Simplified method for generating precise triangular waves
2
Calculations assuming a positive interface charge density Q ss ,iJq = 10 l2 cm~ 2 (Table 1) yield an increase of m, this increase being more pronounced in the case of the lower doped substrate. It should be noted that this increase results in larger values of m in the case of x dm « t S (. This effect is due to the accumulation of an electron layer influenced by the positive charge Q s . s .i- The high electron concentration necessary to compensate for the positive charge Q ssi clamps the Fermi level at the s.s.i. Therefore the question about the nature of the Q ss .i. becomes irrelevant for our calculation. A comparison of the two-dimensional and a one-dimensional calculation of the ld/Vg characteristics is presented in Fig. 2. The one-dimensional calculation is performed assuming the s.o.s structure (Fig. 1) for the solution of Poisson's equation and the current is then calculated following Masuhara et al. s 10-7 10-8 E o 10-9 i t l d .A drai n curre r 10-12 10-T3 V\\ I Fig. 3 Comparison characteristics gate voltage V g ,V K 30 ' 3 of calculated and measured subthreshold 0 3 X 10'° 5 X 10'° experiment m 2-3 30 3-4 With vanishing Q s . s .i. the different results obtained by both methods again illustrate the influence of C ss and C sd (Fig. 1). With Q s . s jJq = 10 12 cm~ 2 deviations of less than 5% are obtained for the low and moderately doped film. The calculations have been compared with experimental results for devices with different technological data. 4 The agreement obtained is shown in Fig. 3. The fast surface state density Nf S is treated as an adjustable parameter. The device parameters are: gate oxide thickness 0-12 /zm, t si = 0-6ixm, (W/L) = 25, initial film doping approximately^ = 2 X 10 1 s cm" 3 , boron ion implantation dose 2 X 10 n cm~ 2 , implanta- tion energy 50 keV. Summary: The influence of specific s.o.s. properties on the subthreshold characteristics has been discussed. In the idealised case of vanishing charge at the s.s.i., s.o.s.-m.o.s.t.s with superior subthreshold behaviour can be obtained. Practically, however, the charge Q ss .i. results in values of m larger than corresponding values of bulk m.o.s.t. Acknowledgments: This work has been partly (D.K.) supported under the technological programme of the Federal Department of Research and Technology of the FRG and partly (W.F.) sponsored by the Fonds zur Forderung der wissenschaftlichen Forschung (project 3358). The authors alone are responsible for the contents. DITMAR KRANZER 9th January 1978 Siemens A G, Werk fur Bauelemente D-8000 Miinchen, Balanstrasse 73, W. Germany WOLFGANG FICHTNER Institut fur Physikalische Elektronik der Technischen Universita't Wien A-1040 Wien, Gusshausstrasse 27-29, Austria References 1 SCHLOTTERER, H., and PREUSS, E.: 'Silicon on sapphire devices: realization and properties'. Invited paper ESSDERC 1977, Brighton, England 2 SWANSON, R. M., and MEINDL, J. D.: IEEE J., 1972, SC-7, p. 146 3 FICHTNER, W.: 'Two-dimensional modelling of s.o.s. transistors', IEEJ. Solid-State & Electron Devices, 1978, 2 (to be published) 4 KRANZER, D., PREUSS, E., SCHLUTER, K., and FICHTNER, W.: IEEE Trans. Electron Devices (to be published) 5 MASUHARA, T., and ETOH, J.: IEEE Trans., 1974, ED-21, p. 799 0013-5194/78/0830-0161 $1.50/0 SIMPLIFIED METHOD FOR GENERATING PRECISE TRIANGULAR WAVES Indexing term: Function generators Kaplan and Tatrash have given a method for the design of triangular-wave generators. The practical implementation of this circuit requires use of analogue multipliers. The aim of this letter is to propose another method which does not need multipliers. In a recent letter, 1 Kaplan and Tatrash have studied the following set of nonlinear differential equations: x = (0//I) sign O) + e[l -//(|x| + \y\)]x y = -0J//i) sign (x) + e[l -/i(M + \y\)]y (1) They have shown that the trajectory in the x,y plane gradually approaches a limit cycle, defined by the equation \x\ + \y\ = \. In steady state, the oscillations x(t) and y(t) are a pair of symmetrical triangular waves, the phases differing by a quarter of period from one another. The fundamental point in the oscillator introduced by Kaplan and Tatrash is the intervention of nonlinear terms (terms in square brackets in eqns. 1), which are responsible for the stabilising mechanism. But it is clear that the sign of these terms is a more important factor than their amplitude, since 162 the amplitude is maintained zero in the steady state. This remark suggests an examination of the following set of differential equations: (2) where Hi holds for sign (x) and JU 2 holds for sign (y), j3 and e are positive constants. The integration of this system is straightforward. Let us put z = lx| + \y | = /ijx + ix 2 y. From the eqns. 2, we obtain and -z) z = 1 — (1 — z 0 ) exp (— let) (3) where z o = juix o + ^2^0 is the value of z at t = 0. Eqn. 3 shows that z tends towards 1, whatever z 0 is, when t becomes infinite. Note that for the set of equations (eqns. 1), the same quantity z is given by the relation - = l - f l - — ) exp(-er) z \ z o j Forming the expression ix 2 x — M1.V and integrating we obtain ELECTRONICS LETTERS 16th March 1978 Vol. 14 No. 6
Transcript of Simplified method for generating precise triangular waves
Calculations assuming a positive interface charge densityQss,iJq = 10l2cm~2 (Table 1) yield an increase of m, thisincrease being more pronounced in the case of the lowerdoped substrate. It should be noted that this increase resultsin larger values of m in the case of xdm « tS(. This effect isdue to the accumulation of an electron layer influenced by thepositive charge Qs.s.i- The high electron concentrationnecessary to compensate for the positive charge Qssi clampsthe Fermi level at the s.s.i. Therefore the question about thenature of the Qss.i. becomes irrelevant for our calculation.
