NATL INST. OF STAND & TECH

AlllOb *171b71

.ICATIONS

MBS1R 83-2643

Wave Form Simulations for

Josephson Junction Circuits

Used for Noise Thermometry

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Washington, DC 20234

January 1 983

UJ

y[3(3 U.S. DEPARTMENT OF COMMERCE

83-2643 NATIONAL 3UHEAIJ CF STANOAflDS

19U3

C.2

RATIONAL BUREAU OF STANDARDS

LIBRARY fLB l 1983

0~F A-CC ~ C

NBSIR 83-2643

. WAVE FORM SIMULATIONS FOR

JOSEPHSON JUNCTION CIRCUITS

USED FOR NOISE THERMOMETRY

F. Sullivan, D. Kahaner, H.A. Fowler,

and J. Knapp-Cordes

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Washington, DC 20234

January 1 983

U.S. DEPARTMENT OF COMMERCE. Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director

WAVE FORM SIMULATIONS FOR JOSEPHSON JUNCTION CIRCUITS

USED FOR NOISE THERMOMETRY

F. Sullivan, D. Kahaner, H.A. Fowler, and J. Knapp-Cordes

Center for Applied Mathematics

National Bureau of Standards

INDEX

Page

Abstract 2

Introduction 3

Historical Synopsis 6

The Algorithm 8

Structure of the Program 8

Input Instructions 9

Graphics Output Choices 10

Scaling of the Equation to the Physical Problem 11

A. Tests of Quiet Cases: 13

Test of Higher Drive Frequency and "Demodulation" 15

Tests at a Low Value of (a - 1) 16

B. Tests of a Physically Noisy "Jump" Case (y - 1.5) 17

Long-Time Behavior 20

Summary of Observations 22

Quiet Cases 22

"Jump" Case 22

Conclusions as to the Accuracy of the Integrator 23

Continuing Directions of the Investigation 24

Acknowledgments 25

Bibliography with Comments 26

Figures 1-37

Appendix A User Guide to the Program CUSHEQ A1

Appendix B Circuit Algebra B1
ABSTRACT

Solutions to the Resistively Shunted Junction (RSJ) Equation:

0 4> + ( 1 + ycos <J)) <{> + sin <J> = a + km sin wt

for small 8 ( = .001) have been approximated using the SDRIVE integration

package. Various graphic displays are used to examine the output, including

plots of <j> , di/dt, and sin$ as functions of time; Poincare diagrams; and plots

of the Lienard coordinate

z = 8 <j> + <{> + y sin (j>

which has a close connection with the "slow manifold", as a function of $ , of

sin <j> , and of time. Integration is performed by separation of the second-order

equation into a coupled pair of first-order equations, and numerically

integrating with respect to time.

Several cases have been examined, for y < 1, representing quiet behavior of

resistively shunted thermometer oscillator devices. The report is an archive

record of program-test data.

A case of "jump"( voltage-spike ) oscillator performance for y = 1.5 has

been simulated in considerable detail, principally as a test of the integrator.

Parallel, independent computer results (Sanders and van Veldhuizen, Amsterdam)

were available for comparison. This case is of considerable mathematical

interest, and values of | y | >1 may also occur in the RSQUID thermometer.

2
INTRODUCTION

We are interested in numerical approximations to the solutions of the

non-linear equation

0 <j) + (1+y cos<f>)<j> + sin <{> = a + kco sin ait ,

which is a model for simulating the behavior of the circuit consisting of a series connection of an inductor, a resistor and a Josephson junction (used as a noise thermometer). Here the variable <\> is the time-dependent

quantum-mechanical phase difference across the Josephson barrier, and the

parameters represent combinations of circuit parameters, as well as the

amplitude and frequency of the applied signal. We concentrate on the conditions

of small 0 as is the case for the noise thermometer.

The value of the parameter y is crucial to the behavior of solutions. If

| Y

| <1

,

we have the so-called "quiet" case. Taking |y|> 1 gives "jump"

behavior. We will examine this dichotomy in more detail later, but for the

present consider the following: If |y|<l, the coefficient of <j> is always

positive and, hence, acts as a damping force. However, for |y|>l this term

1_ changes sign (at cos“l (“y)) and switches from damping to forcing. This

causes rapid "jumps" in the wave form. Physically, the system develops short,

high-amplitude pulses in its voltage output.

For the quiet case |y|<l,0 small and a>l , intuition about the general

solution can be gained by looking at the reduced first-order autonomous

equation

(1+Y cos<j> ) ^ = a - sincj)

<j>(o) = <j3 0

This is integrated directly for t as a function of £ , giving a periodic

motion with period

2tt

/ a ^-l

The next step is to generalize to the first order driven equation for a) > 1

(1+Y coscj> )<p + sin $ = a + kco sin cot

Using non-linear variation of parameters, averaging, and letting y

-* 0,

Sanders I 11

] shows that there is a (winding) solution with period

2tt T =

/ a

z - J0

Z (k)

so long as ct JqOO

>1 Here J 0 is the zero order Bessel function. The Bessel

3
function behavior is borne out by experimental observation. Our numerical

simulations show that the above is a good estimate of the period in the second

order case, so long as 3 is small. The simulation also gives additional

detailed information about the wave form, which is quite smooth for |y|<|.

In case | y

| >1 the situation is very different. In order to see why, first

notice that defining

z = 3b + ( 4> + Y sincfO

gives the Lienard co-ordinate formulation

z = a - sinb + km sin wt

3b= z -(b+Y sine})) .

The curve z = <j> + ysin<j> in the <f> - z plane is called the slow manifold, [ 33

]

because <j> = 0 on this curve. If we take $ modulo 2tt

, the curve has critical

points at the two values of cos- -*- ("7) and an inflection point at <p = tt

4

>

Suppose now that a>l, and that we choose initial values of <j> and z on the slow

manifold (say at (0,0)). The "average" value of z over one cycle is positive,

so that z tends to increase. However, for points (b,z) above the slow

• • manifold, 6 is positive while for points below <j> is negative. Hence,

solution trajectories tend to be attracted to the slow manifold and move slowly up it until the first critical point is reached. On this part of the

trajectory d<p/dz behaves approximately like

(z ~ (?+Y sin) )/S

" • z 3b + (1+7 cost?)b

z ~ (b+Y sin ?)

3 2

b + 3( 1+7 cos)b

Since 3 is small, the denominator is close to zero, and for (z,p) above the

slow manifold db/dz becomes large. The solution "jumps" across to the other

branch of the slow manifold. (This description can be made rigorous). The

form which the jump takes depends very strongly on the initial conditions. It

4
is even possible for the solution trajectory to reach the critical point and

loop back to the first branch before jumping across to the second branch.

The time-dependent behavior of the wave-form in this situation can be seen in

Figs. 23-37 below.

Using an asymptotic expansion, Sanders t 11

] obtains expressions for the time

at which the jump occurs. If k=0 (the autonomous second order equation), and

y is close to 1 , the period of the solution is approximated by T = To + AT

where

2u

To =

/ a ^-1

and AT = 2

2

(1-y)'

In two driven cases which we have examined, with larger values of y-1 (y= 0.5

and y = 1.5), this formula yields too large a value for the observed AT; the

time-averaged limit cycle is better approximated by the Bessel-function

expression mentioned above.

5
HISTORICAL SYNOPSIS

^ 1 The second-order RSJ equation, in autonomous form, was proposed by McCumber [ x

] and Stewart [ 2

] on the basis of the Josephson relations:

V (t) = 34> /3 1- h/2e I sc (t) * I c sin <t>

combined with Kirchoff's laws for the circuit in which the tunneling element is

connected. (Here V and I sc are, respectively, the voltage across and the

supercurrent through the tunneling barrier; <j> is the superconducting-phase

difference; and I c , the critical current). Their objective was to explain the

(time-averaged) dc current-voltage characteristic curve which they observed for

the circuit.

Kamper [ 3 ], and Kamper and Zimmerman [ 4

] have suggested that a point contact

(the Josephson junction) facing an inductive loop in series with a very low

resistor might serve as a unique form of noise thermometer. Soulen and Giffard

[ 5] applied the RSJ model to describe the behavior of the circuit in which the

Bessel function dependence was derived. A recent paper by Gallop [ 6

] surveys

the importance of Josephson SQUID noise thermometry, as an absolute temperature

calibration standard.

Peterson, Soulen, and Van Vechten [ 7 ],[ 8 ],[ 9

] examined a first-order-accuracy

solution to the time-dependent driven equation. (A recent manuscript by Sanders

t 11

] clarifies the same picture for second order.) This provides a partial

description of the frequency perturbations in the relaxation oscillation

observed when the drive amplitude is varied. In addition to the relaxation

frequency dependence on drive amplitude, through a Bessel function expression

(which is expected from the model, and observed experimentally), a weak upward

or downward slope is found experimentally. This may be simulated partially by

expanding the Bessel-function solution to the first-degree equation, and

partially by the inclusion of interactions with the (external) tank circuit.

Puzzling problems remain such as the observed occurrence of "doubled” peaks; R.

Soulen, Private Comm., [ 1

1

A] and it is clear that neglect of the second

derivative leaves the treatment incomplete.

Schlup (1979) [ 10

] has offered a relatively complete model for behavior of the

small-beta autonomous case. The treatment by Sanders [ 1

1

] links the autonomous

relaxation oscillation to a Hopf bifurcation from the solutions on the

zero-voltage step (that is, when the bias-current term alpha has values below

one.

)

Driven Case for beta large

The case of beta large (8» 1 ) has recently been rather broadly treated.

Schlup (1974) [ 12

] provided a numerical solution for the time dependence, in

this the near-Hamiltonian-oscillator case. His paper was the first to include

specific effects from the y term. Belykh, Soerensen, and Petersen I and II

[ i3

, i4

] have provided a survey of possible solutions for the autonomous and

driven cases, respectively.

