Transistors are the cornerstone of modern electronics. Yet, their relatively complex characteristics, allied with often observed great parameter variation, remain a challenge for discrete and integrated electronics. Much of transistor research and applications have relied on transistor models, as well as respective equivalent circuits, to be employed for circuit analysis and simulations. Here, a simple and yet accurate transistor model is derived, based on the Early effect, which involves only the voltage Va and a companion parameter s. Equations are obtained for currents and voltages in a common-emitter circuit, allowing the derivation of respective gain functions. These functions are found to exhibit interesting mathematical structure, with gain values varying almost linearly with the base current, allowing the gains to be well characterized in terms of their average and variation values. These results are applied to deriving a prototypic Early space summarizing the characteristics of transistors, enriched with recently experimentally obtained prototypes of NPN and PNP silicon BJTs and alloy germanium transistors. Though a trade-off between gain and linearity is revealed, a band characterized by small values of Va stands out when aiming at both high gain and low distortion. The Early equivalent model was used also for studying the stability of circuits under voltage supply oscillations, as well as parallel combinations of transistors. In the former case, it was verified that more traditional approaches assuming constant current gain can yield stability factors that deviate substantially from those derived for the more accurate Early approach. The equivalent circuit obtained for parallel combinations of transistors was shown also to closely follow the Early formulation.

"Where there is matter, there is geometry."

J. Kepler

I. INTRODUCTION

Developed in 1947 [START_REF] Riordan | Crystal Fire -The Birth of the Information Age[END_REF], transistors quickly became the cornerstone of modern electronics. Because of their central importance, be it in integrated or discrete circuits, these devices have become the subject of intense and continued research from both theoretical (e.g. semiconductor physics) and practical (e.g. electronic engineering) points of view. There are two main applications of transistors: digital and analog (e.g. [START_REF] Horowitz | The Art of Electronics[END_REF][START_REF] Jaeger | Microelectronic Circuit Design[END_REF][START_REF] Boylestad | Electronic Devices and Circuit Theory[END_REF][START_REF] Sedra | Microelectronic circuits[END_REF][START_REF] Streetman | Solid State Electronic Device[END_REF]). In the former case, transistors are used as switches transitioning between two logical levels, which underlies the area of digital electronics. In the latter case, transistors are typically used as linear amplifiers, characterizing the area known as analog (or linear ) electronics (e.g. [START_REF] Carusone | Analog Integrated Circuit Design[END_REF][START_REF] Casier | Analog Circuit Design[END_REF][START_REF] Hyat | Engineering Circuit Analysis[END_REF][START_REF] Pettit | Electronic Amplifier Circuits: Theory and Design[END_REF][START_REF] Sedra | Microelectronic circuits[END_REF][START_REF] Williams | Analog circuit design: art, science, and personalities[END_REF][START_REF] Williams | The art and science of Analog Circuit Design[END_REF]). Because the many signals in nature exhibit continuous values, being therefore called analog, linear electronics continues to be essential for myriad electronic applications. Here, the challenge is to achieve efficient circuits capable of amplifying * ldfcosta@gmail.com, luciano@if.sc.usp.br signals with the lowest level of distortion. This constitutes a challenging endeavor because transistors are inherently non-linear devices. So, at least three alternative approaches have been typically considered in order to try obtaining linear amplifiers by using transistors: (i) to improve the of the devices; (ii) to find the most linear operation region; and (iii) to develop circuits capable of enhancing linearity. With the introduction of negative feedback in electronics by H. S. Black in 1927 [START_REF] Black | Stabilized feedback amplifiers[END_REF], alternative (iii) became the most standard and commonly applied approach for achieving practical amplifying circuits with improved linearity.

