Instructions for use

Title Some Considerations on the Performance of the Esaki-diode Oscillator

Author(s) Ogawa, Yoshihiko

Citation Memoirs of the Faculty of Engineering, Hokkaido University, 11(5), 527-557

Issue Date 1964-03

Doc URL http://hdl.handle.net/2115/37842

Type bulletin (article)

File Information 11(5)_527-558.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

Some Considerations on

the Esaki-diode

the Performance of

Oseillator

Yoshihiko OGAWA

Contents

Summary I. Introduction ........................... 527 II. The Circuit Equations ............,......,.'. 527 III. The First Approximation...,.....,........... 529 IV. The Second Approximation .........,......... 548 V. The Rising Time of the OscillatiQn ................ 550

VI. Conclusion ..............,............ 553 Acknowledgment .................... ......... 554 Appendix I.....,...............,......... 554 Appendix II .............................. 554 Appendix III ...................... .l ....... 556 ,1 Appendix IV ....,...............,......... 557 References ............,,............･-･.･ 557

Summary

In this paper the hysteresis phenomena and rising time of the oscillation

of the Esaki-diode oscillator were analyzed. Also these results were eompared

with experimental results, from which it can be seen that both results coincide

I. Introduction

It has been frequently observed that considerable hysteresis phenomena

are observed in the operation of the Esaki-diode oscillator. In this paper these

phenomena are explained to be the results of the shift of the bias voltage,

i.e. the origination of the auto-bias voltage, in the state of the oscillation.

This explanation coincides well with the experimental results.2)

Next, in this paper the rising time of the oscillation of the Esaki-diode

oscillator was considered. From this eonsideration it can be seen that the

rising time is very short, which shows that the Esaki-diode oscillator is suitable

for pulsed osciliation.

II. The Circuit Equations

The circuit equations corresponding to the circuit of Fig. 1 are

528

i(v,)i V

Fig. 1.

ClvD iL -i(VD) == C dt

and

2yD+L CIiL +riL=E dt

Combining the equations

d:w;,D + 6 (l+2Z

T=(Oot '

to, - 11VLC

Z= caoL =

2 = (to,Cr>-i --

Now let

XID E!l XJDo + T7,t

and

. i(?7D)Eii(?ZJDo+W,l),

where wD, is the bias voltage

excluded from the circuit of

duced in the Esaki-diode only

be noticed that the d-cfollows' ,

V,l!Ii!V,lo+Z? '

where w,i, is the d-c

vdi- v,io is called the auto-bias

Yoshihiko

p

Thecircuit 6scillator.

(la)

'

' ' (la) and

tt Cli Z'.D)] .E) (2)

tt ' tt

VLIC

,--iVLIC

(3a)

(3b)

supplied to the Esaki-diode when L and C were

Fig. 1, and x,,, is the component of voitage pro-

when the circuit of Fig. 1 oscillates. It must

compopent is included in va. We rewrite wa as

(4)

component of v,i and w is the alternating component of

voltagegeneratedintheosciilatorystate. ･

,

OGAWA

Lr - iL

of Esaki-diode

(lb), we get

+ T)D + 7"i(WD) ==

SomeConsiderationsonthePerformanceoftheEsaki-diodeOscillator 529

In the non-oscillatory state, w,i is zero, and 'Eq. (2) can be written as

X'Do+7"i(2YDo) === E･ , (5) In the oscillatory state, we shall divide the current flowing in the Esaki-

diode into d-c and alternating components as follows;

i(Z'Do+Wtt)i!ii(Wo+V)='ia(Wo+ZJ)+ia(Z7o+V) (6)

where

2Zlo ! 2VDo +27ao

i,,(ve+v):alternatingcomponent . (7) id(2yo+w): d-c cornponent

By using Eq. (6) we can draw the d-c relation of Eq. (2) in the case of steady

state as

2yo+7"i,t(xJe+w)=E. (8)The rest of the relation of Eq. (2) can be rewritten as follows;

`dl2,2.', +z)=ef'(v, dthZ';v,) (g)

'where

e-: 1!2 ' f(w,iS;;?yo)==h(2ria(`vo+zJ)+pa(xJ;vo)illli. (lo)

' ' g(w;?v,)--1+2zd£(V.D) ..-･- ,..

Now we assume that the current i(w) of the Esaki-diode can be approxi-

mated as

i(z,) -:: ih.x,if. (11) fir=O

By considering Eqs. (5), (8), (9) and (11) we can investigate the oscillatory

characteristics of the circuit of Fig. 1.

III. The First Approximationi)

Assuming that the alternating voltage w may be expressed as

where a and ¢ are the functions of T and by substituting Eq, (12) into Eq. (8),

530 Yoshihiko OGAwAwe get (see Appendix I) ' ' z,o+r.Z5=,,tf.,hff(iS,)(,,1,Il12)2-"bs.,w,"r"bam=E. (13)

Especially in the case of a=O, i.e. the non-oscillatory state, we get from the

above equation

5 vDo+rXhifvffo == E. (14) z=oThis equation is identical with Eq. (5).

Now if we assume that the relationship between a and zJ, can be deter-

mined, then from Eq. (13) the source voltage E can be obtained in relation to

the vaiues of a and x,,. On the other hand, the relationship between E and

wDo can be determined from Eq. (14). From these two results we can compare

w, with xiDe with regards to the same value of E. The difference w,-wD, is

the auto-bias vo}tage in the oscillatory state.

