The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems
We investigated numerically synchronization of Josephson junctions inside the transmission line. We have found that due to the resonance behavior of the system there appear the self-induced resonance steps, strong synchronization of junctions on these steps and the inhomogeneous distribution of th...
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irk-123456789-1171232017-05-21T03:03:02Z The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems Grib, A. Seidel, P. Квантовые когерентные эффекты в сверхпроводниках и новые материалы We investigated numerically synchronization of Josephson junctions inside the transmission line. We have found that due to the resonance behavior of the system there appear the self-induced resonance steps, strong synchronization of junctions on these steps and the inhomogeneous distribution of the Joule heat extraction (the selfheating) along the line which can lead to the formation of “hot spots” in the line. The developed model can be applied to explain recent experiments in which these effects were obtained. 2012 Article The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems / A. Grib, P. Seidel // Физика низких температур. — 2012. — Т. 38, № 4. — С. 409–413. — Бібліогр.: 14 назв. — англ. 0132-6414 PACS: 05.45.Xt, 74.50.+r, 85.25.Cp http://dspace.nbuv.gov.ua/handle/123456789/117123 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые когерентные эффекты в сверхпроводниках и новые материалы Квантовые когерентные эффекты в сверхпроводниках и новые материалы |
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Квантовые когерентные эффекты в сверхпроводниках и новые материалы Квантовые когерентные эффекты в сверхпроводниках и новые материалы Grib, A. Seidel, P. The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems Физика низких температур |
| description |
We investigated numerically synchronization of Josephson junctions inside the transmission line. We have
found that due to the resonance behavior of the system there appear the self-induced resonance steps, strong synchronization
of junctions on these steps and the inhomogeneous distribution of the Joule heat extraction (the selfheating)
along the line which can lead to the formation of “hot spots” in the line. The developed model can be
applied to explain recent experiments in which these effects were obtained. |
| format |
Article |
| author |
Grib, A. Seidel, P. |
| author_facet |
Grib, A. Seidel, P. |
| author_sort |
Grib, A. |
| title |
The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems |
| title_short |
The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems |
| title_full |
The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems |
| title_fullStr |
The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems |
| title_full_unstemmed |
The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems |
| title_sort |
influence of standing waves on synchronization and self-heating of josephson junctions in resonant systems |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2012 |
| topic_facet |
Квантовые когерентные эффекты в сверхпроводниках и новые материалы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117123 |
| citation_txt |
The influence of standing waves on synchronization and
self-heating of Josephson junctions in resonant systems / A. Grib, P. Seidel // Физика низких температур. — 2012. — Т. 38, № 4. — С. 409–413. — Бібліогр.: 14 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT griba theinfluenceofstandingwavesonsynchronizationandselfheatingofjosephsonjunctionsinresonantsystems AT seidelp theinfluenceofstandingwavesonsynchronizationandselfheatingofjosephsonjunctionsinresonantsystems AT griba influenceofstandingwavesonsynchronizationandselfheatingofjosephsonjunctionsinresonantsystems AT seidelp influenceofstandingwavesonsynchronizationandselfheatingofjosephsonjunctionsinresonantsystems |
| first_indexed |
2025-07-08T11:41:16Z |
| last_indexed |
2025-07-08T11:41:16Z |
| _version_ |
1837078810209550336 |
| fulltext |
© Alexander Grib and Paul Seidel, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4, pp. 409–413
The influence of standing waves on synchronization and
self-heating of Josephson junctions in resonant systems
Alexander Grib1,2 and Paul Seidel1
1Institut für Festkörperphysik, Friedrich–Schiller–Universität Jena, Jena D-07743, Germany
E-mail: paul.seidel@uni-jena.de
2Physics Department, V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv 61022, Ukraine
Received October 17, 2011
We investigated numerically synchronization of Josephson junctions inside the transmission line. We have
found that due to the resonance behavior of the system there appear the self-induced resonance steps, strong syn-
chronization of junctions on these steps and the inhomogeneous distribution of the Joule heat extraction (the self-
heating) along the line which can lead to the formation of “hot spots” in the line. The developed model can be
applied to explain recent experiments in which these effects were obtained.
