The relation between the power factor and the temperature coefficient of the dielectric constant of solid dielectrics

THE RELATION BETWEEN THE POWER FACTOR AND THE TEMPERATURE COEFFICIENT OF THE DIELECTRIC CONSTANT

-j-rr-THE RELATION BETWEEN -j-rr-THE POWER FACTOR

AND THE TEMPERATURE COEFFICIENT OF THE

DIELECTRIC CONSTANT OF SOLID DIELECTRICS

PROEFSCHRIFT TER VERKRIJGING VAN DEN GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOOGESCHOOL TE DELFT. OP GEZAG VAN DEN RECTOR MAGNIFICUS ]. M. TIENSTRA, HOOG-LEERAAR IN DE AFDEELING DER WEG-EN WATERBOUWKUNDE.VOOR EEN COMMISSIE UIT DEN SENAAT TE VERDEDIGEN OP WOENSDAG 4 JUNI 1947 DES NAMIDDAGS TE 2 UUR

DOOR

JHR MATHIJS GEVERS

NATUURKUNDIG INGENIEUR. GEBOREN TE LAGE-ZWALUWE

•V

BiBLIOTflEEKj

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DEN PROMOTOR

PROF. DR. C. ZWIKKER

Aan de Directie van het N a t u u r k u n d i g L a b o r a t o r i u m dor N.V. P h i l i p s ' Gloeilampcnfabrieken betuig ik o|) deze plaat.s mijn hartelijken d a n k voor de mij verleende medewerking, in het bijzonder voor de t o e s t e m m i n g de v o o r n a a m s t e r e s u l t a t e n van mijn werk t e publiceeren.

Aan mijn Vader

Aan mijn Vrouw

C O N T E N T S

p a g e

P R E L I M I N A R Y R E M A R K S 11 P a r t I. S U M M A R Y O F T H E O R I E S

A) Theories of t h e d e p e n d e n c e of t h e dielectric c o n s t a n t a n d dielectric losses on t h e frequency

1. I n t r o d u c t i o n 15 2. T h e b e h a v i o u r oi a condenser on D.C. voltage . 15

3. T h e bebavi()ur of a condenser subjected t o an

a l t e r n a t i n g field 16 4. T h e after-eireet functions '^'(t) found in t h e

liter-a t u r e 18 5. W a g n e r ' s i n h o m o g e n e i t y t h e o r y 23 6. D e b i j e ' s dipole t h e o r y 25 7. D e b i j e ' s e x t e n d e d t h e o r v 32 8. Ionic theories 33 9. G y e m a n t ' s t h e o r y 34 10. Conclusions 35 B) Theories a b o u t t h e t e m p e r a t u r e coefficient of t h e dielectric c o n s t a n t 35 C) R e l a t i o n b e t w e e n power factor a n d t e m p e r a t u r e coefficient of dielectric c o n s t a n t 37 P a r t I I . S U M M A R Y O F P L B L I S H E D M E A S U R E M E N T S A) Dielectric losses

1. L i t e r a t u r e a n d d a t a on dielectric losses of solids . 38 2. Comparison of t h e results o b t a i n e d b y various

a u t h o r s 40 3. T h e b e h a v i o u r of t a n Ö as a function of frequency . 42

4. T h e b e h a v i o u r of t a n ö a t different t e m p e r a t u r e s . 42 5. Discussion of t h e dielectric properties of solids . . 44 B) T e m p e r a t u r e cocflicient of dielectric c o n s t a n t 6. D a t a on t h e t e m p e r a t u r e coefficient of t h e dielectric c o n s t a n t 50 7. Discussion of t h e d a t a on t h e t e m p e r a t u r e coefficient of dielectric c o n s t a n t . Conclusion 51 T a b l e 1 54

8

P a r t I I I . A N E W T H E O R Y O N T H E D I E L E C T R I C L O S S E S , T H E T E M P E R A T U R E C O E F F I C I E N T O F T H E D I E L E C T R I C C O N S T A N T O F A M O R P H O U S S O L I D D I E L E C T R I C S , A N D T H E R E L A T I O N B E T W E E N T H E S E Q U A N T I T I E S 1. S t r u c t u r e of a m o r p h o u s a n d polycrystallinc solid

dielectrics; a c t i v a t i o n energy a n d r e l a x a t i o n time 57 2. P o l a r i z a t i o n , dielectric c o n s t a n t e' a n d power

factor ( t a n ó) 59 3. Some p r o p e r t i e s of dielectrics h a v i n g an irregular

s t r u c t u r e 63 4. T e m p e r a t u r e coefficient of t h e dielectric c o n s t a n t

a n d of t h e c a p a c i t y 68 5. T h e relation b e t w e e n t h e power factor t a n <) a n d

t h e t e m p e r a t u r e coefficient of t h e capaeitv (or

dielectric c o n s t a n t ) 69 6. T h e dielectric c o n s t a n t , t h e t e m p e r a t u r e eoeflicicnt of t h e c a p a c i t y a n d t h e p o w e r factor of a m i x t u r e of dielectrics 71 P a r t I V . A P P A R A T U S F O R M E A S U R I N G T H E P O W E R F A C T O R A N D T H E T E M P E R A T U R E C O E F F I C I E N T O F T H E C A P A C I T Y O F C O N D E N S E R S A) Description of t h e a p p a r a t u s for m e a s u r i n g t a n d 1. M e t h o d of m e a s u r e m e n t 72 2. D e s c r i p t i o n of t h e m e a s u r i n g a p p a r a t u s 74

3. M e a s u r e m e n t of t h e power factor a t different t e m p e r

-a t u r e s 76 4. A p p l i c a t i o n of t h e electrodes 78

5. Sources of errors: corrections to be applied, accuracy

of m e a s u r e m e n t 79 6. Review' of t h e m o s t i m p o r t a n t p r o p e r t i e s of t h e power factor t e s t i n g a p p a r a t u s 82 7. A c c u r a t e t e s t i n g a p p a r a t u s for 1.5 Mc/s . . . . 83 B) Description of t h e a p p a r a t u s for m e a s u r i n g t h e t e m p e r a t u r e coefficient of t h e c a p a c i t y of condensers 8. M e t h o d of m e a s u r e m e n t 83 9. Description of t h e a p p a r a t u s 84 10. Sources of errors; corrections; accurac^ of t h e

m e a s u r e m e n t s 88 Conclusion 89

P a r t V. R E S U L T S O F T H E M E A S U R E M E N T S O F T H E P O W E R F A C T O R A N D T H E T E M P E R A T U R E C O E F F I C I E N T O F T H E D I E L E C T R I C C O N S T A N T O F D I E L E C T R I C S 1. P o w e r factor a n d t e m p e r a t u r e coefficient at 20 C and 1.5 Mc, s 90 2. T h e dependence on t h e t e m p e r a t u r e of t h e r a t i o A of t h e t e m p e r a t u r e coefficient a n d t h e power factor 91 3. T h e power factor as a fuiu'tion of t e m p e r a t u r e . 93

4. T h e power factor as a function of t h e frequencv . 99 5. InfbuMHH'! of a small a m o u n t of a d m i x t u r e on t h e dielectric losses 103 6. Results of m e a s u r e m e n t s for t h e t e m p e r a t u r e coefficient of t h e c a p a c i t y as a function of t e m p e r -a t u r e -a n d frequency 105 7. Conclusions 105 R E F E R E N C E S 107 S U M M A R Y 110 S A M R N V A T T I N G 112 S Y N T H E S E 111. Z U S A M M E N F A S S U N G 116 S T E L L I N G E N 118

P R E L I M I N A R Y R E M A R K S

I n high-frequency techni(jue solid dielectrics are employed as a m e d i u m in condensers a n d further as insulators for leads subjected to a l t e r n a t i n g voltage, which m u s t be held in a fixed position.

T h e electrical condition of a dielectric s u b s t a n c e is defined by two v e c t o r s , t h e electric field s t r e n g t h E a n d t h e dielectric displacement D. I n a n isotropic m a t e r i a l these vectors are parallel a n d related by t h e e q u a t i o n

D^sE, (1)*)

where e is t h e dielectric c o n s t a n t of t h e m a t e r i a l .

W h e n E is a sine function of t h e t i m e , for e x a m p l e E -^ E^ cos cot,

D changes sinusoidally with t i m e , b u t generally n o t precisely in t h e

s a m e p h a s e as E. I n general the dielectric d i s p l a c e m e n t D will have a small phase difference with respect t o E, so t h a t

D = e* E„ cos {cot~d), (2)

where e* is a p r o p o r t i o n a l i t y factor different from e (only in the case where Ö = 0, is £* e x a c t l y equal to a).

I t is well k n o w n t h a t in this case t h e r e is a dissipation of energy, a "dielectric l o s s " .

