscellaneous Signal Modification Circuitry In addition to the signal modification devices discussed so far in this chapter, there are many other types of circuitry that are used for signal modification and related tasks Examples are phase shifters, voltageto-frequency converters, frequency-to-voltage converters, voltage-to-current converts, and peak-hold circuits The objective of the present section is to discuss briefly several of such miscellaneous circuits and components that are useful in the instrumentation of dynamic systems 16.8.1 Phase Shifters A sinusoidal signal given by v ẳ va sinvt ỵ fị R 16:101ị has the following three representative parameters: va ẳ amplitude v ¼ frequency f ¼ phase angle Input vi R Rc B − Output vo A + C Note that the phase angle represents the time FIGURE 16.24 A phase-shifter circuit reference (starting point) of the signal The phase angle is an important consideration only when two or more signal components are compared The Fourier spectrum of a signal is presented as its amplitude (magnitude) and the phase angle with respect to the frequency Phase-shifting circuits have many applications When a signal passes through a system, its phase angle changes due to dynamic characteristics of the system Consequently, the phase change provides very useful information about the dynamic characteristics of the system Specifically, for a linear constant-coefficient system, this phase shift is equal to the phase angle of the frequency-response function ( frequency-transfer function) of the system at that particular frequency This phase-shifting behavior is, of course, not limited to electrical systems and is equally exhibited by other types of systems including mechanical vibrating systems The phase shift between two signals can be determined by converting the signals into the electrical form (using suitable transducers) and shifting the phase angle of one signal through known amounts, using a phase-shifting circuit, until the two signals are in phase Another application of phase shifters is in signal demodulation For example, one method of amplitude demodulation involves processing the modulated signal together with the carrier signal This, however, requires the modulated signal and the carrier signal to be in phase Usually, however, since the modulated signal has already transmitted through electrical circuitry having impedance characteristics, its phase angle has changed Then, it is necessary to shift the phase angle of the carrier until the two signals are in phase, so that demodulation can be performed accurately Hence, phase shifters are used in demodulating, for example, when demodulating LVDT displacement-sensor outputs A phase-shifter circuit, ideally, should not change the signal amplitude while changing the phase angle by a required amount Practical phase shifters can introduce some degree of amplitude distortion (with respect to frequency) as well A simple phase-shifter circuit can be constructed using resistance (R) and capacitance (C) elements A resistance or a capacitor of such an RC circuit is made fine-adjustable so as to obtain a variable phase shifter An opamp-based phase shifter circuit is shown in Figure 16.24 We can show that this circuit provides a phase shift without distorting the signal amplitude The circuit equation is obtained by writing the current balance equations at nodes A and B, noting, as usual, that the current through the opamp leads © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-57 can be neglected due to high input impedance; thus vi vA dv ¼C A RC dt vi vB vo vB ỵ ẳ0 R R On simplifying and introducing the Laplace variable, we obtain vi ¼ ðts þ 1ÞvA ðiÞ ðv þ vo Þ i iiị and vB ẳ in which, the circuit time constant, t; is given by t ¼ Rc C Since vA ¼ vB as a result of very high gain in the opamp, by substituting Equation ii into Equation i, we obtain vi ẳ ts ỵ 1ịvi ỵ vo ị It follows that the transfer function GðsÞ of the circuit is given by vo tsị ẳ Gsị ẳ ỵ tsị vi 16:102ị It is seen that the magnitude of the frequency-response function Gð jvÞ is pffiffiffiffiffiffiffiffiffiffiffiffi ỵ t v2 lGjvịl ẳ p ỵ t v2 or lG jvịl ẳ 16:103ị and the phase angle of Gjvị is /G jvị ẳ 2tan21 tv tan21 tv or /Gjvị ẳ 22 tan21 tv ẳ 22 tan21 Rc Cv 16:104ị As needed, the transfer function magnitude is unity, indicating that the circuit does not distort the signal amplitude over the entire bandwidth Equation 16.104 gives the phase lead of the output, vo ; with respect to the input, vi : Note that this angle is negative, indicating that actually a phase lag is introduced The phase shift can be adjusted by varying the resistance, Rc : 16.8.2 Voltage-to-Frequency Converter A voltage-to-frequency converter (VFC) generates a periodic output signal whose frequency is proportional to the level of an input voltage Since such an oscillator generates a periodic output according to the voltage excitation, it is also called a voltage-controlled oscillator (VCO) A common type of VFC uses a capacitor