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16.2 Amplifiers

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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.

16-2 Vibration and Shock Handbook

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

ðþvs and 2vsÞ; and a ground lead.

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

(no feedback) conditions

vo ¼ Kvi ð16:1Þ

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 2 vin ð16:2Þ

The open loop voltage gain K is very high (105 to 109) for a typical opamp. Furthermore, the input

impedance, Zi; could be as high as 1 MV and the output impedance is low, of the order of 10 V. Since vo

is typically 1 to 10 V, from Equation 16.1 it follows that vi ø 0 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, 1 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 ¼ Kvip ð16:3Þ

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 1 £ 105. If the saturation voltage is 15 V, determine the

output voltage in the following cases:

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

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

3. 5 mV at the positive lead and 2 2 mV at the negative lead.

4. 2 5 mV at the positive lead and 2 2 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.

vip

vin

(Power Supply)

Inputs Output

vo= K vi

vi Zi

Kvi

−

(a)

(b)

vin

vip

−

FIGURE 16.1 Operational amplifier: (a) a schematic

model; (b) conventional symbol.

Signal Conditioning and Modification 16-3

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:

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 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:

1. Frequency response

2. 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.

TABLE 16.1 Solution to Example 16.1

vip vin vi vo

5 mV 2mV 3mV 0.3 V

2 5 mV 2mV 2 7 mV 2 0.7 V

5 mV 2 2 mV 7mV 0.7 V

2 5 mV 2 2 mV 2 3 mV 2 0.3 V

1 V 0 1V 15 V

0 1V 21 V 2 15 V

16-4 Vibration and Shock Handbook

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:

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

circuitry

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 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

vo ¼ Kv vi ð16:5Þ

in which

vo ¼ 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 ¼ Kiii ð16:6Þ

in which

io ¼ output current

ii ¼ input current

Ki ¼ current gain

Note that voltage follower has Kv ¼ 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) 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 KpPi ð16:7Þ

in which

po ¼ output power

pi ¼ input power

Kp ¼ power gain

Signal Conditioning and Modification 16-5

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

Kp ¼ Kv Ki ð16:8Þ

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

(a)

(b)

(c)

Input

Output

−

ii Input

(Output)

i + o

A −

RL Load

Feedback

Capacitor

−vo /k K

Sensor

Charge

vo vo

Output

Voltage Drop Across Zo = 0

−

+ −

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

16-6 Vibration and Shock Handbook

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 2 vi

Rf ¼

This gives the amplifier equation

vo ¼ 1 þ

􀀏 􀀐

vi ð16:9aÞ

Hence, the voltage gain is given by

Kv ¼ 1 þ

R ð16:9bÞ

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 þ

io ¼ 1 þ

􀀏 􀀐

ii ð16:10aÞ

The current gain of the amplifier is

Ki ¼ 1 þ

R ð16:10bÞ

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:

q þ

Cc þ vo þ

􀀏 􀀐

Cf ¼ 0 ð16:11Þ

From this, we obtain

vo ¼ 2

ðK þ 1ÞCf þ Cc

q ð16:12aÞ

Signal Conditioning and Modification 16-7

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 ¼ 2

Cf ð16:12bÞ

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 2 vB

R ¼

Rf ðiÞ

in which vB is the voltage at B. Similarly, current balance at point A gives

vo 2 vA

Rf ¼

vA 2 vi1

R ðiiÞ

Now, we use the property

vA ¼ vB ðiiiÞ

for an operational amplifier to eliminate vA and vB from Equation i and Equation ii. This gives

vi2

ð1 þ R=Rf Þ ¼ ðvoR=Rf þ vi1Þ

ð1 þ R=Rf Þ

16-8 Vibration and Shock Handbook

vo ¼

R ðvi2 2 vi1Þ 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

(a)

(b)

vi1

Inputs Output

A −

R B

vi2

Output

R4+δ R4

−

vi1

Inputs

vi2

FIGURE 16.3 (a) A basic differential amplifier; (b) a basic instrumentation amplifier.