Now from Eqs. (10) and (12) the values of a and ¢ can be determined,within the !imitation of the first approximation, by the next equations (see

/

if;=-EA,(a) ' (15a)and

' ile=-1+EB,(a) i11 .1'(lsb)

where

Ai(a)--{I71(w,)a+4(T),)a3+FLa5] (16a)

B,(a)--S' (-5+Ah(")] (i6b)

I71,(v,)-=-l}-(1+s2Z.Z5]=,rvtA･2ygy-')--ll-[1+QZtr(w,)l (16c)

4(v,)-g2z.z5-L,(blh.℃,ff-3 (16d) t t tt ,4 == i6 2Zhs･ (16e)It must be noticed that the value g (w,) in the Eq. (16c) is the conductance

cli(v,)1du, at the point of the operating voltage w, of the Esaki-diode.

Some Considerations on the Performance of the Esaki-diode Oscillator 531

AKa)

o

e"IBNS lt

sof6gN

aq as .a

76{53tw

a

℃ts

2

Fig. 2. Four typical curves which may be drawn from the equation AiCa)=O.

' Now considering Eq. (15), it can be seen that the amplitude a is respec-

tively in a damping, growing or steady state coyresponding to the negative,

positive or zero value of A,(a). Moreover there are three steady states which

are stable, half-stable and unstable. The function A,(a) can not take the p}us

value in the region of a sufficientiy large value of a from the nature of the

Esaki-diode, so that IiL or h, must be positive. Noticing this fabt, we can

classify the function A,(a) into four groups which are shown typically with

four curves in Fig. 2. The points at which the function A,(a) crosses with

the axis a can be evaluated from the roots of A,(a)==O. These roots are

and

-a±VF:'-4I711TL .. , (17b) a == 2PL

The nature of stability in steady state can be investigated by determining the

sign of the function

dAde,(")=-[E(v,)+3ITs(z,,)a2+slTLa`} ' (18)

for the real values of a in Eqs. (17). The steady state is stable, half-stable or

unstable according to the minus, zero or plus value of Eq. (18) respectively.

532 Yoshihiko OGAWA Now the origination of oscillation indicates the existence of, at least within

the limitation of the first approximation, some limit cycles in the phase plane

of i(zJD)-v]o. By noticing this fact and investigating the nature of the curves

shown in Flg. 2, we can classify the configuration of the trajectories in the

phase plane into four types as shown below:

I) There is no limit cycle but oply one stable focal point at a==O, in

the phase plane. In this case the oscillation can not occur. This case corre-

sponds to curve 1 of A,(a)-a plane, in Fig. 2.

II) There are one stable focal point and one half-stable or outwardly

stable Iimit cycle. This case corresponds to curve 2, in Fig. 2.

III) There are one stable focal point and two limit cycles. One of these

two limit cycles is unstable and the other is stable and larger than the unstable

Iimit cycle. This case corresponds to curve 3, in Fig. 2,

IV) There are one unstable focal point and one stable limit cycle. This

case corresponds to curve 4, in Fig. 2.

In the first class, there is no real value of a satisfying Eq. (17b) so that

eitherofthenexttwoconditionsmustbesatisfied: ''

F:(woi) (19 a) Il;(woi) > 44

or ' 4(woi)lllO,.E(v,,)ll:O. . (19b)Now the condition that the focal point is stable can be decided from Eq, (18)

by [dAi(a)1dn]c`r.o, which is

or by rewriting

1 -g(woi) :;l 2z･ (20b)Eq. (20b) shows that the foca} point is stable only when ,the magnitude of the

negative conductance of the Esaki-diode at the operating point is not larger

than the resonant conductance of the passive circuit excluding the Esaki-diode.

It must be noticed that both conditions of Eqs. (19) are satisfied by the con-

dition of Eq. (20b),

In the second class, there is only one positive dual-root ai satisfying Eq.

(17b), so that the condition

E(.,,)=FZ(Z'o2), L(,,,,)<o (21) 44

SomeConsiderationsonthePerformanceoftheEsaki-diode.Oscillator 533

must be satisfied. The dual-root ai of this case ･is

H4(wo2) . (22) al =: 24

It must be noticed that the condition (21) also satisfies the condition (20b).

In the third class, there are two positive roots satisfying Eq. (17b) which

are

.,...V-L(Vo3)-VFSi'io3)-4I71(2'o3)IiL (23.)'

and

.,..V-]I71(wo3)+VFgt.a3)im4Fi(Wo3)IiL, . (23b)

where a2<a3.Now a, and a3 must satisfy two inequalities

[dAdt',(")].,='[Jil(wo3)+3L(?yo3)a:+5Ila;}>O (24a)

and

[dAde'(a)].-:-[l71(w,,)+34(w,,)aZ+54ag]<O･ (24b)

3From Eqs. (23a, b) and (24a, b) the inequality

4(2yo3)VF:(w,,)-4Fl(v,3)E, <O (2s)can be deduced. Now it is seen that the inequality (24a) satisfies the ine-

quality C20b). By noticing this and the inequality (25) it can be concluded that

the condition

Fg(wo,) >E(v,,)>O>a(w,,) (26) 44

must be satisfied.in this case. Only one stab}e limit cycle is the one corre-

sponding to a3.

the nienxtthienei:Ztafi.lgSSm' :lli) bfgCgktR.s/rendt :corresponding to a=o is unstabie so that

[dAdu'(a)].=, =: waE(wo4)>O- (27)

