Other Sensing Methods.pdf

OTHER SENSING METHODS

The very wide variety of devices and methods used to measure di¨erent physi-

cal quantities makes it di½cult for any sensor classi®cation criterion to be ex-

haustive. The criterion followed in this book is not an exception. This chapter

discusses some additional sensors and measurement methods that are not based

on any of the measurement principles described in previous chapters. They rely

on semiconductor devices (not just semiconductor materials) or on some radi-

ation modi®ed by the measurand.

9.1

SENSORS BASED ON SEMICONDUCTOR JUNCTIONS

Semiconductor junctions are the basis of self-generating sensors such as some

photoelectric cells (Section 6.4) and of several modulating sensors. The latter,

however, need a current or voltage bias in order to provide a useful output, in

the same way that modulating sensors based on resistance or reactance varia-

tion need voltage or current excitation.

Sensors based on semiconductor junctions are of twofold interest. First, the

large yield of microfabrication processes results in very competitive prices for

them. Second, it is possible to include sensor, signal conditioning, signal proc-

essing, and communication circuits to produce intelligent sensors (Section 8.7).

The journal Sensors and Actuators reports on scienti®c research in sensors (and

actuators) based on semiconductors (in general). The magazine Sensors covers

industrial developments. Reference 1 authoritatively reviews fundamentals and

applications of semiconductor sensors.

501

502

OTHER SENSING METHODS

9.1.1

Thermometers Based on Semiconductor Junctions

The forward characteristic for a diode is temperature-dependent (about ÿ2

mV/ C for silicon diodes), which is usually considered a shortcoming. How-

ever, we can use that dependence to measure temperature or any other quantity

related to a change in temperature (see Problem 9.1). But this dependence is

nonlinear and not repetitive enough for accurate measurements. It is therefore

better to use the temperature dependence of the base±emitter voltage v BE of a

transistor supplied with a constant collector current.

According to the Ebers±Moll model, the collector current for an ideal tran-

sistor is

i C a F I ES e qv BE =kT ÿ 1ÿI CS e ÿqv CB =kT ÿ 1

9:1

where

a F the forward current transfer ratio

I ES the emitter saturation current

q 0:160 aC is the electron charge

v BE the base±emitter voltage

k 1:3807 10 ÿ23 J/K is Boltzmann's constant

T the absolute temperature

I CS the collector saturation current

v CB the collector±base voltage

The product a F I ES is sometimes designated I S . In the active zone, i C g I S .If

in addition we make the collector±base voltage zero, from (9.1) we deduce

v BE kT

i C

I S

9:2

which shows that v BE depends on the temperature, but I S is also temperature-

dependent according to [2]

I S BT 3 e ÿqV g0 =kT

9:3

where B is a constant that depends on doping level and on the geometry but

does not depend on the temperature, and V g0 is the band-gap voltage (1.12 V at

300 K for silicon).

By combining (9.2) and (9.3), we obtain

v BE kT

i C

BT 3 V g0

9:4

503

9.1

SENSORS BASED ON SEMICONDUCTOR JUNCTIONS

If we designate V BE0 the base±emitter voltage corresponding to a constant col-

lector current I C0 at a given temperature T 0 , then we have

v BE kT

i C

I C0

T 0

V BE0 ÿ V g0 T

T 0 V g0

9:5

The relation between v BE and T is therefore nonlinear and depends on the

collector current. To quantify the nonlinearity, we take the derivative with re-

spect to the temperature at a given constant collector current. For i C I C0 we

have

i C I C0 V BE0 ÿ V g0

dv BE

ÿ 3k

T 0

1 ln

9:6

T 0

The ®rst term on the right-hand side is the sensitivity, while the second term

describes the nonlinearity. Their respective values for silicon are about ÿ2:2

mV/ C and 0.34 mV/ C.

Example 9.1 The thermometer in Figure E9.1 uses a diode-connected transis-

tor that at 25 C has v BE 0:595 V and ÿ2:265 mV/ C temperature coe½cient

when the collector current is 100 mA. If I 0 100 mA, design the circuit to obtain

an output range from 0 V to 10 V for a temperature range from 0 Cto100 C.