*The abbreviation RSJ (Resistively Shunted Junction) equation is in common use

for this equation, which may variously appear without the 3 4) , y cos <J> , or k a) sin ojt terms. The term R-SQUID noise thermometer refers to the circuit

described above.

6
Connections of the case y

= 0 and 0 > 1 with bifurcations, chaos, and

oscillator instability have been examined by Packard, Huberman, et al.

[15,16,1?]; in detail by R. Kautz, [ 18 ,1 9

, 20

], and by Pederson and Davidson

[ 2 1

]

,

who looked at amplifier stability over a broad range of frequency and

amplitude parameters. This case has important applications to phase-locked

f requency-to-dc-voltage converters, and to microwave parametric amplifiers.

Driven case for beta small, and gamma non-zero .

The detailed parametric treatment of the RSQUID* noise thermometer has been

begun by Soulen, Park, Seppa, and Van Vechten [ 22

, 23, 24

] . Careful estimates

of realistic parameter ranges have been made by Van Vechten [ 2d ].

Extremely fast transitions between parts of the slow manifold appear at once

if gamma is assumed larger than one. These render the solution of the first

degree equation at best an unsatisfactory approximation to the full wave form.

Cushman [ 26

] has completed a global treatment of existence of periodic

solutions for the second-order autonomous equation, based upon a general model

of flows in phase space. Important results from this case include indications

of the strong global stability of solutions in the autonomous case, for all

values of gamma; and rapid convergence of these to the periodic solution.

Sanders [ 1

] has offered an approach to the ’'jump" ( y > 1 ) cases, which is

inspired by the work of Levi [ 2

8

] , and which proceeds from recent successful

treatments of the van der Pol equation by Grasman and collaborators [ 29

, 30 ].

These techniques, in turn, are based upon Smale's work in differential

topology.

On the basis of this wide spectrum of interest in the mathematical features of

the driven (forced) equation, and its physical consequences as a simulation,

we have been led to an approach which stresses detailed, accurate numerical

integration combined with varied graphics displays.

RSQUID = Resistor - Superconductive Quantum Interference Device

7
THE ALGORITHM

The numerical procedure for integration has been to separate the second-order

equation into two coupled first-order equations:

Yl = Y2 / 8

(1) Y2 - (1 + Y cos Yl)*Y2 /8 + a - sin Yl + kw sin wt

so that Y1 = (j> Yl = 4>

Y2 = 84> Y2 = 6<j>

This is not a unique separation. Another choice for Lienard coordinates comes from noticing that

8c() + ( 1 + ycos $)<j> , is an exact differential.

In that case

let z = 8$ + ( <j> + -ysin $ ) , so that

( 2 )

and

z = a - sin cf> kxu sin wt

8£= z-($+y sin $ )

Although the first choice of Lienard coordinates appears to involve more multiplications, this choice is numerically more stable than the second

because 8 is small and z stays close to (<j>+Ysin <p). However, the second pair

has some independent mathematical interest, as has been mentioned.

Integration is performed by using either SDRIV3 in single precision or DDRIV3

in double precision These programs utilize implicit raultistep

integration formulas, either of Gear or Adams Moulton, to track the solution

components of (1) or (2) to within certain specified local error criteria.

The nonlinear algebraic equations which result can be solved by any of several

algorithms, including Newton's method, in which case the Jacobian of the

system can be given explicitly or computed numerically. Various output

options are also available.

STRUCTURE OF THE PROGRAM

The main program, CU SHEQ/ DOUBLE , simulates the operation of the physical

system by querying for parameters and initial value settings, through the

subroutine OQUERY / DOUBLE ; driving the DDRIVE integrator with its attached FI

(Non-autonomous terms) and JACOBN (Jacobian) subroutines; returning values at

time intervals of S • 2 tt where S is an input real number; and storing

w

the values of t, Yl , Y2 in arrays.

8
These values are written to a file by the subroutine OPRINT /DOUBLE and may

later be graphed in a variety of ways through the program SPREAD / DOUBLE

,

which offers nine options of two-dimensional plotting. SPREAD collects the

values of <j> and cj> , and evaluates sin * and z (to single precision) in one loop.

This functional structure of the subroutines is illustrated in Table I.

INPUT INSTRUCTIONS

The operator is queried for input values of

A = a

B = 0

C = Y

D = 5 = 1 (a parameter coefficient of the sin <j> term, which is kept in reserve

for later approximations)

E = kw

2e

W = a) , the circular frequency of the drive term, scaled to = — RI C

* *

S , a real number, which selects the time intervals of At = S*2 tt at which

u> numerical results will be returned from the integrator; and for initial

choices of

Y1 = <J>

Y2 = J

T = t

TOL = the numerical tolerance for the completed integration. Finally, the

number N (= number of points to be returned) is requested.

* See Appendix B for the circuit
SUBROUTINE

STRUCTURE

OF

THE

PROGRAM

CUSHEQ

OPTIONS

(9)
GRAPHICS OUTPUT CHOICES

The graphics output subroutines can be combined with appropriate choices of S to

produce a very large variety of displays. These will only be briefly described;

they are best illustrated through examples of use.

Option (1): The Poincare map routine, of <Jj vs mod 2ir , is conventionally used

in non-linear analysis to distinguish fixed points, stable and non-stable

orbits, etc. In the present case, selection of S permits the time interval to be

taken as the period of the drive oscillation (S = 1), or any multiple or fraction thereof.

Option (2): As mentioned on p. 9 , z is constructed by SPREAD from the stored

values of and Y2

. The plot of <j> vs z is useful in distinguishing transition

phenomena for the high-speed ''jump'' intervals which occur for the choices of

gamma larger than 1 , where the term y cos <j> can have negative values greater than

1

.

Option (3): $ vs t is the actual time-dependent solution of the equation,

simulating the barrier-phase-difference for the Josephson effect. <J> is plotted

mod 2tt , since otherwise the secular term in its solution would cause it to "ramp"

off the page.

4> vs t simulates the instantaneous time-dependence of the barrier

voltage.

Option (4): The return map of <p at intervals corresponding to the drive

period is a method of exploring both the subharmonic structure of the solutions,

and the jumps between branches of the slow manifold which the system may exhibit.

Option (5): sin <jj vs t simulates the instantaneous flow of supercurrent through

the barrier of the Josephson junction. There are a number of interesting

insights and visual simplifications introduced by the cylinder projection of

viewing which the sine function provides. The physical quantity <j> is not an observable; it represents the argument of a trignometric function which is an observable, and $ can, therefore, be thought of as confined to motion on a cylinder.

Option (6): sin ^ vs z and

Option (7): z vs sin $ illustrate the same effects as Option (2), but in

cylindrical projection, relating z to the (observable) supercurrent.

• Option (8): z vs t collects the values of <jj and <j> as returned by the main

integration, and calculates z , displaying it as a function of time. The

smoothness of this function, when its individual terms are undergoing rapid

derivative changes with repect to time, is a measure of the performance of the

program as a whole.

Option (9): ip/t vs t provides a mechanism for averaging the advance rate of

the quantum-mechanical phase difference, a measure of the relaxation, or limit

frequency

<q> = lim _1 ,p(t)dt

y-HX x

0

It is calculated by computing Yl/t values from the output array, while

suppressing the mod 2tt reduction used in plotting $ , in option (3) above.

10
SCALING OF THE EQUATION TO THE PHYSICAL PROBLEM

Time Scaling

Drive and relaxation frequencies are scaled to the real physical parameters of

the thermometer oscillator circuit through the "equivalent plasma frequency"

2e

2tt uj r = RI r in circular measure , or

v c in direct frequency measure (Hz)

2e

The quantity is known extremely accurately, from voltage stabilization

h

experiments, to have a value of 483.6 MHz/microvolt *.

5 6 and for R = 10“ ohms (the shunt resistor-see Appendix B) with I c = 10” amps

v c = (483.6 x 10

12

) x 10 ”5 x 10 “6 Hz

v c = 4.836 x 10 3 Hz or 4.836 kHz , specifically when the

product RI C is equal to 10-11 volts.

Selected Drive Frequencies

A value for co of 200, scaled to the circuit parameters above,

corresponds to

200 x 4.836 x 10'

J = 967 kHz

Or, a value for u of 800 corresponds to 3.869 Mhz

And, a value for oj of 45.68 is equivalent to

45.68 x 4.836 x 10

3 = 220.9 kHz

These are the values of drive frequency used in the simulations reported

below.

* From high-precision experiments against the U.S. standard volt cells, this

quantity is known at better than one part per million accuracy. The rounded

version to four decimal figures is used here, for computational simplicity.

11
Time-dependent quantities

Time-dependent quantities in the RSJ equation are expressed in voltage units

for simulation purposes:

thus a = 1 for R I c = 10“ ^ volts-dc, in the examples used, and therefore

kw = 1 corresponds to an rf input signal at 10“^ volts-rf amplitude (not rms).

Output <J> = Y2/3 = 1 has the scale (10“® volts) and sin <}> = 1 corresponds to

I s c = I c = 10" 6 amps.

The parameter 3

7 Soulen [ ] estimates a value of the second-derivative coefficient 3 at .001, as reasonable for point contact RSQUID devices. We note that a capacitative value

of 3 as small as .01 has been suggested for experiments on tunnel junctions, byFulton [ 34 ]. This does not provide strong guidance for the case of point

contacts, however, because of the different physical barrier-layer present. The

value .001 is consistent with the curves of Fig 2 and Fig. 5 in McCumber ]. We select beta positive to assure that the second derivative term will have a limiting effect during the " jump" intervals for cases of y greater than one.

