16.7 Linearizing Devices

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Nonlinearity is present in any physical device, to varying levels. If the level of nonlinearity in a system

(component, device, or equipment) can be neglected without exceeding the error tolerance, then the

system can be assumed to be linear.

In general, a linear system is one that can be expressed as one or more linear differential equations. Note

that the principle of superposition holds for linear systems. Specifically, if the system response to an input,

u1; is y1 and the response to another input, u2; is y2; then the response to a1u1 þ a2u2 is a1y1 þ a2y2:

Nonlinearities in a system can appear in two forms:

1. Dynamic manifestation of nonlinearities

2. Static manifestation of nonlinearities

The useful operating region of many systems can exceed the frequency range where the frequencyresponse

function is flat. The operating response of such a system is said to be dynamic. Examples include

a typical dynamic system (e.g., automobile, aircraft, chemical process plant, robot), actuator (e.g.,

hydraulic motor), and controller (e.g., PID control circuitry). Nonlinearities of such systems can

manifest themselves in a dynamic form such as the jump phenomenon (also known as the fold

catastrophe), limit cycles, and frequency creation. Design changes, extensive adjustments, or reduction of

the operating signal levels and bandwidths are generally necessary to reduce or eliminate these dynamic

manifestations of nonlinearity. In many instances, such changes are not practical and we have to

somehow manage with the presence of these nonlinearities under dynamic conditions. Design changes

might involve replacing conventional gear drives with devices such as harmonic drives in order to reduce

backlash, replacing nonlinear actuators with linear actuators, and using components that have negligible

Coulomb friction and that make small motion excursions.

A wide majority of sensors, transducers, and signal-modification devices are expected to operate in the

flat region of the frequency-response function. The input/output relation of these types of devices, in the

operating range, is expressed (modeled) as a static curve rather than a differential equation. Nonlinearities

in these devices will manifest themselves in the static operating curve in many forms. These

manifestations include saturation, hysteresis, and offset.

In the first category of systems (plants, actuators, and compensators), if a nonlinearity is exhibited in

the dynamic form, proper modeling and control practices should be employed in order to avoid

unsatisfactory degradation of the system performance. In the second category of systems (sensors,

transducers and signal modification devices), if nonlinearities are exhibited in the “static” operating

curve, again the overall performance of the system will be degraded. Hence, it is important to “linearize”

the output of such devices. Note that, in dynamic manifestations, it is not realistic to linearize the output

because the response is in the dynamic form. The solution in that case is either to minimize nonlinearities

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© 2005 by Taylor & Francis Group, LLC

through design modifications and adjustments so

that a linear approximation would be valid, or to

take the nonlinearities into account in system

modeling and control. In the present section, we

are not concerned with this aspect. Instead, we are

interested in the linearization of devices in the

second category, whose operating characteristics

can be expressed by static input – output curves.

Linearization of a static device can be attempted

by making design changes and adjustments, as in

the case of dynamic devices. However, since the

response is static, and since we normally deal with

an available (fixed) device whose internal hardware

cannot be modified, we should consider ways

of linearizing the input – output characteristic by

modifying the output itself.

Static linearization of a device can be made in

three ways:

1. Linearization using digital software

2. Linearization using digital (logic) hardware

3. Linearization using analog circuitry

In the software approach to linearization, the output of the device is read into a processor with

software-programmable memory and the output is modified according to the program instructions. In

the hardware approach, the device output is read by a device having fixed logic circuitry that processes

(modifies) the data. In the analog approach, a linearizing circuit is directly connected at the output of the

device so that the output of the linearizing circuit is proportional to the input of the device. We shall

discuss these three approaches in the rest of this section, heavily emphasizing the analog-circuit

approach.

Hysteresis-type static nonlinearity characteristics have the property that the input – output curve is not

one to one. In other words, one input value may correspond to more than one (static) output value, and

one output value may correspond to more than one input value. Let us disregard these types of

nonlinearities. Our main concern is the linearization of a device having a single-valued static response

curve that is not a straight line. An example of a typical nonlinear input – output characteristic is

shown in Figure 16.20(a). Strictly speaking, a straight-line characteristic with a simple offset, as shown in

Figure 16.20(b), is also a nonlinearity. In particular, note that superposition does not hold for an input –

output characteristic of this type, given by

y ¼ ku þ c ð16:93Þ

It is very easy, however, to linearize such a device because the simple addition of a DC component will

convert the characteristic into the linear form given by

y ¼ ku ð16:94Þ

This method of linearization is known as offsetting. Linearization is more difficult in the general case

where the characteristic curve could be much more complex.

