Moulson A.J., Herbert J.M. Electroceramics: Materials, Properties, Applications

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be assumed that the diﬀerence between the temperature T of the resistor and the

temperature T

of its surroundings is proportional to the power being dissipated,

i.e.

T  T

¼ k

UI ¼ k

P ð4:12Þ

where P is the power, U is the voltage drop across the thermistor, I is the current

and k

is a constant depending on the mounting, shape and surface ﬁnish of the

thermistor as well as the ambient conditions. Because the power dissipated is

proportional to 1=k

, this can be regarded as a ‘dissipation factor’. Evidently k

can vary between wide limits, but the following example from experiment gives

the order of magnitude involved. Small rods 5–50 mm long and 3–10 mm in

diameter with copper leads about 10 mm long and 0.5–2 mm in diameter

mounted in air have a heat dissipation per degree Celsius excess over that of the

ambient given by

1=k

¼ g

A þ g

ð4:13Þ

where A is the surface area and d is the lead diameter. If A is in square

millimetres and d is in millimetres, g

 0:04 and g

 10. As indicated by

Eq. (4.13), when the thermistors are very small the heat dissipation is governed

by conduction down the leads rather than by convection currents in the

surrounding air, which have the dominating inﬂuence in the case of larger units.

Substituting Eq. (4.12) into Eq. (4.10) and referring to the resistance of the

device rather than to the resistivity of the material gives

RðTÞ¼R

exp



þ k



ð4:14Þ

RðTÞ can be calculated as a function of P from Eq. (4.14); U and I can then be

derived from the relations

¼ PRðTÞ

¼ P=RðTÞ

ð4:15Þ

P and I are shown as functions of U in Fig. 4.16 and R is shown as a

function of U in Fig. 4.17. The curves represent equilibrium conditions, and it

is evident that no equilibrium can exist above a certain maintained maximum

voltage. If a higher maintained voltage is applied, the current will go on rising

indeﬁnitely until the accompanying high temperature destroys the unit. In

practice there must always be a temperature-insensitive resistor in series with

a thermistor if suﬃcient power to raise its temperature appreciably is to be

applied.

162 CERAMIC CONDUCTORS

TEAMFLY

Team-Fly

The maximum voltage that can be applied to a thermistor, without a runaway

condition being established, can be calculated from the voltage–power relation

obtained by substituting Eq. (4.14) into Eq. (4.15):

U ¼ P

1=2



exp



þ k



1=2

ð4:16Þ

The power PðU

max

Þ corresponding to the maximum voltage U

max

that can be

maintained across the thermistor without runaway occurring can be found by

diﬀerentiating Eq. (4.16) and putting dU=dP ¼ 0. Under the condition that

=ðB  2T

<< 1,

PðU

max

Þ¼

B  2T

ð4:17Þ

The corresponding expression for U

max

can be found by substituting PðU

max

from Eq. (4.17) into Eq. (4.16) to give

max





1=2

ðB  2T

1=2

exp



BðB  2T

ðB  T



ð4:18Þ

TEMPERATURE-SENSITIVE RESISTORS 163

Fig. 4.16 Power and current versus voltage for an NTC thermistor with R

¼ 4:54  10

3

B ¼ 3000 K, T

¼ 300 K and T  T

¼ 30P K.

It is apparent that U

max

increases with the ability of the thermistor to dissipate

heat (proportional to 1=k

). Also, because the exponential term dominates, U

max

decreases as T

increases, as expected.

Some of the properties and uses of thermistors, other than temperature

sensing, can be appreciated from the simple circuit shown in Fig. 4.18. A ﬁxed

voltage U is applied to an NTC thermistor of resistance RðTÞ in series with a load

resistance R

which is invariant with temperature. In this case there is the

complication that, as the thermistor warms up and falls in resistance, the voltage

across it also falls. The situation is analysed as follows:

I ¼

RðTÞþR

ð4:19Þ

and the voltage U

across the thermistor is

¼ RðTÞI ¼

URðTÞ

RðTÞþR

ð4:20Þ

The power P dissipated in the thermistor is

P ¼ IU

RðTÞ

fRðTÞþR

ð4:21Þ

Eliminating P between Eqs (4.14) and (4.21) and rearranging yields

164 CERAMIC CONDUCTORS

Fig. 4.17 Resistance versus voltage under the same conditions as in Fig. 4.16.

fRðTÞþR

RðTÞ



lnfRðTÞ=R

 T



ð4:22Þ

Eq. (4.22) can be used to plot the relationship between U and RðTÞ for given

values of the other parameters. The relationship for B ¼ 3000 K, T

¼ 300 K,

¼ 4:54  10

3

O, R

¼ 50 O and k

¼ 30 K W

1

is shown in Fig. 4.19 as an

example. The current I (Eq. (4.19)) and the temperature of the thermistor

calculated from Eqs (4.21) and (4.12) are also shown. It can be seen from the

ﬁgure that, for U ¼ 50 V, RðTÞ5 O and I  0:9 A and, when temperature

equilibrium has been established, T  T

¼ 127 K (T ¼ 154 8C).