Beta vs omega plane

If we cons^er the plot of beta versus reduced frequency u>/w c which is found

in Kautz [ ] as Fig. 2 of that article, the value 3 = .001 in conjunction

with the line where

wL = 1/ wC , its resulting value

3 “ 1/2 = 31.62

would lie along the lower edge of region VI extended to the upper left,

in Kautz* s log-log beta-omega

plane. Since the values chosen for uj/ w c , namely 200, 800, and 45.68, are above this, our simulations lie up in the region between that lower-edge value

and the line where R = 1/uC

3-1 = 1000

This is equivalent to saying that our results have been limited to a frequency

region above the junction/circuit "equivalent" plasma frequency. Such

inf ormation is helpful when comparing behavior between different families of

Josephson oscillators.

12
Choice of y

y from theory is proportional to I c Le , where I c is of the order of 10-

amps and Le , the inductance of the coupling loop which the Josephson junction

faces, is estimated to be about 2xl0

- ^ henries.

If we take the expression derived by Petersont

y

] for gamma in the RSQUID

circuit

2tt 2e

Y = —- * 2e x I^I C = 2tt x x Lel c

= 2-rr x 483.6 x l0

12

x 2xl0” 10 x 1CT6 = 0.6077

for the above values. Here we see that Le> 0. 33 nanohenry will suffice to

render y > 1. y » as shown by Soulen, Peterson, and Van Vechten [ 7

] can be

adjusted experimentally to fall either below or above 1 . It is a parameter

which can be determined in a coarse manner within the experiment. Of

particular interest are the cases in the transition region as y approaches

1 from below. In some cases there are anomalies in the observed relaxation

frequency as a function of drive amplitude. Previous work has not shown

whether all of these result from an intrinsic wave form of the relaxation

oscillator, or from its coupling to the drive and detection circuits.

A. Tests of Quiet Cases with Smooth Wave Forms

Simulation of the quiet thermometer oscillator has proceeded by examining

several cases for y < 1 • These results can be compared with existing

asymptotic models (Soulen [ 5

, 7

]; Sanders t 11

]) and directly with experiment.

The graphics routines have been used to simulate the waveforms in considerable

detail.

The curves which follow in Figs. 1-10 correspond to the case y = 0.8 , oj = 200, a = 0.5 , which corresponds to a drive frequency of 967 kHz and a relaxation frequency of about 25 kHz. Figs. 11-15 are run with the drive

frequency four times as large; i.e. 3.869 MHz.

Figs. 1-5 — "Test 30"

Parameters

A = 5, B = .001, C = 0.8 ,D = 1.0, E = 382, S = .025, T = 0, TOL = 10- 8

, W = 200, 3000 points

This is a fairly low drive amplitude, simulating a waveform which is

primarily the relaxation oscillation, modulating a "carrier" wave at the

drive frequency.

Fig. 1 shows phi, the solution, as a function of elapsed time t. The

fine-grain vertical oscillations are at the drive, or "carrier" frequency; the

broad stripes conform to the relaxation or limit cycle of the Josephson

oscillator

.

13
The points on Figs. 1, 2, 3 have been returned by the program at the rate of one per 1/40 of the drive cycle. The numerical values have been generated at

double precision (parts in 10 8 ). Phi is plotted modulo 2tt , to prevent the

plot from ramping diagonally off the paper.

Fig. 2 shows the time derivative of phi, simulating the instantaneous

Josephson voltage across the tunneling barrier. Note that this appears to be

primarily amplitude-modulated by the relaxation oscillation, but that the

positive peaks in the envelope are slightly offset in time from the negative

ones. The mean value of phidot is expected to approximate A, or a ; the

time offset is a consequence of the value of C, or y

Fig. 3 shows the sine of phi, simulating the instantaneous super current

through the barrier. This is equivalent to taking the function of Fig. 1 and

wrapping it around a cylinder, whose axis proceeds along the time direction.

The resulting "wrapped ribbon" is viewed from the side; note that the zero at

the bottom of Fig. 1 has now been displaced to the center of Fig. 3. Note

that the ribbon returns down the "far"side of the cylinder with steeper slope

than it rises on the "near" side. A time-smoothed version of the instantaneous

supercurrent would appear to be primarily amplitude-modulated at the

relaxation frequency; Fig. 3 suggests that this would be a somewhat asymetric

sawtooth as a function of time, with a slow rise and a more rapid return to

the negative limit of -1.

Fig. 4 shows z versus phi for these points. 4b and 4c illustrate z vs. sin 4) , to suggest the cylindrical projection which controls the actual values of the

tunneling supercurrent. Fig. 5 shows the composite Poincare map.

Each point in these diagrams can be tracked to corresponding instantaneous

values in Figs. 1,2,3. These methods of display are useful in examining

periodicity and stability properties.

Figs. 6-10 — "Test 38"

Parameters

Identical with "Test 30", except E=1402 , about 3.6 times as large. The

waveforms become more frequency-modulated in character.

Fig. 6 shows phi as a function of time. Note that the fine-grain vertical

oscillations have been replaced by stripes moving along the ribbon, and that some of these stripes appear to cross over each other. Note that the open

bands which appeared before have now been eliminated; the ribbon runs together

or overlaps on itself.

Fig. 7, corresponding to Fig. 2, now shows quite a different form of voltage

signal. The maximum amplitude is nearly uniform, as in a frequency-modulated

waveform. A sixth harmonic of the relaxation oscillation shows up. The

offset of positive- from negative-peaks seen in Fig. 2 is now reflected in the

interior "cluster" detail.

14
Fig. 8. The ribbon winding of Fig. 3 has been widened to the point that it

overlaps itself. At the same time, the "carrier"-f requency detail is now somewhat suppressed, as we should expect if the sidebanding of frequency

modulation is superimposed.

Figs. 9 and 10 correspond to Figs. 4 and 5. Some changes of form are observed.

Test of Higher Drive Frequency and "Demodulation”

Figs. 11-15 — "Test 61"

Parameters

A = 5, B = .001, C = 0.8, D = 1.0, E = 1528 , S = 1.0, T = 0, TOL = 10-4

, W= 800, 3000 points

This is roughly similar to the conditions of Figs. 1-5, except that the

drive frequency has been increased by a factor of four, and the amplitude of

the drive signal increased to produce roughly the same level of amplitude of

the "carrier". In addition, only one point is returned per cycle of the

drive, instead of forty; thus, the drive signal disappears from view in the

output. The horizontal time scale has been compressed by a factor of ten in

plotting.

Fig. 11 shows phi as a function of elapsed time. The profile of the

relaxation oscillation is the dominant detail. In electrical engineering

terms, the "audio” signal has been mathematically "demodulated" from the "rf

carrier" and its sidebands, and is displayed here by itself.

Fig. 12 shows instantaneous voltage. Here we are seeing a sort of

stroboscopic view of one of the traces which can be seen extending through

Fig. 2. It gives only a limited, partial view of the full waveform which

would be experienced if we were seeing all phases of the "carrier" cycles.

Fig. 13 , a correspondingly "stroboscopic" view of the supercurrent could be

correlated with one of the lengthwise threads along the ribbons in Fig. 4 or Fig. 8. Note that this is a continuous smooth path —but in Figs. 4 and 8

there may be crossing of these paths. Whether ribbon twisting can take place

under the proper conditions on C and E (gamma and k omega) is an interesting

problem relating to the frequency stability of the oscillator.

Figs. 14 and 15 are smooth curves, much as expected for a stable system.

15
Tests at a Low Value of (a - 1)

We may select a = 1.2 in order to emphasize the perturbations on the

relaxation frequency which are caused by varying the amplitude of the drive

term. At the same time, the selection of oj = 200 simulates a drive frequency

of 967 kHz, while the chosen value of alpha corresponds to a relaxation

frequency of ~ 5.8 kHz. We now have quite a large band separation between

the drive, or carrier frequency, and the relaxation or limit cycle frequency

of the Josephson oscillator. Figs. 16-22 below are a simulation of this case.

r 5 8 9 l The theory by Peterson, Soulen, and Giffard * > » J , which is based on consideration of the first-derivative term only, gives a value for this

correction:

J 0 2 (k) 1/2

u) re 2_ = a ( 1 - ) x a) c a

2

A series of curves has been run for these values of alpha and omega, over a range of k from 1.91 to 8.65, which is designed to cover the first three zeros

of the Bessel function. The relaxation frequencies, which may be read off to

better than 1% from the plots, do follow the amplitude dependence predicted

from this theory of the first-order equation. This corresponds to the period

of the "winding" solution of Sanders, mentioned in the Introduction.

Figs. 16-18 — "Test 4"

Parameters

A = 1.2 , B = .001, C = 0.5, D = 1.0 , E - 730, S = 0.20 , T = 0 , TOL = 10 - 4 W = 200; 3000 points.

There are five points returned per drive cycle.

Here we expect fully quiet, smooth oscillator behavior. Because of the low

value of alpha, different paths occur for the off-set "strobe" positions

around the drive cycle. This is particularly clear in the "wrapped-cylinder"

paths of Fig. 18.

We note parenthetically that the projection of Fig. 18 is taking the sine of a phi-function which already comprises several low harmonics of the

"fundamental" relaxation frequency. "Sine of sines" implies a Bessel-function

set of harmonic coefficients. In this case, the argument of the Bessel

functions is the drive amplitude. Therefore in Fig. 18 it is not surprising

that we find patterns suggestive of very simple Fourier-component mixes, in

various relative time phases.

The patterns seen in Figs. 17 and 18 bear close qualitative resemblances to

the experimentally "demodulated" audio spectrum of an actual thermometer, when

the first six or eight harmonics of the relaxation frequency are allowed to

pass the audio filter.

16
Figs. 19-21 — "Test 7”

Parameters are the same as in Figs. 16-18 above, except that the drive

amplitude is now increased to 1730, i.e. 2.37 times as large.