PACS: 05.45.Xt Synchronization; coupled oscillators;
74.50.+r Tunneling phenomena; Josephson effects;
85.25.Cp Josephson devices.
Keywords: Josephson junctions, synchronization, high-temperature superconductors.
1. Introduction
Since the experimental proof of the non-stationary Jo-
sephson effect [1] the problems connected to the use of
Josephson junctions as tunable sub-mm oscillators with a
high precision of frequency become important because of
the application of the sub-mm wave band sources in differ-
ent fields of science. The main problem of Josephson junc-
tion oscillators is the phase locking of many junctions to
get appreciable large output power. An easy way to solve
this problem is to use a resonator to perform a phase lock-
ing feedback. Recently, immense progress in experiments
on the detection of synchronized radiation from intrinsic
Josephson junction arrays in high temperature supercon-
ductors (HTSC) was achieved [2–4]. It was supposed [2]
that the array of intrinsic junctions itself can be a geome-
trical resonator. To explain the effect, the hypothesis of the
nonuniform distribution of critical currents of intrinsic
junctions was proposed [2] as well as the hypothesis about
the existence of the plasma resonance in the system [5].
Experiments showed also that the temperature along the
measured samples of intrinsic Josephson junctions can be
strongly inhomogeneous [6] and the so-called “hot spots”
(i.e., places in the sample where the temperature can even
reach the critical temperature of the superconductor) can
be formed [7,8].
The hypothesis about the resonance properties of the ar-
ray of intrinsic Josephson junctions assumes the formation
of standing waves of the electromagnetic field inside this
array. Earlier we developed models of synchronization of
the arrays of junctions shunted by the resonant transmis-
sion line [9] and the resonator [10,11] and found that the
current resonance is responsible for synchronization in
both cases. In the present paper we consider the formation
of a standing wave in the resonant transmission line with
open ends and the influence of it on synchronization of
junctions embedded in the line. We suppose that this struc-
ture can model the arrays of intrinsic junctions which were
investigated experimentally. In our model instead of con-
tinues layers of intrinsic junctions there is a set of “elemen-
tary junctions” which form a two-dimensional array inside
the transmission line (Fig. 1). We believe that this model is
more adequate to the experiments in which the typical di-
mensions of the mesastructure is 300 50× µm and the
structure can contain defects like dislocations and grain
boundaries, whereas the distance between intrinsic Joseph-
son junctions is only about 1.5 nm. For the modeling we
chose the simplest case of only two layers of insulator di-
vided to the “elementary junctions”. The pair of these
“elementary junctions” belonging to different layers forms
a stack. It is known [12] that the distance of electromagnet-
ic interaction of junctions in a stack is infinitive, i.e., each
of the junctions in the stack interacts with any other junc-
tion and this interaction does not decay with the number of
the junction in the stack (i.e., with the “distance” between
junctions). Therefore, main particularities of the electro-
Alexander Grib and Paul Seidel
410 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
magnetic interaction between junctions can be discussed in
the simple system of only two junctions in a stack. In the
present paper we do not consider all features of the devel-
oped complicated model but concentrate on the main three
particularities which were obtained experimentally: (i) that
the measured system has the self-induced resonant steps in
its I–V characteristic, (ii) that the coherent radiation is reg-
istered at these resonant steps and (iii) that there is the
strongly inhomogeneous distribution of the temperature
along the radiating system that leads to the formation of
the “hot spots”. We obtain all these three features in our
model and show that all of them have the same origin that
is the resonant behavior of the system.
2. The model
The example of the high-frequency scheme of the con-
sidered system is presented in Fig. 1. The transmission line
is divided on N cells (in Fig. 1 N = 6, but this number can
be varied). The length of the cell with the index j is 0ξ .
The value of 0ξ coincides with the linear dimension of the
“elementary junction” or the “grain”. The geometry of the
problem is as follows. The x-axis is directed along the
transmission line (see Fig. 1). The first cell at the left edge
of the transmission line has the index 1.j = We call the
value of ( )0 1/ 2jξ − as the “distance” along the transmis-
sion line and will calculate “spatial” distributions of cur-
rents and voltages along this direction. Centers of “grains”
coincide with positions of high-frequency current lines.