The room temperature resistance Rð300Þ of the thermistor, calculated from the

equivalent to Eq. (4.10) and using R

¼ 4:54  10

3

O, is 100 O so that

immediately the switch is closed and before the temperature of the thermistor

has changed the current is 0.33 A and the power dissipated in the load is 5.4 W.

After the thermistor has reached temperature equilibrium with its surroundings,

the current rises to 0.9 A and power dissipation in the load rises to 40 W. Thus a

thermistor can be used to delay the development of full power in a load during

the period it takes to reach its ﬁnal temperature. The penalty is the power

required to maintain the thermistor at temperature, in this case about 4 W. Using

a thermistor with a higher TCR, i.e. a higher B value, would not greatly aﬀect the

ratio of the power in the load when ﬁrst switched on to that at equilibrium, but it

would decrease the power wasted in the thermistor, provided that RðT

Þ is the

same in both instances.

A further consideration in the use of NTC resistors arises from the distribution

of temperature within them. Clearly the inner part of the resistor must be hotter

than the surface region in order to maintain an outward heat ﬂow, but, in

addition, the negative coeﬃcient of resistivity will result in the inner part having

a lower resistance and a higher current passing through it, so that the

temperature gradient between the inner and outer parts will be enhanced. If

the thermistor is grossly overloaded, the results can be catastrophic; for instance,

TEMPERATURE-SENSITIVE RESISTORS 165

Fig. 4.18 NTC thermistor with a series load.

the central core of a rod-shaped NTC resistor may melt. In practical cases a

severe mechanical stress can arise from the temperature gradient and may result

in fracture or the development of internal cracks. Thermal stress due to

diﬀerences in thermal expansion coeﬃcients may also cause the metallization to

separate from the ceramic. The probability of thermal stress failure depends on

the same thermal and mechanical properties as thermal shock failure, which is

discussed in Section 5.3.

At low power levels NTC thermistors are widely used wherever temperature

needs to be controlled or accurately measured, for example in automotive

engines, air-conditioning units, hair dryers, and in the medical ﬁeld for heart

catheters, fever thermometers, etc. They are also used to maintain picture

stability in television receivers by compensating for increases in the resistance of

the beam-focusing coils as temperature rises in the cabinet. The dependence of

resistance on the rate of heat dissipation enables them to be used as indicators

that fuel tanks are ﬁlled to a prescribed level and also in instruments for the

measurement of the velocity of ﬂuids.

Examples of NTC thermistors are shown in Fig. 4.15.

166 CERAMIC CONDUCTORS

Fig. 4.19 Resistance, current and temperature versus voltage for an NTC thermistor with

series load: P ¼ðT  T

Þ=30 W.

4.4.2 Positive temperature coe⁄cient resistors

(PTC thermistors)

PTC resistors could be classiﬁed as critical temperature resistors because, in the

case of the most widely used type, the positive coeﬃcient is associated with the

ferroelectric Curie point.

A typical PTC characteristic is shown in Fig. 4.20. In the instance illustrated

the material has the negative resistivity–temperature characteristic associated

with normal semiconductors up to about 100 8C (AB) and above about 200 8C

(CD), while between these temperatures (BC) there is an increase of several

orders of magnitude in resistivity. The underlying physics of the eﬀect as outlined

below draws on the discussions in Sections 2.6.2 and 2.7.3.

The PTC eﬀect is exhibited by specially doped and processed BaTiO

. Because

the eﬀect is not observed in the single-crystal form of the material its cause must

be assumed to lie in processes associated with grain boundaries. Attention here is

focused on lanthanum-doped BaTiO

(BLT), although other donor dopants

would be satisfactory, e.g. yttrium (A site) or niobium, tantalum or antimony (B

site).