The time needed for the capacitor to be charged to a fixed voltage level will depend on the charging voltage (it is inversely proportional) Suppose that this voltage is governed by the input voltage Then, if the capacitor is made to periodically charge and discharge, we have an output whose frequency (inverse of the charge –discharge period) is proportional to the charging voltage The output amplitude will be given by the fixed voltage level to which the capacitor is charged in © 2005 by Taylor & Francis Group, LLC 16-58 Vibration and Shock Handbook Reference Level vs VoltageSensitive Switch Input −vi C R A Oscillator Output vo − + (a) Output vo(t) T = RC (vs − vo(0)) vi vs vo(0) (b) FIGURE 16.25 T 2T 3T 4T Time t A voltage-to-frequency converter (voltage-controlled oscillator): (a) circuit; (b) output signal each cycle Consequently, we have a signal with a fixed amplitude and a frequency that depends on the charging voltage (input) A VFC (or VCO) circuit is shown in Figure 16.25(a) The voltage-sensitive switch closes when the voltage across it exceeds a reference level, vs ; and it opens again when the voltage across it falls below a lower limit, vo ð0Þ: The programmable unijunction transistor (PUT) is such a switching device Note that the polarity of the input voltage, vi ; is reversed Suppose that the switch is open Then, current balance at node A of the opamp circuit gives vi dv ¼C o R dt As usual, vA ¼ voltage at positive lead ¼ because the opamp has a very high gain, and current through the opamp leads ¼ because the opamp has a very high input impedance The capacitor charging equation can be integrated for a given value of vi : This gives vo tị ẳ v t þ vo ð0Þ RC i The switch is closed when the voltage across the capacitor vo ðtÞ equals the reference level vs : Then, the capacitor will be immediately discharged through the closed switch Hence, the capacitor charging time, T, is given by vs ẳ â 2005 by Taylor & Francis Group, LLC v T ỵ vo 0ị RC i Signal Conditioning and Modification 16-59 Accordingly, T¼ RC ðv vo ð0ÞÞ vi s ð16:105Þ The switch opens again when the voltage across the capacitor drops to vo ð0Þ; and the capacitor again begins to charge from vo ð0Þ up to vs : This charging and instantaneous discharge cycle repeats periodically The corresponding output signal is as shown in Figure 16.25(b) This is a periodic (sawtooth) wave with period T: The frequency of oscillation of the output ð1=TÞ is given by f ẳ vi RCvs vo 0ịị 16:106ị It is seen that the oscillator frequency is proportional to the input voltage vi : The oscillator amplitude is vs ; which is fixed VCOs have many applications One application is in analog-to-digital conversion In the VCO type analog-to-digital converters, the analog signal is converted into an oscillating signal using a VCO Then, the oscillator frequency is measured using a digital counter This count, which is available in the digital form, is representative of the input analog signal level Another application is in digital voltmeters Here, the same method as for ADC is used Specifically, the voltage is converted into an oscillator signal and its frequency is measured using a digital counter The count can be scaled and displayed to provide the voltage measurement A direct application of the VCO is apparent from the fact it is actually a frequency modulator, providing a signal whose frequency is proportional to the input (modulating) signal Hence, the VCO is useful in applications that require frequency modulation Also, a VCO can be used as a signal (wave) generator for variable-frequency applications; for example, it can be used for excitation inputs for shakers in vibration testing, excitations for frequency-controlled DC motors, and pulse signals for translator circuits of stepping motors 16.8.3 Frequency-to-Voltage Converter A frequency-to-voltage converter (FVC) generates an output voltage whose level is proportional to the frequency of its input signal One way to obtain a FVC is to use a digital counter to count the signal frequency and then use a DAC to obtain a voltage proportional to the frequency A schematic representation of this type of FVC is shown in Figure 16.26(a) Frequency Signal (a) Digital Counter DAC Voltage Output Charging Voltage vs Comparator Frequency Signal − (b) FIGURE 16.26 Switching Circuit Capacitor Circuit Switching Circuit Voltage Output Threshold Signal Frequency-to-voltage converters: (a) digital counter method; (b) capacitor charging method © 2005 by Taylor & Francis Group, LLC 16-60 Vibration and Shock Handbook An alternative FVC circuit is schematically shown in Figure 16.26(b) In this method, the frequency signal is supplied to a comparator along with a threshold voltage level The sign of the comparator output will depend on whether the input signal level is larger or smaller than the threshold level The first sign change (negative to positive) in the comparator output is used to trigger a switching circuit that will respond by connecting a capacitor to