16.7.1 Linearization by Software

If the nonlinear relationship between the input and the output of a nonlinear device is known, the input

can be “computed” for a known value of the output. In the software approach of linearization, system

(a) Input u

Output

y

Output

y

0

(b) 0 Input u

FIGURE 16.20 (a) A general static nonlinear characteristic;

(b) an offset nonlinearity.

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composed of a processor and memory that can be programmed using software (i.e., a digital computer) is

used to compute the input using output data. Two approaches can be used. They are

1. Equation inversion

2. Table lookup

In the first method, the nonlinear characteristic of the device is known in the analytic (equation) form:

y ¼ f ðuÞ ð16:95Þ

in which

u ¼ device input

y ¼ device output

Assuming that this is a one-to-one relationship, a unique inverse given by the equation

u ¼ f 21ðyÞ ð16:96Þ

can be determined. This equation is programmed into the read-and-write memory (RAM) of the

computer as a computation algorithm. When the output values, y; are supplied to the computer, the

processor will compute the corresponding input values, u; using the instructions (executable program)

stored in the RAM.

In the table lookup method, a sufficiently large number of pairs of values ðy; uÞ are stored in the

memory of the computer in the form of a table of ordered pairs. These values should cover the entire

operating range of the device. When a value for y is entered into the computer, the processor scans the

stored data to check whether that value is present. If so, the corresponding value of u is read and this is

the linearized output. If the value of y is not present in the data table, then the processor will interpolate

the data in the vicinity of the value and will compute the corresponding output. In the linear interpolation

method, the area of the data table where the y value falls is fitted with a straight line and the

corresponding u value is computed using this straight line. Higher order interpolations use nonlinear

interpolation curves such as quadratic and cubic polynomial equations (splines).

Note that the equation inversion method is usually more accurate than the table lookup method and it

does not need excessive memory for data storage. However, it is relatively slow because data are

transferred and processed within the computer using program instructions that are stored in the memory

and that typically have to be accessed in a sequential manner. The table lookup method is fast. Since the

accuracy depends on the number of stored data values, this is a memory-intensive method. For better

accuracy, more data should be stored. However, since the entire data table has to be scanned to check for

a given data value, this increase in accuracy is derived at the expense of speed as well as memory

requirements.

16.7.2 Linearization by Hardware Logic

The software approach to linearization is flexible in the sense that the linearization algorithm can be

modified (e.g., improved, changed) simply by modifying the program stored in the RAM. Furthermore,

highly complex nonlinearities can be handled by the software method. As mentioned before, the method

is relatively slow, however.

In the hardware logic method of linearization, the linearization algorithm is permanently

implemented in the IC form using appropriate digital logic circuitry for data processing, and memory

elements (e.g., flip-flops). Note that the algorithm and numerical values of parameters (except input

values) cannot be modified without redesigning the IC chip, because a hardware device typically does not

have programmable memory. Furthermore, it will be difficult to implement very complex linearization

algorithms by this method and, unless the chips are mass produced for an extensive commercial market,

the initial cost of chip development will make the production of linearizing chips economically infeasible.

In bulk production, however, the per-unit cost will be very small. Furthermore, since the access of stored

Signal Conditioning and Modification 16-51

© 2005 by Taylor & Francis Group, LLC

program instructions and extensive data manipulation are not involved, the hardware method of

linearization can be substantially faster than the software method.

A digital linearizing unit that has a processor and a read-only memory (ROM), whose program cannot

be modified, also lacks the flexibility of a programmable software device. Hence, such a ROM-based

device also falls into the category of hardware logic devices.

16.7.3 Analog Linearizing Circuitry

Three types of analog linearizing circuitry can be identified:

1. Offsetting circuitry

2. Circuitry that provides a proportional output

3. Curve shapers

We will describe each of these categories now.

An offset is a nonlinearity that can be easily removed using an analog device. This is accomplished by

simply adding a DC offset of equal value to the response, but in the opposite direction. Deliberate

addition of an offset in this manner is known as offsetting. The associated removal of original offset is

known as offset compensation. There are many applications of offsetting. Unwanted offsets such as those

present in ADC and DAC results can be removed by analog offsetting. Constant (DC) error components

such as steady-state errors in dynamic systems due to load changes, gain changes, and other disturbances

can be eliminated by offsetting. Common-mode error signals in amplifiers and other analog devices can

also be removed by offsetting. In measurement circuitry such as potentiometer (ballast) circuits, where

the actual measurement signal is a change, dvo; in a steady output signal, vo; the measurement can be

completely wiped out due to noise. To reduce this problem, first the output should be offset by 2vo so

that the net output is dvo and not vo þ dvo: This output is subsequently conditioned by filtering and

amplification. Another application of offsetting is the additive change of scale of a measurement, for

example from a relative scale (e.g., velocity) to an absolute scale. In summary, some of the applications of

offsetting are:

1. Removal of unwanted offsets and DC components in signals (e.g., in ADC, DAC, signal

integration).