The z-axis is directed along these high-frequency current
lines and the y-axis is perpendicular to the plane of Fig. 1.
The size of the transmission line along the y-direction is
not limited. The size along the z-direction is the sum of
thicknesses of two superconducting layers and two layers
of insulator.
Each of the cells of the transmission line in Fig. 1 has
the capacitance ,C the inductance L and the resistance .R
The two-dimensional array of junctions is notified by in-
dices 1, 2i = and 1,..., 1,l N= − where the index i is for
the upper ( 1)i = or the lower ( 2)i = junction in the high-
frequency electrical line which divides two neighbor cells
and the index l is for the number of this electrical line in
the system. These two junctions in the high-frequency
electrical line we call as a stack of junctions. Each of the
stacks has the source of the direct current. All sources for
all stacks have the same value .bI The direction of the
current in the stack is from the lower junction to the upper
junction. The set of junctions with the indices 1i =
represents the upper layer of insulator divided on “grains”
and the set of junctions with the indices 2i = represents
the lower layer of insulator divided on “grains”, corres-
pondingly. For the simplicity we assume that all “grains”
in both layers have the same size. Boundary conditions of
the problem include the reflection of electromagnetic
waves from ends of the line. These conditions were mod-
eled by means of capacitances 1,C inductances 1L and
resistances 1R at ends of the line.
Let us denote ,i lϕ as the phase difference of the order
parameter across the ith junction in the stack belonging to
the high-frequency current line with the number l. Then the
system of differential equations describing phase dynamics
in the range of the model of the capacitively and resistively
shunted junction [12] is as follows:
0 0 , 0
, , , , 1
,
sin
2 2
i l
i l i l ci l i l b j j
i l
C
I I q q
R +
Φ Φ
ϕ + ϕ + ϕ = − +
π π
,
1,..., 1, ,l N j l= − = (1а)
2 2
0
, 1 ,
1 1
2
,
2
j
j j i l i l
i i
q
Lq Rq
C −
= =
⎛ ⎞Φ
+ + = ϕ − ϕ⎜ ⎟⎜ ⎟π ⎝ ⎠
∑ ∑
2,..., 1, ,j N l j= − = (1b)
2
01
1 1 1 1 ,1
1 1
, 1,
2 i
i
q
L q R q j l
C =
Φ
+ + = − ϕ = =
π ∑ (1c)
2
0
1 1 , 1
1 1
, , 1,
2
N
N N i N
i
q
L q R q j N l N
C −
=
Φ
+ + = ϕ = = −
π ∑ (1d)
where ,i lϕ and ,i lϕ are the first and the second time deriv-
ative of the phase difference, jq is the charge which
passes the inductance of the jth cell, 0Φ is the quantum of
magnetic flux, 0 ,i lC and ,ci lI are the capacitance and the
critical current of the junction, correspondingly, ,i lR is the
resistance of the junction. For simplicity we keep all criti-
cal voltages across junctions , , ,ci l ci l i l cV I R V= = to be the
same for all junctions.
3. Results and discussion
For calculations typical values of parameters for critical
currents and critical voltages of intrinsic Josephson junctions
[1] 15cI = mA, 2cV = mV were chosen. To obtain phase
locking we consider the simplest artificial case of small
spread of critical currents 0.02δ = which is the same in each
stack, so the upper layer of junctions has critical currents
( )1cI −δ and the lower layer of junctions has critical cur-
rents ( )1cI + δ . Values of 1.62L = fH and 0.12C = pF
were obtained to satisfy the value of the velocity of light ob-
C
JJ
JJ
C1
L1 RL
x
yz
R1
C1
L1R1
Fig. 1. The high-frequency scheme of the system which contains
6 cells and 5 stacks of junctions. The system considered in the
paper contains 60 cells and 58 stacks. The x-position of the origin
of coordinates is in the middle of the bottom line of the first cell.