Signal Conditioning and Modification 16-9

differential amplifier. However, since the operational amplifiers are not ideal and since they usually do

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 2 should be vi2 and the voltage at

point 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 continuity

equations

vB 2 vi2

R1 ¼

vi2 2 vi1

R2 ¼

vi1 2 vA

in which VA and VB are the voltages at points A and B, respectively. Hence, we obtain the two equations

vB ¼ vi2 þ

R2 ðvi2 2 vi1Þ

vA ¼ vi1 2

R2 ðvi2 2 vi1Þ

Now, by subtracting the second equation from the first, we have the equation for the first stage of the

amplifier; thus

vB 2 vA ¼ 1 þ

2R1

􀀏 􀀐

ðvi2 2 vi1Þ ðiÞ

From the previous result (see Equation 16.13) for a differential amplifier, we have (with dR4 ¼ 0)

vo ¼

R3 ðvB 2 vAÞ ðiiÞ

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) do 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 do 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:

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 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).

16-10 Vibration and Shock Handbook

The speed of response of an amplifier dictates the ability of the amplifier to faithfully respond to

transient 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 the

bandwidth 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 input

amplitude, 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 the

output amplitude (i.e., the product of the DC gain and the input amplitude). However, for a device that

has a limited slew rate, the bandwidth, or the maximum operating frequency at which output distortions

may be neglected, will depend on the output amplitude. The larger the output amplitude, the smaller the

bandwidth for a given slew rate limit.

We have noted that stability problems and frequency-response errors are prevalent in the open-loop

form of an operational amplifier. These problems can be eliminated using feedback because the effect of

the open-loop transfer function on the closed loop transfer function is negligible if the open-loop gain is

very large, which is the case for an operational amplifier.

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

1. Bias currents

2. Offset signals

3. Common-mode output voltage

4. Internal noise

In analyzing operational amplifiers, we assume that the current through the input leads is zero. This is

not strictly true because bias currents for the transistors within the amplifier circuit have to flow through

these leads. As a result, the output signal of the amplifier will deviate slightly from the ideal value.

Another assumption that we make in analyzing opamps is that the voltage is equal at the two input

leads. However, in practice, offset currents and voltages are present at the input leads, due to minute

discrepancies inherent to the internal circuits within an opamp.

16.2.5.1 Common-Mode Rejection Ratio

Common-mode error in a differential amplifier was discussed earlier. We noted that ideally the commonmode

input voltage (the voltage common to both input leads) should have no effect on the output

voltage of a differential amplifier. However, since a practical amplifier has imbalances in the internal

circuitry (for example, gain with respect to one input lead is not equal to the gain with respect to the

other input lead and, furthermore, bias signals are needed for operation of the internal circuitry), there

will be an error voltage at the output that depends on the common-mode input. The common-mode

rejection ratio (CMRR) of a differential amplifier is defined as

CMRR ¼

Kvcm

vocm ð16:14Þ

in which

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

vcm ¼ common-mode voltage (i.e., voltage common to both input leads)

vocm ¼ common-mode output voltage (i.e., output voltage due to common-mode input voltage)

Note that, ideally, vocm ¼ 0 and CMRR should be infinity. It follows that the larger the CMRR, the

better the differential amplifier performance.

The three types of unmodeled signals mentioned above can be considered as noise. In addition, there

are other types of noise signals that degrade the performance of an amplifier. For example, ground-loop

Signal Conditioning and Modification 16-11

noise can enter the output signal. Furthermore, stray capacitances and other types of unmodeled circuit

effects 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 be

reduced by using suitably connected compensating circuitry, including variable resistors that can be

adjusted to eliminate the effect of unmodeled signals at the amplifier output (e.g., see dR4 in Figure

16.3(b)). Some useful information about operational amplifiers is summarized in Box 16.1.