2. Removal of steady-state error components in dynamic system responses (e.g., due to load changes

and gain changes in Type 0 systems. Note: Type 0 systems are open-loop systems having no free

integrators).

3. Rejection of common-mode levels (e.g., in amplifiers and filters).

4. Error reduction when a measurement is an increment of a large steady output level (e.g., in ballast

circuits for strain-gage and RTD sensors).

5. Scale changes in an additive manner (e.g., conversion from relative to absolute units or from

absolute to relative units).

We can remove unwanted offsets in the simple manner as discussed above. Let us now consider more

complex nonlinear responses that are nonlinear in the sense that the input – output curve is not a straight

line. Analog circuitry can be used to linearize this type of response as well. The linearizing circuit used

will generally depend on the particular device and the nature of its nonlinearity. Hence, often linearizing

circuits of this type have to be discussed with respect to a particular application. For example, such

linearization circuits are useful in transverse-displacement capacitative sensor. Several useful circuits are

described below.

Consider the type of linearization that is known as curve shaping. A curve shaper is a linear device

whose gain (output/input) can be adjusted so that response curves with different slopes can be obtained.

Suppose that a nonlinear device having an irregular (nonlinear) input – output characteristic is to be

linearized. First, we apply the operating input simultaneously to the device and the curve shaper, and

the gain of the curve shaper is adjusted such that it closely matches that of the device in a small range

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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

and other analog devices can be minimized by

including a compensating resistor that can provide

fine adjustments at one of the input leads.

Furthermore, the larger is the feedback signal

level in a feedback system, the smaller is the steadystate

error. Hence, steady-state offsets can be

reduced by reducing the feedback resistance

(thereby increasing the feedback signal). Furthermore,

since a ballast (potentiometer) circuit

provides an output of vo þ dvo and a bridge

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 2 vA

Rc ¼

vA

Ro

or

vA ¼

Ro

ðRo þ RcÞ

vref ðiÞ

Similarly, the current balance at node B gives

vi 2 vB

R þ

vo 2 vB

R ¼ 0

or

vo ¼ 2vi þ 2vB ðiiÞ

vref

(DC Reference)

Input

vi Output

vo

+

R B −

R

R A c

Ro

FIGURE 16.21 An inverting amplifier circuit for offset

compensation.

Signal Conditioning and Modification 16-53

© 2005 by Taylor & Francis Group, LLC

Since vA ¼ vB for the opamp (because of very high open-loop gain), we can substitute Equation i into

Equation ii. Then,

vo ¼ 2vi þ

2Ro

ðRo þ RcÞ

vref 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

to linearize the output of a capacitive transversedisplacement

sensor. In constant-voltage and

constant-current resistance bridges and in a

constant-voltage half bridge, the relation between

the bridge output, dvo; and the measurand (the

change in resistance in the active element) is

nonlinear. The nonlinearity is least for the

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

linear relations are inexact and are not suitable if

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 2 vA

R4 ¼

vA

R2

vref 2 vB

R3 þ

vo 2 vB

R1 ¼ 0

Hence,

vA ¼

R2

ðR2 þ R4Þ

vref

and

vB ¼

R1vref þ R3vo

ðR1 þ R3Þ

Now, using the fact vA ¼ vB for an opamp, we obtain

R1vref þ R3vo

ðR1 þ R3Þ ¼

R2

ðR2 þ R4Þ

vref

DC Supply

vref

Output

vo

+

R3 B −

R A 4

R2

Active

Element

R1

RL

Load

FIGURE 16.22 A proportional-output circuit for an

active resistance element (strain gage).

16-54 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Accordingly, we have the circuit output equation

vo ¼ ðR2R3 2 R1R4Þ

R3ðR2 þ R4Þ

vref ð16:98Þ

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 ¼ ½R2 2 RðR þ dRÞ􀀉

RðR þ RÞ

vref 2 0

or

dvo

vref ¼ 2

1

2

dR

R ð16:99Þ

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

whose gain is adjustable. A typical arrangement for

a curve-shaping circuit is shown in Figure 16.23.

The feedback resistance, Rf ; is adjustable by some

means. For example, a switching circuit with a

bank of resistors (say, connected in parallel

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

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 ¼ 0; thus

vi

R þ

vo

Rf ¼ 0

or

vo ¼ 2

Rf

R

vi ð16:100Þ

It follows that the gain ðRf =RÞ of the amplifier can be adjusted by changing Rf :

Input

vi Output

vo

+

A − R

Rf

Resistance Switching

Circuit

FIGURE 16.23 A curve-shaping circuit.

Signal Conditioning and Modification 16-55

© 2005 by Taylor & Francis Group, LLC

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