I n order t o define a dielectric c o n s t a n t in t h e case of an a l t e r n a t i n g field we n o w use t h e symbolic calculus. F o r a n a l t e r n a t i n g electric field of sine-wave form we m a y write;

£ = E o e i ™ ' , (3) so t h a t (1) becomes D ^ eE^ €>'••'. (4) I n symbolic n o t a t i o n e q u a t i o n (2) becomes ö = £*Eoei(»'-'^). (5) N o w we m a y write formally D = £ £o eJ"", (6) which e q u a t i o n is identical with (5), t h e "dielectric c o n s t a n t " in (6) b e i n g ,

however, complex. F r o m (5) a n d (6) it follows, t h a t

e = e* c~J'^ -=- e* cos ö—je* sin ö. (7) Usually we write

s = s'^jS". (8)

T h e real p a r t e' is called the dielectric c o n s t a n t or p e r m i t t i v i t y ; since usually e" < ^ e', i n s t e a d of t ' , t h e m o d u l u s I e'^ -|- e"^ m a y be used.

T h e i m a g i n a r y p a r t s" is c o n n e c t e d w i t h t h e p h a s e angle ö; from (7) a n d (8) it follows t h a t

^ ' = t a n < 3 . (9) *) Throughout this thesis t h r «•Icctro'^tatic c.g.s. system oi' units is used.

T h e q u a n t i t y t a n d. called t h e power factor*), is a m e a s u r e of t h e q u a l i t y of t h e dielectric.

F r o m t h e complex dielectric c o n s t a n t according to (8) follows a n equi-v a l e n t circuit for a di(dectrie. W e consider a condenser w i t h electrodes fixed on t h e dielectric, w hich is in t h e form of a p l a t e or a t u b e .

Fig. 1. Equivalent eircuit of a romniercial eoridenser

This condenser is e q u i v a l e n t t o a " p e r f e c t " condenser h a v i n g a capaci-t a n c e C„ w i capaci-t h a resiscapaci-tanci^ R in p a r a l l i l { fig. 1), capaci-t h e a d m i capaci-t capaci-t a n c e of capaci-t h i s circuit a t t h e a n g u l a r fretjueney o) being

Y = jo>C,+ \!R. (10)

This a d m i t t a n c e is c(|uivaleut to a complex capacity C. Analogous t o (8) we h a v e

C=C'-jC";

from (10) we o b t a i n

Y - - j,oC - jo> ( C - -yC") ^ jo> {(\,-Jl,o R),

thus (',' -= ('„ a n d C " —- 1/wK, so t h a t according t o (9)

C" 1

t a n d = ~ = Wr~ • (^^) These e q u a t i o n s show t h a t the e q u i v a l e n t circuit of a condenser t h a t

contains a dielectric h a v i n g a c e r t a i n loss angle consists of a condenser with a n ideal dielectric h a v i n g a dielectric c o n s t a n t fi', a n d a resistance in parallel w i t h it.

T h e dielectric loss, t h a t is t h e energy dissipation f)er sc^cond in t h e interior of t h e dielectric, can Ix^ d e t e r m i n e d i m m e d i a t e l y w i t h t h e aid of t h e e q u i v a l e n t circuit.

Let F be t h e effective vabn; of t h e sinusoidal a l t e r n a t i n g voltage, ro t h e a n g u l a r frequency, t h e n t h e energy dissipation p e r second a m o u n t s t o

W = VVR -= V- o>C^ t a n (5 = F2 (uC t a n Ö . (12)

T h e q u a n t i t y C' t a n ó. a n d therefore also e' t a n 3, is a m e a s u r e of t h e dielectric losses a n d of t h e d a m p i n g resistance. I n t h e l i t e r a t u r e t h i s p r o d u c t is called t h e "dielectric-loss f a c t o r " . This q u a n t i t y is i m p o r t a n t especially when a dielectric is used as a n insulating m a t e r i a l . I n this case we h a v e t o select t h e m a t e r i a l t h a t has t h e lowest dielectric losses, i.e. t h e smallest di(dectric-Ioss factor.

W h e n a dielectric is used as a m e d i u m in a condenser a m a t e r i a l will gencu'ally be selected t h a t has a high di(dcctric c o n s t a n t . A low value *) More exactly sin (5 is the power factor; for tan (5 < 0.1 the difference may be neglected.

of t a n d will always Ix- sought; for a condenser c o n t a i n i n g a high-loss dielectric, w h e n c o n n e c t e d in parallel w i t h a circuit, will give rise t o a considerable decrease of t h e i m p e d a n c e of t h e t u n e d circuit and therefore to a decrease of t h e a l t e r n a t i n g voltage on t h a t circuit.

T h e dielectric c o n s t a n t e' a n d t h e power factor t a n d are n o t a b s o l u t e c o n s t a n t s for a n a c t u a l dielectric, b o t h ([uantities depending on several factors, n a m e l y on t h e t e m | ) e r a t u r e , t h e frequency a n d t h e magnitud(! of t h e a l t e r n a t i n g voltagi^, on t h e h u m i d i t y of t h e air, etc. The dependence of t h e dielectric c o n s t a n t on t h e t e m p e r a t u r e is v e r y i m p o r t a n t . This dependence is expressed b y a new q u a n t i t y , t h e t e m p e r a t u r e coefficient of t h e dielectric c o n s t a n t , naniidy t h e relative increase of t h e dielectric c o n s t a n t per degree centigrade t e m p e r a t u r e rise; it is d e n o t e d b y a,-, so t h a t

« ^ • ^ £ ' Ö T - (^^^

I n p r a c t i c e we f r e q u e n t l y use condensers whose electrodes are fixed on t h e dielectric. T h e T.C. *) of the capacity, therefore, has a more practical m e a n i n g t h a n t h e T.C. of t h e dielectric c o n s t a n t . T h e T.C. of t h e c a p a c i t y , d e n o t e d b y ac, is t h e rfdativi^ increase of t h e c a p a c i t y of" a condt^nscr per degree centigrade t e m p e r a t u r e rise, so t h a t

Usually wlien t h e t e m p e r a t u r e varies t h e electrodes of a solid-dielectric condenser move w i t h t h e diehuitric. T h e T.(". of t h e c a p a c i t y t h e n exceeds t h e T.C. of t h e dielectric c o n s t a n t bv an a m o u n t ecpial to t h e t h e r m a l e x p a n s i o n coefficient a; * * ) .

I n m o s t cases therefore

«c = 'V +

«/. (15)

Usually <ii <^ a,', a n d therefore very often ac «a «,•.

T h e stability of t h e g e n e r a t e d frequency of a high-frequency oscillator with respect to t e m p e r a t u r e v a r i a t i o n s is d e t e r m i n e d b y t h e T.C. of t h e c a p a c i t y . A s s u m i n g tlu^ self-inductance L t o be insensitive to t e m p e r a t u r e c h a n g e s , we find b y differentiation of or'' LC -= 1

2Av _ 2Au) __AC V a.) C

Therefore t h e increase of frequency z\v a t a t e m p e r a t u r e rise of one degree a m o u n t s t o

Av = ~l,vac, (16)

a n d c o n s e q u e n t l y it is p r o p o r t i o n a l to t h e T.C. of t h e c a p a c i t y . I n m a n y cases w h e n selecting a dielectric we wUI prefer a m a t e r i a l h a v i n g a small T.C. of t h e dielectric c o n s t a n t .

*) Abbreviation for temperature coefficienL

''*)From C = eOjAnd, the capacity of a parallcl-platc condenser containing a dielectric with dielectric constant e', surface 0, thickness d, it follows that

For a number of solid dielectrics we measured both quantities, tan d

and ac, within a wide range of temperatures and frequencies. Measurements

have shown that in general dielectrics having a large T.C. also have a

large value of tan 0, whereas dielectrics having a small T.C. also have a

small value of tan d. A reinarkable fact is that the ratio of the T.C. of the

capacity to the value of tan d at a given temperature and frequency is nearly

the same for most solid dielectrics. At 20 °C and in a wide range of frequencies

tlie following relation holds:

(If' ^ ac = 0.06 tan d.

It is impossible to explain this relation from the existing theories.

Stimu-lated by our experimental results. Dr. du P r é developed a new theory,

whicli was worked out in more detail by us. This theory explains the results

of measurements very satisfactorily.

In part I of this thesis a summary is given of several well-known

theo-ries concerning the causes of the dielectric losses and the temperature

coefficient of the permittivity of ionic crystals. It will be shown that

these theories are not able to explain the previously mentioned relation

between the power factor and tht; T.C. The existing theories must

there-fore be leplaeed bv a new one.

Part II gives a critical summary of the available data on tan d and the

temperature coefficient, restricted to solid diehictrics and to radio

fre-quencies. As far as possible the causes of the dielectric losses are mentioned.

From this summary it follows that the data found in the literature are not

always reliable and are insufficient to check our relation between T.C.

and tan d. It was necessary, therefore, to repeat all measurements of

tan d and the T.C. as a function of tem()erature and frequency.

