Nonlinear Dynamics of Electronic Systems 2012, Wolfenbüttel
On the affine nonlinearity in circuit theory Emanuel Gluskin The Kinneret College on the Sea of Galilee and Ort Braude College (Carmiel), Israel.
[email protected]; http://www.ee.bgu.ac.il/~gluskin/
Abstract—Affine dependence represents a well definable and most commonly met nonlinearity to which, however, usually, no attention is paid. In fact, this nonlinearity exists only in system theory and is linearity in analytical geometry, which is explained by the fact that circuit's power-activity has no direct analogy in the geometry. We also associate ([1], these Proceedings) the affine nonlinearity (ANN) of some 1ports, obtained via deriving and mutually connecting Thevenin's equivalents, with an analysis of the "structural complexity" (i.e. "entropy") of a system. The Thevenin's contraction is thus used in [1] as some repeatable operation associated with an (algebraic) ideal in a circuit modeling of the growing of some biological structures.
I.
INTRODUCTION: THE AFFINE MAP (NONLINEARITY)
The present work discusses the affine nonlinearity (ANN) in circuits, and also supports the attempt [1] of introducing the idea of structural entropy for some lumped circuits (active 1-ports) having this nonlinearity. The point of this support is that some changes in the structure can be detected by measurement of the nonlinearity. In simplest algebraic concept, ANN appears because of an independent (a priori given) additive term in a map (equation), which is not a system input, directly or indirectly, i.e. cannot be changed. For a and b constants, in the map x y, given as
y = ax + b ,
(1a)
'b' makes the map "affine". Using linear operators Lˆ1 and Lˆ 2 , we also have, because of a known (fixed) function form:
ψ (t ) , that x (t ) → y (t ) given in the Lˆ1 y (t ) = Lˆ2 x (t ) +ψ (t )
(1b)
is an affine map. The nonlinearity is readily seen, since the simultaneous replacements x → kx
and y → ky , with any k ≠ 1 , violates (1b) by the error (1 − k )ψ (t ) . More generally, for (1a), the equality N
N
m =1
m =1
y ( ∑ km xm ) = ∑ km y ( xm )
(2)
is not satisfied for ∀{{k m },{xm }} , but only when N
∑ km = 1
m =1
(3)
(as if {kp} have the sense of some probabilities), because while b ≠ 0 appears in the left-hand side of (2), k1b + k 2b + ... + k N b appears in the right-hand
side, and the same for (1b) with ψ (t ) instead of b. The reason for a system-science test of linearity to fail for (1a,b) is that 00 is not satisfied by this map. Though in analytical geometry, affine dependence, associated there with a straight line, is considered as linearity [even for a = 0 in (1a) this is a straight line, but think about saturated amplifier not transferring signal's information …], and the term "linear affine map" can be used, in system science (and thus also in linear functional analysis or in operator theory) all maps that do not realize 00 are nonlinear. This categorical exclusion in "systems" of the affine map from all linear maps, which we accept, means, hoever, that the affine nonlinearity is an important nonlinearity. The reasons for this importance, physical, irrelevant to analytical geometry, are: the aspect of power flow (the activeness of the circuit), and the simplification of a structure when an ANN 1port is derived. The latter because just when the initially multi-port system is redefined as a 1-port, it becomes affine and nonlinear. A simple argument for the affine nonlinearity also is that since battery, having its voltage independent of its current, can be (Section II) sometimes seen as a voltage hardlimiter, i.e. as a strongly nonlinear element, its inclusion (when deriving Thevenin's equivalent) in a linear circuit makes this circuit nonlinear.
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When seeing in (1a) some voltage-current (v-i) characteristic of a 1-port, we have parameter b as evidence of the activeness of the 1-port. Conversely, the activeness means nonlinearity. II.
THE AFFINE CIRCUIT
Equation
v = ∓ Ri + E .
(4)
is both the direct rewriting of (1a) in the standard electrical engineering notations, and the equation of the well-known Thevenin's 1-port [2,3], shown in Fig. 1(A). When the 1-port has a passive load, (4) is v = E − Ri , because it is natural then to direct the current out of the 1-port. However, the externally connected circuit can be active, with a stronger battery supplying energy to our 1-port (see [2]), and then the direction of the actual energy flow suggests directing the current into the 1-port, which gives v = E + Ri . In the latter case, the battery can be replaced by a (passive) voltage hardlimiter. "Thevenin's 1-port" v
a
Though in Fig. 2(b), one can change ETh as a parameter, by changing {E1, E2, E3} that are the ports in the initial linear configuration (a), if Thevenin's equivalent (b) is realized, then the initial circuit (a) with {E1, E2, E3} already does not exist. However, in fact, it is only important that when we start to see (a) as a 1-port; then also no one of {E1,E2,E3} in (a) by itself can be changed, otherwise it would be a voltage input, i.e. a port. The nonlinearity is thus basically associated with definition of ports. See [2,4] for this axiomatic point. E1
E2
E3
The chosen port
4-port
The same port
ETh
1-port
(a)
(b)
Fig. 2: (a) A schematic linear 4-port (each arrow is a port), with 3 inputs, which is turned by the "Thevenin's contraction" into a nonlinear (ANN) 1-port (b). "ETh" cannot be changed; otherwise, it would be a port.