a fixed charging voltage This will charge the capacitor The next sign change (positive to negative) of the comparator output will cause the switching circuit to short the capacitor, thereby instantaneously discharging it This charging –discharging process will be repeated in response to the oscillator input Note that the voltage level to which the capacitor is charged each time will depend on the switching period (charging voltage is fixed), which is in turn governed by the frequency of the input signal Hence, the output voltage of the capacitor circuit will be representative of the frequency of the input signal Since the output is not steady due to the ramp-like charging curve and instantaneous discharge, a smoothing circuit is provided at the output to remove the noisy ripples Applications of FVC include demodulation of frequency-modulated signals, frequency measurement in mechanical vibration applications, and conversion of pulse outputs in some types of sensors and transducers into analog voltage signals 16.8.4 Voltage-to-Current Converters Measurement and feedback signals are usually Input Output Current io transmitted as current levels in the range of to Voltage R R vi 20 mA rather than as voltage levels This is B + particularly useful when the measurement site is Load P RL not close to the monitoring room Since the − A R measurement itself is usually available as a voltage, R it has to be converted into current by using a voltage-to-current converter (VCC) For example, pressure transmitters and temperature transmitFIGURE 16.27 A voltage-to-current converter ters in operability testing systems provide current outputs that are proportional to the measured values of pressure and temperature There are many advantages to transmitting current rather than voltage In particular, the voltage level will drop due to resistance in the transmitting path, but the current through a conductor will remain uncharged unless the conductor is branched Hence, current signals are less likely to acquire errors due to signal weakening Another advantage of using current instead of voltage as the measurement signal is that the same signal can be used to operate several devices in series (for example, a display, a plotter, and a signal processor simultaneously), without causing errors through signal weakening due to the power lost at each device, because the same current is applied to all devices AVCC should provide a current proportional to an input voltage without being affected by the load resistance to which the current is supplied An operational-amplifier-based voltage-to-current convert circuit is shown in Figure 16.27 Using the fact that the currents through the input leads of an unsaturated opamp can be neglected (due to very high input impedance), we write the current summation equations for the two nodes, A and B, thus: vp vA vA ¼ R R and vi vB v vB þ P ¼ io R R Accordingly, we have 2vA ¼ vP © 2005 by Taylor & Francis Group, LLC ðiÞ Signal Conditioning and Modification 16-61 and vi 2vB ỵ vP ẳ Rio iiị Now, using the fact that vA ¼ vB for the opamp (due to very high gain), we substitute Equation i into Equation ii This gives v 16:107ị io ẳ i R in which io ẳ output current vi ¼ input voltage It follows that the output current is proportional to the input voltage, irrespective of the value of the load resistance, RL ; as required for a VCC 16.8.5 Peak-Hold Circuits Output Unlike a S/H circuit that holds every sampled value Input Voltage Follower of the signal, a peak-hold circuit holds only the Signal Diode + + Peak Value vi largest value reached by the signal during the − (Output) vo − monitored period Peak holding is useful in a variety of applications In signal processing for v Reset shock and vibration studies, what are known as Switch response spectra (e.g., a shock response spectrum) are determined by using a response spectrum analyzer that exploits a peak holding scheme Suppose that FIGURE 16.28 A peak-holding circuit a signal is applied to a simple oscillator (a singledegree-of-freedom second-order system with no zeros) and the peak value of the response (output) is determined A plot of the peak output as a function of the natural frequency of the oscillator, for a specified damping ratio, is known as the response spectrum of the signal for that damping ratio Peak detection is also useful in machine monitoring and alarm systems In short, when just one representative value of a signal is needed in a particular application, the peak value is a leading contender Peak detection of a signal can be conveniently done using digital processing For example, the signal may be sampled and the previous sample value replaced by the present sample value if and only if the latter is larger than the former By sampling and then holding one value in this manner, the peak value of the signal is retained Note that, usually, the time instant at which the peak occurs is not retained Peak detection can be done using analog circuitry as well This is, in