The new theory al>out dielectric losses and the variation of the

permitti-vity upon teiTiperature changes is explain(^d in part I I I . From this theory

the above-mentioned relation follows in a natural way. The negative T.C.

of some dielectrics {e.g. TiO^ and polystyrene) is explained from the fact

that these solids either have a considerable value of the permittivity or

(and) a high value of the thermal expansion coefficient.

Another conclusion from our theory, which is confirmed by experiment,

is the existence of a similarity principle.

In part IV a description is given of the apparatus used by us for

mea-suring tan d and «c of condensers at fre({uencies ranging from 100 kc/s

up to 40 Mc/s, and at temperatures ranging from —180 °C up to + 1 5 0 °C.

The sources of errors and the magnitude of the different possible accidental

and systematic errors are considered in detail; in connection with this the

accuracy attainable is discussed.

Finally in part V the results obtained with these apparatus are given. It is

shown that these results may be explained quite satisfactorily with the

aid of the new theory. Some interesting special cases are discussed briefly.

I. S U M M A R Y O F T H E O R I E S

A) Theories of the dependence of the dielectric constant and dielectric /o.s.scs

on the frequency

1. Introduction

E v e r since v o n S i e m e n s ' s d i s c o v e r y ' ) t h a t energy is produced in a condenser to which an a l t e r n a t i n g voltage is applied, a t t e m p t s havi' been m a d e t o explain this ])henomenon. We shall give a s u m m a r y of the most i m p o r t a n t theories developed on this subject in chronological order, giving p a r t i c u l a r a t t e n t i o n to those p a r t s thereof which we need for use in t h e new t h e o r y , given in p a r t I I I .

T h e n o t a t i o n s used will often difft'r from those used in t h e original ))ubli-cations, in order t o show t h e corres|)ondence b e t w e e n t h e ((uantities employed in t h e different theories.

I n th(; l i t e r a t u r e a n u m b e r of excellent s u m m a r i e s are to be found, a m o n g others those of H a r t s h o r n ^ ) , J a c k s o n ^ ) , Y a g e r * ) , S c h u p p ' ' ) , b u t these s u m m a r i e s sometimes give too m u c h a n d sometimes too little for our p u r p o s e .

After a t r e a t m e n t of t h e b e h a v i o u r of a condenser on direct current voltage we shall discuss a n d c o m p a r e with each o t h e r t h e \ arious theories of t h e dielectric losses a n d t h e dielectric c o n s t a n t .

2. The behaviour of a condenser on D.C. voltage

To exj)lain t h e h e a t p r o d u c t i o n in a condenser t o which an a l t e r n a t i n g voltage is applied, initially t h e properties of dielectrics were studied when a c o n s t a n t voltage was s u d d e n l y applied t o a parallel-plate condenser con-taining t h e dielectric u n d e r t e s t . Especially t h e charging c u r r e n t was mea-sured as a function of t h e t i m e .

T h e t o t a l charging c u r r e n t m a y be divided i n t o t h r e e p a r t s :

1) t h e normal charging c u r r e n t , which is d e t e r m i n e d b y t h e m a g n i t u d e of t h e p o t e n t i a l difference, t h e capacity of t h e condenser, t h e resistance and self-induction of t h e leads;

2) t h e normal conduction current, whicli is i n d e p e n d e n t of t h e t i m e , b u t depends on t h e surface resistance of t h e dielectric;

3) t h e after-effect current *) which is an a n o m a l o u s charging c u r r e n t , decreasing w i t h time; in t h e cas(^ of some dielectrics this c u r r e n t equals zero after a v e r y long t i m e . I n t h e ease of solid dielectrics this c u r r e n t is reversible, i.e. when t h e condenser is short-circuited t h e same c u r r e n t flows as when it is charging, b u t in t h e opposite direction: t h e " a b s o r b e d c h a r g e " is l i b e r a t e d d u r i n g discharge.

Now this after-effect is due to t h e dielectric losses; tlie h e a t loss in t h e t r u e resistance in parallel n o t being r e c k o n e d as dielectric h e a t loss. I n t h e follo-*^ Also called absorption current

wing, instead of the current, we shall consider the dielectric displacement D

which is related to the current by

4 .T ö«

When a constant field E is applied at the instant tg, it appears that at the

same instant tg the displacement jumps by an amount D {tg). Then the

dis-placement increases gradually towards a final value ü(=o) (fig. 2). The

dielectric is then said to show an afier-ejfect.

a~)

D(fo

mmt-to)

to

_--*^

Kig. 2. Displacement D as function of lime t, when a constant electric field is applied at the instant I.,.

This term is used in analogy to the elastic after-effect occurring when

material is suddenly loaded by a force; the elongation then reaches an

instantaneous value AIQ, increasing gradually towards the final value

zj/co-That part of the displacement which depends on time is usually expressed

by a formula indicating the deviation with respect to the final value.

From fig. 2 we see that at an arbitrary instant t the displacement D(t)

is giv(!n by

D{t) = DH ^k DMf{t~t,) t>t„, (17)

whereas D = 0 when t < t^.

In eq. (17) f^ is the time at whicli the constant potential difference is

applied to the condenser. When t = oo,/(t—t^) = 0; assuming/(t—tg) = 1

when t = tg, it follows that

_ D(co)-Ö(to)

" D{t,)

The time function kf{t) is called after-effect function and often

written 0(t) .

(18)

3. The behaviour of a condenser subjected to an alternating field

In deriving the equations for the dielectric constant and tfie power

factor as functions of the frequency we make use of H o p k i n s o n ' s ^)

principle of superposition, generally valid in the case of solid dielectrics as

shown by many measurements.

Fig. 3. Division of tlie applied electric field inl^o rectangular impulses.

An alternating electric field E,, ej'"' is supposed to have been applied for an infinitely long tiini^ to a condenser containing a solid dielectric. Let the curve be divided into a series of rectangular impulses of width At and height Eg e^'"' (fig. 3) (assuming that we may apply this complex form to a real phenomenon).Calling f|| the dielectric constant for (o ^= 0, ECO *) the same for oj ^ oo, the contribution of the impulse starting at the instant T to the displacement at the instant /, according to the preceding s<etion, amounts to

AD(t) -- «0 E^eJ"" -fo-^ / ( t - T ) Ege>"~]B„ E^eJ-'-ke^ / ( ( - T - / I T ) E^eJ'-[ =

= fc£^ E,e^-']f{t-r-Ar)^(t~T)[^k^^ E,eJ''>^'^^^ At. We obtain the total displacement at the instant t by summation of the contributions of all impulses, adding the upwards impulse at time t. If

1T->0, the summation becomes an integration:

D{t) = £co E„ ei'"' + ke^ ƒ E„ eJ- M j ^ dr . (19)

Assuming by definition that D(f) = e ÊQ e^'"', we obtain from e<j. (19): e = £=0 + k E^ ƒ e-J'"('~') ^ ^ dr .

-K dr This equation may be written in the form

where 0(t) = kf(t), and A: -= (e^—EX>)IECO, in accordance with (18). According to (20) the dielectric constant e is in general com])lex. Usually e is composed of a real and an imaginary part, as follows:

E^e'-je". (21)

The minus sign has been chosen in order to be able to write

tan d Ö < 7ii2 (22)

where d is t h e loss angle of t h e dielectric; t a n d is called t h e power factor. Dividing e i n t o t w o p a r t s according to (21), we find

e' = £a= 1 - ƒ cos wt - ^ ^ d t (23) \ 0 df /

a n d

£" = - f c o ƒ sin o,t — - - ^ At, (24) 0 At

while t a n 6 m a y be found from E' a n d E" according t o (22). E q u a t i o n s (23) a n d (24) show t h e dependence of t h e real a n d i m a g i n a r y p a r t s , respecti-vely, on t h e frequency. T h e result will d e p e n d on t h e after-effect function 0 ( t ) . T h e q u a n t i t y d 0 ( f ) / d ( is p r o p o r t i o n a l t o t h e after-effect c u r r e n t , w h e n a c o n s t a n t p o t e n t i a l difference is applied t o t h e condenser.

F r o m t h e foregoing it follows t h a t t h e after-effect c u r r e n t gives rise t o a d e p e n d e n c e of t h e dielectric c o n s t a n t on t h e frequency a n d to t h e a p p e a -r a n c e of a loss angle d a n d , in consequence, an ene-rgy dissipation in t h e interior of t h e dielectric.

4. The after-effect functions 0{t) found in the literature 1) Pellat's theory ' )

P e l l a t was t h e first who i n t r o d u c e d an after-effect function as follows:

0{t) = ke-'i'. (25)

I n this e q u a t i o n k is a c o n s t a n t r e p r e s e n t i n g t h e t o t a l fractional dispersion of t h e dielectric c o n s t a n t :

k=^^^-- (26)

ÊCO

I n (25) T is a t i m e c o n s t a n t which is usually called t h e relaxation time of t h e after-effect.

