108 I CHAPTER 2

3. The high- and low-pass RC filters are passive circuits used to block undesired fre-quencies from data signals.

4. Operational amplifiers (op amps) are a special signal-conditioning building blockaround which many special-function circuits can be developed. The device wasdemonstrated in applications involving amplifiers, converters, linearization circuits,integrators, and several other functions.

PROBLEMS

Section 2.22.1 Derive Equation (2.1) for general circuit loading.2.2 The unloaded output of a sensor is a sinusoid at 200 Hz and 5 V rms amplitude. Its

output impedance is 2000 + 6007. lf a0.22- pF capacitor is placed across the out-put as a load, what is the sensor output rms voltage amplitude?

Section 2.32.3 A sensor resistance varies from 520 to 2500 O. This is used for R1 in the divider of

Figure 2.4, along with R2 : 500,f} and % : 10.0 V. Find (a) the range of the di-vider voltage, Vp and (b) the range of power dissipation by the sensor.

2.4 Prepare graphs of the divider voltage versus transducer resistance for Example 2.2and Problem2.3. Does the voltage vary linearly with resistance? Does the voltageincrease or decrease with resistance?

2.5 Show how the bridge offset equation given as Equation (2.7) can be derived fromEquation (2.6).

2.6 Derive Equation (2.t0) for the bridge circuit Th6venin resistance.2.7 A Wheatstone bridge, as shown in Figure 2.5, nulls with R1 : 227 O, Rz : M8 O,

and R3 : I4l4 '0. Find Ra.

2.8 A sensor with a nominal resistance of 50 O is used in a bridge withRr : Rz : 100 f,), V : 10.0 V, and Rr : 100-,f} potentiometer. It is necessary toresolve 0.1-O changes of the sensor resistance.a. At what value of R3 will the bridge null?b. What voltage resolution must the null detector possess?

2.9 A bridge circuit is used with a sensor located 100 m away. The bridge is not leadcompensated, and the cable to the sensor has a resistance of 0.45 O/tt.The bridgenulls with R1 :3400O,R2 :3aA5 O, and Rl: 1560O. What is the sensorresistance?

2.10 The bridge in Figure 2.5 has Rr : 250 O, Rr : 500 O, & _ 340 f,1, and V --1.5 V. The detector is a galvanometer with R6 : 150 O.a. Find the value of R2 that will null the bridge.b. Find the offset current that will result if R2 : 190 O.

2.ll A current balance bridge, shown in Figure 2.8, has resistances of Rr :Rz : 1k f,l, Ra : 590 f,), Rs : 10 Cl, andV : 10.0 V.

-IANALOG STGNAL CONDTTTONTNG | 10e

a. Find the value of R3 that nulls the bridge with no current.b. Find the value of R, that balances the bridge with a current of 0.25 mA.

2.12 A potential measurement bridge, such as that in Figure 2.9, has V : 10.0 V,

Rr : Rz : Rr : 10 kO. Find the unknown potential if the bridge nulls withRa : 9'73 kO'

2,13 An ac Wheatstone bridge with all arms as capacitors nulls when Cr : 0.4 pF,Cz : 0.31;r.rF, and C3 - 0.27 pF. Find Ca.

2.14 The ac bridge of Figure 2.48 nulls with R1 : I kO. Rz : 2 kO, R: : 100 O, and

Lz : 250 mH'a. Find the values of Ro and La.

b. If the circuit is excited by a 5-V rms, I -kHz oscillator, find the offset voltagefor Lo: 510 mH.

c. What are the amplitudes of the in-phase and quadrature (90') components ofthe offset voltage?

2.15 Develop a low-pass RC filter to attenuate 0.5 MHz noise by 97Vo. Specify the crit-ical frequency, values of R and C, and the attenuation of a 400-Hz input signal.

2.16 A low-pass RC filter has f. : 3.5 kHz. Find the attenuation of a 1-kHz signal.

2.17 A high-pass RC filter must drive 120 Hz noise down to l%o. Specify the filter criti-cal frequency, values of R and C and the attenuation of a 30-kHz signal.

2.18 A high-pass filter is found to attenuate a I -kHz signal by 20 dB. What is the criticalfrequency?

2.19 Design a band-pass filter with critical frequencies of 100 Hz and l0 kHz, respec-tively. Use a resistance ratio of 0.05. Draw a semilog graph like that in Figure 2.21

showing voltage output to input from l0 Hz to 100 kHz.2.20 A sensor output needs to feed an amplifier with a l0-kO input impedance. There is

significant noise in the range of 4 to 5 kHz. The data spectrum lies below 200 Hz.

FIGURE 2.48ac bridge for Problem 2.14.