The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 411
tained for the high-temperature superconductor [1] under the
condition of the size of the junction 0 1ξ = µm. For the
boundary conditions we chose 1L L= , 1C C= and 1 0R =
(the transmission line with the open ends). We also intro-
duced small Joule losses in the line 4( 10R −= Ohm). To
demonstrate the main results of the model in the present pa-
per we use small values of the McCumber parameters of
junctions 2
0 0 0(2 ) / 1.C c jI R Cβ = π Φ = Though these values
are smaller than those for high-temperature superconductors
0( 10),Cβ ≈ we checked that main results of the model are
saved for them too.
We investigated the system which contains 58 stacks. In
Fig. 2 we present the I–V characteristics for the first stack
from the left side of the system. The I–V characteristic con-
tains resonant steps. Note that these steps are self-induced,
i.e., they appear without the external irradiation but only as
the result of the electromagnetic interaction of junctions
inside the transmission line. It was shown [9] that the reson-
ances in the system of Josephson junctions loaded by the
transmission line appear when the averaged over time vol-
tage across the junction becomes equal to the value
/ (2 ),rn cV nV〈 〉 = χ where n is integer, χ is the length of
the system in units of wavelengths of the ac current in the
line and the sign ...〈 〉 means averaging over large time.
Standing waves in the line appear at these resonant voltages.
For the given parameters of the model the resonance voltage
is equal to 1 0.63 .r cV V〈 〉 = The centers of steps in Fig. 2
match values of 0.63rn cV n V〈 〉 = ⋅ exactly. It is necessary
to note that I–V characteristics of stacks along the transmis-
sion line can be quite different from the I–V characteristic
shown in Fig. 2 because the resonant conditions for them are
not satisfied. For example, for junctions in the centre of the
line there exist only even numbers of the steps.
Junctions radiate coherently at the resonant steps. It is
seen from Fig. 2 that they have the equal averaged voltages
near the upper edges of steps that is characteristic for phase
locked states. It is known [12] that in-phase locking is de-
termined by the negative value of the imaginary part of the
connection coefficient Yi,l of the electromagnetic interac-
tion between junctions. For two junctions loaded by the
transmission line with the open end the connection coeffi-
cient has tangential particularities at the resonant steps due
to the current resonance [9], so the strongest phase locking
is expected at upper edges of the resonant steps where
Im(Yi,l) < 0 and the absolute values of Im(Yi,l) are maximal.
The behavior of I–V characteristics in Fig. 2 is in full
agreement with this consideration. We checked also the
presence of phase locking numerically considering the time
dynamic of phase differences across junctions and proved
that 14 stacks at the left end and 14 stacks at the right end of
the transmission line radiate coherently. In inset of Fig. 2 we
show the spatial distribution of the mean square root value
of the ac current flowing through stacks. It is seen that the
largest values of this current are at ends of the line. This
current synchronizes junctions at ends of the line.
Distributions of voltages in stacks along the transmis-
sion line are shown in Fig. 3 for different values of the bias
current which increases with the step 0.02 .cI There is one
large irregularity of voltages in the center of the line in the
vicinity of the voltage corresponding to the main resonant
mode, two irregularities in the vicinity of the second reso-
nant mode etc. These irregularities indicate the standing
waves which appear at different dc currents.
Fig. 2. I–V characteristics of individual junctions for the first stack
from the left side of the transmission line plotted in normalized
units: ,1 ,1 /i iV V〈ν 〉 ≡ 〈 〉 and / .b b ci I I= The plot of ,1 ( )i bf i〈ν 〉 =
is shown only for the increasing bias current. The I–V curve for the
junction with smaller critical current is shown by the solid line.
Points of the I–V characteristic marked by arrows A, B, C are dis-
cussed in the paper. Inset: the spatial distribution of the mean square
root (m.s.r.) value of the ac current flowing through stacks at
1.14bi = (the point A in the I–V characteristic).