P e l l a t did n o t give a n y physical significance to this after-effect function. Electrically e q u a t i o n (25) m e a n s t h a t t h e v(docity with which t h e after-effc^ct c u r r e n t A0{t)IAt a p p r o a c h e s t h e final value is p r o p o r t i o n a l to t h e d e v i a t i o n of t h e i n s t a n t a n e o u s value from t h e final v a l u e .

S u b s t i t u t i n g (25) in e q u a t i o n s (23) a n d (24) a n d i n t e g r a t i n g , we find t h e real a n d i m a g i n a r y p a r t of t h e dielectric c o n s t a n t a n d t h e power factor

«' = ^»(i + r + ^ J ' (27)*)

F r o m t h e conditions t h a t £ ^ £g in t h e case where co = 0 a n d £

*) Substituting (25) in the complex equation (20), we obtain £ = t^ [1 + kl(\-\-ju)T)\.

This form, obtained in D e b i j e ' s theory, can be split up into real and imaginary parts according to £ = n'—^je", giving (27) and (28).

Fig. 4. The behaviour of E' and tan S as functions of CUT.

In fig. 4 E' and tan è are plotted against frequency; tan è has its maximum

value for mx = 11 + A;, the maximum value being A/21 1 -f fe; the inflexion

|)oint of the curve representing E' is situated at MX = I 1/3 = 0.577.

In the case of low frequencies tan 6 is proportional to the frequency; in

the case of high frequencies ( « T ^ l ) tan (5 is inversely proportional to

the frequency.

For the dependence of tan d on the temperature P e l l a t made no

sug-gestions.

The results of v o n S c l i w e i d l e r ' s experiments on solid dielectrics

did not agree with the simple equations (27) and (29). An extension of

P e l l a t ' s after-effect function was, therefore, given by v o n S c h w i - i d l e r .

2) von Schiveidler's theory^)

In order to adapt theory better to experiments, v o n S c h w e i d l e r

introduced an after-effect function consisting of an infinite number of

exponential terms. The function 0(t) contains an infinite number of

relaxation times x„, each ridaxation time Xn to be present in a density

G{xn), the density function G(xn) being defined so that

1 G (r„) == 1.

0

According to v o n S c h w e i d l e r the after-effect function becomes:

0(t) = ^-0^^~ Ï: G{xn) e-'^'". (30)

£co 0

It may be shown that e' and E" are obtained by the summation of an

in-finite number of terms analogous to (27) and (28):

== G ( T „ )

e' = £=0 + (fu -£=c ) ^ r X T ^ T ^ ' (-^^^

E"^{E,-E^)ËG{x.)~^-y (32)

The result of introducing an infinite number of discrete relaxation times is

that e' and fi" are much less dependent on the frequency, since e" consists

of a superposition of a number of terms having their maximum at different

frequencies; consequently tan ó may be nearly constant in a wide range of

frequencies.

F u r t h e r v o n S c h w e i d l e r sup])osed t h a t t h e r e is a continuous

distri-bution of r e l a x a t i o n t i m e s . (Jwing t o this t h e s u m m a t i o n s in (30), (31)

a n d (32) become i n t e g r a t i o n s , w i t h t h e result: 0 ( / ) = ' ^ » ~ ^ " r G ( T ) e - ' / ' d T , (33) e=o c") £' = eoo + (fio-fcc ) ƒ f ^ U d r , (34) £ " - ( £ o - ^ c o ) T G ( T ) - ^ . d r , (35) 0 1 -f- (I) T / / t a n o ^ —, •

If £g—Ceo is very small c o m p a r e d with e^. , t h e n a p p r o x i m a t e l y :

t a n (5 -= ^ — fCix) ~'^-,-, Ax . (36)

£=0 I) 1 +

'""T-lii t h e formidae (33) to (3()) t h e d i s t r i b u t i o n function G(x) is defined

CO

so t h a t J G(x) Ax = 1. u

V o n S c h w e i d l e r gave t h e following |)hysical i n t e r p r e t a t i o n of t h e a b o v e a s s u m p t i o n s . A dielectric consists of two kinds of molecules: a) molecules whose ions or electrons wdi<>n displaced from their position of equilibrium v i b r a t e a b o u t t h a t ])osition with definite n a t u r a l periods of v i b r a t i o n . These molecules give rise t o t h e n o r m a l i n s t a n t a n e o u s displacement, also p r e s e n t in t h e case of a perfect dielectric;

b) molecules wliose ions m a k e heavily d a m p e d , aperiodic, m o t i o n s . If a c o n s t a n t p o t e n t i a l differ<'nee is s u d d e n l y applied t o t h e dielectric, these charges a p p r o a c h a new equilibrium position in such a w a y t h a t their deviation from i t decreases w i t h t i m e according t o e"'!'', x being t h e t i m e c o n s t a n t or r e l a x a t i o n t i m e of t h e aperiodic charges. These molecules give rise t o t h e after-effect.

T h e physical i n t e r p r e t a t i o n of P e l l a t ' s e q u a t i o n s is t h a t all t h e vi-b r a t i n g charges of t h e t y p e vi-b ) h a v e t h e s a m e r e l a x a t i o n time x. T h e charges of t h e t y p e a) give rise t o £ „ , t h e charg(!S of t h e t y p e b) give a n e x t r a c o n t r i b u t i o n t o t h e dielectric c o n s t a n t , being g r e a t e r t h e lower t h e fre-q u e n c y . A t low frefre-quencies t h e l a t t e r charges can still be displaced, a l t h o u g h their m o t i o n is heavily d a m p e d . At higher frequencies, however, these charges are unable t o follow t h e field; c o n s e q u e n t l y , t h e dielectric c o n s t a n t decreases with increase of frecjuency.

T h e physical i n t e r p r e t a t i o n of v o n S c h w e i d l e r ' s e q u a t i o n s is t h a t each of t h e charges of t h e ty])e b) i n a v h a v e its own r e l a x a t i o n t i m e . T h e t o t a l difference £g—£^ is p r o p o r t i o n a l t o t h e n u m b e r of molecules of t h e t y p e b ) .

3) WagJiers distribution function

V o n S c h w e i d l e r did n o t a s s u m e a n y special form for t h e d i s t r i b u t i o n function G{x). I n order t o o b t a i n concrete; r e s u l t s , K. W. W a g n e r ^ ) in-t r o d u c e d a d i s in-t r i b u in-t i o n law for in-t h e in-t i m e c o n s in-t a n in-t s . W a g n e r assumed

t h e r e l a x a t i o n times t o be d i s t r i b u t e d according to a p r o b a b i l i t y func-tion ^''). T h e density G(x) with which t h e l e l a x a t i o n times are grouped a b o u t X, m a y be r e p r e s e n t e d by the e({uation:

G ( T ) d T : - e~('''"'i')'Aln(xft) (37)

which is W a g n e r ' s d i s t r i b u t i o n function, being a function of t In ( T / T ) where 6 is a d e n s i t y p a r a m e t e r ; the greater b, t h e more dense is the grou-ping a b o u t In T. This function, which is G a u s s ' s p r o b a b i l i t y function, is illustrated b y fig. 5.

' ^ / f . , «

Fig. 5. W a g n e r ' s dislribution function G ( T ) .

F o r t h e form of this distributiem function W a g n e r gave the obvious i n t e r p r e t a t i o n t h a t t h e r e is an infinite; n u m b e r of inde|)endent causes d i s t u r b i n g t h e existence of a sinjile rcdaxation time x.

S u b s t i t u t i n g t h e d i s t r i b u t i o n function (37) in e q u a t i o n (33) we obtain t h e following expression for t h e after-effect function:

0(t) = ^ - - J ^ - 1 / c - C ' i - H ' e~'" A{b In T/T)

£co I Jl J

I ^ 0

(38)

B y s u b s t i t u t i o n of G{x) in (34) a n d (36) we o b t a i n the dispersion of t h e dielectric c o n s t a n t a n d t a n i), respeetivedy:

I .

These integrals cannot be evaluated in finite form; W a g n e r therefore

gave families of curves representing E' and tan ö as functions of WT in terms

of different values of b.

In the case of 6 = co the equations (39) and (40) are transformed into

the expressions (27) and (29) for the case of a single relaxation time. In the

case of very small values of b we obtain the following limiting values of E'

and tan è, respectively:

fn+^o

(41)

t a n (5 = I 2,-1 6

(42)

It is seen from eq. (41) that in the case of small values of 6 the dielectric

constant approaches the arithmetic mean of £fl and £» in a wide range of

frequencies, independently of the value of b. Equation (42) shows that in

that case tan b is proportional to b.

The physical interpretation is as follows: if the distribution of relaxation

times is very flat, i.e. if all relaxation times are present in the same proportion,

then the dielectric constant and the power factor are independent of the

frequency in a wide range of frequencies, the power factor being

propor-tional to the number of particles having the rtdaxation time x.