fact, the basis of analog spectrum analyzers A peak-holding circuit is shown in Figure 16.28 The circuit consists of two voltage followers The first voltage follower has a diode at its output that is forward biased by the positive output of the voltage follower and reverse-biased by a low-leakage capacitor, as shown The second voltage follower presents the peak voltage that is held by the capacitor to the circuit output at a low output impedance, without loading the previous circuit stage (capacitor and first voltage follower) To understand the operation of the circuit, suppose that the input voltage, vi ; is larger than the voltage to which capacitor is charged (v) Since the voltage at the positive lead of the opamp is vi and the voltage at the negative lead is v; the first opamp will be saturated Since the differential input to the opamp is positive under these conditions, the opamp output will be positive The output will charge the capacitor until the capacitor voltage, v; equals the input voltage, vi : This voltage (call it vo ) is in turn supplied to the second voltage follower which presents the same value to its output (gain ¼ for a voltage follower), but at a very low impedance level Note that the opamp output remains at the saturated value only for a very short time (the time taken by the capacitor to charge) Now, suppose that vi is smaller than v: Then, the differential input of the opamp will be negative, and the opamp output will be saturated at the negative saturation level This will reverse bias the diode Hence, the output of the first opamp will be in open circuit, and as a result the voltage supplied to the output voltage follower © 2005 by Taylor & Francis Group, LLC 16-62 Vibration and Shock Handbook would still be the capacitor voltage and not the output of the first opamp It follows that the voltage level of the capacitor (and hence the output of the second voltage follower) would always be the peak value of the input signal The circuit can be reset by discharging the capacitor through a solid-state switch that is activated by an external pulse 16.9 Signal Analyzers and Display Devices Vibration signal analysis may employ both analog and digital procedures Since signal analysis results in extracting various useful bits of information from the signal, it is appropriate to consider the topic within the present context of signal modification as well Here, we will introduce digital signal analyzers Signal display devices also make use of at least some signal processing This may involve filtering and change of the signal level and format More sophisticated signal display devices, particularly digital oscilloscopes, can carry out more complex signal analysis functions such as those normally available with digital signal analyzers Oscilloscopes as well are introduced in the present section, though they may be treated under vibration instrumentation Signal-recording equipment commonly employed in vibration practice includes digital storage devices such as hard drives, floppy disks, and CD-ROMs, analog devices like tape recorders, strip-chart recorders and X–Y plotters, and digital printers Tape recorders are used to record vibration data (transducer outputs) that are subsequently reproduced for processing or examination Often, tape-recorded waveforms are also used to generate (by replay) signals that drive vibration test exciters (shakers) Tape recorders use tapes made of a plastic material that has a thin coating of a specially treated ferromagnetic substance During the recording process, magnetic flux proportional to the recorded signal is produced by the recording head (essentially an electromagnet), which magnetizes the tape surface in proportion to the signal variation Reproduction is the reverse process, whereby an electrical signal is generated at the reproduction head by electromagnetic induction in accordance with the magnetic flux of the magnetized (recorded) tape Several signal-conditioning circuitries are involved in the recording and reproducing stages Recording by FM is very common in vibration testing Strip-chart recorders are usually employed to plot time histories (that is, quantities that vary with time), although they also may be used to plot such data as frequency-response functions and response spectra In these recorders, a paper roll unwinds at a constant linear speed, and the writing head moves across the paper (perpendicular to the paper motion) proportionally to the signal level There are many kinds of strip-chart recorders, which are grouped according to the type of writing head that is employed Graphic-level recorders, which use ordinary paper, employ such heads as ink pens or brushes, fiber pens, and sapphire styli Visicoders are simple oscilloscopes that are capable of producing permanent records; they employ light-sensitive paper for this Several channels of input data can be incorporated with a visicoder Obviously, graphic-level recorders are generally limited by the number of writing