�i,1
�1,1
�2,1
A
B
C
3
2
1
0 1 2 3
6
4
0 10 20 30 40 50
i b Distance, m�
m
.s
.r
.
cu
rr
en
t,
A
m
Fig. 3. The distributions of averaged voltages
( )
1, 2,[( ) / ] / 2a
l l cl V V Vν = 〈 〉 + 〈 〉v along the transmission line at
different values of the bias current which increase with the step
0.02 .cI Distributions which correspond to currents 1.14 cI and
1.58 cI (points A and B in Fig. 2) are marked by the bold dashed
line. The distribution which corresponds to the current 1.24 cI
(the point C in Fig. 2) is marked by the bold dotted line.
2.0
1.5
1.0
0.5
0
0 10 20 30 40 50
Distance, m�
�
l(
)
a
v
Alexander Grib and Paul Seidel
412 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4
In the following consideration we discuss values of the
averaged over time instant power in the stack lP =
( )
2
0
, 1
12 i l b l l
i
I q q +
=
Φ
= ϕ − +
π ∑ with j = l. This value is
responsible for the Joule heat extracted from the gi-
ven stack l. We also introduce the value of
22
0
, ,
1 2l i l i l
i
S V
=
⎛ ⎞Φ⎛ ⎞= ϕ −⎜ ⎟⎜ ⎟⎜ ⎟π⎝ ⎠⎝ ⎠
∑ that is the square of the
ac voltage over the lth stack. The value of lS is proportion-
al to ac power of coherent radiation of junctions in the lth
stack [10,12]. If junctions in the stack radiate coherently
with the zero phase shift, the value of lS increases up to
,4 ,i lS where
2
0
, , ,2i l i l i lS V
Φ⎛ ⎞= ϕ −⎜ ⎟π⎝ ⎠
is the value that
is proportional to ac power of the individual junction in the
stack; if junctions radiate with the phase shift
2
π then
0lS = and if the phase shift is random then ,2 .l i lS S=
In Figs. 4,a and 5,a we plotted distributions of Pl along
the transmission line for the bias currents corresponding to
the upper edges of steps (these points in the I–V cha-
racteristic are shown in Fig. 2 by arrows A and B and the
corresponding distributions of voltages are shown in Fig. 3
by bold dashed lines). The distributions of lS for the
same bias currents are plotted in Figs. 4,b and 5,b. It is
seen from Figs. 4, 5 that in the spatial places of the line
where Pl has a maximum, the value of lS decreases, and
the maxima of lS appear in spatial places where values of
Pl are minimal.
The described effect is the main result of the present
paper. Its physical meaning is as follows. The emitted
power of synchronized radiation of junctions in the system
is maximal at the self-induced resonant steps (more pre-
cisely, at upper edges of steps). At the same time, the
standing wave is formed at the resonant steps. Note that in
this standing wave there is the increase of averaged on
time voltages in the center of the line. Junctions cannot
radiate coherently in the places where the standing wave
exists because the averaged voltages over junctions have
different values along the standing wave. Therefore, the
parts of the line where junctions radiate coherently and the
parts where the standing wave appears should be in spatial-
ly different places of the line.
It is seen from Fig. 4,a that due to the standing wave the
value of Pl can increase as much as 1.5 times in some spa-
tial places of the transmission line. This means that the
Joule heat extraction is larger in these places, so the tem-
perature in these places increases. In the model [13] the
value of Pl is proportional to the difference lTΔ between
the temperature of the overheating of the stack and the
temperature of the cryostat: l l lT PΔ = γ , where the coeffi-
cient of proportionality lγ can be roughly estimated from
the Fourier law of the stationary heat flux: / ,l lZ Aγ = Δ λ
where lZΔ is the characteristic distance along the z-di-
rection at which the temperature changes, λ is the thermal
conductivity of the high-temperature superconductor and A
is the cross-section of the junction. The conditions of the
experiment [2] claim that the main heat sinks are the free
surface of the mesa and the volume of the high-tem-
perature superconductor single crystal. The influence of
the surface on the increase of the temperature cannot be
calculated in the ranges of the Fourier law, therefore we
considered only heat flux into the volume of the supercon-
ductor and therefore our estimate of the temperature of
overheating is quite rough. Supposing 71 10lZ −Δ = ⋅ m for
a stack of two junctions placed at the distance 0.1 µm from
the surface and substituting parameters of the model
1210A −= m2 and λ = 8 W/(m·K) [14] into the expression
Fig. 4. Distributions of Pl (a) and Sl (b) along the transmission
line for the bias current 1.14b cI I= corresponding to the point A
in Fig. 2.