Electron acceptor states in the grain boundary together with nearby ionized

donor states give rise to an electrical double layer, as shown in Fig. 4.21. In

consequence, conduction band electrons moving up to a grain boundary from

the interior of a grain are confronted by a potential barrier of height f.To

obtain an expression for f the simplifying assumption is made that the positive

charge density is constant out to a distance d from the grain boundary, where it

falls to zero, as shown in Fig. 4.22. It is also assumed that the potential varies

TEMPERATURE-SENSITIVE RESISTORS 167

Fig. 4.20 Typical characteristic of PTC thermistor material.

only with the distance x from the grain boundary. The barrier height is found by

ﬁrst integrating the one-dimensional form of Poisson’s equation

¼

ð4:23Þ

in which V and r are respectively the electrostatic potential and the charge

density at x and e is the permittivity. If the boundary condition E ¼dV=dx ¼ 0

at x ¼ d is assumed and V ¼ 0 is arbitrarily ﬁxed at x ¼ 0, the potential V

x ¼ d is given by

¼r

ð4:24Þ

168 CERAMIC CONDUCTORS

Fig. 4.21 Electrical double layer at a grain boundary.

Fig. 4.22 Assumed positive charge density distribution in the vicinity of a grain boundary.

From Fig. 4.22 rd ¼

jej where N

is the surface density of acceptor states near

the grain boundary and jej is the magnitude of the electronic charge. Therefore

the height f of the barrier to an electron becomes

f ¼eV

8en

ð4:25Þ

where n ¼ r=jej is the volume density of donor states in the grain.

The probability that electrons are able to surmount the barrier is measured by

the Boltzmann factor expðf=kTÞ, leading to the following proportionality for

the resistance R

of a grain boundary:

/ exp





ð4:26Þ

Because BLT is ferroelectric, above its Curie temperature e ¼ C=ðT  yÞ (see

Eq. (2.90)), where C is the Curie constant and y is the Curie–Weiss temperature.

Therefore

/ exp



8nkC



1 



T > y ð4:27Þ

The PTC eﬀect is seen to have its origins in the resistance of the grain boundary

region which increases exponentially with temperature above the ferroelectric–

paraelectric transition temperature. It therefore depends on the number of grain

boundaries per unit volume of ceramic, i.e. on the microstructure, and of course

on the acceptor and donor state densities N

and n.

It is evident that the double layers at the grain boundaries constitute Schottky

barriers which are similar in some respects to those formed in VDR resistors. In

accord with this it is found that the resistivity–temperature relation of PTC

material is voltage sensitive. The low-temperature resistivity may be reduced by a

factor of 4 by an increase in applied ﬁeld from 1 to 80 kV m

1

, and the ratio of

maximum to minimum resistivities, above and below T

, may be reduced from

ﬁve to three orders of magnitude.

Because the PTC eﬀect is partly a consequence of the steeply falling

permittivity just above the ferroelectric transition temperature, why does the

resistance not similarly increase with a strongly decreasing permittivity as

temperature falls below the Curie temperature? Below T

the material is

ferroelectric with each grain comprising domains terminating on grain

boundaries. A discontinuity in polarization at the grain boundary necessitates

a polarization charge at the grain boundary whose sign depends upon the nature

of the discontinuity. This surface charge partially cancels the double-layer eﬀect

and removes, or at least reduces, the barrier in places. The barriers throughout a

polycrystalline ceramic may be short circuited by such a process so that the

material as a whole has low resistivity. The eﬀect will depend upon details of the

TEMPERATURE-SENSITIVE RESISTORS 169

domain conﬁguration, the magnitude of the polarization and the density of

surface acceptor states. The lack of precise information about these precludes the

possibility of predicting electrical properties in this temperature region.

In its essentials the model serves as a sound basis for understanding the PTC

eﬀect, but little is known regarding the nature of the acceptor states. It has been

proposed that they arise because the ceramic does not achieve thermodynamic

equilibrium during processing. On this basis a somewhat speculative mechanism

for the formation of acceptors is outlined below.

At the sintering temperature the excess charge of La

is largely compensated

by the promotion of electrons into the conduction band (see Eq. (2.57)). On

cooling, some of the electrons are replaced by V

0000

(see Eq. (2.59)):

4TiO

þ O

ðgÞþ4e

Ð 3Ti

þ V

0000

þ 8O

þ ‘TiO

’ ð4:28Þ

It can be seen from Eq. (4.28) that the interchange of electrons and vacant cation

sites requires the presence of oxygen gas and the formation of TiO

in a phase

separate from that of BaTiO

. Oxygen is available at the surface of the grains

through diﬀusion along the grain boundaries. The extension of the reaction to

the interior of a grain requires the diﬀusion of oxygen ions through the crystal

lattice, which is a much slower process than grain boundary diﬀusion, especially

as the concentration of oxygen vacancies is minimal in the presence of donor ions