Next there is one positive single-root satisfying Eq. (17b) which is

534 Yoshihiko OGAWA ' .,=V-llFli(wo4)+VIZi2?'3t.o4>-4E(wo4)Ei. ' '(2s)

'The root a, must satlsfy the inequality

' [dAdn'(")].,=]:m{-l71(voi)+3L(vo4)aZ+5Lat]<O･ (29)

The inequality (27) satisfies the inequality (29), so that the single condition is

1l(wo4) <O (30 a)or

1 -g(Vo4)>2z･ - ･ (30b) It is self-evident that the oscillation occurs in the circuit of Fig. 1 when

the condition (30b) is satisfied. But even when this condition can not be

satisfied, it is possible, as seen in class III), that the oscillation originates only

when the value of the initial condition of a is larger than a,. This phenomena

can be illustrated physically as follows: the oscillation originates when the

magnitude og the average negative conductance g(a;w,) of the Esaki-diode

one ofoscillationis s

r-rlT.

Ivivl.-

--- [

l I l 1 [ l I 1 1-- -1

elL

eg(asY

t

Il

s --) 5C,( jco)

s

l

1

1-l

1

I

,

ti

1

t

L

c

r

.-l

I

1

ll

I

l

1

-i

s il ! Fig. 3. Equivalent circuit of Esald-diode

oscillator shown in Fig, 1.

by considering only the r-f component.

Yo(ju):=go(tu)+foo(tu)

= r,+7(A(DL), +j'co(C-

The adniittance of the Esaki-diode g(a, ip

approximation is permitted. The relation

r-f voltage z, can be shown by using

through cycie not smaller than the resonant con-

ductance lf2Z of the circuit even if

the magnitude of the negative con-

ductance at the operating point, g(zJ,), is smaller than lf2Z.

The above conclusion becomes

more clear by analyzing the steady-

state amplitude and frequency of the

circuit shown in Fig. 1 as follows.

We will rewrite the circuit shown

in Fig. 1 as the one shown in Fig. 3

The admittance y,(ju) in Fig. 3 is

r2 +ll I.,L)2 ]' (31)

; wo) is conductive so far as the first

between the r-f current i and the

g(a, ip; v,) as

SomeConsiderationsonthePerformanceoftheEsaki-diodeOscillator 535

i-g(a, ¢; z),)･v. (32)It must be noticed that the conductance g(a, ¢; wo) is the function of the phase

¢ or normalized time T. Now if we assume that the oscillation can occuronly when the next relation is satisfied between the conductance ye(j'tu) and

the average conductance g(a; v,) of g(a, gb; wo) through one cycle of oscillation;

g(a; v,)+yo(j'to) == O, (33)or by rewriting

g(a;w,)+ '" =-O (34a) r2+((ic)L)2

and

r2+((DL)2

From Eq. (34b) the angular frequency can be,obtained as

to -- vic ･Vi- Oi2. . (3s)

The above value coincides with the one obtained from Eq. (15b) within the

limitation of the first approximation.

Next using by Eq. (35), Eq. (34a) can be rewritten as follows;

1 g(a; z,o)+ 2z =- O･ (36)This is the relation deciding the r-f amplitude.

Now g(a;w,) can be calculated as follows;

' g(a; wo) == .1.. Sl"i(acosgb; z,,)cossbdip

= .la Si"tg.,hir(wo+acosgb)rcosgbdip

= .S.,KhffVok'-i + (h3. + 4h4wo + 10h,v:) ･ -ii'-a2 + -l} h,a`

' =g(vo)+ 22z {L(v,)a2+4a`]. (37a)

Eliminating g(a; z,,) from Eq. (36) and Eq. (37a), we can obtain the same

condition obtained from A,(a)-::O (see Eq. (･16a)).

It is interesting to rewrite Eq. (37a) as

536 Yoshihil<o OGAWA -[g(a;wo)-g(w,)}--Q2z[4(w,)a2+I}a`}.' '(37b)

If

.2<-4("o), (3s) 4

then the right hand side of Eq. (37b) is positive, so that･ the magnitude of the

average negative conductance becomes larger than that of the negative con-

ductance in the operating point. Noticing this fac.t and Eq. (36), we canconclude that the oscillations can originate even when the inequalities

1 >-g(v,) (39) -g(a; wo) > Qz

are satisfied at the r-f amplitude a satisfying the inequality (38), The above

condition coincides with that of the hard oscillation shown in Eq. (26').

Now we will take the numerical examples:

h, - O.3216858856 × lo-3

h, - O.715310156s × lo-i

h. =- -O.9813351377 × 100

h,-O.5063747180×10i .. , h, -= -O,1153375937 × 102

h, -= O.9760796488 × loi

and

C-350pF., L==:1ptH, (40b)where hi (i--O-v5) are the coefficients used in Eq. (11), and C and L are the

capacitance and the inductance shown in Fig. 1. The static current-voltage

characteristics approximated by Eq. (40a) is compared with the one measured,

in Fig. 4.

Now it is convenient to define the next function:

e(?y,) -: 6z [2E(w,)-i- FjiXWIo)), (4i)

By rewriting Eq. (16c), g(2y,) is

' g(wo)==6z'{2Fl(wo)Tl]- . .(16c')'

'For the above numerical example, 6(w,) and g(z,,) are shown in Fig. 5. The

Some Considerations on the Perforrnance of ･the Esaki-diode Oscillator 537

i(v)

2.2

2.

1.

1.

1.4

1.

1.

o.