All the above mentioned three approaches to enhance linearity when using transistors share a common, critical aspect: they all rely critically on the availability of effective transistor models, representations and equivalent circuits. As a consequence, several modeling and equivalent circuits have been proposed and used in linear electronics (e.g. [START_REF] Carusone | Analog Integrated Circuit Design[END_REF][START_REF] Casier | Analog Circuit Design[END_REF][START_REF] Hyat | Engineering Circuit Analysis[END_REF][START_REF] Jaeger | Microelectronic Circuit Design[END_REF][START_REF] Pettit | Electronic Amplifier Circuits: Theory and Design[END_REF][START_REF] Streetman | Solid State Electronic Device[END_REF][START_REF] Williams | Analog circuit design: art, science, and personalities[END_REF][START_REF] Williams | The art and science of Analog Circuit Design[END_REF]). An alternative transistor modeling approach was reported recently [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] that relies on the Early effect, discovered by J. M. Early in 1952 [START_REF] Early | Effects of space-charge layer widening in junction transistors[END_REF][START_REF] Heinz | Solid State Physical Electronics[END_REF][START_REF] Streetman | Solid State Electronic Device[END_REF]. This effect is characterized by the variation of the charge carrier portion of the base with the base-collector voltage. An immediate consequence of this phenomenon in junction transistors is that the characteristic isolines of transistors (indexed by the base current) will not cross the collector voltage axis at V C = 0, but at a further away negative value V C = V a , which corresponds to the Early voltage. Though these facts have been known for a long time, they have not often considered for modeling transistors, except for a few works relating the Early voltage with the transistor output resistance R o , especially in the case of FET devices (e.g. [START_REF][END_REF]).

The methodology described in [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] uses the Early effect to derive a complete transistor model characterized by two parameters: the Early voltage V a and a proportionality parameter s relating the characteristic isoline angles θ with the modulating base current I B . Perhaps the key element in this alternative type of transistor modeling consists in the fact that it was experimentally found for hundreds of small signal BJTs [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Characterizing germanium junction transistors[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] that θ = sI B , i.e. the isoline angles are directly proportional to the base current. This linear relationship not only simplifies the Early modeling approach, but also leads to the specially important property that both the parameters involved in the Early model result completely independent of the transistor/circuit operation in the space (V C , I C ). This contrasts sharply with the fact that in the more traditionally applied models, the two involved parameters current gain β and output resistance R o vary with the normal operation of the circuit. This has implied that, given a specific transistor, it is impossible to characterize it by a single parameter setting (β, R o ), being necessary to consider maximum or average values of these parameters along the operation space, which is not accurate because the parameters of the transistor characteristic isolines tend to vary extensively even in normal operation. It should be recalled that several other transistor models have been employed (e.g. [START_REF] Gray | Electronic Principles: Physics, Models and Circuits[END_REF][START_REF] Boylestad | Electronic Devices and Circuit Theory[END_REF][START_REF] Sedra | Microelectronic circuits[END_REF][START_REF] Streetman | Solid State Electronic Device[END_REF]), some of which aimed at providing more complete representations by incorporating a larger number of components.

In spite of its recent introduction, the Early modeling approach [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] has already been successfully applied to several issues in electronics, including the characterization of real-world NPN and PNP silicon [START_REF] Da | An Early model of transistors and circuits[END_REF] and germanium [START_REF] Da | Characterizing germanium junction transistors[END_REF] junction devices, allowing the derivation of a prototypical atlas of device characteristics in which NPN exhibit lower parameter variations than the PNP counterparts. In addition, it was found that both the NPN and PNP transistors tended to have the same average β while differing markedly in V a and s values, parameters that had not been usually considered. It has also been verified that the total harmonic distortion of transistors depends only on the parameter s, being independent of the Early voltage V a . Actually, the intrinsic adherence of the Early model to the geometrical structure of transistors, allied with the simplicity and accuracy of the proposed basic mathematical representation, paves the way to several advances in understanding, designing, and applying transistors in integrated (e.g. [START_REF] Carusone | Analog Integrated Circuit Design[END_REF][START_REF] Gray | Analysis and design of analog integrated circuits[END_REF]) and discrete circuits. The geometry of device operation constitutes a particularly promising perspective because not only it promotes better and more intuitive understanding of the potential behavior of a device (often in a very complete way), but it also provides the scaffolding for deriving more effective and intuitive mathematical representations, models, and respective equivalent circuits. Indeed, graphical approaches have been extensively used since the beginnings of electronics, and many of the textbooks from 50's to 70's (e.g. [START_REF] Alley | Electronic Engineering[END_REF][START_REF] Fontaine | Diodes and Transistors[END_REF][START_REF] Gronner | Transistor Circuit Analysis[END_REF][START_REF] Pettit | Electronic Amplifier Circuits: Theory and Design[END_REF][START_REF] Shea | Transistor Audio Amplifiers[END_REF][START_REF] Tinnell | Electronic Amplifiers[END_REF][START_REF] Zimmermann | Electronic Circuit Theory: Devices, Models, and Circuits[END_REF]) are characterized by extensive and systematic use of graphical presentations and developments. Perhaps the progressive shift to numeric-computational modeling of device operation along the 70's and 80's has shifted a little bit this paradigm, but graphical approaches remain, nevertheless, an interesting resource to be considered, exhibiting potential for enhancing the available simulation resources. So, its is not that graphic representations of device operation should be replaced by numericcomputational simulation approaches, but that it should be incorporated as a valuable and useful concept that could contribute to advances in simulation approaches.

