Dynamic Exponent-based Electronics F.C.M. Kuijstermans1, A. van Staveren1 , P. van der Kloet2, F.L. Neerho 2 , C.J.M. Verhoeven1 Delft University of Technology, Faculty of Information Technology and Systems, 1 Electronics Research Laboratory, 2 Laboratory of Circuits and Systems, Mekelweg 4, 2628 CD Delft, the Netherlands tel. +31 (0)15 278 1183 fax. +31 (0)15 278 5922
[email protected]
Abstract | In this paper dynamic exponent-based the analysis and synthesis of nonlinear electronics. electronics is introduced as a good candidate for Some challenges are: signal-dependent poles, signalovercoming some of the limitations of classic linear- dependent noise and noise-dependent transfers. We based electronics. However, the use of the expo- try to cope with these challenges by using a linear nential relation as a primitive for the synthesis of approximation of the nonlinear circuit. electronic circuits implies that we have to deal with time-varying This approximation is obtained by linearizing the beanalysis and synthesis of dynamic nonlinear circuits. The linear time-varying approximation ap- havior of the nonlinear system around its signal depears to be an accurate and convenient description pendent bias trajectory. It allows us to generalize the to this aim. We show how this approximation can eigenvalue and pole concept from linear systems to be used to calculate the time-varying eigenvalues nonlinear systems, and might enable us to extrapoof a nonlinear circuit and how these can be used to late our frequency-space synthesis methods to nondetermine stability by means of Lyapunov and Flo- linear circuits. This generalized eigenvalue and pole quet exponents. We elucidate these concepts by an example of a simple class-B stage, of which we cal- concept is described in more detail in the accompanyculate the time-varying eigenvalues and the Floquet ing paper by van der Kloet et al. [6]. exponents. We show that for nonlinear circuits In this paper we will rst explain that using the exthe time-varying eigenvalues, which determine sta- ponential relation is a good candidate for overcoming bility, no longer coincide with poles appearing in some of the limitations of linear-based electronic cirfrequency-space transfer functions, as was the case cuits, and we will give some thoughts on exponentfor linear systems. based topologies and the the circuit demands which Keywords | Exponential function, nonlinear dy- follow from these. Then we give a brief overview of our namic circuits, stability, Floquet exponents. approach for the stability analysis of dynamic nonlinear circuits, which is based on a linear time-varying approximation of the nonlinear circuit. Finally we I. Introduction elucidate these concepts by an example of a simple In the design of classic linear-based electronic circuits class-B stage, of which the signal-dependent stabilwe are approaching the limits of what is theoretical ity will be described by means of dynamic eigenvalues possible, given a certain IC-process. An evolutionary and Floquet exponents. We show that these dynamic improvement of these linear-based circuits only leads eigenvalues no longer coincide with poles of frequenceto slight improvements. Using linear design methods space transfer functions, as was the case for eigenvalwe can only obtain big advances in performance by ues and poles of linear circuits. migrating to more advanced processes or technologies, II. Exponential-based electronics which is very expensive. Another option is the use of entirely new nonlinear circuit concepts. For linear-based electronics the maximal obtainable We will introduce exponent-based electronics as one dynamic range is order of 140dB. The upper limit of possible new concept. However, the use of the expo- the dynamic range is due to the nonlinearity of the nential relation as a primitive for the synthesis of elec- active devices. We use a linearization of the transtronics inherently implies that we have to deal with fer function of the devices around a bias point, which