For the dependence of E' and tan è on temperature, W a g n e r gave

only some qualitative considerations: the predominant relaxation time x

decreases rapidly with increase of temperature, the constant b depends very

little on temperature; if there is any dej)endence, then it increases in all

cases with increase of temperature.

, ' ' ^3

Fig. 6a. E ' ^ / ( T ) at different frequencies (fOa > oj,) according to W a g n e r (Ann. Phvs. I.pz., 40, "817, 191,3).

Fig. 6a shows cjualitatively the de|>endeiice of E' on ti;mperature at two

frequencies, as computed by W a g n e r with the above-mentioned

as-sumptions. Fig. 6b indicates qualitatively the behaviour of tan ó as a

function of temperature. At cox «s 1, tan d reaches a maximum, being

displaced towards higher temperature at higher frequency, since r

de-creases with increase of temperature.

*" / •f:i706

Fig. 6b. tan d ^ f{T) at different frequencies ('Oj > o)^) according to Wagner. According to fig. 6 b , a b o v e a c e r t a i n t e m p e r a t u r e t a n d decreases with increase of t e m p e r a t u r e , b u t for m o s t of t h e solid dielectrics used t a n ê increases with increase of t e m p e r a t u r e , according t o published d a t a a n d also according to our own m e a s u r e m e n t s .

T h e complicated formulae do n o t allow a n y p r e d i c t i o n a b o u t the be-h a v i o u r off' a n d t a n d a t different t e m p e r a t u r e s . I t is, tbe-herefore, impossible to derive a relationship betw een t h e t e m p e r a t u r e coefficient of t h e dielectric c o n s t a n t a n d t h e p o w e r factor t h a t holds in a wide range of frequencies a n d t e m p e r a t u r e s .

5. Wagner's inhomogeneity theory

I n 1873 M a x w e l l ' ^ ' ^ ) gave a physical e x p l a n a t i o n of t h e cause of t h e dielectric after-effect which h a s been used a n d e x t e n d e d in t h e case of alter-n a t i alter-n g voltage b y W a g alter-n e r .

M a x w e l l derived t h e c o n d u c t i v i t y of a m e d i u m containing small c o n d u c t i v e spheres. Additionally he c o m p u t e d t h e residual charge *) of a dielectric consisting of layers of different dielectric c o n s t a n t s a n d conduc-tivities w h e n a c o n s t a n t voltage is applied t o it ^^).

X,

Fig. 7. Wagner's two-layer condenser.

I n s p i r e d b y M a x w e l l ' s a s s u m p t i o n a b o u t t h e inhomogeueous s t r u c -t u r e of a dielec-tric, W a g n e r ^ ' * ) p o i n -t e d o u -t -t h a -t -t h e v a r i a -t i o n of -t h e *) Tfic residual charge is the charge stored on the boundary layers when a constant

dielectric c o n s t a n t a n d power factor as a function of t h e frequency m a y be e x p l a i n e d in the- case of a condenseer c o n t a i n i n g a dielectric consisting of t w o layers in series h a v i n g different values of t h e dielectric c o n s t a n t s a n d conductivities (fig. 7). M e v e r ^ " ' ) was t h e first w h o derived t h e depen-dence of fi' and t a n cb on t h e frequency in this case. To this two-layer denser a single r e l a x a t i o n t i m e x m a y be ascribed, de^pending on t h e con-s t a n t con-s E\, B'2, Aj, Xo. T h e bebaviemr of E' a n d t a n 6 of t h e condencon-ser m a y be repreesented b y t h e e q u a t i o n s (27) a n d (29). I n t h e case where E\JX-^^ =

E'^/X,,, T = 0 a n d t h e after-effect diminishes.

W h e n t h e condenser c o n t a i n s several diffe'rent layers t h e calculations are very difficult; t h e r e s u l t however is t h a t E' a n d t a n d change m u c h m o r e slowly with the- frequency.

W a g n e r ' ^ " ) c o m p u t e d also t h e b e h a v i o u r of a diedectric according t o M a x w e l l ' s first assum])tion. T h e basic s u b s t a n c e of t h e s t r u c t u r e is con-sidered t o be a dielectric of diede'ctric c o n s t a n t Ë\ a n d c o n d u c t i v i t y A,. D i s t r i b u t e d t h r o u g h o u t its m a s s a r e a numbe*r of small spheres of elielectric c o n s t a n t s'o and c o n d u c t i v i t y X.,, which are; so far a p a r t t h a t each sphere d i s t u r b s t h e fiedd of t h e applied voltage only j u s t as m u c h as if t h e o t h e r s were a b s e n t . This diedectric showing t h e after-e'ffect m a y be characterized b y a single r e l a x a t i o n time- x.

I n this case t h e depe-nele'iice e)f E' a n d t a n (5 on frequency is represented again b y t h e eejuations (27) a n d (29), in which x a n d k depend on t h e q u a n t i t i e s E\, £'.,, Aj, X., a n d the' v o l u m e concentratie)!! p of t h e small sphe;res.

I n t h e simple case t h a t £'j = E'.,, X^ = 0, X., ^ X one o b t a i n s : £^3 = Ë\,

k = ip, X ^ 'iE\IX.

I n order t o a d a p t t h e t h e o r y b e t t e r to t h e results of e x p e r i m e n t s , W a g n e r exte;nded t h e calculations, a s s u m i n g t h a t t h e r e is a n infinite n u m b e r of kinds of spl!(;res, all w i t h diffe;re;!it c o n d u c t i v i t y a n d v o l u m e conce'iitration. W a g n e r furtfier supposed t h a t t h e l e l a x a t i o n times of t h e sphe-res are d i s t r i b u t e d a b o u t a pre'dominant redaxatie)n t i m e x according t o a p r o b a b i l i t y function. T h e results of his calculations h a v e a l r e a d y been given b y t h e e q u a t i o n s (38), (39) a n d (40). T h e dielectric c o n s t a n t a n d power factor v a r y very little with fre-quency.

W a g n e r ' s extension of M a x w e ; i r s t h e o r y , assu!ni!ig t h a t a condenser c o n t a i n s several k i n d s of inhomogeneities, is k n o w n u n d e r t h e n a m e e)f

inhomogeneity theory.

T h e two-laver-dielectrie t h e o r y deies n o t agree w i t h m o s t e x p e r i m e n t s ^^). I n t r o d u c i n g suitable de-pe-ndences e)f t h e redaxation times on t e m p e r a t u r e , it is possible to e-xplain q u a l i t a t i v e l y t h e b e h a v i o u r of some solid dielectrics. W a g n e r assumes t h e possibility t h a t t h e c o n d u c t i v i t y varies w i t h t e m p e -r a t u -r e acco-rding t o t h e law A ^^ Ag e"^, whe-re T signifies t h e absolute t e m p e r a t u r e , a a c o n s t a n t d e p e n d e n t on t h e s u b s t a n c e , Ag t h e c o n d u c t i v i t y a t T ^ 0 °K. Conseque-ntly, the; relaxatie)!! t i m e is t h e n g o v e r n e d b y T = Tg e~"^. I n t h e case e)f a b r o a d d i s t r i b u t i o n of r e l a x a t i o n t i m e s , however, t h i s dependence on the» teMnperature does ne)t show itself in t h e e q u a t i o n

(42) fe)r t a n d.

T h e i n h o m o g e n e i t y t h e o r y h a s , however, an i m p o r t a n t merit: it has led t o t h e insight t h a t inliO!nogene;ities consisting of particles h a v i n g a definite c o n d u c t i v i t y give rise t o dielectric losses. I n order to o b t a i n a

low-loss dielectidc t h e diele-ctric should n o t c o n t a i n c o n d u c t i n g p a r t i c l e s . 6. Debije's dipole theory

A l t h o u g h D e b i j e ' s dipe)le t h e o r y h a s n o v a l i d i t y for solid diele'ctrics, we shall t i e a t this t h e o r y briefly, since it gives a n insight into t h e effect of p e r m a n e n t dipoles prese;nt in a diedectric. A t t h e s a m e tiine it gives us t h e e)pportunity t o explain t h e cause e)f t h e dielectric pe)la!dzation.

If a c o n s t a n t electric fiedd E is applied t o a dielectric, it is said to b e

polarized, i.e. t h e positive charge's of a molecule or a t o m are a t t r a c t e d t o t h e

n e g a t i v e electrode, t h e ne'gative charges (the electrons) to t h e positive electrode. T h e molecules a n d a t o m s , c o n s e q u e n t l y , become dipoles. F o r this reason t h e polarizatie)n of t h e mejiecules a n d a t o m s is called

defor-mation polarization. It in (^t^talAishcA w i t h o u t a p p i c c i a b h ' t i m e delay and

therefore gives rise t o t h e i n s t a n t a n e o u s d i s p l a c e m e n t .