(mA)

O.6

o.4

O.2

2

o

'

BA

2

AB

A

B

4

4080120 160200240280520560Fig. 4. CQmparison between the

the measured one (A), of

approximatedEsal<i-diode.

v-i characteristic

v(B) and

(mv)

538

一ξ，　。9　（てy）

　100

与・1♂

　　ロ　　10

　　コ　5x10

　1♂

5x↑σ3

　竃σヨ

5x1♂

　r410　40

Yoshihiko　OGAwA・

L．．．

・一ｨ

／

／

ア

！

、．q亀

噂し．

グ

－

、

、　1張

誘

／ノ渚∠

／7 @　　　　　　　　’

四9

礁 7　．

乙∠

／ 7．9

　　　80　　　　　　「書20　　　　　睾60　耳73　　200　　　　　　240

Fig．5．　一ξalxl　－9　Plotted　as　a　fullction　of　70．

　　280　　三500　320　　　　　三560

　　　　　　　　　％　（颯V）

Hatchedτegiolls　show凡＞0．

与oo

SomeConsiderationsonthePerformanceoftheEsaki-diodeOscillator 539

regions satisfying the inequality ,. '

Iib(w,)<O, (42a)or

-2h,-V4h:-10h3hs<.,<-2h4+V4h:-10h3hs, (42b) h, h, ' 'iZ,S.h)lll:,,".usa,:.n,g)h.ee.,(:.liikM,,// i/9:/ ?q.i,l.//Z/LLI)5.,lf.h..,E.onditions (iga), (igb), (2i), (26) and

i (19 a') -4(2yo) < 2z ,

1 -g(Vo)$2z-, L(2yo)IO, (lgb,)

1 , Ih(w,)<O, .(21') -6(wo) == Qz

'e(z,o)>6z>-g(?ye), o>L(z,,) (26')

and

1 -g(?'ek 2z･ (30a')In Table 1 the above conditions are all collected' and compared with the

number of curve in Fig. 2. We wlll show one example of using Fig. 5 arid

Table 1 in the case of 112Z ==2×10'"3es. First draw the straight line parallel

to the horizontal axis corresponding to 2×10-3cr. Next compare this linewith values of -e(v,) and rg(w,). By refering Table 1, the conclusion

TABLE 1.

Classifiednumber I II III IV V

-e-11(QZ) ,- o +

-g-11(QZ) -,o - +,o

F, +,o - 'tttt

CurvenumberinFig,2corre-sponclingtotheabovecondition,

1 1 2 3 4

540 Yoshihiko OGAwAshown in Table 2 can be dedcued.

TABLE 2.

we (mV)

wo 5 71

71 < wo S. 219

219 < ve < 264

264

264 < wo < 315

z,o l 315

classified No.

II

VIV

III

I

II

state of oscillation

no osc.

free osc.

hard osc.

half stable osc.

no osc.

no osc,

It is plausible to think that there is a hysteresis phenomena with regard

to the characteristics of the r-f amplitude a versus the operating voltage v,

of the Esaki-diode as far as t'he hard oscillation is concerned. In the above

example, by increasing vo the oscillation disappears for w,==264 mV. Next for

decreasing w, from this value it is observed, however, that the oscillation does

not occur for v,>219 mV and it occurs at vo=:219 mV.

It is questionable, however, to believe that the above conclusion is right.

Because in the above conclusion, the fact that d-c current-voltage characteris-

tics of the Esaki-diode in the osciliatory state differs from the one in the static

state is neglected. This difference is shown in Fig. 6. The static characteristic

can be drawn by using Eq. (11) which is

5 i(vo) = : h"woff･ ff.=o 1../Those in the oscillatory state can be drawn by considering Eq. (8) and Eq.

(13); namely d-c current ia(vo+℃J) is

iti == 1?511=,,tll.i,h-(,1.<)("1:192) '2U"'6mwge-""aO", (43)

where the reiation between the r-f amplitude a and the operating voltage vo

can be obtained from Eq. (17b). From Fig. 6 the following interesting con-

clusions can be introduced: First, in the state of curve B the oscillation

disappears continuously with increasing wo at the point Q, where the curve B

meets with the curve A which shows the static characteristic. This fact means

that the hysteresis phenomena can not occur. But in the state of the curve C

the oscillation disappears discontinuously at point 2z with increasing we and

the operating point of the Esaki-diode jumps to point QE. Next, for decreasing

v, from this value the operating point of the Esaki-diode follows the static

f

Some Considerations on the Performance of the Esald-diode Oscillator

id

2.2

2.Q

1.8

1,6

1･4

1 .2

1.0

O.8

o.6

O.4

O.2

Fig.

o

(mA)

tt

RPsP,

tt.

T

P,

'

./

'

E BA

R,

R,

z1

R,

Rt,s1,

lltQ,

illt

QS `ltt

Q;QX

,

4o 8o 120 160

6. Some i(i-v characteristics of a Esaki-diode;

B.vE: the characteristics in the oscillatory

27.57 9, C: 9Z= 150 9, r= 19.05 9, D: 9Z= 500 9, r=5.714 9)

200 240 280 v (mV)A: the statie characteristic,

state (B: QZ=:1009, r==

300 2, r=9.524 9, E: QZ=

'

541

542 Yoshihiko OGAWAcharacteristic line A to the point R where it shifts on the curve C and the

oscillation occurs again. The same hysteresis phenomena appear in the case

of the curves D and E.