At this particular amplitude the fundamental Fourier coefficient J^(k) must be

at a high value. Also, the paths overlap from one relaxation cycle onto the

next, which is only slightly the case in test 4 above.

Fig. 22 illustrates the perturbation of the Josephson relaxation frequency by

the amplitude of the drive signal, which is one of the key problems in

physical variation, for which an answer is sought.

The crosses are estimated numerical values of the relaxation frequency sealed

to the critical frequency wc , obtained from the 2tt - crossings of phi vs t , in the series of which Figs. 16 and 19 are examples. Scatter of these points

is about 6 parts in 10^; it comes partly from numerical roundoff of the

single-precision integration, and partly from interpolation errors in

estimating the 2tt crossing points.

The circles are points taken by interpolating between curves of Peterson [8,

esp. Fig. 2), which is actually calculated for a = 1.25 , rather than u = 1.2,

the value for which Figs. 16, 19 ... have been run. Note that the curves in

Peterson's article are plotted on a squared scale; here we show frequencies

directly. The curve described by the circles has been normalized to the first

two maxima in the crosses — this scale is shown at the right.

We conclude from this comparison that the positions vs k of the first two

maxima and minima, and the magnitude of the swings between, show agreement

within the available limits of error. In broader terms, the second-order

simulation appears to be agreeing in detail with the predictions of the firstorder model, which in turn are known to model a portion of the observed

experimental behavior.

3 . Tests of "Jump" Case(y ^ 1.5) with Discontinuous Waveform Derivatives

The case of gamma larger than one can occur in R-SQUID noise thermometers. This comes about by increasing the pressure on the Josephson point contact, so that

the critical current I c increases, for the same value of exterior Le into

which the junction faces; gamma is proportional to the product of I c and Le

.

The first-derivative term in the equation causes rapid transitions when (1 + y cos $) changes sign. These rapid transitions, on a greatly compressed time

scale, give a very rigorous test of the integrator. While cos «> is negative,

•

b rises or falls very quickly to an extreme value, then turns around and

returns quickly until the solution is "caught" again as the y cos -j> term reaches

-1 , and the system reverts to a "slow manifold" solution.

The problem of "jumps" of this type has already been studied for the case of the

Van der Pol oscillator [ 23

, 3a

] "Drops" and "slices" occur, as presented by

Littlewood [ 3i

, 32 ]. Our approach has been primarily through graphics

techniques, using the integrator program to simulate the physical evolution of

17
the system as nearly as possible. We have simulated a description through the

"slow manifold" approach, which was suggested by Cushman [ 33 ]. We have had

available to us a set of independent calculations by Sanders and Van Veldhuizen,

[ 27

] using a different Shampine-Gordon integrator, which confirm critical

details of the program performance, especially for these "jump" details.

The parameter values are chosen as follows:

0 = .001 , y = 1.50 , k = 1.0 , a = 11.0 oj = 45.6800 , ku = 45.6800

S has been chosen in a variety of ways, to display different details.

This simulates a Josephson thepiometer with shunt resistor value R = 10“ 5 ohms,

with critical current I c = 10” amps, driven at 220.9 KHz and at a drive

amplitude k = 1.0, which places its operation in the "very weakly driven" part

of the Bessel-function plot for relaxation frequency (below the first maximum).

We note that this choice of parameters has the following special features, which

are not common to the best operating thermometer devices: very low drive, or carrier amplitude; drive frequency near the fourth harmonic of the Josephson

relaxation oscillation frequency, which appears to lead to phase locking between

the two oscillations, and sharp voltage "spiking" associated with the

negative-resistance intervals when (l+ycos<j)) is negative.

Thus, a value for oj of 45.68 for this choice of R and I c would correspond to

45.68 x 4.836 = 220.9 kHz

the observed average relaxation frequency is 101/421 of that value; this

corresponds roughly to the ratio between a =11 and cj = 45.68.

There is, however, a perturbation associated with the amplitude factor k for

the drive term. The theory by Peterson, Soulen, and Giffard gives a value for

this correction to the predicted relaxation frequency

(1 - J0

2 /(ll) 2

) 1/2

where the argument of the Bessel function J0 (k) is 1 in the case we have

chosen. The corrected, predicted relaxation frequency would then be

53.0651 kHz; we find in simulation the value 52.9970. The reason for this

discrepancy of 0.13% is not clear — the observed value is lower than the

predicted.

Results of the Integration

When started from <}> =0, dtp /d t =0 , a point which is off the stable

trajectory, the solution evolves quasi-periodically for about 1000 drive

cycles, then settles into the stable periodicity, reproducing itself to about

2 parts in 10 , in both $ and dtp /dt • It is not clear whether failure to

cycle more closely than this represents a fluctuation associated with phase

locking, in the physical system. Both exact and approximated Jacobian

routines have been tried at varying levels of precision, with similar results.

18
We find the numerical solution, in double precision, to be multiply periodic,

with a prominent subharmonic at 421 times the drive period 2tt/w , and at 101

times the relaxation period. This determines the average relaxation

frequency, from the repetition rate, to be 101/421 x 45.6800 very precisely

(to the order of parts in 10 ). There is a family of nearly-periodic

structures appearing ; "quasi-subharmonics"( ? ) reflected in the patterns of

modulation of the high-frequency spikes in d$/dt.

It is not clear at this early stage of the analysis, whether this rational

subharmonic relation is a consequence of frequency-locking of the relaxation

oscillation, via subharmonics, to the nearest subharmonic of the drive

frequency. We note that for the parameters chosen, the drive frequency is a little more than four times the relaxation frequency, and hence " band

separation" is incomplete. Also, the observed relaxation frequency is offset

towards a lower value than that suggested by the simple theory, e.g., there

may be a "pulling" of the relaxation frequency by the frequency locking.

Operation of the program at single precision, in other parameter ranges,

indicates that the spikes originate with the increase of y to about one; when

y reaches a value about 1, at this drive amplitude, the spikes may be negative

in sign as well as positive; at 1.5 they can occur in multiples as well as singly.

Fig. 23 $ , the superconducting phase-difference across the tunneling device,

is shown modulo 2tt as a function of time. Note that on the up-going

transitions between steps, in the relaxation oscillation, only a relatively few

points are caught in the § transition region near tt . The function is a continuous staircase moving upward off the plot.

Fig. 24 $ , the instantaneous voltage, is shown on a very much reduced

scale. There are sharp positive-and negative-going excursions (to positive 800

and to negative 600) during the very brief time intervals when $ is near tt . Other plots indicate that its mean value lies around 4 to 5 on the stable

manifold points. Evidently the spikes contribute a substantial part of the

mean voltage ~ 10 or 11.

Fig. 25 sin<£ , the instantaneous current in the junction loop, is displayed

for the same points. Roughly, it appears as a sinusoid at the drive

frequency, added to a linear ramp from the relaxation oscillation. (Note that

this ramp is not derived from the numerical "chopping", modulo 2tt , of Fig.l,

and is smoothly continuous across those drops); the drops in Fig. 25 represent

quite drastic reversals of the supercurrent direction through the junction,

including some "extra" reversals going to the current limits, associated with

the large spikes in the plot of the instantaneous voltage. In the case of the

Josephson point contact facing into an inductance, the phenomena represent

pulses of supercurrent sharpened by the inductance. Closer examination

reveals that Fig. 25 actually "wraps" Fig. 23 around a cylinder, with the

"jump" regions of very rapid current reversal occurring on the " far" side, as viewed in projection.

19
Figs. 26 and 27 z = 6 <j> + <j> + y sin <j> the coordinate which was suggested as an alternative separation variable, is shown plotted against 4> , for 2 sets of points taken at time intervals corresponding to tt/w and 0.1 tt / u> from the time zero of the sin cot driving term, respectively. (a modified type

of Poincare plot). Two branches of the slow manifold are seen; the points

corresponding to tt/co wobble upward on the left-hand branch to the maximum, then

jump across the central "unstable" zone. The points corresponding to 0.1 tt/(o .... wobble downward on the right-hand branch, then jump across to the left

[with a lower minimum slope than the up-going jump points].

Figs. 28, 29, 30 and 31 display the same data as a function of sin <j> . We note

that <p appears to be a cyclic function of z, modulo 2u ; however we can only

conclude from this that it is a continuous function of z.

Figs. 28 and 29 are representative of fully stabilized cycles in the solution.

Figs. 30, 31, and 32 are data which include the early transient behavior as the

system is started from (0,0) at t=0.

Fig. 33 which shows z as a function of t, should be compared against Fig. 23

and 25. It will be noted that the drops in Fig. 33 are only those associated

with the arithmetic operation of reducing <j> to values modulo 2 tt . z in this

plot is simply the sum of terms phi, phidot, and sin <j> , evaluated from the

returned <j> and cj> of the integrator program, with <j> expressed modulo 2 tt .

z as a function of t evolves smoothly as a linear ramp, with an added sinusoid

at the "carrier" frequency. The slope of the ramp reflects the relaxation

frequency. This smooth evolution takes place through the " jump"

discontinuities in the time derivatives of the other functions plotted; its

behavior confirms the general internal consistency of those solutions.

Thus z vs t is like Fig. 23 in portraying the continous evolution of a "rotating" solution; however, its rate is uniform, and does not exhibit the

cycle of the relaxation oscillation.

Long-Time Behavior

Figs. 34 and 35 illustrate Long-time or Very Slow Behavior

If the results illustrated in Fig. 3 and Fig. 5 are extended by plotting

only one point per drive cycle, and greatly compressing the time scale (3000

drive cycles, or about 12 milliseconds, is the duration of the plot shown here)

the result is an interesting "stroboscopic imaging" of the long-term motion.

After the first 1000 drive cycles, the motion is fully periodic and the diagram

repeats itself to parts in 10 .