=-

110 | CHAPTER 2

Design a low-pass filter for use between the sensor and the amplifier that reducesthe data by not more than I Vo.By how much is the noise reduced?

2.21 A telephone line will be used to carry measurement data as a frequency-modulatedsignal from 5 to 6 kHz. The line is shared with voice data below 500 Hz, and switch-ing noise occurs above 500 kHz. Design a band-pass RC filter that reduces the voiceby 80Vo and the switching by 90Vo. Use a resistance ratio of r : 0.02. What is theeffect on the passband frequencies?

2.22 A single line is multiplexed to carry sensor signals in a frequency range belowI kHz and communication signals ranging from 10 to 50 kHz. There is a large noisecomponent at 4.5 kHz from a turbine in the plant. Design a twin-T notch filter forthe 4.5-kHz noise. Evaluate the effect on sensor and communication signals.

Section 2.52.23 Show how op amps can be used to provide an amplifier with a gain of +100 and an

input impedance of 1.5 kO. Show how this can be done using both inverting andnoninverting configurations.

2.24 Specify the components of a differential amplifier with a gain of 22.2.25 Using an integrator with RC : l0 s and any other required amplifiers, develop a

voltage ramp generator with 0.5 V/s.2.26 Signal-conditioning analysis shows that the following equation must relate output

voltage to input voltage:

Vu, : 3.35 V" - 2.68

Design circuits to do this using (a) a summing amplifier and (b)amplifier.

2.27 Adifferential amplifierhas Rz : 470 kO and R, : 2.7 kO. When V,the output is 87 mV. Find the CMR and CMRR.

2.28 Derive Equation (2.41) for the instrumentation amplifier of Figdre 2.37.2.29 Design an instrumentation amplifier like that of Figure 2.37 with switch-selectable

gains of l, 10, 100, and 1000. Show the complete circuit using 741opamps, pin con-nections, and input offset adjustment.

2.30 A control system needs the average of temperature from three locations. Sensorsmake the temperature information available as voltag€s, 7r, V2, and V3. Develop anop amp circuit that outputs the average of these voltages.

2.31 Use an inverting amplifiea an integrator, and a summing amplifier to develop an out-put voltage given by

rVo,,: l}V^ * 4 J V,ndt

2.32 Develop a voltage-to-current convener that satisfies the requirement .I : 0.0021 %.If the op amp saturation voltage is t12 V and the maximum current delivery is 5 mA,find the maximum load resistance.

a differential

: Vt: 2.5Y

\

ANALOG StcNAL COND|T|ON|NG | 111

Section 2.62.33 AbridgecircuithasRl : R:: R+: l20Oand l/: l0.0V.Designasignal-

conditioning system that provides an output of 0.0 to 5.0 V as Rj varies from 120 to140 O. Plot Vo,,, versus R.,. Evaluate the linearity.

2.34 Develop signal conditioning for Example 2.2 so an output voltage varies from 0 to5 V as the resistance varies from 4 to 12 kO.

2.35 Develop signal conditioning for Problem 2.3 so the output voltage varies from 0 to5 V as the resistance varies from 520 to 2500 f,), where 0 V corresponds to 520 O.

2.36 A sensor varies from I to 5 kO. Use this in an op amp circuit to provide a voltagevarying from 0 to 5 V as the resistance changes.

2.37 A process signal varies from 4 to 20 mA. The setpoint is 9.5 mA. Use a current-to-voltage converter and a summing amplifier to get a voltage error signal with a scalefactor of 0.5 V/mA.

2.38 Sensor resistance varies from 25 to 1.5 k0 as a variable changes from c,.,.,1n to cmax.

Design a signal-conditioning system that provides an output voltage varying from-2 to +2V as the variable changes from min to max. Power dissipation in the sen-sor must be kept below 2.5 mW.

2.39 A pressure sensor outputs a voltage varying as 100 mV/psi and has a2.5- k,f) outputimpedance. Develop signal conditioning to provide 0 to 2.5 V as the pressure variesfrom 50 to 150 psi.

2.40 A system is needed to measure flow, which continuously cycles between 20 and 30gallmin with a period of 30 s. The required output is a voltage varying from -25 to+2.5 V for the cycling flow range. The sensor to be used has a transfer function of\@ volts, where Q is in gaVmin, and an output impedance of 2.0 kf,). Tests showthat the output of the sensor has 60 Hz noise of 0.8 V rms. Design a signal-conditioning system, including noise filtering, and evaluate your design as follows.a. Plot output voltage versus flow, and comment on the linearity.b. Determine the noise on the output as percent FS.