We must also consider the hysteresis phenomena with regard to thecharacteristics of the r-f amplitude a versus the source voltage E instead of

the operating voltage x,, of the Esaki-diode as shown above. We will plot the

characteristics of w, and vD, versus E which can, be obtained from Eqs. (13)

and (14) respectively. These are shown in Fig. 7, Fig. 8, Fig. 9 and Fig. 10 /.for some cases of 2Z and r. For example, it can be seen from Fig. 8 that

l, VetVpo

200

18o

16o

140

120

3OO

80

6o

4o 4o

'' Q'

tt

QVDe L

Vo

l

Ni,

, Q

16"

162

160

158

VDO

VoVo

.t

tn

Q

R187.0187.4187.8188

,

Vo

Voe

P

80 12o

Fig. 7. Lvo

of

t60 200 240

and vDo plotted as a function of E

QZ=100 9 and r=27.57 9.

a8o

for

Ethe

520(Tttv)

case

S60

Some Considerations on the Penfbrniarice, of the Esald-diode Osci 11ator 543

Vo eVbe

280

260

240

220

200'

18o

160

140

120

1Oo

Bo

6o

(mv)

lyo

4o

'-I

.t. tt

tt .. .'

tt

'

VDe

'

Ve'

'L

' '

'i'

'

'

'

Ve

Vpe

'

'

1

P 't.t

/

l'

eo 120

Fig.& vo of

160

ancl vDo plotted

2Z4509 and

200 240 280 Eas a .function of E for the

r=19,059.'

･ 5EO (mv)

case

S6o

544 Yoshihilco' OGAWA '

% tS{)e

2eo

1

260

.240

220

200

180

160

140

120

1OQ

8o

60

4o

(mv)

.t.t

t

ttttt .1/

tttttt t.

Q'Q

.tt.t

:

:

VDe

Vo

a

R

/

'

Vo

t"

Vp'e.

/

'

P

4o 80

Fig. 9.

120

woof

and

2Z

160

vDo plotted

== 300 2 and

200 240 280 E.as a function of E for the

'r==9.5249.

32o (mv)

case

J60

Some Considerations on the Performance of the Esaki-diode Oscillator 545

vb ,v6,

280

260

240

220

200

l80

(mv)

i60 t

140

120

1OO

8o

60

40

Qt

Q

..tt/

'

' '/t

265

I '

'

-265 VDO1iQ

E VeoVe

'I''

'261

'

Ve

t.tt

26-･26626!

'

R

t.

lt.

'

VoVpe

P

4o 8o

Fig.

120

10. vo of

and

2z

160

vDo plotted

= 5009 and

200 240

as a function

r= 5,714 9.

280 520 E (mV)of E for the case

560

-/･

546 Yoshihitgo OGAWA -

by increasing E the oscillation disappears discontinuously at the point 2 where

'the operating voltage of the Esaki-diode jumps to the point 2', and that for

decrea$ing E from this value, the operating voltage of the Esaki-diode follows

the wDo-line and the oscillation can not occur till the point R. The difference

(wo-x,Do) is the auto-bias voltage 6f- the Es-aki-diode ih the ogcillatory state.

This value is much smaller-in the case of Fig. 10 when compared with the

case of Fig. 8. In such cases, the operating voltage of the Esaki-diode might

accidentally jump from some point on the wDe-line to another point on the

v,-line by the external noise-vo}tage affecting the Esaki-diode directly. In this

case, the oscillation might occur unexpectedly.

It is .interesting to note the case of Fig. 7, where there are two values

of vo corresponding to one value of E in the narrow range of it. In this case,

by increasing E the oscillation disappears discontinuousJy at point 2, and,the

operating point of the Esaki-diode jumps to the point 9'. Next for decreasing

E from this value, the operatirig point follows the wD,-line, and the oscillation

occurs at point 2" so that the operating point jumps -to the point 2"'. '

Finally we will plot the relations of the r-f amplitude a･ versus 2y, and

the same value versus E in Fig. 11 and Fig. 12 respectively, which have been

also utiiized in the above analysis. Fig. 13 ls the comparisondbetweeri the

experimental and theoretical values of the r-f apaplitude a versus E in the case

of QZ=3009 and r==9.524 9. So far as the r-f amplitude is concerned, both

curves coincide approximately in the region of the large value of E though

x

a CmV)

uo

200

160

120

80

4o

o

QZ=10oo

'oo

t.

soo'

o

150

l

1111Iltl41

::Y 11ll11vv11

1 lt' li11AA1)1)lb

'I 111111III

llIlll1111Ib

iYiII1:I1[n

6o 80 1oo 120 140 z6o

Fig. 11. The r-f amplitude a,as a

180

function of

200

vo for

220

many

240 260 v. (tr(V>

cases of 91

28o

/

Some Considerations on the Performarice of the Esaki-diode Oscillator 547

a (mV)

eno

200

160

120

80

4o

' ' o

1. ttt

'

QZ=1000tt '

'

500L

' tt'd 3oo

'l

1

'r' t..tt '

i-l200' f

1'

1

1'

lt ': 1 :1) 11 d t

1

'

' 't'Ia

l 1III 11 l ' '

Sl 1 l 1 11I* ' l I 1

t loo 1 I 11'1

'd v , Y

''1l"l ･r" Iilt

tlII Y vlll11i 1 1 11

4 tl1.l1･L

IIII

Si lllfl

'

LIL lt, 11 i 1 , it1 1 l Ilo o 100120 u i 1 o 200220 ajo 2 2eo

E (piV)

Fig. 12. Thefor

r-f. amp]itude

many cases' ofa asQZI

a function of E

a (mV)

t

'

'

ap

oo

60

20

.t

B

-1ltFII ti'

'

lttlte,l'

8o

A tl1]IIIAAIl:

"*lltl

it11lttl

4o'o'

IV11llIt

11'i 11tltijl1

d6oso1oo120!40160ISO 20022024026Fig. 13. Comparison between the theoretical

(B) values of the r-f amplitude a as

the same value of QZ and r whichrespectively.