R (RTh)
v(i) = - R i + E
+ +
i E (ETh)
v
RLi
E A
i 0
b (A)
(B)
Fig. 1(A,B): (A) The common Thevenin's circuit (a non-ideal voltage source) realizing the affine nonlinearity. (B) The resistive (affine) v-i characteristic of the 1-port. The passive load characteristic and the "work-point" 'A' are also shown.
Thus, this very common circuit realizes the affine nonlinearity. Since, according to Thevenin's theorem [3], such a circuit equivalently replaces numerous more complicated circuits, ANN can be associated with a structural simplification, expressed, first of all, in reduction of the number of the ports. It is very essential (see also [4]) that in the Thevenin equivalent only one port is left. Since we have a single port which is not controlled by (nor is the terminals of) ETh, we are not allowed to change ETh during any test of linearity. Thus, ETh is an analogy of b in (1a).
In the opposite process, an affine 1-port can be turned into a linear multi-port having some sources as inputs. In this sense, the affine nonlinearity is "reversible", but not in a unique way, while the addition of inputs in the opposite process means an increase in entropy of the system, -- a point for [1]. By simply showing activity and nonlinearity, ANN can explain the essence of some other nonlinearities. For instance, consider the solar cell characteristic, Fig.3. i(v) i = v/RL
0
v
Fig. 3: Solar cell nonlinear characteristic – an inverted and shifted (because of absorption of solar radiation, coming via a physical, power, but not any electrical, input) diode characteristic [5]. The load's characteristic, as in Fig. 1(B), is also shown. For the power features, it is obviously important that 0 → 0 is not the case (i.e. the "affine kernel" i(0) ≠ 0 ).
207
Such a characteristic could, in principle, be straightened, or positively (oppositely) curved, while the "affine essence", i(0) ≠ 0 , of the connection of the activity with nonlinearity would remain, i.e. the affine component of the cell's nonlinearity is the main component. That is, in i(v) = i(0) + [i(v) − i(0)] the first term is the main one.
WHEN USUAL NONLINEARITY BECOMES ANN: A HARDLIMITER (HL) RESISTIVE ELEMENT IN THE SERIES HL-L-C CIRCUIT III.
The following steady-state dynamic equation for i(t) had been studied:
L
di 1 + Asign[i (t )] + ∫ i (t ) dt = v (t ) = U ξ (t ) . (5) dt C
Here ξ (t ) is a T-periodic, ξ (t + T ) = ξ (t ) , given function or wave-form (in practice, usually sinusoidal), normed in some way. Equation (5) plays an important role [6,7] in a nonlinear theory of fluorescent lamp circuits with L-C ballasts, fed at regular line frequencies, while this equation can also be modeled using two zener-diodes, connected back to back instead of the lamp. For some range of the ratio x = A / U ,
x ≡ A / U ∈ [0, x max ] (or U ∈ [U min , ∞) ) , (6) it is possible [6,7] to have i(t) as a zerocrossing function. Using Fourier series, we then have the rectangular (usually square) time-function (wave) A sign[i(t )] as
F (t ,{tk }) = Aζ ({t − tk ( x )}) ,
(7)
ζ is some known function, and tk = tk (mod T ) are the zerocrossings of i(t) belonging to a period of ξ (t ) . There may be 2
where
(usually), or 4, etc., such zerocrossings. generally, tk = tk ( x ) = tk ( A / U ) is obvious. Remarkably, for the condition
ωo = 2ω
That,
(8)
where ω ≡ 2π /T is the basic frequency of ξ (t ) ,
tk ( x ) ≡ tk (0), ∀k , where {tk (0)} relate to the function io(t) defined by the linear equation
L
dio 1 + ∫ io (t ) dt = ξ (t ) , dt C
(9)
which is (5) for x = 0. (If in (5) A = 0, then any change in U > 0 does not influence {tk } , obviously.) Thus, employing the zerocrossings of i(t) obtained when A = 0 (i.e. without the hardlimier, or when it is short-circuited), or when U → ∞ , and providing by means of ωo = 2ω the "constancy" of {t k } observed when U is changed, we have in (5), rewritten as
L
di 1 + F (t ,{tk }) + ∫ i (t ) dt = U ξ (t ) , (10) dt C
the resistive term (with A > 0 )
F (t ,{tk }) = Aζ ({t − tk ( x )}) ≡ Aζ ({t − tk (0)}) (11) as a completely (independently) known timefunction. Then, after rewriting (5) as
L
di 1 + ∫ i (t ) dt = U ξ (t ) − Aζ ({t − tk (0)}) , (12) dt C
with the right-hand side completely known, it is not easy to see the nonlinearity. However, a physical paradox (important by itself for the concept of ANN) with the seeming linearity is well-felt, because in order to obtain this "linear" situation, only the series capacitor is adjusted in practice, yielding ωo = 2ω . The same strongly nonlinear fluorescent lamp remains in the fixture, and it is strange that because of the change of a linear element, this strongly nonlinear circuit becomes a linear one. In view of the above discussion, one sees, however, that also for tk ( x ) ≡ tk (0) , i.e. for
Asign[i (t )] fixed, equation (5) or (12) is nonlinear. This is the case when the "zerocrossing nonlinearity" becomes ANN, because with this fixed term the equation
and ωo ≡ ( LC ) −1/ 2 , {tk } can be constant for the bounded U or x. This means that
208
L
di 1 + F (t ,{tk (0)}) + ∫ i(t ) dt = U ξ (t ) (13) dt C
realizes the affine map
ξ → i as
ˆ )(t ) + F (t ,{t (0)}) = U ξ (t ) (Ti k
(14)
where Tˆ is the linear operator of the LC circuit. Compare (13) also with (1b). In order to allow Thevenin's equivalent be analogy of an element of a biological structure [1], we introduce now a description tool, -- originally algebraic, but in the present version not absolutely static, permitting [1] some dynamic considerations.