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is not valid for large signals. For bipolar transistors this nonlinearity is the exponential relation between base-emitter voltage and collector current. This exponential relation however is measured to be accurate and predictable for more than 200dB of collector current. Therefore a bene cial use of this exponential relation may lead to systems with a dynamic range higher than 140dB. Another important issue in present-day circuit design is power consumption. If we use the intrinsic inputoutput relations of the devices, we don't need to operate them around a bias point. All current consumed by a circuit can be used as signal current, and we obtain the ultimate low-power solution. Therefore we expect exponent-based electronics to be very power ef cient. A global overview of the topology of a circuit ex dx dt
ex
ln (x)
∫x dt
Fig. 1. Global topology of an exponent-based circuit
III. Analysis and synthesis of dynamic nonlinear electronics
The use of the exponential relation as a primitive for the synthesis of electronic circuits inherently implies that we have to deal with analysis and synthesis of dynamic nonlinear circuits. In dealing with this challenge it is favorable to extrapolate our knowledge and expertise in the synthesis of linear electronics to concepts for the synthesis of nonlinear electronics. In literature numerous methods for the analysis of nonlinear circuits can be found, e.g. by means of Volterra series [7] or Harmonic Balance [8]. All these methods have in common that they lead to a description of the circuit which has little resemblance to the linear description we are used to. They are really only suited for (numerical) analysis and are dicult to use in the synthesis of new circuits. An exception is the linear time-varying approximation [13]. Using this approximation we can describe the dynamic behavior of nonlinear circuits by time-varying eigenvalues, and we might be able to extrapolate our frequency-space synthesis methods from linear circuits to nonlinear ones. In this section we will rst describe the basic approach used in the linear time-varying approximation of nonlinear systems. Then we will show how this approximation can be used to describe the dynamic behavior of nonlinear circuits by time-varying eigenvalues, and relate these to Lyapunov and Floquet exponents. This method is described in more detail in the accompanying paper by van der Kloet et al. [6]. A. Linear time-varying approximation The linear time-varying approximation of a nonlinear circuit is obtained by linearizing the behavior of the nonlinear circuit around its signal dependent bias trajectory. We begin with the di erential algebraic equation (DAE) which can be written down for any circuit composed from capacitors, inductors, resistors and other electronic elements. This equation has the branch voltages and currents as variables, in which the currents through inductors and the voltages over capacitors yield dynamic equations. The other variables yield algebraic equations, and they can be eliminated from the DAE, resulting in the well known state-space description
based on the exponential relation is given in gure 1. At the input of the circuit we see one or more exponential functions, which translate the input signals from the linear space to the exponential space. All signal processing and frequency selectivity is done in the exponential space, where we have a high bandwidth and dynamic range. This is in contrast with translinear circuits [3], where the signal processing is done in logarithmic space. At the output of the circuit we will need on or more logarithmic-type functions to translate the signals back to the linear world, for which we probably need some kind of feedback con guration. The limited bandwidth and dynamic range of this con guration is no problem, since at this point we only need the bandwidth and dynamic range of the output signal. The exponential functions at the input of the circuit will generate a number of harmonics of the input signals. For correct operation we will probably need to process signals up to the nth harmonic of the input signals, where n is determined by the required accuracy of the transfer. Therefore the bandwidth redx1 quired inside the exponential space will be larger than = f (x1 ; u1 ; t) (1) dt the bandwidth of the input and output signals. The development of SiGe-technology is making this band- Here x1 represents the vector of all inductor currents and all capacitor voltages, u1 represents external width available in standard processes [9].