By having access to effective transistor models, devices operating in the so-called "linear" regime and having forward-biased base-emitter junction and inverselybiased base-collector junctions can be conveniently substituted by their respective equivalent circuits, providing a better understanding of the circuit characteristics and promoting possibilities for further enhancements. The derivation of transistor equivalent circuits can benefit greatly from the Early model [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] intrinsic simple geometry fully compatible with transistor operation, characterized by radiating isolines. The main objective of the current work is to develop such an equivalent circuit for the Early model of transistors, and illustrate its theoretical and practical application potentials with respect to the characterization of important electronic properties of transistor-based circuits, including current, voltage and power gains as well as distortion. In addition, the proposed equivalent model is also applied to the analyses stability of circuits in presence of power supply oscillations and parallel combinations of BJTs.

This article starts by revising a more traditional transistor modeling approaches based on the current gain β and output resistance R o , and proceeds by presenting the Early modeling approach and deriving respective equivalent circuits, from which current and voltage equations describing the behavior of a simplified common-emitter circuit configuration are derived. These equations are then employed to quantify respective current and voltage gains, exhibiting an interesting mathematical structure that implies that, at least for the considered circuit and parameter configurations, the gains vary in very nearly to linear fashion with the modulating current base I B . This fact allows the gains to be effectively quantified in terms of their respective average and variation values. These results allowed the derivation of a prototypic Early space characterizing a trade-off between gain and linearity, as well as incorporating prototypic groups of NPN and PNP silicon and germanium devices. The potential of the Early equivalent circuit is further illustrated with respect to applications to the study of stability induced by voltage supply oscillations as well as for the characterization of parallel combinatios of transistors.

II. REAL-WORLD TRANSISTOR CHARACTERISTIC SURFACES

Real-world transistors are characterized by several specific features regarding their electronic behavior, and these features tend to vary from one device to another. Figure 1 illustrates the characteristic surface, represented in terms of its constituting isolines, each one indexed by a respective base current I B , of a real-world PNP small signal transistor. This characteristic surface was obtained experimentally by using a microprocessed acquisition system [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF]. The operation space corresponds to the Cartesian coordinate system V C × I C , where V C is the voltage current measured with reference to the ground and I C is the collector current (positive sign corresponding to current entering the device). The characteristic surface in this figure was obtained for the operation region defined by V C,min = 0V , V C,max = 10V , I C,min = 0mA and V C,max = 15mA. The transistor was set in a simplified common emitter configuration [START_REF] Alley | Electronic Engineering[END_REF][START_REF] Gray | Electronic Principles: Physics, Models and Circuits[END_REF][START_REF] Horowitz | The Art of Electronics[END_REF][START_REF] Ryder | Electronic Systems and Circuits[END_REF], with inversely biased base-collector junction and forward-biased base-emitter junction, with a resistive load R L = 670Ω attached between the collector and the external voltage source V CC = 12V . For simplicity's sake, all current and voltage values of PNP transistors are represented by inverse values, so as to keep the operation space in the first quadrant, therefore achieving a unified approach with respect to NPN devices.

Interestingly, characteristic surfaces such as that shown in Figure 1 incorporate all information necessary to characterize the operation of the respective device, except in cases where it has reactive components. The most important region of the characteristic surface is that where the I B indexed isolines tend to be straight and reasonably spaced one another. This region is often called the "linear" operation region. The narrow vertical region next to the I C axis corresponds to the transistor saturation and are normally avoided during linear transistor operation. Similarly, the narrow horizontal strip next to the V C axis, often called the transistor cut-off region, is FIG. 1: The characteristic surface, represented by base current-indexed isolines, of a real-world PNP small signal transistor.

The cut-off and saturation regions are excluded from the "linear" region of operation. Observe the progressively increasing slopes of the isolines with I B .

characterized by coalescence of isolines and is similarly not taken into account for linear transistor operation.