W h e n t h e r e are permanent dipoles p r e s e n t in t h e dielectric their electric fields neutralize each o t h e r w h e n t h e r e is n o e x t e r n a l applied field, t h e dipole axes being o r i e n t a t e d in all possible directions owing t o t h e r m a l a g i t a t i o n . An e x t e r n a l c o n s t a n t field a|)])lie'el te) t h e dielectiic causes these p e r m a n e n t dipoles t o b e orie'utate'd to the fie'ld, fe)r which a r o t a t i o n of t h e whole dipole; mole'cule is needed. F o r this re-ason t h e polarization cause'd b y t h e orien-t a orien-t i o n of orien-t h e p e r m a n e n orien-t di[)oles is called orienorien-taorien-tion polarizaorien-tion. W h e n a dipe)le' mole;cule' is be;ing e)rie!!tated, a frictional force oppejses t h e orien-t a orien-t i o n , giving rise orien-t o orien-t h e aj'orien-ter-effecorien-t.

T h e effect of t h e polarization is, t h a t a t t h e positive; eh'Ctrode t h e r e is a n e g a t i v e charge, a t t h e negative' electrode t h e r e is a pe)sitive charge, say I per s q u a r e cm. I is called t h e polarization.

Since t h e q u a n t i t y D/4.T e-ejuals t h e charge per s q u a r e cm of t h e electrode, we find on t h e one h a n d

D==eE, (43)

where E is t h e die;lecti'ic ceuistant of t h e diedectric a n d E is t h e a|)plied elec-tric field, a n d on t h e o t h e r banel

fl-47ï/=E. (44) I n t h e case of a n isotre)pic m a t e r i a l , e q u a t i o n s (43) anel (44) m a y be

c o m b i n e d , giving

4.T ƒ - E ( s - l ) . (45) T h e t o t a l polarization / consists of t w o p a r t s : ƒ j , t h e deformation

polari-zation, a n d Jg, t h e e)rientation p o l a r i z a t i o n , so t h a t we m a y write

J = Jj -f Jg = nj /Jrf + n„ Pa , (46) where

nd = n u m b e r of i n d u c e d dipe)les [)er u n i t of v o l u m e , Uo = n u m b e r eif p e r m a n e n t dipoles per u n i t of v o l u m e , pd -^ polarization p e r " d e f o r m a t i o n " dipole,

ƒ>„ = polarization per " p e r m a n e n t " dipole.

W e shall now calculate t h e ce)ntribution of t h e i n d u c e d and p e r m a n e n t dipoles t o t h e dielectidc c o n s t a n t , t h e former giving rise t o £33 , t h e dielectric c o n s t a n t a t v e r y high freejuencies, t h e l a t t e r t o t h e difference- fig—E^ , £g being t h e dielectric c o n s t a n t w h e n t h e frequency a p p r o a c h e s zero.

a) Contribution of the deformation polarization to s

Accoreling t o e'epiatie)n (46) t h e deformation polarization a m o u n t s t o

11 = nd Pd . (47)

/ j indicates not e)iily t h e polarization charge; per square cm, b u t also t h e induced dipole- me)ine!!t per u n i t of v o l u m e ; pd indicates, therefore, also t h e i n d u c e d dipole- me)!nent per molecule or atom. This q u a n t i t y pd is propor-t i o n a l propor-to propor-t h e elecpropor-tric force /'' acpropor-ting on propor-tlie induced dipole; we m a y wripropor-te consequently

Pd - ad E, (48)

where a,; is t h e de-f'e»rmation pe)larizability p e r i n d u c e d dipole.

According to C l a u s i u s a n d M o s o t t i , t h e inte;rnal or acting field is defined b v

E = £ + '^|'/. (49)

F r o m t h e l a t t e r thre-e eepiatie)ns a n d (45) we find £ — 1 4.T

£ — 2 ^ - 3 " " " " • (^^^ I n the- absence eif perinane-nt dipoles, £ is the te)tal eliele-ctric c o n s t a n t ;

whe-i! the-re are alse) per!nai!e-!it elipoles prese-nt in the- dielectric, £ is t h e die-lectric c o n s t a n t a t very high frequencie-s, d e n o t e d b y f^o, since t h e defeirmation polarization gives lise te) t h e i n s t a n t a n e o u s displacement.

Solving for £ (here ^^ £co ) in e q u a t i o n (50) we o b t a i n

e = o = l + 4 . T -, (51) 1 ^ Mj ad

This e-ejualie)!! is eifte-n writte-n in the; feillowing fe)rm:

£ce = 1 + ^7TX<I , (52) whe-re /d is the susceptibility of the- de-formation polarizatie)!!, de-fined b y

lid ad , c „ ,

1 — ^ iidcu

F r o m e q u a t i o n (50) we iininediately derive C l a u s i u s - M o s o t t i ' s law: t h e molar polarization P is defined b y t h e polarizability per grammolecule, multiplied by 47r/3, therefore

where iV is A v o g a d r o ' s n u m b e r .

A s s u m i n g all particle;s t o h a v e t h e same polarization pre)perties, nu is t h e t o t a l n u m b e r of particles in t h e unit of v o l u m e , while

M N Q Ud

where M is t h e molecular weight, p t h e d e n s i t y of t h e s u b s t a n c e . Using (54) a n d (55), (50) becomes

I f f

— 1

p M E ^ L ^ (56)

O £co + 2

which is C l a u s i u s - M o s o t t i ' s law concerning t h e electronic a n d atoniic ])olarization.

b) Contribution of the orientation polarization to E

WheMi p„, which is p r o p o r t i o n a l t o t h e i n t e r n a l field F, is the- polarization of a p e r m a n e n t dipole, t h e polarization p e r u n i t of v o l u m e /., is

I^ = n„ p„ = n„ a„ F, (57)

where; n» is t h e n u m b e r of p e r m a n e n t dipoles per u n i t e)f ve)lu!ne', a„ t h e o r i e n t a t i o n polarizability p e r dipole.

I n 1912 D e b i j e i**) c o m p u t e d t h e q u a n t i t y a„, assiiniing all dipoles to h a v e t h e sanie dipole m o m e n t ^ . D e b i j e further a s s u m e d t h a t t h e dipole c o n c e n t r a t i o n n„ is so small t h a t t h e r e is no i n t e r a c t i o n b e t w e e n t h e dipoles. A t m o d e r a t e field s t r e n g t h s D e b i j e o b t a i n e d /^ (58) "° ^ 3 AT where- k is B o l t z m a n n ' s c o n s t a n t = 1.38 X 10"^" e-rgs/elcgree a n d T is t h e absolute t e i n p e r a t u r e in degrees K e l v i n .

F r o m (58) it is seen t h a t t h e o r i e n t a t i o n polarizability a„ is d e p e n d e n t on t h e t e m p e r a t u r e , as c o n t r a s t e d w i t h t h e deformation polarizability ad, which in t h e first a p p r o x i m a t i o n is i n d e p e n d e n t of t h e t e m p e r a t u r e .

I n additie)!! t o t h e o r i e n t a t i o n t h e p e r m a n e n t dipoles are i n d u c e d t o t a k e on a slightly larger dipole m o m e n t , as a consequence- e)f deformation.

\ s s u m i n g t h e defe)r!nation polarizability of t h e perinane-nt dipe)le;s t o be

ad, t h e t o t a l contributiem of t h e p e r m a n e n t dipoles to the; polarization

a m o u n t s t o

I = no F (cid + «o) = noF (ad + ^ ^ ) . (59)

W h e n all t h e molecules h a v e t h e s a m e dipole m o m e n t , as for instance in a gas, we find for t h e molecular p o l a r i z a t i o n P:

N (ad + 5 ^ ) . (60)

e — J- *^ M c L i "

This e-quation gives occasion t o t h e following reniarks:

1. e q u a t i o n (60) only holds for rarefied gases, since all !nole;cules are assu-m e d to be p e r assu-m a n e n t dipoles which do n o t i n t e r a c t ;

2. t h e dielectric c o n s t a n t £ occurring in t h e left-hand side of (60) is n o t e q u a l t o £co, since t h e o r i e n t a t i o n is affected b y friction, w hich is due t o t h e after-effect of t h e polarization;

3. formula (60) provides t h e possibility of d e t e r m i n i n g the- dipe)le m o m e n t

ju b y measui'ing M , Q a n d t h e dielectric c o n s t a n t £ (at c o n s t a n t v o l t a g e ) ,

as functions of t h e t e m p e r a t u r e T.

dilute solution of a polar liquid (1) in a !!0!!-pe)lar liepiid (2). I n this case t h e !ne)le-cular polarization P becomes ^'):

f 12 ^ r '^ a

whe-ie/j a n d ƒ2 re[)re-sent t h e mole fractions of (1) a n d (2), M,.^ the- !ne)le-cular weight a n d Oy^ t h e d e n s i t y of t h e m i x t u r e ,

ay = deformatie)n polarizability of t h e n o n - p o l a r solvent,

a^ — t o t a l polarizability of t h e polar molecules, as m e a s u r e d in t h e gasee)us state'.

c) The behaviour of a rarefied gas, consisting of pernianenl dipoles, or of a

dilute solution of a polar in a non-polar liquid under alternating voltage.