heads possible (typically, one or two), but visicoders can have many more input channels (typically, 24) Performance specifications of these devices include paper speed, frequency range of operation, dynamic range, and power requirements In vibration experimentation, X –Y plotters are generally employed to plot frequency data (for example, PSD, frequency-response functions, response spectra, transmissibility curves), although they also can be used to plot time-history data Many types of X– Y plotter are available, most of them using ink pens and ordinary paper There are also hard-copy units that use heat-sensitive paper in conjunction with a heating element as the writing head The writing head in an X –Y plotter is moved in the X and Ydirections on the paper by two input signals that form the coordinates for the plot In this manner, a trace is made on stationary plotting paper Performance specifications of X –Y plotters are governed by such factors as paper size; writing speed (in./sec, cm/sec); dead band (expressed as a percentage of the full scale), which measures the resolution of the plotter head; linearity (expressed as a percentage of the full scale), which measures the accuracy of the plot; minimum trace separation (in., cm) for multiple plots on the same axes; dynamic range; input impedance; and maximum input (mV/in., mV/cm) © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-63 Today, the most widespread signal recording device is in fact the digital computer (memory, storage) and printer combination This and the other (analog) devices used in signal recording and display make use of some signal modification to accomplish their functions However, we will not discuss these devices in the present section 16.9.1 Signal Analyzers Modern signal analyzers employ digital techniques of signal analysis to extract useful information that is carried by the signal Digital Fourier analysis using FFT is perhaps the single common procedure that is used in the vast majority of signal analyzers As we have noted before, Fourier analysis will produce the frequency spectrum of a time signal It should be clear, therefore, why the terms digital signal analyzer, FFT analyzer, frequency analyzer, spectrum analyzer, and digital Fourier analyzer are to some extent synonymous as used in the commercial instrumentation literature A signal analyzer typically has two (dual) or more (multiple) input signal channels To generate results such as frequency response (transfer) functions, cross spectra, coherence functions, and cross-correlation functions, we need at least two data signals and hence a dual-channel analyzer In hardware analyzers, digital circuitry rather than software is used to carry out the mathematical operations Clearly, these are very fast but less flexible (in terms of programmability and functional capability) for this reason Digital signal analyzers, regardless of whether they use the hardware or the software approach, employ some basic operations These operations, carried out in sequence, are: Antialias filtering (analog) Analog-to-digital conversion (i.e., single sampling) Truncation of a block of data and multiplication by a window function FFT analysis of the block of data We have noted the following facts If the sampling period of the ADC is DT (i.e., the sampling frequency is 1=DT) then the Nyquist frequency fc ¼ 1=2DT: This Nyquist frequency is the upper limit of the useful frequency content of the sampled signal The cutoff frequency of the antialiasing filter should be set at fc or less If there are N data samples in the block of data that is used in the FFT analysis, the corresponding record length is T ¼ N·DT: Then, the spectral lines in the FFT results are separated at a frequency spacing of DF ¼ 1=T: In view of the Nyquist frequency limit, there will be only N=2 useful spectral lines of FFT result Strictly speaking, a real-time signal analyzer should analyze a signal instantaneously and continuously as the signal is received by the analyzer This is usually the case with an analog signal analyzer However, in digital signal analyzers, which are usually based on digital Fourier analysis, a block of data (i.e., N samples of record length T) is analyzed together to produce N=2 useful spectral lines (at frequency spacing 1=T) This is, then, not a truly real-time analysis For practical purposes, if the speed of analysis is sufficiently fast, the analyzer may be considered real time, which is usually the case with hardware analyzers and also modern, high-speed software analyzers The bandwidth B of a digital signal analyzer is a measure of its speed of signal processing Specifically, for an analyzer that uses N data samples in each block of signal analysis, the associated processing time may be given by Tc ẳ N B 16:108ị Note that the larger the B; the smaller the Tc : The analyzer is considered real-time if the analysis time ðTc Þ of the data record is less than the generation time T ẳ NãDTị of the data record Hence, we