The isolines in the linear region are mostly straight lines with slopes that increase with I B . Recall that the slopes of these isolines correspond to 1/R o , where R o is the output resistance of the collector. Observe also that the isolines bend at the cut-off region, but are very nearly straight otherwise. NPN transistors have similar characteristic surfaces, except for the fact that the isolines tend to have smaller slope [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF]. Because the characteristic surfaces of real-world transistors are somewhat complicated, simplified versions are typically adopted for obtaining mathematical characterization of the transistor behavior, as well as respective equivalent circuits. A traditional approach to modeling the characteristic surface, as well as the respectively derived equivalent circuit, is revised in the next section.

III. A TRADITIONAL APPROACH TO TRANSISTOR MODELING

Figure 2 shows a simple NPN transistor characteristic surface in the (V C , I C ) circuit operation space, in which all the equispaced I B -indexed isolines are assumed to have the same slope 1/R o , therefore being parallel one another. This configuration is normally used assuming that the base-collector junction is reversely biased while the base-emitter junction is forward biased, hence the input port can be approximated by a simplified diode model as in this figure, where R i stands for its internal resistance (observe that the ideal diode in this figure can be omitted under the above hypotheses).

Though it is know that, for most real-world transistors, the slopes actually increase with I B , this fact is not considered in this model for simplicity's sake. As a matter of fact, an even simpler transistor characteristic is sometimes used in which all equispace isolines have null inclination, implying Ro = ∞. Despite these simplifications, such approaches remain interesting, as they provide insights about how more idealized transistors would behave, and can also be used in introductory didactic syllabuses.

FIG. 2: The geometrical set-up defined by characteristic surface of a simplified NPN transistor in which all equispace isolines defined by I B have the same slope 1/Ro being, therefore, parallel one another. The load line defined by having a load resistance L in series with a voltage source V CC are also included. The state value (V C , I V ) of the transistor is restricted to excursioning along the load line.

In this section, we review the analysis of the electronic properties of the ideal transistor represented by the characteristic surface in Figure 2. First, we obtain the current and voltage equations for the input and output ports, and then use these equations in order to derive the current, voltage and power gain, and discuss linearity.

The simplified characteristic surface in Figure 2, together with the assumed simplified diode model, yields the equivalent circuit shown in Figure 3. The ideal diode is included for the generality's sake, but can be omitted under the adopted transistor base-emitter forward bias. Observe that the I B -indexed isolines are obtained by incorporating the constant current source I s in parallel with an internal output resistance R o (Norton equivalent).

FIG. 3: The simplified transistor model respective to the NPN transistor characteristic curve in Figure 2 and the ideal diode model adopted for representing the reversely biased base-collector junction.

We consider a simplified common emitter circuit configuration as in Figure 4. Here, a load resistance R L is attached between the external voltage source V CC and the transistor output (collector). It can be easily verified that this pair of components external to the transistor define a straight line in the (V C , I C ) space as in Figure 2. This line is commonly called load line. The circuit operation is now restricted to take values (V C , I C ) along this load line. While the resistance R L corresponds to the actual load of the transistor, the voltage source V CC is required because the transistor, being a passive device, can not deliver power into the load, but only control an external voltage or current source according to the input current I B . That is why transistors operating as amplifiers are sometimes called "valves". First, we approach the base-emitter junction, which we assumed to operate as a simplified diode. For this input port, we have:

V B = V r + R i I B (1)
Now, we proceed to the output port. By defining the transistor current gain β = I R /I s = constant and applying Kirchhoff's and Ohm's laws (e.g. [START_REF] Hyat | Engineering Circuit Analysis[END_REF]), we have that:

I C (I B ) = V CC + βR o I B R o + R L (2) V L (I B ) = R L I C = R L V CC + βR o I B R o + R L (3)
P L (I B ) = R L V CC + βR o I B R o + R L 2 (4)
When R 0 → ∞, the transistor output stage becomes a perfect current source and we have:

I C (I B ) = βI B (5) V L (I B ) = R L I C = R L βI B (6)
P L (I B ) = R L (βI B ) 2 (7)
The AC current, voltage and power gains at the resistance load are commonly defined, respectively, as:

a i (I B ) = I C (I B ) I B Q (8) a v (I B ) = V L (I B ) V i (I B ) Q (9) a p (I B ) = P L (I B ) P i (I B ) Q = a i (I B )a v (I B ) (10)
Strictly speaking, the AC gains are defined respectively to an operating point Q = (V C,Q , I C,Q ). This effectively implies R i → 0. In addition, because of the constant slope of the I B -indexed isolines in the simplified transistor model under consideration, the choice of this point becomes immaterial and we have:

