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vout =
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RF RF vsensor + 1 + RS RS
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Vref
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Substituting the expression for vsensor into the equation above, we nd that: vout = = RF RF (18 + 01 cos( t)) + 1 + RS RS Vref Vref
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RF RF RF (01 cos( t)) (18) + 1 + RS RS RS
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Since the intent of the design is to remove the DC offset, we require that or Vref = (18)
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RF RS
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RF RF (18) + 1 + RS RS
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Vref = 0
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= 1714 V
VS+ R R RP + Vref
Comments: The presence of a precision voltage source in the circuit is undesirable,
because it may add considerable expense to the circuit design and, in the case of a battery, it is not adjustable The circuit of Figure 1217 illustrates how one can generate an adjustable voltage reference using the DC supplies already used by the op-amp, 2 resistors, R, and a potentiometer, Rp The resistors R are included in the circuit to prevent the potentiometer from being shorted to either supply voltage when the potentiometer is at the extreme positions Using the voltage divider rule, we can write the following expression for the reference voltage generated by the resistive divider: Vref = R+ R 2R + Rp VS+ VS
R VS
12
Operational Ampli ers
If the voltage supplies are symmetrical, as is almost always the case, one can further simplify the expression to: Vref = R+ R 2R + Rp VS+
Note that, by adjusting the potentiometer, Rp , one can obtain any value of Vref between the supply voltages
FOCUS ON MEASUREMENTS
Sensor Calibration Circuit
In many practical instances, the output of a sensor is related to the physical variable we wish to measure in a form that requires some signal conditioning The most desirable form of a sensor output is one in which the electrical output of the sensor (for example, voltage) is related to the physical variable by a constant factor Such a relationship is depicted in Figure 1218(a), where k is the calibration constant relating voltage to temperature Note that k is a positive number, and that the calibration curve passes through the (0, 0) point On the other hand, the sensor characteristic of Figure 1218(b) is best described by the following equation: vsensor = T + V0
vout vsensor
V0
T ( C)
0 (b)
T ( C)
Figure 1218 Sensor calibration curves
It is possible to modify the sensor calibration curve of Figure 1218(b) to the more desirable one of Figure 1218(a) by means of the simple circuit displayed in Figure 1219 This circuit provides the desired calibration constant k by a simple gain adjustment, while the zero (or bias) offset is adjusted by means of a potentiometer connected to the voltage supplies The detailed operation of the circuit is described in the following paragraphs As noted before, the nonideal characteristic can be described by the following equation: vsensor = T + V0
Part II
Electronics
vsensor RS
VS+ + Vref VS vout
Figure 1219 Sensor calibration circuit
Then, the output of the op-amp circuit of Figure 1219 may be determined by using the principle of superposition: vout = = RF RF vsensor + 1 + FS RS Vref
RF RF ( T + V0 ) + 1 + RS RS
Vref
After substituting the expression for the transducer voltage and after some manipulation, we see that by suitable choice of resistors, and of the reference voltage source, we can compensate for the nonideal transducer characteristic We want the following expression to hold: vout = RF RF T + 1 + RS RS Vref RF V0 = kT RS
If we choose RF =k RS and Vref = RF /RS V0 1 + RF /RS
then vout = kT Note that Vref V0 if RF RS 1
and we can directly convert the characteristic of Figure 1218(b) to that of Figure 1218(a) Clearly, the effect of selecting the gain resistors is to change the magnitude of the slope of the calibration curve The fact that the sign of the slope changes is purely a consequence of the inverting con guration of the ampli er The reference voltage source simply shifts the DC level of the characteristic, so that the curve passes through the origin
12
Operational Ampli ers
Practical Op-Amp Design Considerations The results presented in the preceding pages suggest that operational ampli ers permit the design of rather sophisticated circuit in a few very simple steps, simply by selecting appropriate resistor values This is certainly true, provided that the circuit component selection sati es certain criteria Here we summarize some important practical design criteria that the designer should keep in mind when selecting component values for op-amp circuits Section 126 explores the practical limitations of op-amps in greater detail 1 Use standard resistor values While any arbitrary value of gain can in principle be achieved by selecting the appropriate combination of resistors, the designer is often constrained to the use of standard 5 percent resistor values (see Table 21) For example, if your design requires a gain of 25, you might be tempted to select, say, 100-k and 4-k resistors to achieve RF /RS = 25 However, inspection of Table 21 reveals that 4 k is not a standard value; the closest 5 percent tolerance resistor value is 39 k , leading to a gain of 2564 Can you nd a combination of standard 5 percent resistors whose ratio is closer to 25 2 Ensure that the load current is reasonable (do not select very small resistor values) Consider the same example given in 1 Suppose that the maximum output voltage is 10 V The feedback current required by your design with RF = 100 k and RS = 4 k would be IF = 10/100,000 = 01 mA This is a very reasonable value for an op-amp, as you will see in Section 126 If you tried to achieve the same gain using, say, a 10- feedback resistor and a 039- source resistor, the feedback current would become as large as 1 A This is a value that is generally beyond the capabilities of a general-purpose op-amp, so the selection of exceedingly low resistor values is not acceptable On the other hand, the selection of 10-k and 390- resistors would still lead to acceptable values of current, and would be equally good As a general rule of thumb, you should avoid resistor values lower than 100 in practical designs 3 Avoid stray capacitance (do not select excessively large resistor values) The use of exceedingly large resistor values can cause unwanted signals to couple into the circuit through a mechanism known as capacitive coupling This phenomenon is discussed in 15 Large resistance values can also cause other problems As a general rule of thumb, you should avoid resistor values higher than 1 M in practical designs 4 Precision designs may be warranted If a certain design requires that the ampli er gain be set to a very accurate value, it may be appropriate to use the (more expensive) option of precision resistors: for example, 1 percent tolerance resistors are commonly available, at a premium cost Some of the examples and homework problems explore the variability in gain due to the use of higher and lower tolerance resistors
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