E (7gv).

(A) and experimental

a-function of E with

are 300 2 and 9.524 9

548 Yoshihiko OGAwAthey' do not coincide in the region of the, smal} value of E which may be due

to the existenpe of the higher harmonics. So far as the hysteresis phenomena

are concerned, both curves coincide approximately, except for the region of

the small valqe of E where a slight hysteresis phenomena exists in the experi-

mental curve which can not be illustrated in the scope of the first approxi-

matloll.

IV. The Second Approximationi)

'It is convenient to utilize the following analysis for obtaining the second-

approximated solution.

We put the first-approximated r-f voltage w of the Esaki-diode as

.?v = a cos ip

¢ =, tot+9･

Th'e current flowing into the Esaki-diode -by this voltage can be expanded into

Fotirier series as

ind- i(a cos gb; w,)=ii.(a; vo) cosngb, (44) 7t=!o

where i.(a; z,o) are shown in Appendix III.

Now by the current component i.(a; vo) cosn¢, the voltage

g,(p)-ii,,(a;v,)cosnip (45)is produced at both ends of y,(.2b) (p=-cnydt) shown in Fig. 3.

Put

y,(1'9)-ly,(9)le""<O'=g,(9)+j'b,(9), (46)

where

Izlo(9)l ==:

g,(9) -

-9fbo(9) -t

cos ¢ :=: Vg:

. sm¢== Vg:

(45) can be

Vgg(9)+bi(9)

1-r2 + (S?L)2

C- L 1 r2+(9L)2J

g, (9)

(9)+b:(9)

bo(9)

then Eq.

(9) + bZ (9)

rewrltten as

'

(47)

SomeConsiderationsonthePerformanceoftheEsaki-diodeOscillator 549

yii(j'9)i.(a; z,,) cos nip

= v,liiSfZi)(.) cosl"tut+ngmip(nto)}, (4s)

so that the formula of w in the second,approximation is ,//

t. tt t lti ･ w==acos.¢+,ii.l,vg:li'ii:IS-:,ln,,)'cos[ngb-¢(n(v)l ,

'

=-acos¢+A, (49) ' t tt ttt A!7ii#ivggll',i.(glZ:)(n.,)'cos[n¢-ip(nto)].

Generally Eq. (49) is utilizable only if"'

vgg(i"ggl:ll.of<a,n:¥i･ /.. (so)

The current i can be calculated more precisely by using Eq. (49), and can

be expanded into Fourier series as

i = i(a cos gb + A)

!Fi(acosip)+A･i'(acos¢) 5 = Z i.(a) cos n¢+4(a) 71==O

+,;l.I,{L,.(a)cosn¢+4.(a)sinnip], , (sl)

tt ttwhere L.(a)' and 4.(a) afe showri in Appendix IV. '' Now if we consider that the fundamental component of the right handside of Eq. (51) is equal to the fundamental component of the current flowing

into the linear eiement y,(1'tu), then we have

/. . ' -y,(]'Q))acos(cat+g)==-1y,(to)]acos{tot+g+¢(w)}

' :-ii(a)cos(cat+g)+4i(a)cos(tot+g)'

+4i(a)sin(tot+,p),

Lso that

-]ye(to)]acosip(to)-i,(a)+4,(a) (52a)

.t ' lyo(to)Ia sin¢(ca) -= 4,(a), (52b)where(seeAppendixlv) e

550 .YoshihikoOGAwA -･ IL!(a)='li'L,,£.,ly,(i..)t'dZd'3t£")･cosip(nto) '(s3a)

and

4i(")===.Z.,,,ly,(1..)l'Mi(")sin¢(nto)･:'-.. ' (s3b)

From Eqs. (52a, b) and'Eqs. (53a, b) the r-f,'amplittide a and the angular

frequency to can be calculated in the scope of the second approximation.

V. The Rising Time of the Oscillation

From Eqs. (15a) and (16a), . ' do --ELdT. (54) =lilt-k{lg)-(Wo) a+ 4(Ve) aa+as

'4 4Hereafter, we restrict the oscillatory condition to the case where only one

stable oscillation exists. This condition is (see Eq. (27))

iNow put

.2.. -4(Vo)+VFZ<Vo)-4Fi(Z'o)FL, a,>o (56a) 24

and ' bz=.Fk(w,)+VFg(wo)-4Iil(wo)"IiL, b,>o. (s6b) 2IL 'Eq. (54) can be rewritten by using Eqs. (56a,. b) as

du -= -EE,(lc･ a(a2-a:)(a2+b:)

Byintegratingthisequation, ' '

a =C,exp[-EF,(w,)T],' (57) (a?-a2)"bi(b42+a2)M2

where Co is the integra} constant, and mi and m2 are

b;･ Ml = 2 (aZ + bZ) ' a: ･ 77Z2=2(a,2+b42): '

SomeConsideratiensonthePerformandeof..theEsaki-diodeOscillator 551

Now let the r-f amplitude at T=O be a,, and this at T=T be

a=:rp(r).au,.llrp(T)>"O=rp,. (58) , a4

Then ' '

(aZ-a:)Mi,(bZ+a:)m2 , 'Eq. (57) can be rewritten by using Eq. (58) and (59) as

a:mirmi(1-rp2)7,},(bz+.xrp2)",, = Co eXP["E171Tl, (s7')

or

T== -:i7i (in i -miinlilZi2 mm2inlil[lilZi'ii22.M:]･ (s7")

The first term of the right hand side in Eq, (57") shows the initial state in

the rising of the oscillation. This can also be obtained with the linear-appro-

ximated differential' equation of the circuit shown in Fig, 1. The second term

of the right hand 'side in Eq. (57") shows the latter state in the rising of the

oscillation. This,shows the saturated characteristics in the oscillation. The

last term of th'e right hand side in Eq, (57") is a corrective term and is of

little importance.