need Tc , T © 2005 by Taylor & Francis Group, LLC 16-64 Vibration and Shock Handbook or N ,T B or N , N·DT B or ,B DT ð16:109Þ In other words, a real-time analyzer has a bandwidth greater than its sampling rate A multichannel digital signal analyzer can analyze one or more signals simultaneously and generate (and display) results such as Fourier spectra, power spectral densities, cross spectral densities, frequencyresponse functions, coherence functions, autocorrelations, and cross correlations They are able to perform high-resolution analysis on a small segment of the frequency spectrum of a signal This is termed zoom analysis Essentially, in this case, the spectral line spacing, DF; is decreased while keeping the number of lines (N), and hence the number of time data samples, the same That means the record length T ẳ 1=DFị has to be increased in proportion, for zoom analysis 16.9.2 Oscilloscopes An oscilloscope is used to observe one or two signals separately or simultaneously Amplitude, frequency, and phase information of the signals can be obtained using an oscilloscope In this sense, the oscilloscope is a signal modification as well as a measurement (monitoring) and display device Both analog and digital oscilloscopes are available A typical application of an oscilloscope is to observe (monitor) experimental data such as vibration signals of machinery as obtained from transducers They are also useful in observing and examining vibration test results, such as frequency-response plots, PSD curves, and response spectra Typically, only temporary records are available on an analog oscilloscope screen The main component of an analog oscilloscope is the cathode-ray tube (CRT), which consists of an electron gun (cathode) that deflects an electron ray according to the input-signal level The oscilloscope screen has a coating of electron-sensitive material, so that the electron ray that impinges on the screen leaves a temporary trace on it The electron ray sweeps across the screen horizontally, so that waveform traces can be recorded and observed Usually, two input channels are available Each input may be observed separately, or the variations in one input may be observed against those of the other In this manner, signal phasing can be examined Several sensitivity settings for the input-signal-amplitude scale (in the vertical direction) and sweep-speed selections are available on the panel 16.9.2.1 Triggering The voltage level of the input signal deflects the electron gun proportionally in the vertical (y-axis) direction on the CRT screen This alone will not show the time evolution of the signal The true time variation of the signal is achieved by means of a sawtooth signal that is generated internally in the oscilloscope and used to move the electron gun in the horizontal (x-axis) direction As the name implies, the sawtooth signal increases linearly in amplitude until a threshold value then suddenly drops to zero, and then repeats this cycle again In this manner, the observed signal is repetitively swept across the screen and a trace of it can be observed as a result of the temporary retention of the illumination of the electron gun on the fluorescent screen The sawtooth signal may be controlled (triggered) in several ways For example, the external trigger mode uses an external signal from another channel (not the observed channel) to generate and synchronize the sawtooth signal In the line trigger mode, the sawtooth signal is synchronized with the AC line supply (60 or 50 Hz) In the internal trigger mode, the observed signal (which is used to deflect the electron beam in the y direction) itself is used to generate (synchronize) the © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-65 sawtooth signal Since the frequency and the phase of the observed signal and the trigger signal are perfectly synchronized in the last case, the trace on the oscilloscope screen will appear stationary Careful observation of a signal can be made in this manner 16.9.2.2 Lissajous Patterns Suppose that two signals, x and y; are provided to the two channels of an oscilloscope If they are used to deflect the electron beam in the horizontal and the vertical directions, respectively, a pattern known as Lissajous pattern will be observed on the oscilloscope screen Useful information about the amplitude and phasing of the two signals may be observed by means of these patterns Consider sine waves x and y: Several special cases of Lissajous patterns are given below Same frequency, same phase: Here, x ¼ xo sinðvt ỵ fị y ẳ yo sinvt ỵ fị Then we have x y ¼ xo yo which gives a straight-line trace with a positive slope, as shown in Figure 16.29(a) y y x x (a) (b) y y x x (d) (c) y y x wy (e) yo yintercept wx = y x wy = wx x wy = wx FIGURE 16.29 Some Lissajous patterns: (a) equal frequency and in-phase; (b) equal frequency and 908 out-ofphase; (c) equal frequency and 1808 out-of-phase; (d) equal frequency and u out-of-phase; (e) integral frequency ratio © 2005 