The upward "return tracks" record the intervals at which extreme values of

voltage spikes recur (the subharmonic 421/101).

20
Fig. 35 is the same type of plot, but calculated with y

= 0.5 instead of 1.5.

The "stroboscopic path" is greatly smoothed, and more nearly sinusoidal. In

this case, voltage "spiking" is not present. The maximum value of phidot

encountered in this run is about 14.

Careful examination of this diagram reveals that the pattern at the right hand

end is starting to repeat, after time = 330.4 on the time scale of the plot.

Examination of printout shows that this doubly-periodic solution is

reproducing itself to within 3 or 4 parts in 10 .

The repetition period corresponds to 2402 drive cycles, or 577 periods of the

relaxation oscillation. The high rational "winding ratio" may be used to

estimate the numerical value of the relaxation period, averaged over the long

repetition period and expressed in units of 2t:/w

Trel = 2402 = 4.16291

577

For comparison, the "autonomous" relaxation period expressed in the same units

w = 45.680 = 4.1527273

11.0

when corrected by the Gif f ord-Soulen-Peterson-Sanders factor

a

= ( .9975775)' 1 /a^-J 0

Z (k)

for a = 11 k = 1 yields the value 4.16281.

We conclude from this that the "autonomous" relaxation period, corrected by

this Bessel-function formula, agrees accurately with the average relaxation

period seen in the simulated solution of the second-order equation, for y =0.5

Figs. 36 and 37 show confirmation between a result from Sanders, on a problem of

critical variation, and the same answer as calculated by this program. For this

test the frequency w is set at 45.68128 (slightly different from the values used

in the preceding series of figures), and the initial value of $ is varied in

steps of less than 2 parts in 10^.

The initial value of phi is chosen as follows:

a) 6.374423

b) 6.374424

c) 6.374425

d) 6.374426

e) 6.374427

f) 6.374428

This variation produces changes of the order of 1 in the output waveform (phi versus time), which is that corresponding to a triple crossing of the region

where 1 + y cos <p < o. 21
In Fig. 37 the horizontal lines represent the values of <j> for which 1 + y cos<{)

vanishes

Summary of Observations

These tests of quiet and "jump" cases have produced a variety of graphical and

numerical output. We summarize here a few of the more interesting results.

Quiet cases

o For y < 1 ( at least as high as 0.8) all waveforms are smooth,

regardless of the amplitude of the drive signal. There is a gradual

transition resembling that from amplitude-to frequency-modulation, as the drive amplitude is increased.

o A useful memory device is to think of <f> as an argument function

confined to move on a cylinder. Sin <j> , the instantaneous

supercurrent, is then visualized as a projection on a plane parallel to

the axis of the cylinder.

o A type of mathematical "demodulation” or separation of the

low-frequency relaxation oscillation from the high-frequency "carrier"

is easily achieved by making the return interval of the integrator

program coincide with the period of the drive signal.

o Tests at a low value of (a-1 ) give good agreement with the

Bessel-function dependence of relaxation frequency (period) on drive

amplitude coefficient k .

"Jump" case

o As expected from first-order theory, the simulation exhibits "jumps"

between parts of the slow manifold. These rapid transitions can be

accurately traced out in full detail, provided by the second-order

equation. They are accompanied by sharp voltage pulses.

o The solution is observed to be doubly periodic at high precision, with

the relaxation oscillation at the 421/101 subharmonic of the drive

frequency. Since the drive signal has been selected near the 4th

harmonic of the relaxation frequency, we suspect that phase-locking is

occurring because of the strong overlap between the "audio" and

"carrier" bands. Thus it may be difficult to compare the observed

repetition period of the relaxation, with a theoretical prediction

based solely upon values of a , k , and y . Unlike the quiet cases which

were considered above, the solution may have a relaxation frequency

which wanders from a constant value during the "long" 421 repetition

period, while averaging to a mean value which is frequency-locked to

the drive frequency.

o Plots of z versus phi exhibit sharp details of the "up-going" and

"down-going" transitions between parts of the slow manifold. If phi is

shown in cylindrical projection, the connection with current reversals

going to the critical limits can be made clear.

22
o Plots of z versus time exhibit the uniform advance of a smooth function,

apparently sinusoidal at the drive frequency, atop a highly linear ramp;

this in spite of the fact that its three constituent terms show

discontinuities in their derivatives. This suggests the possibility of

further simplifications, in pursuit of a numerically simple routine for

evaluating relaxation period (frequency).

Conclusions as to the Accuracy of the Integrator

o Figs. 36 and 37 demonstrate the independence of the solution method from

details of algorithm, programming, and machine; and also the high

sensitivity of both NBS's and Sanders’ algorithms to critical changes in

initial conditions. The figures are an illustration of how small changes

in the initial conditions can generate dramatic changes in the behavior of

the wave form as <j)(t) traverses the region where 1+y cos $ < 0. This is an instance of the changes from ’dips' to 'slices' which were studied by

Littlewood for the case of the van der Pol equation.

A tolerance of 10” is needed to produce Figs. 36 and 37. Accumulation

of error is a well known phenomenon in numerical solution of o.d.e.'s.

The accumulation of error eventually gives a completely wrong answer. An

accumulated error of one part in 10^ would alter the traversal of the 1+y cos cp < 0 region (as can be seen in Figures 36 and 37).

o The system is not "stiff". Considerations of accuracy restrict the step

size. We obtained the most efficient integrations by using the Adams

Moulton integrator coupled with Newton's method for the nonlinear

equations. However, given sufficient machine precision any competent

modern integration will correctly follow these solutions. This was confirmed by Sanders, et al , who obtained some similar results using

entirely different software.

o The observed doubly-periodic cycling of the "jump" case for gamma =1.5 is

followed to parts in 10^ for the observed output values of phi and phidot,

after an initial "settling" period of about 1000 drive cycles. We

associate this "settling" period with the approach of an "off-manifold"

solution to the asymptotically stable form.

o The results just mentioned were obtained only at double-precision 10”^

tolerance requirement. Improving the tolerance to parts in 10-*-0 did not

produce distinguishable improvements. By contrast, single precision

(parts in 10^) produced "time-slippage" which threw off the accuracy of

the long-time-period cycling. Although general features of the

single-precision runs were similar to the double-precision ones, they

clearly lacked the full detail and accuracy. (We note that errors in phi

and phidot are cumulative with time, so that an error of one part in 10^

per cycle can accumulate to one part in 10^ after 1000 cycles).

These results are in accord with expectations, since the double-precision

integrator is expected to perform distinctly better in the rapid

transitions between the "slow manifold" and the "jump" intervals.

For the "quiet" cases where gamma is less than 1, the distinction in

behavior between single- and double-precision routines is expected to be

less sharp, since they do not have to adjust to the "jump” transitions.

23
CONTINUING DIRECTIONS OF THE INVESTIGATION

o With the reliability of the integrator now confirmed by severe tests, we wish to apply it to some major unsolved problems of the experiments. One

such is the behavior of

<<£> = lim _1 /

T <£(t)dt

X-xx> 'p O

vs k for a driven y>l case. Sanders t 11

] gives an estimate of the period

T 0+AT for a y>l autonomous case, and the formula

2tt w

T = >V- JQ

Z (k)

for a y<l driven case.

We are trying to learn more about the observed anomalous "doubling" in

the driven y>l case. We had to have an utterly reliable integrator

before attempting this.

Foremost in interest will be to run with a wide range of drive amplitude

values, up to k = 8 or 9, for values of gamma between 1 and 2; the output

quantity of interest will be phi divided by t-t 0 , evaluated to a part in

10^, for comparability with the experimental observations, which have

shown examples of "doubling” at the parts-in-lO^ level, not in accord

with a simple Bessel expression.

o Return-mapping techniques will be tried, in order to examine questions of

frequency stability, especially the possibility of chaotic behavior.

Chaotic, or erratic behavior of the observed relaxation frequency would

be expected to interfere with the desired high-fidelity operation of the

Josephson thermometer as a voltage-to frequency converter of thermal

(Nyquist) noise.

o An asymptotic expansion formula derived by Sanders L 11

] may provide a key

to developing Fourier analysis of the "quiet” waveforms, such as Figs.

16-21. This will be explored in further detail.

o The programs will be rendered transportable (with consideration being

given to its possible future use on smaller machines than the Univac

1100/82). Jill Novotny is currently installing it on the Catholic

University DEC 10.

24
ACKNOWLEDGMENTS

We wish to record the contributions of numerous colleagues.

R. Soulen and D. Van Vechten of the Temperature Standards Division, NBS

formulated the RSJ equation in the present context, to analyze their

experimental results. Soulen, Van Vechten, and R. Peterson (of the

Electro-magnetic Technology Division, NBS Boulder) developed first-orderaccurate solutions for the equation in first order. R. Rehm and H. Baum, of

Mathematical Analysis Division and Center for Fire Research, NBS, suggested a two-timing approach to the solution of the first-order equation, by a singular-perturbation expansion.

A. Deprit of Mathematical Analysis Division, recognizing the similarity

between this equation and the Van der Pol oscillator, proposed to R.H. Cushman

(of Rijksuniversiteit Utrecht and a guest worker in Mathematical Analysis

Division) that some classical techniques of nonlinear dynamics be tried on the

problem. Cushman has developed a global theory for the phase flow of the

autonomous second-order equation. This has led to the important suggestion

that the Lienard coordinate z be used to examine the stability effects of the

driven equation. Cushman outlined the problem to Jan Sanders (Vrije

Universiteit , Amsterdam), who has proceeded independently with an asymptotic

analysis. In conjunction with van Veldhuizen (Mathematisch Centrum, Amsterdam)

Sanders has developed an integrator and a mapping approach to the unstable

cases. We are indebted to Sanders for permitting us to see his computations

and his Ms. on asymptotics, before publication. Sanders's outline for the

asymptotics is consistent with the earlier proposal by Rehm.