APPENDIX: THE THEVENIN'S
EQUIVALENT AS AN IDEAL OF THE SET OF CIRCUITS REDUCIBLE TO SUCH AN EQUIVALENT. (A PREPARATION FOR [1].)
Mathematical "ideal" [8] has a property which can be named "absorption" and which seems to be suitable for the description of growing structures in biology and for some respective circuit modeling. In the usual terms of abstract sets and their elements, we let Q ⊂ P , and if for q ∈ Q and p ∈ P , qp ∈ Q , then Q is an ideal of P . When thinking about applications that are not purely algebraic or numerical, we have, however, a problem with the multiplication involved. Indeed, since Q ⊂ P , it should be that for the physical dimensions ("[.]") of the elements, [ q] = [ p ] . Thus,
The binary nature of the new qp operation obtains then [1] information meaning; an element q must be present as a template of the required construction in circuits. A similar meaning of q in qp appears in biology side, because of the physical interactions (via molecular electrical fields) with the new p, naturally taking place when Q is created from the added by feeding molecules. Only the completed operation qp results in a new q; this is a dynamic aspect, not given by the usual "static" (arithmetic) ideals. In some cases, associated with commutative ring theory [8], elements of "arithmetic" ideals have the structure qo p + nqo , n ∈ . (16) This is "principle ideal" completely created by one element qo (i.e. q = qo p , ∀q ). Term nqo in (16) has a simple meaning in our physical "ideal". Namely, "n" is the number of the parallel branches of the Thevenin type in a 1-port composing Q. (Consider in [1] the transfer from qo + qo p shown in Fig.2 to nqo shown in Fig.4.) It appears furthermore [1] that for the whole 1-port structure of such Q, we have ANN ~ n, and thus for n >> 1 ANN is proportional to entropy [1] of the structure.
REFERENCES
[ qp] = [ q] ≠ [ q] , an hence qp ∈ Q is problematic.
[1]
One can try to avoid this problem by replacing the product qp by γ qp with a constant factor γ for
[2]
2
which [γ ] = [ q]−1 . However, γ can hardly be universal, i.e. it can be determined only in the concrete context of an engineering problem being considered. Even when taking some zerocrossing functions (a subset of more general time-functions whose product with a zerocrossing function again gives a zerocrossing function) as the ideal, we do not avoid the scaling problem of physical and engineering constructions, if the detection of the zerocrossings (say, using electronic comparators) is needed for an (any) technical application of the zerocrossings. In view of the modeling needs, we replace ([1] for the details) the numerical multiplication involved in the definition of ideal by an operation of a construction, just keeping the product notation qp.
[3] [4]
[5]
[6] [7]
[8]
209
E. Gluskin, "Electrical ANN Circuits (1-ports) and Schrödinger's 'What is life?'", these Proceedings. E. Gluskin, "An Extended Frame for Thevenin (Norton) Theorem", IEEE CASS Newsletter, vol. 5, issue 6 (Dec. 2011). Ch.A. Desoer & E.S. Kuh, "Basic Circuit Theory", McGraw Hill, Tokyo, 1969. E. Gluskin, "An application of physical units (dimensional) analysis to the consideration of nonlinearity in electrical switched circuits", Circuits, Systems, and Signal Processing, Vol. 31, No. 2 (April 2012), pp. 737-752. J.L. Wyatt, Jr, and Leon O. Chua, "Nonlinear Resistive Maximum Power Theorem, with Solar Cell Application", IEEE Transactions on Circuits and Systems, Vol. CAS-30, no. 11, (November 1983) pp. 82-828. E. Gluskin, “On the theory of an integral equation”, Advances in Applied Mathematics, 15(3), 1994 (305-335). E. Gluskin, “A zerocrossing analysis”, Progress in Nonlinear Science, Proc. of the Int’l. Conf. dedicated to the 100th Anniversary of A.A. Andronov, Mathematical Problems of Nonlinear Dynamics, Nizhny Novgorod, Russia, I, 2-6 July 2001 (241-250). Found also at arXiv:0806.4853. A.N. Kolmogorov and S.V. Fomin, "Elements of the Theory of Functions and Functional Analysis", Graylock Press, Rochester, 1957.