by Taylor & Francis Group, LLC 16-66 Vibration and Shock Handbook Same frequency, 908 out-of-phase: Here, x ¼ xo sinvt ỵ fị y ẳ yo sinvt ỵ f ỵ p=2ị ẳ yo cosvt ỵ fị Then we have x xo ỵ y yo ẳ1 which gives an ellipse, as shown in Figure 16.29(b) Same frequency, 1808 out-of-phase: Here, x ẳ xo sinvt ỵ fị y ẳ yo sinvt ỵ f ỵ pị ẳ 2yo sinvt ỵ fị Hence, x y ỵ ẳ0 xo yo which corresponds to a straight line with a negative slope, as shown in Figure 16.29(c) Same frequency, u out-of-phase: x ¼ xo sinvt ỵ fị y ẳ yo sinvt ỵ f ỵ uị When vt ỵ f ẳ 0; y ẳ yintercept ¼ yo sin u: Hence, sin u ¼ yintercept yo In this case, we obtain a tilted ellipse as shown in Figure 16.29(d) The phase difference u is obtained from the Lissajous pattern Integral frequency ratio: vy Number of y-peaks ¼ vx Number of x-peaks Three examples are shown in Figure 16.29(e) vy ¼ ; vx vy ¼ ; vx vy ¼ vx Note: The above observations are true for narrowband signals as well Broadband random signals produce scattered (irregular) Lissajous patterns 16.9.2.3 Digital Oscilloscopes The basic uses of a digital oscilloscope are quite similar to those of a traditional analog oscilloscope The main differences stem from the manner in which information is represented and processed “internally” within the oscilloscope Specifically, a digital oscilloscope first samples a signal that arrives at one of its input channels and stores the resulting digital data within a memory segment This is essentially a typical ADC operation This digital data may be processed to extract and display the necessary information The sampled data and the processed information may be stored on a floppy disk, if needed, for further processing using a digital computer Also, some digital oscilloscopes have the © 2005 by Taylor & Francis Group, LLC Signal Conditioning and Modification 16-67 communication capability so that the information may be displayed on a video monitor or printed to provide a hard copy A typical digital oscilloscope has four channels so that four different signals may be acquired (sampled) into the oscilloscope and displayed Also, it has various triggering options so that the acquisition of a signal may be initiated and synchronized by means of either an internal or an external trigger Apart from the typical capabilities that are possible with an analog oscilloscope, a digital oscilloscope can automatically provide other useful features such as the following: Automatic scaling of the acquired signal Computation of signal features such as frequency, period, amplitude, mean, root-mean-square (rms) value, and rise time Zooming into regions of interest of a signal record Averaging of multiple signal records Enveloping of multiple signal records FFT capability, with various window options and antialiasing These various functions are menu selectable Typically, first a channel of the incoming data (signal) is selected and then an appropriate operation on the data is chosen from the menu (through menu buttons) Bibliography Bendat, J.S and Piersol, A.G 1971 Random Data: Analysis and Measurement Procedures, WileyInterscience, New York Brigham, E.O 1974 The Fast Fourier Transform, Prentice Hall, Englewood Cliffs, NJ Broch, J.T 1980 Mechanical Vibration and Shock Measurements, Bruel and Kjaer, Naerum de Silva, C.W 1983 Dynamic Testing and Seismic Qualification Practice, D.C Heath and Co., Lexington, KY de Silva, C.W., and Palusamy, S.S., Experimental modal analysis — a modeling and design tool, Mech Eng., ASME, 106, 6, 56 –65, 1984 de Silva, C.W., The digital processing of acceleration measurements for modal analysis, Shock Vib Dig., 18, 10, 3–10, 1986 de Silva, C.W 1989 Control Sensors and Actuators, Prentice Hall, Englewood Cliffs, NJ de Silva, C.W 2000 Vibration — Fundamentals and Practice, CRC Press, Boca Raton, FL de Silva, C.W 2004 Mechatronics — An Integrated Approach, CRC Press, Boca Raton, FL de Silva, C.W., Henning, S.J., and Brown, J.D., Random testing with digital control — application in the distribution qualification of microcomputers, Shock Vib Dig., 18, –13, 1986 Ewins, D.J 1984 Modal Testing: Theory and Practice, Research Studies Press Ltd., Letchworth, UK Meirovitch, L 1980 Computational Methods in Structural Dynamics, Sijthoff & Noordhoff, Rockville, MD Randall, R.B 1977 Application of B&K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum, Denmark © 2005 by Taylor & Francis Group, LLC ... gain © 2005 by Taylor & Francis Group, LLC 16: 7Þ 16- 6 Vibration and Shock Handbook It is easy to see from Equation 16. 5 to Equation 16. 7 that Kp ẳ Kv Ki 16: 8ị Note that all three types of amplification... the positive lead and negative lead grounded V at the negative lead and positive lead grounded © 2005 by Taylor & Francis Group, LLC 16- 4 Vibration and Shock Handbook TABLE 16. 1 vip mV 25 mV... Taylor & Francis Group, LLC 16- 14 Vibration and Shock Handbook Example 16. 2 Input impedance, Zi , and output impedance, Zo ; can be represented schematically as in Figure 16. 4(a) Note that vo is