Whe'i! a c o n s t a n t electric field is s u d d e n l y applie'd t o a condenser con-t a i n i n g a gase'e)us dieleccon-tric consiscon-ting of permane'iicon-t dipoles or a dilucon-te solution e)f a pe)lar liejuid in a n o n - p o l a r liquid, t h e deforniation polarization follows almost i n s t a n t a n e o u s l y , only lagging b e h i n d a t optical frequencies. T h e o r i e n t a t i o n pe)larizatioii h a s a t i m e lag, however, a l r e a d y a t m u c h lower frequencies, due to t h e friction which opposes t h e m o v e m e n t s of t h e dipoles. D e ' b i j e '*) calculated t h e influence on t h e b e h a v i o u r of such a dielectric of t h e after-effect of t h e o r i e n t a t i o n polarization u n d e r an a l t e r n a t i n g electric fie-ld E -^ E„ eJ'"' .

D e b i j e assumes the' dipoles to be sniall spheres, rotating due to t h e eh'ctric fie'ld in a m e d i u m with viscosity )]. T h e frictional force acting on t h e r o t a t i n g sphere is p r e s u m e d to be p r o p o r t i o n a l t o t h e velocity of r o t a t i o n a n d to a frictional resistance coefficient 1^, as coinpute'd fro!n S t o k e s ' s friction law:

C = 8.T n r\ (62) where r is t h e r a d i u s of t h e spherical dipole.

to ~.^T

Fig. 8. Variation of molecular polarization P, when a constant electric field is applied to a condenser at the instant (Q (according to D e b i j e ) .

W e n o w consider a condenser c o n t a i n i n g a gaseous dielectric, of which all particles h a v e a dipole m o m e n t f». A t t h e i n s t a n t fg a c o n s t a n t voltage is applied. At t h e same i n s t a n t t h e molecular polarization P j u m p s t o t h e value Pco in a g r e e m e n t with t h e deformation polarization. Gradually t h e t o t a l

polarization increases t o w a r d s t h e final value Pg according to t h e expo-n e expo-n t i a l curve (see fig. 8):

P(t) = 0 ( 0 ---- ( P g - P . 0 ) ( 1 - e - '^°), (63)

where Pg—Px, is t h e o r i e n t a t i o n polarization of t h e dipoh's a n d x is t h e r e l a x a t i o n time of t h e after-effect, which m a y b e c o m p u t e d from

(T is t h e average t i m e in wdiich e'ach dipole r e t u r n s for t h e 1/e t h p a r t fre)m t h e directed position t o t h e e;quilibrium position, w h e n t h e applied voltage is s u d d e n l y switched off).

A t r o o m t e m p e r a t u r e x is e)f the; order of lO"'^^ t o 10^^'' se'c (r ^i 10^^ cm,

ij ; ^ 10-2 t , j 1 0 - 4 ) .

W h e n a n alternating field E ^ EQ e^'"' is applied, the' polarization P as a function of t h e frequency becomes:

p = p- + (p^-p-)rTWx' (^^)*)

assuming t h e superposition law t o be' valid.

F r o m (65) i t is se;en t h a t Pg is the; polarization a t o) — 0, P^o t h e polari-zation a t w ^ o o .

Since Pco is t h e defeirmation jiolarization w e n i a y write; ae;coreling t o (54) a n d (56):

M £oo " 1 471

Pa= = - ^ r - 9 = .y Nad. 6 6

{> fioo + ^ J

At t h e freejuency w --= 0 t h e t o t a l polarization Pg becomes, according t o (60) M f i g - 1 47T ^ ^ ^ , lu^

e «0

Finally we- obtain t h e oi-ientation polarization P g — P » fre)!n the; eejuations (66) a n d (67):

Making use of t h e h'ft-hand e q u a t i o n s of (66) a n d (67), a n d ele-fining for t h e general case

P - ^ ^ , (69)

Q E + 2

where P a n d consequently £ m a y be complex, e q u a t i o n (65) beco!nes:

e + 2 £ „ " + 2 ^ ^£g + 2 £ „ + 2 M + jcor' ^ '

F r o m this e q u a t i o n t h e co!nplex dielectric c o n s t a n t £ m a y be- solved for:

£g . Eco

So + 2 Soi +2 £g + £ c o ; w T ' ,

' = " ï ~ 7 ^ r - = "T+7o>T^' (^^)

i^o=-;^-2=y^(-+3W)- '''^

+ 2

*) This equation may be derived in analogy with the derivation given in section 3 by substituting equation (63) for 'l>(t). Instead of the displacement J) we here consider the polarization P.

e.-1-2

Fre)m (71) we finei t h e real a n d i m a g i n a r y p a r t s of t h e complex dielectric c o n s t a n t , according to £ = E'—JE": ,2 ^'2 t g T^ Ceo £co CC T a n d 1 + c.y^ T'2 = Ceo + fig £a 1 + oj'^ T'^ e - Ifio-Sco; J - ^ ,„2 ^'2 T h e pe)wer factor t a n f5 becomes

t a n Ó = (f„-fco)

(73)

(74)

(75)

Fig. 9. f' and tan fi as functions of ojr' according to Debije.

Fig. 9 illustrate;s the- variatie)!! of fi' a n d t a n d as functions of t h e frequency.

T h e dielectric c o n s t a n t decreases w i t h increasing frequency: a t v e r y low freque-ncies t h e dipoles are o r i e n t a t e d b y t h e electric field w i t h o u t t i m e lag, fi' h a v i n g its !naximu!n value £g. I n this case t h e o r i e n t a t i o n of t h e dipoles is n o t a c c o m p a n i e d b y a n y loss of energy, t a n d c o n s e q u e n t l y being low. A t very high frequencies t h e dipoles are n o t able t o follow the- v a r i a t i o n s of t h e field, whereas t h e n o r m a l t h e r m a l a g i t a t i o n is still ])rese!it. I n this case t h e dielectric c o n s t a n t is defined by t h e deformation polarization alone, which t a k e s place w i t h o u t time lag * ) ; a t t h e sa!ne time t a n d, which is p o r p o r t i o n a l to t h e loss of energy per period, alse) has a small v a l u e . A t interme-diate fre-queneies t h e dielectric c o n s t a n t decreases from £g t o £o3, while a t t h e frequency defined b y cnx' =» 1 **) t h e o r i e n t a t i o n has its m a x i m u m phase difference with respect t o t h e electric field, t a n d h a v i n g its m a x i m i ü n v a l u e .

C o m p a r i n g D e b i j e ' s formulae (73) a n d (75) w i t h P e l l a t ' s a n d W a g n e r ' s equations for the- case of a single re-laxation t i m e , it is striking t o find t h a t these fe)r!!!ulae- are; identical, t h e only differe-nce being t h e m e a n i n g of t h e r e l a x a t i o n t i m e : i n P e l l a t ' s a n d W a g n e r ' s e q u a t i o n s

*) That is to say towards frequencies of the light.

**) Acluaify at CUT' = 11 -f fc = I EO/^» ' where fc = (£„—£«, )/«„ according to section 4.

t h e r e l a x a t i o n t i m e x of t h e displacement D occurs, w bile- in D e b ij e's forniulae t h e r e l a x a t i o n t i m e x' occurs, which is r e l a t e d t o the- r e l a x a t i o n t i m e of t h e polarization P by e q u a t i o n (72).

As r e m a r k e d above, D e b i j e ' s the;ory n o t only holds in t h e case of rare-fied gases, b u t also in t h e case of a dilute solution of a polar s u b s t a n c e in a n o n - p o l a r solvent. T h e e q u a t i o n s (73), (74) a n d (75) r e m a i n valid where £g a n d £» signify t h e dielectric c o n s t a n t s a t low a n d high frequencies, respectively, w i t h respe-ct to t h e solution.

T h e e q u a t i o n s concerning E' a n d t a n ó are valid also for m i x t u r e s of dipole gases a n d dilute solutions of polar liquids in non-|)olar liquids ^ ' ) .

Tlie dependence of t h e r e l a x a t i o n time on t h e t e m p e r a t u r e is m a i n l y d e t e r m i n e d b y t h a t of t h e viscosity rj, according t o (64). In the case of gase;s a n d liquids t h e fe)Ilowing relation holds ^^):

so t h a t t h e dependence of re-laxation t i m e on t h e temperature- is give-ii by

el'I'''

T = T g ^ - ' (76) whe-re- Tg a n d /? are c o n s t a n t s d e p e n d i n g on t h e s u b s t a n c e .