Now we define the rising time r, of the oscillation as i Ts =:: [T]v--･o,g-[T]v==o.i･ (58)Then by using Eq. (57"), T, is ' ' ･ ' Ts= E'1 (ln9+miln ¥'g9 -m,In li?ig3s8iil:). (sg)

By rewriting Eq. (59), - - T,Z-mp,Z=I"(v,, QZ), (60) 'where ' r(X'o, 2Z)= giz-+2g(.,) [2-i97+i･650mi-m2ln li2ig:7,:].

(61)

tlll8tg,gh.at.,',"di :.."d,.M/,2.la.r,:...bO.tfh E",I c(t6is?,s. gf wo and 2Z)･ The next relation

552 ･ YoshihikoOGAWA

r(Y,,QZ) :(bhtn)

1Oooo

500o

JOo

100 .I 60 ,-80 100 120 140 16o 180 200 '220 240

' /'1' IC, (MV) ･･' Fig.14.r('vo,QZ)plottedasafunctionofvo.

fE==l!2, 2==:r-'VLIC

ca,Z-=1/C, Z-:=VLIC, tu,-11VLC

Ts ==: CVots

E(?yo)=t[1+2Zg(v,)] ,

l"(w,, 2Z) are plotted in Fig. 14 as a function of 2yo in some cases of QZ,

If QZ, Z and to, are determined, then three circuit elements L, C and 7'

can be decided. Assume that 2Z and tu, are decided. Then the larger Z is,

the shorter the rising time of the oscillation is. Next assume that Zi and' tu,

are decided. Then the larger the resonant impedance 2Z is, the shorter the

rising time is. Lastly assume that eZ, Z and tu, are all decided. Then the

rising time is the shortest in the neighborhood of v,T:120 mV,

We will take one numerical example:

.' L-1ptH, C::::350p.E, r-5.7149 ' wo=120mV to, = 5.35 × 10' rad/sec (A = 8.5 MC)

Z-53.59, 2Z=5009. 'In this case, by referencing Fig. 14,

QZ 1OO

2 o

}oo5oo

1OOO

1

'

Some Considerations on the Performance of the Esaki-diode Oscillator 553

Fig. IS. The initial state of oscillation of the Esaki-diocle oscillator:

I"=1#IL C=350p.F. and r=69.

- r(w,, QZ) - %4- 2.z -i･36cycle･ (63)From Eq. (63) it can be seen that the rising time of the Esaki-diode oscillator

is fairly short. The experimental results of the rising time using the same

circuit constants shown above is shown in Fig. 15. This photograph supports

the analysis which was conducted previously.

VI. Conclusion

The hysteresis phenomena of the Esaki-diode oscillator are observed experi-

mentaly in two regions of bias voltages, i.e. small bias voltages and large bias

voltages. The latter can be illustrated from the IIirst approximated analysis of

the non-linear differential equation, the results of which adequately coincide

with the experimental results. But the former can not be illustrated within

the scope of the first approximation. However it may be illustrated from the

second approximation.

It can be seen from both the analysis and the experiments that the rising

time of the oscillation of the Esaki-diode oscillator is considerably short. This

result shows that the Esaki-diode oscillator is suitabled for pulsed oscillation.

554 ' '''tYoshihilcoOGAwA

, i-･'.//i .t4Fcknowledgment .i

The author wishes to'thank Prof'. Teiichi Kurobe and his staff of the

departMent'df eleq.trdnic engineering,.Hokkaido Ufiiversity, for their valuable

suggestions and constructive criticisms.

. Appendixl ' ttt t tt 5 i(?yo + a cos ¢) = E hu(?v, + a cos ¢)'ff Ar.,,o =;=-Alil.I,,tlS.,hA'z]8K-"b)a"'(7Kn)cos"bgb

. "b"1"6on, ==,II5..I,,t/(.Ii,., t?., h.fi'v:"rrM)a"t(:Sn)((-l}7)"b"i(ew)cos(m-2n)¢

' ' +(.','IZ2)gi',t], K277z, m>2n (Al).

' ' /, ../

6R:==(2i `.i,;i. .O.d,d. '(A2)

' '

i,,(w,+acosip)=.tg.,,tf.i,h.z,sfi'-m)a"e(liFt)(7;z7fZ2)g;;:

=.tF.,h.A-won'+-l}-.*.,(5)hA･?y8A'r2)a2+-g-t/t.,({il)hA-w8'"-4)a` (A3)

' 'and

[ia(?Yo+aCOsgb)],.,,,=(t?.,Khl-.z,].E"-i)a+-2-.*.,("E:)hff2yEA'-3)a3

+-Z-h,a5]cos¢==Ci(a;w,)cosip.. , (A4)

Appendik IIi)

fl(a cos ip,., -a sin ¢) == -9ri.(w,+a cos ip) +g(a cos ¢)a sin ip. .