Determine the temperature error at 0 C because of the op amps' o¨set voltages

when the op amps are at ambient temperature of 30 C. If the resistors have

1 % tolerance, determine the error due to their standardized value and their

tolerance.

Figure E9.1 Thermometer based on the temperature coe½cient of the base±emitter

junction of a diode-connected transistor.

504

OTHER SENSING METHODS

The output voltage will be

ÿ v BE R 2

R 1

1 R 2

R 1

v o I 0 R 0

where the base±emitter voltage is

v BE T0:595 V ÿ2:265 mV= CT ÿ 25 C

The conditions to ful®ll at 0 C and 100 C are

ÿ 0:595 V ÿ2:265 mV= C0 C ÿ 25 C R 2

R 1

1 R 2

R 1

0V I 0 R 0

ÿ 0:595 V ÿ2:265 mV= C100 C ÿ 25 C R 2

R 1

1 R 2

R 1

10 V I 0 R 0

which lead to the equation system

ÿ0:6516 V R 2

R 1

1 R 2

R 1

0V10 ÿ4

AR 0

ÿ0:4521 V R 2

R 1

1 R 2

R 1

10 V 10 ÿ4

AR 0

We obtain R 2 =R 1 44:15. If R 1 1kW, we need R 2 44:1kW and R 0

6371 W.

The output voltage because of o¨set voltages is

ÿ V io1 R 2

1 R 2

R 1

v o 0V io2

R 1 45:15V io2 ÿ 44:15V io1

Because of power dissipation, op amps will raise their temperature above 30 C.

Nevertheless, the OP07A has its o¨set voltage speci®ed after warm-up. There-

fore, we need to consider only the temperature di¨erence from 25 C ambient

temperature in data sheets to 30 C actual ambient temperature. In a worst-case

condition, for the output op amp we have

V io2 25 mV 0:6 mV= C30 C ÿ 25 C28 mV

For the ®rst op amp, the worst-case condition is to have an equal but opposite

initial voltage (ÿ25 mV ) and the typical drift (instead of the maximal drift as

supposed for the output op amp). Hence,

V io2 ÿ25 mV 0:2 mV= C30 C ÿ 25 Cÿ24 mV

505

9.1

SENSORS BASED ON SEMICONDUCTOR JUNCTIONS

and

v o 045:15 28 mVÿ44:15ÿ24 mV2:3mV

which implies an error of about 0:02 C.

If we do not trim each resistor, we have errors because calculated values may

be di¨erent from standard resistor values, and also because of resistor toler-

ance. If R 1 1kW the closest standard values for R 2 and R 0 with 1 % tolerance

are R 2 44:2kW and R 0 6:34 kW. We will therefore have zero and sensitiv-

ity error. The worst-case situation because of tolerance will happen when R 0

and R 2 have their minimal value and R 1 is maximal. At 0 C we will have

A6:34 kW0:99 1 44:2 0:99

1 1:01

v o 010 ÿ4

ÿ0:6516 V 44:2 0:99

1 1:01 ÿ0:4V

which implies an error of ÿ4 C. At 100 C, the same resistors would yield

A6:34 kW0:99 1 44:2 0:99

1 1:01

v o 10010 ÿ4

ÿ0:4251 V 44:2 0:99

1 1:01 9:4V

Therefore, the sensitivity would be 98 mV/ C instead of 100 mV/ C.

The nonlinearity of the base±emitter voltage and the requirement for a col-

lector current that must be kept constant with time and temperature make this

solution unattractive. The usual alternative consists of using two bipolar tran-

sistors whose emitter current densities have a constant ratio.

A method for that uses two identical transistors supplied by di¨erent collec-

tor currents (Figure 9.1a). If both sensors are at the same temperature, the dif-

ference between the respective base±emitter currents is

v d v BE1 ÿ v BE2 kT

I C1

I S1 ÿ kT

I C2

I S2

9:7

If both transistors are assumed identical, we have I S1 AI S2 and

v d kT

I C1

I C2

9:8

Therefore, if I C1 =I C2 is constant, v d will be proportional to T, without re-

quiring any current source to be kept constant. It is su½cient to have this ratio

between both current sources constant. In Figure 9.1a, I C1 =I C2 2, so that

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