a i = βR o (R o + R L ) (11) a v (F ) = βR L R o R i (R o + R L ) ( 12
)
a p (I B ) = R L (βR o ) 2 R i (R o + R L ) 2 (13)
So, we have that all the considered gains do not depend on I B , V C or I C , as would be expected as a consequence of adopting equispaced isolines. For the further simplified case when R o → ∞, we have:

a i = β (14) a v (I B ) = β R L R i (15)
a p (I B ) = β 2 R L R i (16)
IV. THE EARLY MODEL APPROACH Now, we revisit the circuit configuration discussed in the previous section, but adopting a simple Early model [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] instead of the just considered current-source model. Figure 5 depicts the characteristic surface underlying the adopted Early model, defined in terms of a set of I B -indexed isolines radiating from the same point along the V C = V a axis. Observe the varying inclination and spacing of the I B -indexed characteristic isolines along the load line, which account for a more accurate and realistic representation of the transistor intrinsic electronic properties, as can be immediately inferred by comparing the diagram in Figure 5 with the experimental characteristic isolines of the real-world transistor in Figure 1. Though the "fan"-like radiation may initially appear as a complication, it will turn out that this is not really so. In addition, this geometrical organization provides a much more accurate representation of the behav-ior or real-world transistors than the previous simplified approach, as it allows the increasing slopes of the nonequally spaced isolines to be taken into account [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF]. Observe that these slopes correspond to the inverse of the output resistance R o , and also that tan(θ) = 1/R o . In addition, the two parameters involved in the Early modeling, namely the Early voltage V a and the proportionality parameter s, remain constant throughout the operation space, i.e. these two parameters do not depend on either V C or I C (which is not verified for more traditional approaches based on β and R o ). These key properties of the Early approach, that constitute the main motivation of the present work, are summarized in the following box:

In the Early model, the characteristic surface of realworld transistors, allowing for varying slope of nonequally spaced characteristic isolines indexed by the base current I B , is represented in terms of only two constant parameters (V a and s) that provide a very comprehensive characterization of the device operation.

The possible reason why such an Early model approach [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] was been adopted earlier is because the relationship between θ and I B did not seem to be known. However, at least for several types of small signal transistors, it has been experimentally verified recently [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] that β = sI B , allowing a substantially simple and effective transistor model to be developed and applied [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF]. Therefore, only two elements underly the Early model: the hypotheses that θ = sI B , and the fact that the all the I B -indexed isolines intersect at a same value V a along the V C axis. The Early model is immediately applicable equally to NPN and PNP models, and in this work all PNP currents and voltages are shown with negative values, for simplicity's sake.

The equivalent circuit of the graphical construction shown in Figure 5 can now be easily derived as illustrated in Figure 6. The input port is modeled in the same way as in the previous section. However, in the Early approach, the output port of a transistor is now represented as voltage source with fixed value V a corresponding to the Early voltage, in series with a variable output resistance R o (I B ) = 1/tan(sI B ), where s is the proportionality constant constituting the second parameter in the Early model [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF]. Rarely, if ever, equivalent circuits employ variable resistances (or conductances), but there is no intrinsic shortcoming in this type of approach. On the contrary, the I B -controlled resistance of the isolines provides a natural represented by the geometrical organization of the transistor operation. Observe also that though a respective Norton version of the equivalent cir-cuit proposed for the Early modeling would be possible, it would imply both output resistance and current source to vary with I B , while the voltage source in the adopted Thevenin configuration remains constant with I B as well as with I C and V C . The presented Early equivalent circuit can be understood as having similar complexity to the varying current source model in Figure 3. However, it is typically much more accurate than that model as allowed by the variable resistance representation of the progressively more inclined, non equally space isolines found in most real-world transistors. It could also be argued that the voltage-based (Thevenin) approach is probably more intuitive than the more traditional current-based (Norton) counterpart.

Figure 7 shows the same circuit as in the previous section, but with the transistor represented in terms of its Early model. Since the base-emitter is forward biased and the base-collector is inversely biased, the two diodes can be conveniently omitted.

Because the input port is modeled in the same ways as in the previous section, Equation 1 is verified also for this circuit configuration. The output current, as well as V L and P L at the output stage can be immediately determined by applying Kirchhoff and Ohm's laws as:

I C (I B ) = V CC -V a R L + Ro(I B ) (17) V L (I B ) = R L V CC -V a R L + Ro(I B ) (18)
P L (I B ) = R L V CC -V a R L + Ro(I B ) 2 (19)
where R o (I B ) = 1/tan(sI B ). These expressions are hardly more complicated than those obtained for the more traditional, and less precise, modeling approached addressed in the previous section.