' (A5) .,,-

Now put･

' 1(z>(w)[i::･Sl'g(z,)clw, ' . (A6)

then

Some Consiclerations on the Performance of the Esaki-cliode Oscillator

di (w) =- Sg(i+ 2z `,i,i((.2',o.'.V)) ]clv

-zJ+QZi(w,+v). (A 7)From Eq. (A7), ¢(acos¢) can be obtained as

5 di(acos¢)=acosip+2ZZhfit(w,+acosip) lr=.o " t' 7rt-'t"67n 5I' .2 ==aCOS¢+21.Z.=,,,Z,=, ,Z,=, hfi･wSre-"b)･L z

,..m(£:)((ml}m)O'LLi('.'Z)cos(m-2n)cb+(.',112).gli3]･ (A8)

Bypartialdifferentiatingdi(acosip)withrespecttoip,,, . 1.

7]b-1-tim 6op(ao;OS¢) = -asinip-2z.iz,,£,=, tF., hffz,S-"'m)'

･a"b(5t)(m-2n)(-;-)M-i(7iZ)sin(m-2n)ip

=06di(£".C.O,S¢9)-6("6C¢OS¢):-g(acos¢)-(-asinip). (Ag)

/.From this equation .... . 7n,-1-'Oetb g(acosip)asinip=asin¢+2Z.i=,,S,--, tF.,'hffvg"rM)'

･a"b(.K,)(m-2n)(-ll-)M7i(7tZ)sin(m-2n)¢. ' (Alo)

The component of sin¢ of Eq. (AIO) is .. [g(asinip)asin¢],,.,==(a+2ZCi(a;'v,)}sin¢

== S, (a; T,,) sin ¢, (A ll)where

S,(a;℃,)==a+2ZCi(a;v,)

By using the above equations, A,(a) and B,(a) of Eqs, (15a, b) can be

culated as

' ' A,(a)=:-21.S2,nf(acosip,-asin¢)sinipd¢ '

555

cal-

556

and

Eqs.

From

and

Yoshihilce OGAWA

== - i. Sin[g(a sin ¢)a sin ¢],,. ,,'sin ip de

= -LS, (a; ℃o) 2

Bi (a) = - i Sirtf(a cos ip, -a sin ¢) cos¢d¢

"= ;l.l'." S:n'[ia(wo+"cosgb)],.,v,'cosgbdip

.. 2'A c,(a;wh')･ ' 2a

(A12a, b) are the same with Eqs. (16a, b) respectively.

Appendix M

Eq. (Al),

io(a)=.*.,hirwSr+-llTiil5]=,({;)hx-v8ff"2)a2

+mlllm,N5=,(5)hfi-wsff-`'a`,

ii(a)==al,#..,Khnw8ffrri)+-2-.21.],(5)h.r?y8irm3)a2

. +-g-h,a4], ,i

i2(a) = a2(-ll} jP5=,(5)hAr T,8"'-2) + -ll-ti.,({i:)h-tz,Sk'-")az]

i3(a) = a3It.i=,(Il )hff wS'r-3) + 156 h,a2] ,

i4 (a) -: -l}-p5fi,(5)h.wEn-4)a4

' i h,a5. i,(a) : 16

'

,

(A 12 a)

(A 12 b)

(A 13 a)

'

(A 13 b)

(A l3 c)

(A 13 d)

(A 13 e)

'(A 13 f)

SomeConsiderationsonthePerformanceoftheEsaki-diodeOscillator 557

Appendix IV

6i("oC.OSip)[=i'(acos¢)･cos¢==n,.,dZi't(a)-cosn¢, (A14a)

Oi(aoC¢OS ip) = i, (a cos ip)･(-a sin ¢) == - ,il.l,nin(a)'sin nip･

(A 14 b)

By using Eq. (A14a),

IL.(a) = l f2"A･i'(a cos ¢)･cos ¢dip

n Jo

=:: 3,l]., 1yiiZ(;fi)1 ' tSiil?' (a cos ¢)'cos ip･cos[n¢- ip(nw)]dip

=. ,pu., lzli,"(l",Zx ･ -; Sl'T .E., CII'iiri(a) cos Kcb[cos .gb ,o, ip (.,,)

- sin n¢ sin ip (n to)] d¢

-: jl,ll., lyiit(ia,Z)] ･ cli2itl(a) ･.., ip(..)

=,Z,.,-ll-'ty,(l,.)l'dZik(")'cos¢(nto), (Aisa)

and by using Eq. (A14b),

lk.(a) = -l;-SinA･i' (a cos ¢) sin ¢clip

= -l;- 3,lll., ]iii(Sgi)1 ' Sinz'(a cos ¢) sin ip[cos n¢ cos ip(ntu)

-sinnipsinip(nw)]clip

= ,Z,., ty, (l,.)1 ' 'Zii,(") 'sin¢(ntu)･ (A is b)

References

e 1) H. H. BOTOJIKI)BOB I'I !O, A. MPITPOnOJIbCKL'IM: "Asymptotic methods in analysis of non-linear vibrations".

2) T. Kurobe, Y. Ogawa and S, Kondo: "On the hysteresis phenomena of the Esalci-

diode oscillator". The records of the General Meeting of I.E,, I.E.C,, I,I. and

I.T.E, of Japan; No. 1635, April, 1963,