(Fig. 2.13). Therefore it is possible that, on rapid cooling to room temperature,

0000

will be at a higher concentration in the surface layers of the grains where they

will act as the acceptors postulated in Fig. 4.21. The ‘TiO

’ may form part of an

intergranular phase based on Ba

(see Fig. 5.41) which would require the

diﬀusion of V

0000

into the bulk of the grains, but it is unlikely that such a highly

charged entity would be particularly mobile, so providing a further reason for a

higher V

0000

concentration at the grain surface. It has been found empirically that

PTC properties are improved when the acceptor ion Mn

is present (at about the

0.05 cat.% level) in the intergranular region. The overall increase in resistivity

during the transition is made larger and the resistivity at lower temperatures is

reduced.

The PTC eﬀect is distinguished from the majority of other critical temperature

eﬀects in the ease with which the critical temperature can be shifted by altering

the composition. The replacement of barium in BaTiO

by strontium lowers the

critical temperature by 4 8C per percentage atomic replacement, whilst

replacement by lead raises the critical temperature by 4.3 8C per percentage

atomic replacement (see Fig. 2.47). Since the critical temperature for BaTiO

120–130 8C, it is a simple matter to prepare ceramics with PTC regions anywhere

between 100 8C and þ250 8C, although the highest temperature coeﬃcients are

found in barium titanate compositions without major quantities of substituents.

The fabrication route for PTC thermistors is typical of that employed for

modern electroceramics except in so far as special attention is given to

maintaining high purity and to the ﬁring schedule.

170 CERAMIC CONDUCTORS

The basic composition is usually derived from oxides or carbonates, e.g.

BaCO

, SrCO

, TiO

,La

etc., which are mixed in a polyethylene-lined ball-

mill using mullite or agate balls and deionized water. Alternatively, and if

economically viable, the mix can be synthesized from organometallic

compounds, usually in conjunction with soluble inorganic salts.

The mix is dried and calcined (1000 8C) when the semiconducting ceramic

is formed. The calcine is ball-milled, in a similar mill to that used before, to a

size of about 1 mm. At this stage other dopants and binders can be added, e.g.

MnSO

and polyvinyl alcohol, or the mix might be blended with PbTiO

the device is to have a high switching temperature; the special treatment given to

the lead compound is necessary because of the high volatility of PbO. The slurry

is then granulated, usually by spray drying, when it is ready for pressing into

discs.

Sintering at about 1350 8C in air and the subsequent cooling stages have to be

carefully controlled since this is when the barrier-layer characteristics are

established. The conditions must be such as to allow a barrier layer of optimum

thickness (0.1–1 mm) to form, while the grains grow, ideally uniformly, to the

optimum size, normally about 50 mm.

After the sintering stage, electrodes are applied, usually either by electroless

nickel plating or by painting or screening on specially adapted silver paint. Leads

are then soldered to the electrodes when, for many applications, the device is

complete; in other cases it may be encapsulated in epoxy or silicone resins.

Examples are illustrated in Fig. 4.15.

A typical current–voltage relation for a PTC body in thermal equilibrium is

shown in Fig. 4.23. At low voltages (EF) the relation is approximately ohmic

(Fig. 4.20, AB); then, as the temperature of the thermistor reaches the regime of

steeply rising resistance (Fig. 4.20, BC), its temperature rises only slowly with

increasing voltage and the current falls to give a correspondingly slow increase in

power dissipation. If the increase in voltage is suﬃcient to bring the temperature

above the region of rising resistance (Fig. 4.20, CD), the temperature coeﬃcient

becomes negative and a rapid increase in current and temperature results.

If the rate of heat dissipation from the thermistor is changed, the location of

FG in Fig. 4.23 will shift. The temperature of the element changes only slightly,

but the power changes to a level corresponding to the new rate of heat

dissipation. If the voltage is kept constant the current becomes a measure of the

rate of heat dissipation, and this relation is used in a number of devices to sense a

change in environment. There is a marked change in heat dissipation when a

probe at thermal equilibrium in air is plunged into a liquid at the same

temperature as the air. Devices for indicating the level of liquids in tanks are

based on this change.

The relative constancy in the temperature of a PTC device, despite changes in

both the voltage supply and ambient conditions, when it is maintained on the

steeply rising limb of its resistance–temperature characteristic has led to its use as

TEMPERATURE-SENSITIVE RESISTORS 171