Fig. 10. e' = / ( T ) at a certain frequency (i> according to Debije's theory. F r o m e q u a t i o n (73) we m a y c o m p u t e the temperature coefficient of the

dielectric constant fi', m a k i n g use of t h e dependence of fig, fico a n d x on t h e

t e m p e r a t u r e , which m a y be calculated from t h e molecular polarizations Pg, Pco (equations (67) a n d (66)) a n d frt)!n e q u a t i o n (76), resjie-e;tivedy. T h e e-xpiessions o b t a i n e d are very complicated. Fig. iO illustrates the ejualitative be-haviour of fi' a t different t e m p e r a t u i e s as m a y be valiel feir m a n y polar liquids (e.g. glycerine). At low t e m p e r a t u r e s t h e dielectric ce)nstant in-creases a n d c o n s e q u e n t l y also t h e resistance against t h e dipole o r i e n t a t i o n . . \ t high t e m p e r a t u r e s t h e dielectric c o n s t a n t decreases again, since t h e d i s o r i e n t a t i o n due to t h e r n i a l a g i t a t i o n gains on t h e effect due to t h e decreasing viscosity. A t low t e m p e r a t u r e s t h e temperature- coefficient of e' is positive, a t high t e m p e r a t u r e s negative.

M a n y results of m e a s u r e m e n t s cari-ied o u t on fi' a n d t a n () as functions of t e m p e r a t u r e a n d frequency are ejualitatively in agre-ement with D e b i j e ' s dipole t h e o r y ^o), a l t h o u g h in this t h e o r y m a n y simplifying a s s u m p t i o n s are m a d e . Especially t h e applications of the- internal field

according t o C l a u s i u s - M o s o t t i anel of S t o k e s ' s law of friction a r e d i s p u t a b l e .

F r o m t h e m e a s u r e d m a x i m u m v a l u e of t a n d t h e r e l a x a t i o n time could be calculated, a n d from this t h e r a d i u s of t h e dipole molecules by m e a n s of (64).

I n m a n y cases, however, a curve re-lating t a n 6 a n d co was o b t a i n e d t h a t has a m u c h flatter shape t h a n acce)rding t o D e b ij e-'s t h e o r y . There-fore D e b i j e ' s t h e o r y h a s been extende-d by several a u t h o r s for t h e case of non-spherical dipoles, w h e r e b y mo!e re-laxation time-s are i n t r o d u c e d , causing a b r o a d e r c u r v e relating t a n d — f(cü).

I n t h e case of ne)n-dilute solutions the dipoles i n t e r a c t . T h e inffuence of this interactitni e)n t h e polarizatie)!!, the dispersion of t h e dielectric c o n s t a n t , a n d the- die-lectric losses h a v e been i n v e s t i g a t e d b y V a n A r k e l a n d S n o e k 21), S p e n g l e r ^ ^ ) ^ O n s a g e r ^ s ) , C o h e n H e n r i q u e z 2*), while B o t t e l i e r 2'') gave a s u m m a r y of t h e new formulae concerning t h e dielectric c o n s t a n t .

7. Debije's extended theory'^^')

T h e e x t e n d e d tl!e;ory m a y be applie-d te) highly viscous liquids a n d , with some restriction, te) solids. I n this tlie;ory t h e interactie)!! of t h e dipoles h a s b e e n t a k e n into consideration. F r o m X - r a y p h o t o g r a p h s it has been s h o w n t h a t in highly viscous liquids a epiasi-crystalline s t r u c t u r e is p r e s e n t . Consequently the-re- m u s t exist a c e r t a i n coupling betwe-en t h e molecules. W h e n t h e dipoles h a v e a high conce-ntration in this liepiiel, each dipole possesses a preferre-d o r i e n t a t i o n a t a n y i n s t a n t . T h e i n t e r a c t i o n of t h e dipoles is defined b y a c e r t a i n energy q (^^Behinderungsenergie"). If all dipoles are equal a n d t h e quasi-crystalline liquid is h o m o g e n e o u s , t h e n to be o r i e n t a t e d b y the- applied field E, all dipoles h a v e to s u r m o u n t t h e same energy </.

T h e r e s u l t e)f this is, t h a t t h e orie-iitatie)i! pejlarizabilitv a„ does n o t equal /«'•^/3AT', as should be t h e case according to t h e simple t h e o r y of D e b i j e ,

b u t t h a t

- - 3 T > ( A y ' (^")

whe-re R (qjkT) is a function of f//AT h a v i n g t h e following fe)rni:

I n t h e case of strong i n t e r a c t i o n , t h a t is for q^kT,

SO t h a t according t o (77) t h e o r i e n t a t i o n polarizability be-comes:

« 0 = 3 ^ ^ (80) F r o m this e q u a t i o n it is seen t h a t in the- case of highly viscous liquids and

of liquids h a v i n g a high dipole c o n c e n t r a t i o n t h e o r i e n t a t i o n polarizability «„ is independent of the temperature.

polari-zability b y ÊCO = 1 + 47T X'I fe-quation (52)). I n t h e s a m e m a n n e r we m a y define t h e susceptibility / „ , deie- te) t h e o r i e n t a t i o n pe)lai-izabilitv, ac-cording t o

£„' - 1 -h 4,T7„, (81) in which EJ is t h e c o n t r i b u t i o n e)f t h e o r i e n t a t i o n polarization t o t h e

die-lectric c o n s t a n t ; io, anale)ge)us te) (53), is r e l a t e d t o a„ b y t h e expression

_ /lo a„

^° ^ \ 4jr '

1 - - „ - Ho «o

where ;;„ is t h e n u m b e r e)f dipole-s pe-r unit e)f v o l u m e .

Frenn e q u a t i o n s (80) a n d (82) it is se-en t h a t in t h e first ajiproximation t h e Susceptibility y^„ of the- e)!ie-i!tatie)ii polarizatie)n alse) is independent of

temperature a n d is inversely pre)portie)nal t o t h e " a c t i v a t i o n " e^nergy q.

This p r o p e r t y will be used in p a r t I I I .

D e b i j e did n o t a p p l y his exte-nde;d theory to t h e be-haviour of e' a n d t a n (3 as functions of freepieney a n d te-mperature. b u t it is easy t o under-s t a n d t h a t the- e q u a t i o n under-s (73) a n d (75) will n o t be c h a n g e d . He)we;ver. t h e epiantity £g will have- a eliffere-nt depe-ndence on t h e t e m p e r a t u r e .

D e b i j e ' s e x t e n d e d thee)ry has be-en confirmed b y several a u t h o r s in t h e ease of conce;!itrated solutions of a pe)lar liquid in a non-polar s u b s t a n c e a n d alse) in t h e case of highly viscenis liquids. Measure-inents h a v e been carried o u t b y S p e n g l e r 22). P l o t z e - 2 " ) , F i s c h e r a n d K l a g e s 2**). I t is obvious t h a t this e x t e n d e d theeiry can be applied in a first a p p r o x i m a t i o n to solid dielectrics.

A p r a c t i c a l conclusion from D e b i j e ' s dipole t h e o r y is t h e following:

t o o b t a i n a low-los^; dielectric we- niust preve-nt t h e piese-nee e)f dipole-s. 8. Ionic theories

Several a u t h o r s h a v e trie-d t o explain t h e dielectric afte-r-etfe-et a n d die-lectric losses b y a s s u m i n g t h e presence of ions in t h e die-die-lectric.

M u r p h y a n d L o w r y 2'') a s s u m e a solid dielectric t o h a v e an irregular s t r u c t u r e c o n t a i n i n g m a n y holes. (Jn t h e beiundaries of these he)les ions are

adsorbed t o a thickness e)f several ate)mic laye;rs; in t h e interstice-s/ree ions

m a y be ])rese!!t. If an electric fiehl is applied t o t h e dielectric, t h e adsorbed ions are displaced, a n d t h u s t h e dielectric is polarized, t h e ions being able te) m o v e freely in t h e adsejrption layer. To each ionic l a y e r as wedl as te) t h e free ions we m a y attribute- a relaxation time x.

T h e ions are adsorbed with intensities r a n g i n g betwee-n w ide- limits, t h e r e l a x a t i o n t i m e of t h e ions incre-asing w i t h increase of the- stre-ngth e)f

ad-so!-ptioii. As in v o n S c h w e-idle-r's t h e o r y , we liave a large n u m b e r of dift'erent r e l a x a t i o n times T c o n t r i b u t i n g t o t h e after-effect function (when a c o n s t a n t field is s u d d e n l y applied) te) an e x t e n t of G ( T ) <^ '''. I t is easily seen t h a t , concerning t h e behavie)ur of fi' a n d t a n ö as functions of t h e fre-q u e n c y , we o b t a i n t h e same- efre-quatie)ns as in v o n S c h w e i d l e - r ' s t h e o r y (e-e{uation (34) and (36)).

B o n i n g * ' ) also explains t h e after-effect by a s s u m i n g the e-xistence of ions in t h e dielectric. H e also distinguishes t w o t y p e s of ions: adsorbed ions a n d conduction ions, t h e l a t t e r h a v i n g a charge opposite in sign t o t h a t of the- a d s o r b e d ions. T h e adseube-el ions are ne)t able- to me)ve u n d e r t h e