Modelo de estudio

The following three types of analog linearizing hardware are discussed now:

1. Offsetting circuitry

2. Circuitry that provides a proportional output 3. Curve shapers

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, 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 the results of ADC and DAC 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 as well 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 small change δvo of a steady output signal v_o, the measurement can be completely masked by noise. To overcome this problem, first the output should be offset by −v_o, so that the net output is δvo and not v_o + δvo. Subsequently, this output can be conditioned through filtering and amplification. Another application of offsetting is the additive change of the scale of a measurement from a relative scale to an absolute scale (e.g., in the case of velocity). In summary, some 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 with 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-gauge and RTD sensors)

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

We can remove unwanted offsets in the simple manner as discussed earlier. Let us now consider more complex nonlinear responses that are nonlinear, in the sense that the I/O curve is not a straight line.

Analog circuitry can be used to linearize this type of responses 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 a transverse-displacement capacitive sensor. Several useful circuits are described later.

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 with an irregular (nonlinear) I/O characteristic is to be linearized.

First, we apply the operating input simultaneously to both the device and the curve shaper, and then adjust the gain of the curve shaper such that its output closely matches that of the actual device in a small range 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 has to be adjusted 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 by 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: In this manner, we can replace a nonlinear device by a linear device (curve shaper) within a multicomponent system without greatly sacrificing the accuracy of the overall system.

2.9.2.1 Offsetting Circuitry

Common-mode outputs and offsets in amplifiers and other analog devices can be minimized by includ-ing a compensatinclud-ing resistor, which will provide fine adjustments at one of the input leads. Furthermore, the larger the magnitude of the feedback signal in a control system, the smaller the steady-state error.

Hence, steady-state offsets can be reduced by decreasing the feedback resistance (thereby increasing the feedback signal). Furthermore, since a ballast (potentiometer) circuit provides an output of vo + δvo

and a bridge circuit provides an output of δvo, the use of a bridge circuit can be interpreted as an offset compensation method.

The most straightforward way of offsetting a nonlinear device is by using a differential amplifier (or a summing amplifier) to subtract (or add) a dc voltage to the output of the device. The dc level has to be variable so that various levels of offset can be provided with the same circuit. This is accomplished by using an adjustable resistance at the dc input lead of the amplifier.

An op-amp circuit that can be used for offsetting is shown in Figure 2.47. Since the input vi is con-nected to the negative lead of the op-amp, 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 op-amp, we will have a noninverting amplifier.

The dc voltage vref provides the offsetting voltage. The compensating resistor Rc is variable so that dif-ferent values of offset can be compensated for 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 op-amp (because of very high input impedance). Hence

v v

FIGURE 2.47 An inverting amplifier circuit for offset compensation.

Similarly, the current balance at node B gives v v

R

v v

R

i- B + o- B

= 0

or

vo= - + 2 (2.118)vi vB

Since vA = vB for the op-amp (because of very high open-loop gain), we can substitute Equation 2.117 in Equation 2.118 to get,

v v R

R R v

o i o

o c

= - + ref

+ 2

( ) (2.119)

Note the sign reversal of vi at the output (because this is an inverting amplifier). This is not a problem because the polarity can be reversed at input or output when 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 2.119. This term can be adjusted by picking the proper value for Rc so as to com-pensate for a given offset in vi.

2.9.2.2 Proportional-Output Hardware

An op-amp circuit may be employed to linearize the output of a capacitive transverse-displacement sensor. We have noted that in the constant-voltage and constant-current resistance bridges and in the constant-voltage half bridge, the relation between the bridge output δvo and the measurand (change in resistance in the active element) is nonlinear in general. The lowest nonlinearity is with the constant-current bridge and the highest is with the half bridge. As δR is small compared with R, the nonlinear relations can be linearized without introducing large errors. However, the linear relations are inexact, and are not suitable if δR cannot be neglected in comparison to R. Then, 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. This approach is illustrated previously (see Figure 2.43c).

Another way is to use the op-amp circuit shown in Figure 2.48. This should be compared with the Wheatstone bridge shown in Figure 2.43a. In Figure 2.48, R1 represents the only active element (e.g., an active strain gauge).

DC supply

v_ref Output

v_o +

– R₃ B

R₄ A

R₂ Active element

R₁

R_L Load

FIGURE 2.48 A proportional-output circuit for an active resistance element (strain gauge).

First, let us show that the output equation for the circuit in Figure 2.48 is quite similar to Equation 2.91. Using the fact that the current through an input lead of an unsaturated op-amp can be neglected, we have the current balance equations for nodes A and B:

v v

Accordingly, we have the circuit output equation

v R R R R

This relation is quite similar to the Wheatstone bridge equation (Equation 2.91). The balance condition (i.e., v_o = 0) is again given by Equation 2.92.

Suppose that R₁ = R₂ = R₃ = R₄ = R in the beginning (hence, the circuit is balanced), so that v_o = 0.

Next, suppose that the active resistance R₁ is changed by δR (say, due to a change in strain in the strain gauge R₁). Then, using Equation 2.120, we can write an expression for the resulting change in the circuit output as

By comparing this result with Equation 2.93, we observe that the circuit output δvo is proportional to the measurand δR. Furthermore, note that the sensitivity (1/2) of the circuit in Figure 2.48 is double that of a Wheatstone bridge (1/4) with one active element, 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.

2.9.2.3 Curve-Shaping Hardware

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 2.49. 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 a weighted-resistor DAC) can be used to switch the feedback resistance to the required value. Automatic switching can be realized by using Zener diodes as well, which will start conducting at certain voltage levels. In both cases (i.e., external switching by switching pulses and auto-matic switching using Zener diodes), amplifier gain is variable in discrete steps. Alternatively, a poten-tiometer may be used as R_f so that the gain can be continuously adjusted (manually or automatically).

The output equation for the curve-shaping circuit shown in Figure 2.49 is obtained by writing the current balance at node A, noting that v_A = 0. Thus, (v_i/R) + (v_o/R_f) = 0; or,

v R

R v

o f

= - i (2.122)

It is seen that the gain (Rf/R) of the amplifier can be adjusted by changing Rf.