A comparison of the two-dimensional and a one-dimensionalcalculation of the ld/Vg characteristics is presented in Fig. 2.The one-dimensional calculation is performed assuming thes.o.s structure (Fig. 1) for the solution of Poisson's equationand the current is then calculated following Masuhara et al.s
10-7
10-8
Eo 10-9
it l d.
Adr
ain
cur
rer
10-12
10-T3
V\\
I
Fig. 3 Comparisoncharacteristics
ga te voltage Vg,V K30'3
of calculated and measured subthreshold
03 X 10'°5 X 10'°experiment
m2-3303-4
With vanishing Qs.s.i. the different results obtained by bothmethods again illustrate the influence of Css and Csd (Fig. 1).With Qs.sjJq = 1012cm~2 deviations of less than 5% areobtained for the low and moderately doped film.
The calculations have been compared with experimentalresults for devices with different technological data.4 Theagreement obtained is shown in Fig. 3. The fast surface statedensity NfS is treated as an adjustable parameter. The deviceparameters are: gate oxide thickness 0-12 /zm, tsi = 0-6ixm,(W/L) = 25, initial film doping approximately^ = 2 X 101 s
cm"3, boron ion implantation dose 2 X 10ncm~2, implanta-tion energy 50 keV.
Summary: The influence of specific s.o.s. properties on thesubthreshold characteristics has been discussed. In theidealised case of vanishing charge at the s.s.i., s.o.s.-m.o.s.t.swith superior subthreshold behaviour can be obtained.Practically, however, the charge Qss.i. results in values of mlarger than corresponding values of bulk m.o.s.t.
Acknowledgments: This work has been partly (D.K.)supported under the technological programme of the FederalDepartment of Research and Technology of the FRG andpartly (W.F.) sponsored by the Fonds zur Forderung derwissenschaftlichen Forschung (project 3358). The authorsalone are responsible for the contents.
DITMAR KRANZER 9th January 1978
Siemens A G, Werk fur BauelementeD-8000 Miinchen, Balanstrasse 73, W. Germany
WOLFGANG FICHTNER
Institut fur Physikalische Elektronik der Technischen Universita't WienA-1040 Wien, Gusshausstrasse 27-29, Austria
References1 SCHLOTTERER, H., and PREUSS, E.: 'Silicon on sapphire devices:
realization and properties'. Invited paper ESSDERC 1977, Brighton,England
2 SWANSON, R. M., and MEINDL, J. D.: IEEE J., 1972, SC-7,p. 146
3 FICHTNER, W.: 'Two-dimensional modelling of s.o.s. transistors',IEEJ. Solid-State & Electron Devices, 1978, 2 (to be published)
4 KRANZER, D., PREUSS, E., SCHLUTER, K., and FICHTNER, W.:IEEE Trans. Electron Devices (to be published)
5 MASUHARA, T., and ETOH, J.: IEEE Trans., 1974, ED-21, p. 799
0013-5194/78/0830-0161 $1.50/0
SIMPLIFIED METHOD FOR GENERATINGPRECISE TRIANGULAR WAVES
Indexing term: Function generators
Kaplan and Tatrash have given a method for the design oftriangular-wave generators. The practical implementation ofthis circuit requires use of analogue multipliers. The aim ofthis letter is to propose another method which does not needmultipliers.
In a recent letter,1 Kaplan and Tatrash have studied thefollowing set of nonlinear differential equations:
x = (0//I) sign O) + e[l - / / ( |x | + \y\)]x
y = -0J//i) sign (x) + e[l - / i (M + \y\)]y(1)
They have shown that the trajectory in the x,y planegradually approaches a limit cycle, defined by the equation\x\ + \y\ = \. In steady state, the oscillations x(t) and y(t)are a pair of symmetrical triangular waves, the phases differingby a quarter of period from one another.