Richard Kautz (Electromagnetic Technology Division, NBS Boulder) has published

two papers treating the parameter space for the large-beta case. Using a Runge-Kutta integrator, he has examined bifurcations and chaos for the driven

oscillator in the large-beta (quasi-harmonic-oscillator) case. These papers

convinced us of the usefulness of an approach through a capable integrator

routine.

R. Henry of the Physics Department, Bucknell University, has provided us with

a pre-publication copy of a manuscript on mapping technique, applied to the

frequency stability of voltage-controlled oscillators. This provides very

important insights, using a simple and powerful approach.

Subsidized computer time on the NBS UNIVAC 1100/82 system has been generously

supported by the National Engineering Laboratory and the Center for Applied

Mathematics, as a portion of their Competence development budget for FY 1982.

We are indebted to Jill Novotny for running out many of the plots which are exhibited here, and for composing a compact User's Guide for the CUSHEQ/DOUBLE

program.

25
BIBLIOGRAPHY WITH NOTES

This listing omits many basic references to the methods of nonlinear

dynamics, which are utilized in the solution techniques which have been

followed.

It includes only the sources for Josephson-eff ect devices and theory

which seem to us to be most directly pertinent to the problem at hand. Since

1963 there is a very large literature of Josephson devices in other

applications. Closely related to the problem of the second-degree RSJ

equation is the Sine-Gordon equation with damping term; a literature has

proliferated on the "fluxon" or "kink" solutions to that equation. The

Sine-Gordon equation is the partial differential equation which is appropriate

for a Josephson junction which is space-distributed: i.e. it is being operated

at a high enough frequency that its surface area must be considered to have

parts which are out of synchronism with each other.

The Josephson name is attached to (1) the basic electrodynamic relations

of the superconducting tunneling effects; (2) several differential equations

which have been attached to these problems; (3) some other unrelated problems

in very-low temperature phase changes. Readers encountering titles in this

field for the first time should be careful which is meant.

1. D.E. McCumber, Effect of ac Impedance on dc Voltage-Current Char’s of

Supercond. Weak-link Junctions, J. Appl. Phys. 39, 7, 1968, pp. 3 113-18

Develops the beta parameter for both capacitive and inductive loadings.

Capacitance.

2. W.C. Stewart, Current-voltage Characteristics of Josephson Junctions, Appl.

Phys. Letters 12, 8, 1968, pp. 277-280

Formulates RSJ Equation for autonomous case. Capacitance, Dissipation.

3. R.A. Hamper, Survey of Noise Thermometry, in Proc. 5th Symp. Temperature,

Its Meas't & Control in Sci. & Ind. , Washington, 1971, pp.

4. R.A. Hamper and J.E. Zimmerman, Noise Thermometry with the Josephson

Effect, J. Appl. Phys. 42, 1971, pp. 132-136

26
5. R.J. Soulen Jr. and R.P. Giffard, Impedance and Noise of an rf-biased

RSQUID operated in the non-hysteretic regime, Appl. Phys. Lett. 32, 1978,

pp. 770

Discusses low-frequency variation of alpha causing thermometer frequency

spectrum.

6. J.C. Gallop, The Impact of Superconducting Devices, Metrologia 18,2, 1982,

pp. 67-92 (esp. p. 86)

Discusses present temperature ranges and level of precision for Jos.

thermometers

.

7. D. van Vechten, R.J. Soulen, Jr., and R.L. Peterson, Behavior of the dc

Impedance of an rf-biased resistive SQUID, in "SQUID *80", H.D. Hahlbohm and

H. Lubbig. ed. , Berlin (de Gruyter) , 1980, pp. 569-584

Presents examples of the most difficult unexplained experimental

anomalies.

8. R.L. Peterson, Mathematical Modeling of the Impedance of a Resistive rf

SQUID, J. Appl. Phys. 52, 12, 1981, pp. 7321-26

Solution for average period in heavy dissipation, by a Bessel function

expansion.

RSJ Equation, Electronic Experiment.

9. R.L. Peterson and D. Van Vechten, Modeling the Impedance of a Josephson-junction Noise Thermometer, Phys. Rev. B24 , 6, 1981, pp. 3588-91

DC Voltage approximated from period by a Bessel-function Expansion.

RSJ Equation, Experiment, Computer Simulation.

10. W.A. Schlup, Relaxation Oscillations of a Josephson Contact, ZAMP 30,

1979, pp. 851-857

Autonomous equation is solved; limit cycles are examined in detail..

11. J. Sanders, The Josephson Equation; an Example of Asymptotics: MANUSCRIPT

TO BE PUBLISHED, Springer Mathematics Lecture Notes Series, July 1982, pp.

Seen in hand-drafted form.

12. W.A. Schlup, I-V Characteristics & Stationary Dynamics of a JJ including

Interference Term, J. Phys. C7 , Solid State Physics, 1974, pp. 736-748

Includes Cosine term; large beta microwave case; model includes

normal-current flow.

13. V. Belykh, N. Pedersen and O.Soerensen, Shunted Josephson Junction Model

I. The Autonomous Case., Phys. Rev. B16, 1977, pp. 4853-59

A first extended treatment of parameter space.

RSJ Equation, Nonlinear Dynamics, Poincare Diagrams.

14. V. Belykh, N. Pedersen, and 0. Soerensen, Shunted Josephson Junction Model

II. The Non-autonomous Case., Phys. Rev. B16, 1977, pp. 4860-71

Various Attractors are presented for parameter regions.

RSJ Equation, Nonlinear Dynamics, Poincare Diagrams.

27
15. N. Packard, Modelling Turbulent Phenomena, w. appl. to Josephson

Junctions, Talk at NBS Temperature Division, 1981 (May 27).

Incl. Movie of Topological progression along Str. Attr .Manifolds

.

Strange attractors. Topology, Mappings.

16. B.A. Huberman, J.P. Crutchfield, and N.H. Packard, Noise Phenomena in

Josephson Junctions, Appl. Phys. Lett. 37, 8, 1980, pp. 750-751.

Power vs. Frequency Bifurcation Diagram.

Noise, Strange attractors, Power Spectrum.

17. B.A. Huberman and A.B. Zisook, Power Spectra of Strange Attractors, Phys.

Rev. Letters 46, 10, 1981, pp. 626-628.

Shows integrated noise spectrum in the chaotic regime.

Noise, Strange attractors, Power spectrum.

18. R.L. Kautz, On a Proposed Josephson-ef f ect Voltage Standard at Zero

Current Bias, Appl. Phys. Lett. 36,5, 1980, pp. 386-388.

Rationale and experiment for a stripline device.

Electronic Experiment, RSJ Equation, Stability.

19. R.L. Kautz, The AC Josephson Effect in Hysteretic Junctions: Range and

Stability of Phase Lock, J. Appl. Phys. 52,5, 1981, pp. 3528-41.

Numerical modeling of Weakly Dissipative RSJ Equation.

Stability, Bifurcations, Noise.

20. R.L. Kautz, Chaotic States of rf-biased Josephson Junctions, J. Appl.

Phys. 52,10, 1981, pp. 6241-46.

Full details of Bifurcation in Parameter Space.

Strange Attractors, Bifurcations, Spectral Analysis, Numerical

Simulation.

21. N.F. Pederson and A. Davidson, Chaos and Noise Rise in Josephson

Junctions, Appl. Phys. Lett. 39, 10, 1981, pp. 830-832.

Frequency-Amplitude Parameter Plane for zero dc voltage, in amplifier

applications

.

RSJ Equation, Computer Simulation, Chaos, Stability.

22. G. Park, PRIVATE COMMUNICATION, Manuscript Notes, addressed to R. Soulen

and D. Van Vechten, 1980.

Treats tank circuit of thermometer in interaction with the RSQUID

loop.

23. H. Seppa and R.J. Soulen, Jr., Modeling the dc Impedance of an rf-driven

R-SQUID, Post-deadline paper HY 15, American Physical Society Meeting, March,

1982.

Models resonance at tank-circuit frequency, by feedback around to the

RSQUID loop.

24. R.J. Soulen, Jr., H. Seppa, and D. van Vechten, The Effect of Additive

Noise & Bandpass Filters on the Performance of a JJ Noise Therm.

,

Post-deadline Paper HY 15, American Physical Society Meeting, March, 1982.

Isolates and models a weak dependence on gain and on filter bandwidth.

28
FIG. 4

Z VS PHI

FIG. 4b

Z VS SIN OF PHI

FIG. 4c

SAME AS 4b, WITH

AXES EXCHANGED
FIG.

5

COMPOSITE

POINCARE MAP

PHIDOT VS PHI

o

o

XCL. FOR ALL 40

START TIMES

1000. 0 -|

aoo.o600.0-

400.0

200. 0

-'

0.0

f'- /"« -

*- :

**

-* ; : * * * \ *

-

• : : - - - - - 777" » 7 -v-

: ; ;

.** : * • , O'. N . ^ *i: , . -200.0 - VV.Vvv-"-". » -

M00.0-

-€00.0-

-aoo.o-

""v'

1000.0
FIG. 6

PHI VS TIME

FIG.

PHIDOT VS TIME

I-* oQ

r

a.

FIG. 8

SIN OF PHI

VS TIME

TI*E

I

|

I

|
2500.

0

2000.0 FIG. IO

1500.0 COMPOSITE

1000.0 POINCARE MAP

500.0 o

2 o.o X

-500.0 CF. FIG.

5

1000.

0
I

FIG.