Box 16.1

OPERATIONAL AMPLIFIERS

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 lead

grounded and the output in open circuit

* Output impedance ¼

Voltage between output lead and ground in open circuit

Current through that lead

with 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 the

output 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 gain

Common-mode gain

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

16-12 Vibration and Shock Handbook

16.2.6 Component Interconnection

When two or more components are interconnected, the behavior of the individual components in the

overall system can deviate significantly from their behavior of each component when they operate

independently. The matching of components in a multicomponent system should be done carefully in

order to improve system performance and accuracy, particularly with respect to their impedance

characteristics. 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 at

each interface in order to realize the devices’ rated performance level. One adverse effect of improper

impedance matching is the loading effect. For example, in a measuring system, the measuring instrument

can distort the signal that is being measured. The resulting error can far exceed other types of

measurement error. Loading errors will result from connecting a measuring device with low input

impedance 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 an

additional dynamic load that will modify the actual acceleration at the monitoring location. Similarly, a

voltmeter 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, thereby

significantly affecting digitized data.

Another adverse effect of improper impedance consideration is inadequate output signal levels, which

can make signal processing and transmission very difficult. Many types of transducers (e.g., piezoelectric

accelerometers, impedance heads, and microphones) have high output impedances in the order of a

thousand megohms. These devices generate low output signals, and they require conditioning to step up

the signal level. Impedance-matching amplifiers, which have high input impedances (megohms) and low

output impedances (a few ohms), are used for this purpose (e.g., charge amplifiers are used in

conjunction with piezoelectric sensors). A device with a high input impedance has the further advantage

that it usually consumes less power (v2=R is low) for a given input voltage. The fact that a low input

impedance device extracts a high level of power from the preceding output device may transpire to be the

reason 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 as

the ratio of the open-circuit (i.e., no-load) voltage at the output port to the short-circuit current at the

output port.

Open-circuit voltage at the output is the output voltage present when there is no current flowing at the

output 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 very

low-impedance ammeter is connected at the output port.

The input impedance, Zi; is defined as the ratio of the rated input voltage to the corresponding

current through the input terminals while the output terminals are maintained as an open

circuit.

Note that these definitions are associated with electrical devices. A generalization is possible that

includes both electrical and mechanical devices; one must interpret voltage and velocity as across

variables, and current and force as through variables. Then, mechanical mobility can be used in place of

electrical impedance in the associated analysis.

Signal Conditioning and Modification 16-13

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, vi and vo are

represented by their respective Fourier spectra. The

corresponding transfer relation can be expressed in

terms of the complex frequency-response (transfer)

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

conditions:

vo ¼ Gvi ð16:15Þ

Now, consider two devices connected in cascade,

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

that the following relations apply:

vo1 ¼ G1vi ðiÞ

vi2 ¼

Zi2

Zo1 þ Zi2

vo1 ðiiÞ

vo ¼ G2vi2 ðiiiÞ

These relations can be combined to give the overall input/output relation

vo ¼

Zi2

Zo1 þ Zi2

G2G1vi ð16:16aÞ

We see from Equation 16.16a that the overall frequency-transfer function differs from the ideally

expected product ðG2G1Þ by the factor

Zi2

Zo1 þ Zi2 ¼

Zo1=Zi2 þ 1 ð16:16bÞ

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

Zo1=Zi2 p 1; this deviation becomes insignificant. From this observation, it can be concluded that, when

frequency-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 the

input 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. (Note

that the impedance of a capacitor is 1=ðjvCÞ and, hence, at zero frequency there will be an infinite

impedance.) If the input lead of a device has a series capacitor, we say that the input is AC coupled and, if

the output lead has a series capacitor, then the output is AC coupled. Typically, an AC-coupled amplifier

has a series capacitor both at the input lead and the output lead. Hence, its frequency-response function

will 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 an

AC-coupled amplifier, stability problems are not very serious.

Input vo Output

vo = Gvi

vi Zi

+ Zo

− −

−

− −

(a)

(b)

vo1 vi Zi1

Zo1

G1 G2

v Zo2 Zi2 vi2 o

FIGURE 16.4 (a) Schematic representation of input

impedance and output impedance; (b) the influence of

cascade connection of devices on the overall impedance

characteristics.

16-14 Vibration and Shock Handbook

16.3 Analog Filters

Unwanted signals can seriously degrade the

performance of a vibration monitoring and

analysis system. External disturbances, error components

in excitations, and noise generated

internally within system components and instrumentation

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

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