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sources and t is the time. This state-space description can be used to calculate the bias trajectory xb of the state variables as function of the external sources ub . The state-space description however is the result of an abstraction from electronic reality by mathematic modeling, and disturbances in the parameters of f and in the sources can give variations in both state-space and modeled input. If we denote these variations as x respectively u, we can model them by substituting x1 = xb + x and u1 = ub + u in (1). Linearizing this equation around xb and ub yields d(xb + x) = dxb + dx dt
dt
@f @x
dt
= f (xb ; ub ; t) + x + @@uf
x=xb ;u=ub
u
(2)
x=xb ;u=ub
of Ax are given by
1 (t) = a11 + l(t)a12 2 (t) = ?l(t)a12 + a22
in which l = l(t) is a solution of the Riccati equation l_ = ?a12 l2 ? (a11 ? a22 )l + a21 (6) We mention again that the matrix elements aij are functions of time. Linear independent solutions of (4) are given by 1 e1 and m e2 (7) l 1 + ml with 4
i (t) =
Z
t
i ( )d (i = 1; 2)
Since xb is the solution of (1) for u1 = ub , we obtain the following variational equation for the variations x and with m = m(t) a solution of in the state and the variations u in the sources m_ = (1 ? 2 )m + a12
x = Ax(xb ; ub; t ) x + Au (xb ; ub ; t ) u (3) The matrix Ax is the Jacobian of f with respect to the state-space vector in its dynamic point of operation and Au the Jacobian of f with respect to the input vector. In studying stability it suces to deal with u = 0. d dt
This is described in the next subsection and in more detail in [6]. In noise problems however, as is demonstrated in [12] we have to deal with the complete equation (3). In that case u represents the noise sources (the signal sources are represented by ub ). B. Stability analysis/Floquet-exponents The stability of a periodic solution xb of the nonlinear system (1) for a periodic source ub can be examined by means of Floquet theory [11]. The stability is determined by the time-varying eigenvalues and eigenvectors of the variational equation (3) with u = 0. A systematic method to obtain the eigenvalues and eigenvectors for second order linear time-varying systems is presented in [6]. We start with the second order state-equation d dt
x1 = a11 a12 x2 a21 a22
x1 , x_ = A (t)x x x2 (4)
in which the elements of Ax and x are real periodic functions of the time t. Then the dynamic eigenvalues
(5)
0
(8) (9)
We apply the de nition of the Floquet-exponent with point of operation xb ; ub of [1] to the modes (7) of (4). If we write these modes as Ai (t)[exp i (t)] then
i = Re[T ?1 i(T )] (i = 1; 2)
(10)
with T the period of the periodic solution. The periodic solution is stable if all Floquet exponents are negative. For linear time-invariant (LTI) systems the eigenvalues i (t) are independent of time, so the Floquetexponents i work out to the real part of the eigenvalues, which for LTI systems are equivalent to the system poles. Thus for LTI systems the stability criterion (10) simpli es to the familiar criterion that all system poles should have negative real parts. In [6] also a relation is given between the dynamic eigenvalues (5) and a time-varying pole-concept as introduced in [5]. The dynamic eigenvalues are shown to be mathematically equivalent to the 'right poles', as the author of [5] calls them. This however does not mean that dynamic eigenvalues are equivalent to poles in the classic linear sense. The dynamic eigenvalues are equivalent to roots of a generalized characteristic polynomial, and stability is determined by them, but their signi cance in frequency plots is not studied until now. In the next section we will illustrate the use of the linear time-varying approximation in a simple example
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of a class-B stage. It will be shown that the dynamics of this stage can be described by a rst order nonlinear di erential equation. In that case the algebra of this section simpli es considerably. From the nonlinear di erential equation we obtain the variational equation
x_ = ax(t)x The dynamic eigenvalue is simply given by (t) = ax(t) and a time-varying mode of (11) equals exp[(t)] in which 4
(t) =
Z
0
t
( )d
(11)
Iout
+ Iin
Vin
-
Fig. 2. Simple class-B stage
We use the Gummel-Poon model for the bipolar tran(12) sistors [4]. Both transistors are used in the forward region (collector-base junction is reverse biased) and we neglect second order e ects such as leakage currents, Early e ect and high-level injection. The transistors (13) are excited by a current source and the output current is sensed, so the ohmic resistances in series with base, emitter and collector have no e ect. Ib (14) b c I /B