9

Z VS PHI

FIG. 9b

T~ VS

SIN OF PHI

-2.0 J

1.0

1

FIG. 9c u.3i i'J SAME AS 9b '

i 1 WITH AXES

,

4

} '«.J EXCHANGED "

t

' , 'O/a.l

-0.4-

-o.o

-o.l-j

!

i -I.C-*
7.0 413.5 Msec

FIG. II

PHI VS TIME

ONE POINT PER

DRIVE CYCLE

& oV O'* TIME
FIG. 14

Z VS PHI

FIG. 15

POINCARE MAP

PHIDOT VS PHI

f—

oa

x

a.
6.0

FIG. 16

PHI VS

TIME

*H 172 /JSEC |-< 1 TIME

li

FIG.

22

RELAXATION

FREQUENCY

VS

AMPLITUDE

OF

THE

DRIVE

SIGNAL

o
7.0

(.0

I Trtl l-«

FIG. 23

PHI VS TIME “

f\ :

: V

i.o3. 0

-

2.0

1 . 0 -

V

0.0

A. A* t

79. Z yUsec

T1FE

. 000. 0

-

FIG.24

PHIDOT

VS TIME

8 203. 0 -!
r.On

N

6.0

5.0-

4.0-

3.0-

2.0

1.0 J

FIG. 26

Z VS PHI

START AT

T = ^/uO

//

/

/

/

/

/

/

/

/

/

0.0-1 1 1 1 r O <0 O O, O

<5* V V V

-r f T

<*> O V> <V •>* O' PHI

7. °1

O V> V o>' 55

<** •o

•O'

N

6.0

5.0-

4.0 H

3.0 i

2.0

1.0

FIG. 27

Z VS PHI

START AT

T =0.1

/

/

/

I / /

/

o.o- /

/

/

<5 O >> O' O' V 'V O M

•O' <V w

<* «» o

<o' •o V
1.0 -|

o.a0 . 8

-

OM -

0 . 2

-

FIG. 29

SAME DATA,

PLOTTED AS

-0.8

- 1.0 Z VS SIN PHI,

AXES REVERSED

—i

1 1 1 1 1

1—

y

1 0 *» "

• o o *> ff «jf V V <o <o‘ V

1 /

s+ s

' ./ V

* /

/
SIN

PHI
Z VS TIME

FOR DATA OF 50 ‘ FIGS. 30-31

3.0'

2. 0

-

1 . 0

-

0. 0

-

••

7.0-

6.0- FIG.33

Z VS TIME

FOR DATA

OF FIGS. 23-25

4.0 -
1.0

0. 8

-

FIG. 34

SIN PHI

VS TIME

f = 1.5,

LONG

TIME

SCALE

-o.s- ...

' ...

•

j J. '.V. / . ... . .....

’ “ 1-^ 13.58 msec

FIG. 35

SAME,

FOR

i = 0.5

!

I
TI rc
FIG. 37 SANDERS’S RESULTS

Y ( TO ) = 6.374423 DY ( TO 1 = 10.0000 ' Y ( TO ) = 6.374424 OY(TO)= 10-0000

Y t TO ) = 5.374427 OY(TO)= 10.0000 Y ( TO ) = 5.374428 OY(TO)= 10.0000
_

i * s 4 I I | 8 ' 11 "WI7”^I

ii
b-0 cn

£ CD

.3

w

•i—

i

—i 3

O

p

Sh

a ^

J* w

bfl

O <4H

%

°

CD

a

1 H

as

co

3)

u

<

*

•H

T3G

0)n

cd

' L

0

U

a

0)r<

0

4J

0)

X3

•-H o

Sh

0)

Cfl 3)

to

1—o

co3

o

Q\

O

UJX

toX

o

®a

towX

o

cx • a x

10 0)

c. >

C3 1010 c

0 <0 3 - *0 4-c 0) c N > > •»- 44 <0 (0 o

D *—

«

c (/) a

a 04 Q. o T3 c

44 O — 0) o 4-» 0

3 < 3 4-* T3 .Q (0 4-*

• « o H- (0 a (0 c c c. > a *— c_ 0 <D * CD — i

d> X 0) *0 d o 4-» aj

4-1 *-» T3 to 4-» 0 JZ a 4-» to 0) c (1) n c

E o» c II •- 13 U) o

c. c C» > c o «- CL 4-* T3 (0 1

— o

<4- 44 to UJ «3 — > c 4-1 il c — to *— 0) D A3 o L. 4-*

C- (0 to a 0 c. C0 3 (D jC c • 0 X • L. *—

«

10 0 W) 4-» c c. o a Q- c o >T3 QJ 0 o 04 i-4 0) 0) T3 3 w- c w -» I r to -C C

0 C- Q to > > CL 4-* — 4-» D 0) c_ O <0 X X O (/) >

c.

to

c.oa

£ l—

0)

E -i

to co

C. X

X o

o a

L N

aoUJ

a) x

oj x

x E E 3

3 C

3

U

a>X

0)

a)c c —

o —

W c

3 O

O

®

X • UJ <

to ** X

to < u

to

®

E - - -XX w CL Q. a to

C- ‘

- x

tJ to w

• C Q —*

C VC. to X O Oh-X

U to Q. C id to A

111 A < >

*-« I— — t— — < to

ffl c. X o

E *- 3X O

o o o

2 t-01

C. X

O -M

-4- to X

— 3

•W C. <4- O 01 o

C 3

to to

c_ c ai 0 0 *-• 10

C. 0) c

01 x —

X 4-> X

4-> C. ® >4- O X o 3—0

c. — o

10 -M

ai X 0)E > —

U -M

to

C_ +4 3 C. U (0 U *-> < to

— CN CO *7 in tO f* CO Cl O

3

O>

— ci n

- c

a

toE

(0c

*>c

toE

to

to

toE

0

to

• <

< ht- <

< a

a *x

x a

a uj

LLi tO

l/> O

O X

X

® *J

• — c x — Cl O **- E

••

4J ® X

to to — c X i-1 ® —

to 3 >x c o X toll) —

0>X® o

3 10 ©•••••

u ® to to c — cn n <r in Of) C. X —

x x o —

®

a

x o c —

a to

- c

UJ O

a. O CN > > X X

X X

a cl Q !0

c.

- O o

(_> o

4-< c.

- c o

00 -i x — E

- 3 t^<tozo — u «< <n

a o o o

2 2

c. c. o o

® X

X c w X o

to to

<4- C. 0-0to w

to ® ® X ® « E

to > —

c u «

— to X c. -

C 3 to O V to U *«

in

a a c. a> o o — O £ u) 3 ^ 4-» o • H- (0 03 — —

c. C. o cr * CD • CD <

to 4-« <34 C H- to 4-* a 44 03 tO c c

u — CD — CO . C z: c to — a

Q. 4-* o UJ

UJ 03 4-> r in 44 (/> L- (1) c. 44 <T — o

co a 03 “0 o 4-* o CO o

- c *-4 (0 44 ID c.

» a

44 o

to —

c

0)

, 44c— to

c_ »-•

id co

o • • I- — 04

co

c

toE r» ®— O

to —

a

E

to to

a a u < <

*o E c z < a a

c 03 D 0 < >— z z

03 U) 0 c Ai < 3 3

u — u. Q cc cr £ 03 u 1

* • 4

03 to - ZL X X

0 to - • — a a a

C. — o b* UJ UJ UJ UJ

44 —

1

— to to to to

Ifl o c. 2! O O O O

c c 0 —

•

"0 0 3 3

CO — m cr - o o • L. 3 3 a < < < o o o • • • o • * — a «— Z 5 o O a uj o O O *— o — 3 >* LO in to to o 0 0 cr to < < < 3

o o o o o o o o 44 o ^

ntjintflofflci- a

to a

l. x U I-

— of-> 00 (300000 C 00000000000 C 000000 OOOOOOOOOOOOOOOOOOOOOOOOOOaO co ^ o _ o(ro ^ ir)0p_ ;30 oooooooooo-(Nri 9 ititflMacio-(vn 7 ^o^ 3 ^o nnonnnnn ,,, C tf 7 <nr 79 ifliiiininifliiiu)'jiin — CNCOTUOtCr^XOl — — — — — — — — — — 04040404 CNO4 t 4 t.-i^

AI
I H' 05

0)

XJ• —

-“4 3o ©x c 44

— • • •

05 05

a>

X L> 4- 3 c — a c •

<D 4-*

G

X a> .4 4-

X c

05 1)

a 4-*X 4^ o 0 • 4-* — X G 0) G 0 — tfl 05 3 CO x f^» X 0) G CO — Q) 3 x E o • c. • E IT} u E CD o a CO 0 0) 05 05 — o £ -

co

X z 05 05 Q) vu 3 0 44

C (J

A 05 t. 0

a> 0) 4-1 4-»

w +J E 44

— 0 05 44 0 c. E E 0 0) 4“ X G c a> 3 X X X G •*- u

to

G 05 a>

O

C 0 X

X 05 4-

— c c C c 0) c E a 4-* — • 44 X 0) 05 > 05 3 X — 0 44 3 0) 4- G a c 4-* > 44 X X X X O a c

LU CN c k—X

u

C 3 0)G • G 44

—

uG 44X o 0)

—

-J 0) (1) z o X 44 0) E E E c. • O 3 44

X 05 05

X

C -

CD 6 E LU 0)

—

G a> c 0)

C

O L.X CD

— o X

X

a UJ

—

—

Z>

a <u cc 4-

—

44 CJ 44—

O C.G t—C 3 44

— 05 44

X —J 0 o C — LU 44 w G E G a «4- v4- o 0 0) 05 3 G 44 t—

G CD c. — Q +- 0; u. E 3 0 X 0) X o c 0) • z 44 44 Q • t+- O 0 a \ c o> u. G O c 4-< *— 4-4 E — G X C_ UJ 44 C 44 44 G 05 O H- O o E •—

X 05 0) a> O 4-4 4-< 0) -J c. — CD 05 G <4 G a C. LU E O Q CT> CT 0 3 c <15 — 05 (3 G X 05 X X X E c 0) 0) \ 3 3 X O 05 o o — 3 E G C X — c c o 3 3 44 44 X o 44 X • 0 O O

uo *—

G

x

a

— C.