The fundamental point in the oscillator introduced byKaplan and Tatrash is the intervention of nonlinear terms(terms in square brackets in eqns. 1), which are responsible forthe stabilising mechanism. But it is clear that the sign of theseterms is a more important factor than their amplitude, since
162
the amplitude is maintained zero in the steady state. Thisremark suggests an examination of the following set ofdifferential equations:
(2)
where Hi holds for sign (x) and JU2 holds for sign (y), j3 and eare positive constants.
The integration of this system is straightforward. Let us putz = lx| + \y | = /ijx + ix2y. From the eqns. 2, we obtain
and
-z)
z = 1 — (1 — z0) exp (— let) (3)
where zo = juixo + ^2^0 is the value of z at t = 0. Eqn. 3shows that z tends towards 1, whatever z0 is, when t becomesinfinite. Note that for the set of equations (eqns. 1), the samequantity z is given by the relation
- = l - f l - — ) exp(-er)z \ zoj
Forming the expression ix2x — M1.V and integrating we obtain
ELECTRONICS LETTERS 16th March 1978 Vol. 14 No. 6
a second relation between x and y
(4)
and from eqns. 3 and 4, the explicit solutions for x and y
2x = 2fjL2&t + x0 — Ati/i2>'o + /Ji [1 + (z0— l ) exp (— 2et)Y
2y = —2/ii/3f + yo (5)
In the steady state the variable z is equal to 1 and thetrajectory in the x-y plane is the rhombus \x\ + \y\ = 1 ;x andy represent symmetrical triangular waves against time, with aperiod T = 4/j3. The x wave is delayed by a quarter of periodin respect to t he^ wave. Fig. 1 shows a limit cycle for e = 0-05and Fig. 3a shows the x and y waves for e = 0-5. Thesecurves have been obtained by making use of an analoguecomputer.
y
Fig. 1 Limit cycle for e = 0-05
Although both sets (eqns. 1 and 2) exhibit the sameproperties in the steady state it is interesting to note that thepractical implementation of the system (eqns. 2) does notneed multipliers. The operations introduced by /ii and n2 areeasily realised using zero-crossing detectors and analogueswitches.
In the transient state the properties of the oscillatorsgoverned by eqns. 1 and 2 are different, especially if theparameter e is large. For the Kaplan and Tatrash equations, itmay be shown that the vector OM, where M is the point withco-ordinates x,y, always moves in a clockwise direction.
A simple method for studying the transient process is toobserve that the slope of the trajectory y'/x, calculated fromeqns. 2, remains constant if z is a constant. The sign of theslope gives the orientation of the displacement of the repre-sentative point M with the increase of time (arrows in Fig. 2).
In Fig. 2 are shown different trajectories for e = 1 and |3 = 2and different initial conditions. If the point representative ofthe initial conditions is taken in the unshaded regions, thevector OM turns around the origin in the clockwise direction.But if we consider a trajectory which starts from an initialpoint in a shaded region, say, in the region containing thepoint C, the phenomena are different. Firstly, the point Mfollows the direction of the arrow and would go in the fourthquadrant, but since y is negative above the x axis and positivebelow the axis, the representative point slides along the axisto the point B, on the boundary of the shaded region. Afterthat, the vector OM again moves in the clockwise direction.
x,y
Fig. 3 Results
In order to generate other periodic waves, eqns. 2 may begeneralised as follows:
-z)x =
where 4> is a strictly positive function. For the new set ofequations relation 3 is valid. As an example, let us put thaty is an unsymmetrical triangular wave. The steady statecondition M,+ \y\ = 1 and a graphical construction showsthat the function <J> must be 4> = /3(l — an\), a<\. Fig. 3bshows the result obtained, the times TR and 7> being given bythe relations
Fig. 2 Trajectories for e = 1 and 0 = 2
ELECTRONICS LETTERS 16th March 1978 Vol.14 No. 6
ROBERT GENIN 13th January 1978CLAUDE JEZEQUELJEANNINE GENIN
Faculte des Sciences et Techniques6, avenue Le Gorgeu29283 Brest cedex, France
Reference1 KAPLAN, B. Z., and TATRASH, Y.. 'New method for generating
precise triangular waves and square waves', Electron. Lett., 1977,13, pp. 71-73
0013-5194/78/0846-0162 $1.50/0
A NEW TECHNIQUE FOR DEVICE COMPATIBLEANODIC OXIDATION OF GaAs
Indexing terms: Anodisation, Gallium arsenide. Semiconductortechnology. III- V semiconductors
A simple and effective new technique has been demonstratedwhich improves markedly the chemical and electrical propertiesof heat-treated anodic oxide films on GaAs. This techniqueshould facilitate the use of anodic oxides in innovative GaAsdevices.
Several methods, including thermal oxidation,1 plasma oxida-tion,2 chemical vapour deposition,3 and anodic oxidation4
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