c f Ic Stability is determined by the Floquet exponent = Cdiff Cjcap Dbe ? 1 T (T ); the periodic solution xb is stable if < 0. e The dynamic eigenvalue however will be shown to be di erent from the linear pole that we obtain if we Fig. 3. Simpli ed Gummel-Poon model linearize the system in each point of the dynamic bias With these simpli cations we end up with a transistor solution ('frozen time approach'), as also remarked in model in which we can distinguish three main e ects [2]. (see gure 3). The instantaneous transfer from baseemitter voltage to collector current is modeled by the IV. Example: class-B stage base-emitter diode Dbe and a current controlled curIn this section we elucidate the use of the concepts of rent source Ic . The e ect of charge storage in the base the previous section by an example of a simple class-B is modeled by the di usion capacitor Cdi . The efstage. First we give a description of the circuit and fect of charge storage in the base-emitter depletion rededuce the rst-order nonlinear di erential equation gion is modeled by the junction capacitor Cjcap. Dbe , which determines its dynamic behavior. Then we will Cdi and Cjcap all contribute to the base current Ib , solve the di erential equation for a periodic input sig- the collector current Ic is determined by the currentnal to obtain the signal-dependent bias trajectory of controlled current source. the circuit. Finally we will use the linear time-varying If we analyze the instantaneous behavior of the entire approximation to obtain the dynamic eigenvalue of stage , we obtain the following relations between outthe circuit and calculate the Floquet exponent to de- put current Iout , input voltage Vin and instantaneous termine the stability of the periodic solution. input current Iin?D A. Circuit description Vin Vin Iout = Is(e Vt ? e? Vt )2Is sinh( VVin ) (15) Our example circuit is a simple class-B stage. It cont sists of a NPN and PNP bipolar transistor connected Vin I Vin I ? out s in parallel, as depicted in gure 2. The stage is ex(16) Iin?D = Bf (e Vt ? e Vt ) Bf cited by the current source Iin and we will examine the dynamic behavior of the stage in terms of the out- Here Is depicts the transport saturation current and put current Iout . The transistors are not biased at a Bf the current-gain factor of the transistors and Vt quiescent current, so we obtain an exponential-type the thermal voltage. relation between input voltage and output current for The di usion capacitors contribute a current Iin ?Cdi to the input current which equals positive and negative input currents.
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Iin ?Cdi = f dIdout t
(17)
Iin ?Cjcap = Cjcap (Vin ) ddVtin
(18)
IinCjcap = p C2 Vt 2 dIdout t Iout + 4Is
(19)
Here f depicts the forward transit time of the transistors. The Junction capacitors contribute a current Iin ?Cjcap to the input current, which equals Here Cjcap (Vin ) equals the sum of the junction capacitors of both transistors as function of Vin . If we approximate the junction capacitors by a constant capacitor C equal to Cjcap (0) and use (15) we get
The total input current equals the sum of the instantaneous input current and the currents owing into the di usion and junction capacitors. Therefore we can add (16), (17) and (19) to obtain the total input current Iin as function of Iout
Iin
!
out + + p C Vt = IBf dIdout f 2 2 t (20) Iout + 4Is
B. Dynamic bias trajectory The rst step in analyzing the nonlinear di erential equation (20) consist of nding a time-varying bias trajectory for a periodic input Iin = A sin(!t). We rewrite (20) into the standard state-space description dIout dt
= f (Iout ; Iin ; t) = ? Iout + A sin(!t) = pBfC Vt + f I 2 +4I 2 out
s
(21)
original di erential equation (21) and proved to be very accurate, the deviation of the derivative of Iout was smaller than one percent. In the following paragraphs an approximated bias trajectory is calculated for high frequencies and large and small amplitudes of the input signal. In the next section these are substituted in the variational equation obtained from the original di erential equation (21) to compute the dynamic eigenvalues and Floquet exponents for these two regions of operation. B.1 Di usion capacitors If we ignore the in uence of the junction capacitors and the instantaneous term in (21), that is we substitute C = 0, we obtain a linear di erential equation for the output current Iout : !t) dIout = ? BfIout + A sin( (22) dt f f This equation can be solved using standard linear analysis, and we obtain a steady state dynamic bias trajectory 2 !t) + Ioutb (t) = ? A f ! Bf2 cos( 2 2 1 + ! Bf f + A Bf 2sin(2!t)2 (23) 1 + ! Bf f In this equation we recognize the amplitude and phase of the familiar frequency-space transfer function for (22). B.2 Junction capacitors If we neglect the in uence of the di usion capacitors ( f = 0) and instantaneous behavior (Iout =Bf = 0), we obtain the nonlinear di erential equation p A sin( !t ) Iout 2 + 4Is 2 dIout = (24) dt
CV
t To keep the nonlinear calculus manageable we consider three regions of operation of the output stage. We can use the method of separation of variables to For low frequencies the stage behaves instantaneously solve this di erential equation: as given by (15) and (16), we needn't analyze this dIout !t) dt , p = A sin( case any further. For high frequencies and large am2 2 C Vt Iout + 4Is plitudes of the input current the di usion capacitors I A sin(!t) dt , dominate and we can ignore the in uence of the juncd arcsinh out = 2Is C Vt tion capacitors in the determination of the bias trajec A cos( !t ) +C , I tory. For high frequencies and small amplitudes the = ? arcsinh 2out 1 Is C Vt ! junction capacitors dominate and we can ignore the A cos( !t ) in uence of the di usion capacitors. These approxiIout (t) = 2Is sinh ? C V ! + C1 mations were checked by a numerical analysis of the t
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For high frequencies the integration constant C1 will equal zero (Vin and Iout contain no DC-terms), so we obtain the dynamic bias trajectory A cos( !t ) (25) I (t) = 2I sinh ? outb
s
C Vt !