6
5
4
0
4
2
0
10 20 30 40 50
10 20 30 40 50
S
l,
1
0
V
–
6
2
P
l,
1
0
W
–
5
I Ib c= 1.14 a
b
Distance, m�
Fig. 5. Distributions of Pl (a) and Sl (b) along the transmission
line for the bias current 1.58b cI I= corresponding to the point B
in Fig. 2.
S
l,
1
0
V
–
6
P
l,
1
0
W
–
5
I Ib c= 1.58
Distance, m�
0 10 20 30 40 50
1.5
1.4
1.3
1.2
6
4
2
a
b
0 10 20 30 40 50
The influence of standing waves on synchronization and self-heating of Josephson junctions in resonant systems
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 4 413
for ,lγ we obtain from Fig. 4,a 30 0.79lT =Δ ≈ K near the
center of the transmission line and 10 0.52lT =Δ ≈ K near
the edge of the line. Note that in our calculations of lTΔ
heat fluxes from junctions are added to each other. In the
system of several hundred junctions in the stack the self-
heating can cause the “hot spot” in the center of the sample
for the first harmonic, see Fig. 4,a or two “hot spots” for the
second harmonic (see Fig. 5,a), whereas the surrounding
junctions will radiate coherently. The detail calculations can
be made with the use of the proposed model. Note that for
the description of the experiments [2–4,6–8] with the given
width of the sample it is necessary to calculate distributions
of voltages and currents also along the y-axis.
It is seen from Fig. 3 that in the vicinities of the second
and higher harmonics of the resonance there is also the de-
crease of the averaged over time voltages. In Fig. 6 we plot-
ted distributions of values of Pl and Sl for one of the bias
currents which corresponds to such a decrease (it is marked
by the arrow C on the I–V characteristic in Fig. 2 and the
corresponding distribution of averaged over time voltages is
shown by the bold dotted line in Fig. 3). The values of Pl
decreases in the places of the line where the averaged on
time voltage decreases (Fig. 6,a), so there should be the de-
crease of the temperature of overheating. Values of Sl are
much smaller than those obtained at the upper edge of this
step, so the coherent radiation in this case is small.
Conclusions
In the present paper we obtained straightforward the
standing wave in the transmission line with Josephson
junctions and analyzed synchronization of junctions under
the influence of the standing wave. In our model we ob-
tained self-induced resonant steps on I–V characteristics of
junctions, explained their appearance and found resonant
voltages. We showed that strong synchronization of junc-
tions with different critical currents appears at these reso-
nant steps due to the current resonance. We showed that
only junctions which are close to ends of the line are syn-
chronized at resonant steps. The averaged on time voltage
of junctions in the center of the line increases and forms
the standing wave. Due to strong extraction of Joule heat
(the self-heating) the temperature of junctions in the center
of the line can exceed the temperature at ends as much as
1.5 times that can lead to the formation of the “hot spot” in
the center of the line, whereas at the same time junctions at
ends of the line radiate coherently. Thus, we showed that
three particularities noted in experiments [2–4,6–8], name-
ly the self-induced resonant steps on I–V characteristics,
strong coherent emission at these steps and the appearance
of “hot spots” have the same origin that is the resonance
behavior of the system at certain bias currents.
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Fig. 6. Distributions of Pl (a) and Sl (b) along the transmission
line for the bias current 1.24b cI I= corresponding to the point C
in Fig. 2.
S
l,
1
0
V
–
6
2
P
l,
1
0
W
–
5
Distance, m�
I Ib c= 1.24
0 10 20 30 40 50
0 10 20 30 40 50
9.0
8.5
8.0
1.5
1.0
0.5
0
a
b
|