OC 0 < —

— 3 a 05 o > — •0) 4- < > >

=5 o < a

—

X 05

—

4-4 k— 3 X c C > C or - X *4-

GO 3 4- UJ cj 05 >- 05 3 c > E O c < O G — 05 4-4 G • G CD 44 X 4- E cr

— E < c a> G X — — x — — Q — O 4-4 0) 05 3 — G5 X 3 CO a. a 3 m o Q G X 05 C 4— cr 05 — — c_ X c X c O 44 O 3 X 3 C CN CO 6 O

*-« 0) 4-*

X

X

G

G < 4-4 V4_

G

X 0 05 05 4-» CDn c G <N 44O o 4-

a 0 d o X E 0) E 3 k- 4-4 CL 0 c E X — — 4-» Q O > 4- 0 u- CN E LO C. IT} z < CL a> — > E 44 LO G > G LU 4-4 C» 05 3 *4 o G CN X 3 *» 0 G a < — UJ 3L E — — O a x X in 0) —

0 e a X3 G O V4- X X X CJ m

cr < LO 05

—— 0 o G

X

CG o X c 05

O c X 0 XX c — k- 05 c 0 O' 2 — c

UQ o E 4-* 3 > QO c. 4-»

X 4-4 d) 05J£ c CG 0) 0) c • z g E 0 < - 0 * X G c h- X LU 0 3 X G «4- G >4- 44 X 05 5 •-« E 3 CO X cr — — X X • C X c — cr 0 in <d a a X E c *-4 G 44 3 u 44 • cr 3 

a u. 3 -

CL Q.

w

u 3 < o X E 4-4 - 05 3 3 - - X * a

C 0 o * m

LU LU c G

X 0) k—X 0) - 3 > a> c 0 L- c *— 44 > 0 c u CN 05

X E a) in C/5 0) (1) o C/5 o c k— 4-4 O O CL c > 05 0 4- g — 44 0 0) 05 a X CL ro 3 o O •—

£ W — x 3 k— LO C/5 o 44 X X a 44 O E — X Q a n 44 CN 44 UJ C CT ~D X • 05 X 05 — 05 4-# C- CC X G < a CJ — 05 E .j O c_ G 4- -J > 44 05 LO O

>- 4- CD

G •4-

X < s X

— c c 3 3 O

G C.

OC 4- 3 O 44

G 05 a>

OO JC

hQ CL z © 05

— 44 4-4 C5 kC k— 4-4 k-

— 3 0 C o C- 0) «4»

— o 44

— 0) 44

X c. in o a o *-« Q 3 — G X LO 3 ceo a 3 > o 44 _Q o o > o c C 3 3 a <o X < o u_ E O 4— o> X 4-» 0 - X > X <D 0) 05 a X E 05 — CN a 05 3 05 0 k—

<a

O

O

XG c 4-* k— >- 05 CL

a cl 3 X Z'

X V4. > 3 3 X c.O '4- > O

X 0 05

a ID CT

i/)

X

w 4- 44

a 05A 0 A 0 H- 44 k- •—

4-4 LO

C k- 0 CN •—

44 -44

u 44 05

X

XdC

05 3 05

x

X • < • a i > 1 i t UJ

A 1-

O o 05 E G CL • • c_ 05 4-4 A 4^ UJ u. — 4-» LO • o • • • • oz — — cn

Q.

CO

<0

Sh

0)

0] D

O 0 OOOO

O

O

O 0 0 0 OO 0 0 O

OOOO 0 0 OO 0 0 0 0 0 0 0 O

O

O 0 OO

O 0 O

O 0 4- CN CO <3* If* CO

xC5 O «— CN CO ID CO CD 05 0 —

CN CO <T 10 0 CO 0 O

— CN CO in LO CD 3}

IT) 13 LO

O <3 LO LO LO (O CO LO r*w

X CO CO CO

a CO COA

OOC5 0 a> CD 05 05 G> O) c

A2

000

When

It

Is

done

plotting.

It

will

pause

for

you.

010

When

you

have

finished

staring

at

It,

copying

dat

020

from

It,

and/or

making a

hard

copy.

THEN

AND

ONLY

030

THEN

hit

return

and

It

will

prompt

you

for

more

040

graphs.
APPENDIX B CIRCUIT ALGEBRA TANK

CIRCUIT from dc

and rf feed

1 'rO

GSW

M

RSQUID

THERMOMETER

LOOP

dc

Point Contact

(parasitic) capacitance of this

Resistance associated with the quasiparticle conduction

Single coupling loop (inductance at nanohenry level)

Shunt Resistor, typically of silicon bronze

B1
Josephson relations in the point-contact branch (an active device)

Vj = 1/2it • h/2e • d^/dt instantaneous Josephson emf (la)

I sc

= I c

s;*- n ‘Kt) = tunneling supercurrent (lb)

Current division:

Ij[ n

3 Ij_ + Ie

at input terminal to RSQUID loop (2)

II = Ij + I c + I2 at input to point contact (3)

Displacement current in capacitance

I c = C • dVj /dt (4)

Voltage closure around the emf’s of the main RSQUID loop

Vj = M dl rf /dt + L dl 1 /dt + R^Ie (5)

We assume that the tank circuit functions as a constant-current rf source,

and that the dc bias feed comes from a constant-current source, so that

Iin = nl r f + I dc n < 1 , where I r f = irf sin ut (6)

Here I r f is the rf current resonating in the tank circuit, and n represents the fraction of this which is diverted into Ij_ n

We substitute into (5) and combine these relations into a single equation for

the emf’s around the main RSQUID loop. With regrouping of terms, we obtain

the full equation for <j> as a function of time.

1/ 2tt • h/2e { - L Cj d<{> 3 /dt 3 + [ RgCj - L/Rj J d$ 2

/ dt 2

+ [ 1 + Re /Rj - Ic L cos 9] d(j>/dt } + Re-^-c s:i- n 9

R^ I dc + [ M a) + n Re j i rf sin ut (7)

B2
The third derivative term, and the second term within the second bracket are neglected. We take four new definitions of parameters:

0 E Re Cj - L/Rj

Y = - Ic L

a = (Rg Idc )/( Re Ic ) = Idc / Ic dc bias

[M + n Re ] irf

kcu = rf drive amplitude

ReI c

and note that there is no loss of generality in assuming y positive, except

that we must recall this reversal of sign when re-interpreting emf’s on the

inductance L. This achieves the expected positions of the coefficients.

If in addition we scale the frequency m to the plasma frequency of the

RSOUID loop, which is set by the coefficient of the sine term (the product of

the critical current I c and the shunt resistor R€

):

2tt oj c

= 2e/h • Re

I c =1

we obtain the greatly simplified general form for the RSJ equation of the

RSQUID loop.

3<J> + (1+ycos !p)(f> + sin <j) = a + ka) sin cot (8)

Here we have tacitly assumed that Re /Rj << 1 , so that in fact we neglect entirely the quasiparticle tunneling current through the point contact

at the very low temperature of the RSQUID thermometer. This enables us to

drop the subscript on Rg and to write it as R in the scaling description

starting on p. 11.

When, in addition, we assume a positive value of .001 for 0 , we are implicitly forcing a "smoothing" time constant upon the system response during

the short "jump" intervals:

T RC = Re C ~ 0

• 2tt/co c = .21 microseconds

This detailed circuit model was derived in a slightly different manner by D.

van Vechten (private communication, 1980) and has been partially published in

references (7), (8), and (9).

B3

N BS-114A (rev. 2 -ec) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

SHEET (See instructions) NBSIR 83-2643 January 1983

4. TITLE AND SUBTITLE

WAVE FORM SIMULATIONS FOR JOSEPHSON JUNCTION CIRCUITS

USED FOR NOISE THERMOMETRY

5. AUTHOR(S)

F. Sullivan, D. Kahaner, H. A. Fowler, and J. Knapp-Cordes

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 1 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

N/A

10.

SUPPLEMENTARY NOTES

| | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most si gn i fi cant information. If document includes a si gn i fi cant bibliography or literature survey, mention it here)

Solutions to the Resistively Shunted Junction (RSJ) Equation:

B 4> + ( 1 + ycos $)<}> + sin p = a + kio sin ut

for small S ( = .001) have been approximated using the SDRIVE integration package,

/arious graphic displays are used to examine the output, including plots of <p , dcp /dt

,

and sin$ as functions of time; Poincare diagrams; and plots of the Lienard coordinate

• z = B<}> + <j)+Y sin p tfhich has a close connection with the "slow manifold", as a function of tp

, of sin <p , and

Df time. Integration is performed by separation of the second-order equation into a coupled pair of first-order equations, and numerically integrating with respect to time.

Several cases have been examined, for y < 1, representing quiet behavior of

resistively shunted thermometer oscillator devices. The report is an archive record of

orogram-test data.

A case of "jump"( voltage-spike ) oscillator performance for y= 1.5 has been

;imulated in considerable detail, principally as a test of the integrator. Parallel,

Independent computer results (Sanders and van Veldhuizen, Amsterdam) were available for

lomparison. This case is of considerable mathematical interest, and values of j y j >1 may

also occur in the RSQUID thermometer.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; ana seoarate key words by semicolon s) Josephson Junction, electronic oscillator, differential equation, numerical

integration, Poincare maps, relaxation oscillator

13.

AVAILABILITY

| Vj Unlimited

j&jj For Official Distribution. Do Not Release to NTIS

j j Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

14. NO. OF

PRINTED PAGES

IS. Pr $

Order From National I echnical Information Service (NTIS), Springfield, VA, 2216 1

USCOMM-OC 6043-P30