SU PPLEMENTARY PROBLEMS

Figure 2.49 shows a system proposed as a scale for weighing. The basic sensor is a re-sistor, R',,, that linearly converts weight to resistance; for 0.00 lb, it nominally has a re-sistance of I l9 O, and at299lb it has a resistance of 127 O. The bridge offset voltage isamplified by a differential amplifier and sent to a DVM whose voltage, by design, willequal the weight (i.e., a weight of 134 lb should resulr in a voltage of 1.34 V so the DVMwill read 134\. Neat, huh?

The purpose of the resistor combination in the bridge is to allow resetting the bridgeto zero using the variable resistor, R.-. This will allow compensation for changes of Rp, orany of the other resistors for that matter. The next three questions are related to thissystem.

112 | CHAPTER 2

Ii

FIGURE 2.49Circuit for supplementary problems.

S2.1 Consider first only the bridge circuit in Figure 2.49.a. What value of R. will be required to null the bridge at 0.00 lb?

b. For what minimum and maximum values of Rs' can the bridge be nulled

using R.?c. What offset voltage, A% results when the weightis299lb (assuming the

bridge is nulled at 0.01b)?52.2 Let us consider the amplifier of Figure 2.49 next.

a. Find the gain, K so that the DVM indicates the weight but in volts (i.e., when

the weight is 299 lb, the voltmeter reads 2.99 Vs).b. Provide the circuit for a differential amplifier that can provide this gain. Specify

the resistors in the amplifier circuit. Show how the amplifier inputs are

connected to bridge points a and b to give the right output polarity.

S2.3 Let's evaluate how well the system in Figure 2.49 works and propose a change.

a. First suppose the weight is 150Ib. What will the DVM read? Therefore, what is

the error in lb? Suppose the scale is exact at 0.00 lb and 299lb but has errors at

150Ib. Why is there an error?

b. Change the gain so the reading is exact at 150Ib. Now specify the error at 0 lb(which no one weighs) and299lb (which few people weigh).

c. Prepare a plot of voltage reading versus weight.52.4 Figure 2.50 shows how a single wire can be used to carry measurement data at the

same time. This is done by modulating two widely different carrier frequencies withdata and using filters to extract the data at the receiving end. Suppose one data chan-

nel is a modulated signal of 1.0 to 1.5 kHz and the other is a modulated signal of 50 to

55 kHz. Design the extraction filters using simple RC filters such that the data loss

0.00-to 2.99-V DVM(decimal suppressed)

All resistors in ohms (O)

ANALOG STGNAL COND|TION|NG | 113

FIGURE 2.50System for Problem 52.4.

is restrictedto0.T0T (3 dB, or 50Vo power) in the data signals. How much amplitudecrossover results? (What is the amplitude ratio of the 50-55 kHz in the l-1.5-kHzchannel and vice versa?)

S2.5 A humidity sensor resistance varies linearly from 250 kO to 120 kO as humidityvaries from 0Vo to l0OVo. Power dissipation in the sensor must be kept below100 pW. Design analog signal conditioning to provide a voltage of 0.00 to 1.00 V as

the humidity varies from 0Vo to l0OVo.

52.6 In some cases, we need amplifiers that have high gain when the input voltage islow and decreasing gain as the input voltage increases. So this is an amplifierwhose gain depends upon the input voltage! An op amp circuit such as that inFigure 2.51 can provide this response. Here diodes are used to isolate feedbackresistors via their forward voltage drop until the output voltage rises above pre-defined levels. For the circuit in Figure 2.51, assume the forward voltage drop ofthe diodes is 1.4 V (i.e., they do not begin to conduct until the voltage across themis above 1.4 V). Prepare a plot of output voltage versus input voltage for this cir-cuit. (Hint: When diode D1 begins to conduct, the output must be -1.4 V and thiseffectively just puts the 100-kO resistors in parallel.)

FIGURE 2.51Nonlinear amplifier using diodes for Problems52.6 and 52.7.

Signal 2

100 ko D1

114 | CHAPTER 2

1.4

1.3

1.2

l.lI

0.9

0.8

Voltage 0.7

0.6

0.5

0.40.3

0.2

0.1

0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Pressure (psi)

FIGURE 2.52Voltage versus pressure for Problem 52.7.

52.7 Circuits such as Figure 2.51 can also be used to provide some degree of lineariza-tion. To see this, consider a sensor whose output voltage varies nonlinearly with in-put pressure by the equation V(p) : 0.035p2. This response is shown in Figure2.52. Now, assume the sensor voltage is provided as input to the circuit in Figure2.51. Determine the output voltage over the pressure range, and plot Vou, versus p.

You will see that the resulting voltage is more nearly linear.