C. Linear time-varying approximation As explained in the last paragraph of section III, we can examine the stability of a dynamic bias trajectory of a rst-order nonlinear system by determining the eigenvalue of the variational equation (11). For our stage we obtain the homogeneous variational equation from (21): dIout dt
with
= aIout (t) Iout
?1
+ t Bf pIoutC 2V+4 + f Is 2 b b ? Iout Bf + A sin(!t) C Vt Ioutb
-3.74e+008 -3.76e+008 -3.78e+008 -3.8e+008 -3.82e+008 -3.84e+008 -3.86e+008 -3.88e+008 0
out ; Iin ; t) aIout (t) = @f (I@I out Iout=Ioutb ;Iin =A sin(!t)
=
(26)
-3.72e+008
1
2
3 phi
4
5
6
Fig. 4. Dynamic eigenvalue as function of phase of the input signal (A = 1mA , ! = 1 106rad =s)
1=(Bf f ), which is also the pole following from linear analysis. This is to be expected, since the di erential equation (21) can be approximated quite accurately by the linear di erential equation (22) for large in+ 2 put currents (the di usion capacitors dominate). The ? 2 + 4Is 2 32 t pIoutC 2V+4 I + out f b 2 time varying eigenvalue deviates from the linear pole Is b for the zero-crossings of the input signal ( = 0, and (27) 2) only, since for small currents the junction capacitors can no longer be neglected. The dynamic eigenvalue simply equals aIout (t) and the If we use (28) for A = 1mA and ! = 1 106 rad=s, we Floquet-exponent equals: obtain a Floquet exponent of ?3:878 108 , almost Z T Z T to the linear pole 1=Bf f of ?3:885 108 . This = T1 ( )d = T1 aIout ( )d (28) isequal to be expected, since the Floquet exponents are 0 0 identical to system poles for linear systems. in which T = 2=! is the period of the input signal. C.2 Floquet exponent for junction capacitors C.1 Floquet exponent for di usion capacitors We use the approximated bias trajectory (25) in (27) We use the approximated bias trajectory (23) in (27) to compute the dynamic eigenvalue for small input to compute the dynamic eigenvalue for large input currents . In this region of operation the di usion TABLE I capacitors dominate. We have used an input signal Transistor parameters of the DIMES-01 process with an amplitude A of 1mA and a radial frequency ! of 1 106 rad=s. The transistor parameters used are parameter value summarized in table I, these are the parameters of our Is 18aA in-house DIMES-01 process [10]. In gure 4 we show Bf 117 the eigenvalue as function of the phase = !t for one f 22ps period of the input signal. We see that during most C (0) 46 fF jcap of the period the eigenvalue is a constant equal to
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8e+007 6e+007 4e+007 2e+007 00
1
2
3 phi
4
5
6
-2e+007
For linear systems the approximation simpli es to the concepts we know from classic linear analysis. For nonlinear systems however the dynamic eigenvalues no longer coincide with poles of frequency-space transfer function and the physical interpretation of dynamic eigenvalues remains quite unsatisfactory. A better interpretation would help in getting higher level models to be able to do synthesis. This will be the topic of future research. We will also investigate the use of the linear time-varying approximation for nonlinear noise analysis. Acknowledgments
-4e+007 -6e+007
We would like to thank STW for sponsoring this research.
-8e+007
References
Fig. 5. Dynamic eigenvalue as function of phase of the [1] L.Y.Adrianova, Introduction to linear systems of di erential input signal (A = 0:1A , ! = 1 107rad =s) equations , American Math. Soc., Providence, 1995.
currents . For an input amplitude A of :1A and a radial frequency ! of 1 107 rad=s we obtain a dynamic eigenvalue as shown in gure 5. From the gure we see that in this case the dynamic eigenvalue is strongly varying with time. It even has a positive value for almost half of the period. In this case the dynamic eigenvalue is not easily related to a pole following from linear analysis: using a frozen time approach we would get a pole which is always negative (the di erential input resistance and input capacitance of the stage are always positive). We see that for strongly nonlinear systems the dynamic eigenvalue is very di erent from the classic linear pole. If we use (28) for A = :1 and ! = 1 107 rad=s, we obtain a Floquet exponent of ?0:439. Since is negative we conclude that the approximated dynamic bias trajectory is stable. V. Conclusions and future work
We have introduced dynamic exponent-based electronics as good candidate for overcoming some of the limitations of classic linear-based electronic circuits. Its use however implies that we have to deal with analysis and synthesis of dynamic nonlinear circuits. The linear time-varying approximation appears to be a good instrument to this aim. It allows us to calculate time-varying eigenvalues of a nonlinear system, which can be related to time-varying poles. Stability can be determined by Lyapunov and Floquet exponents.
[2] R. Bellman, J. Bentsman and S.M. Meerkov, Stability of fast periodic systems , IEEE Trans. AC, Vol. 30(3), pp. 289-291, 1985. [3] D.R. Frey, Exponential State Space Filters: A Generic Current Mode Design Strategy, IEEE Trans. on Circuits and Systems I, Vol. 43(1), pp. 34-42, 1996. [4] I. Getreu, Modeling the Bipolar Transistor, Elsevier, Amsterdam, 1978. [5] E.W. Kamen, The poles and zeros of a linear time-varying system, Lin. Algebra and its Appl., Vol. 98, pp. 263-289, 1988. [6] P. van der Kloet, F.L. Neerho , F.C.M. Kuijstermans, A. van Staveren and C.J.M. Verhoeven, Generalizations for the eigenvalue and pole concept with respect to linear timevarying systems, Proceedings of ProRISC/IEEE Workshop CSSP, Mierlo, The Netherlands, November 27 & 28 1997. [7] T. Larsen, Determination of Volterra transfer functions of non-linear multi-port networks, Int. J. Circuit Theory and Applications, Vol. 21(2), pp. 107-131, 1993. [8] M.S. Nakla and J .Vlach, A piecewise harmonic balance technique for the determination of the periodic response of nonlinear systems, IEEE Trans. on Circuits and Systems, Vol. 23(2), pp. 58-91, 1976. [9] L.K. Nanver, E.J.G. Goudena, C. Visser, H.W. van Zeijl and J.W. Slotboom, High-frequency SiGe HBT's with implanted emitters, Proc. ESSDERC'96, pp. 468-472, 1996. [10] L.K. Nanver, E.J.G. Goudena and H.W. van Zeijl, DIMES-01, a baseline BIFET process for smart sensors experimentation, Sensors and Actuators Physical, Vol. 36(2), pp. 139-147, 1993. [11] D.H. Sattinger, Topics in Stability and Bifurcation Theory, Springer Verlag, Berlin, 1973. [12] L. Weiss and W. Matthis, Nonlinear non equilibrium thermodynamics of nonlinear electrical networks, Proc. ECCTD'97, pp. 185-190, 1997 [13] M.Y. Whu, On the stability of linear time-varying systems, Int. J. Systems Sci., Vol 15(2), pp.137-150, 1984.
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