The current, voltage and power gain are now given as:

A i (I B ) = (V CC -V a )tan(sI B ) I B (R L tan(sI B ) + 1) (20)
A v (I B ) = R L (V CC -V a )tan(sI B ) (R L tan(sI B ) + 1)(V r + R i I B ) (21)
Though this last pair of equations is slightly more sophisticated than those obtained for the more traditional modeling approach discussed in the previous section, this is so because now they account for the important fact that all gains are indeed functions of I B . This is particularly important because this dependence directly implies that the current, voltage and power amplifications will necessarily be non-linear in a real-world circuit employing a real-world transistor.

The AC version of the obtained current and voltage gains (the power gain will be henceforth omitted as it can be directly obtained as a p = a i a v ) can now be calculated as:

a i (I B ) = - V a tan(sI B ) I B (R L tan(sI B ) + 1) ( 22
)
a v (I B ) = - R L V a tan(sI B ) R i I B (R L tan(sI B ) + 1) (23)
It is interesting to rewrite the above equations as:

a i (I B ) = -V a tan(sI B ) I B (R L tan(sI B ) + 1) ( 24
)
a v (I B ) = - V a R i R L tan(sI B ) I B (R L tan(sI B ) + 1) (25)
so that we can now define the two functions g() and h() that play the role of kernels of the transistor operation:

g(I B , s, R L ) = tan(sI B ) I B (R L tan(sI B ) + 1) (26) h(I B , s, R L ) = R L tan(sI B ) I B (R L tan(sI B ) + 1) (27)
These functions are critically important because they define the linearity of the transistor amplification in the considered circuit configuration. Both g() and h() depend only of the base current I B , the proportionality parameter s, and the load resistance R L . The current and voltage gains can be easily obtained by multiplying g() and h() by -V a and -V a /R i , respectively, as implied by Equations A.1 and A.2.

Figure 8 illustrates the behavior of g() in terms of I B , and Figure 9 depicts the function h() in terms of these same two variables for 0 ≤ I B ≤ 100µA and R L = 670Ω. For simplicity's sake, we henceforth set V a = -100 and s = 2, which is typical of small signal real-world NPN devices [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF]. Both cases assume s = 2 and 50Ω ≤ R L ≤ 1kΩ, but similar behaviors are observed for other typical parameter value configurations. R L = 0, when g() = 0), but now in an almost perfectly linear fashion. Indeed, the relative variations of g() and h() with I B both increase with the fixed R L value. Given the almost linear variations of g() and h(), they can be effectively characterized in terms of their average and variation values. The former of these are immediately given as: The gain variations are all important for characterizing the linearity of the transistor operation (observe that increased variations will imply larger distortions in the amplification), so that it is worth defining them in terms of the relative maximum excursions, i.e.:

δ ai I B,max I B,min = g(I B,min ) -g(I B,max ) I B,max -I B,min (30)
δ av I B,max I B,min = h(I B,min ) -h(I B,max ) I B,max -I B,min (31)
where g() and h() are immediately calculated by using Equations 26 and 27, respectively. Interestingly, these variations are directly related to the total distortion implied by the respective amplifications. In particular, full linear operation would be achieved if the variations were null.

The nearly linear variations of the current and voltage gain with I B indicate that, at least for the considered parameters and variable ranges, the properties of those gains can be effectively quantified in terms of the respective average and variation values.

We are now in a position to study the overall AC properties of the considered transistor/circuit configurationnamely the current and voltage gains, as well as the overall implied distortions -in terms of all possible choices of the involved parameters, i.e. V a , s, R i and R L . We limit our discussion to the current gain because an analogue behavior is observed for the voltage gain. Figure 10 illustrates the average current gain, obtained by multiplying g() by -V a .

FIG. 10: The current gain averaged for 0 ≤ I B ≤ 100µA for the considered circuit and parameter configurations. Respective isolines are also shown. Though this result was obtained with respect to R L 670Ω, Va = -100 and s = 2, similar structures are obtained for other typical configurations. Observe that the maximum average gain is achieved at the upper left-hand side of the Early space, decreasing steadily towards both the Va and s axes.

The average current gains tend to reach a peak at the top lefthand side of the Early space (V a , s) and the lowest values near the V a and s axes. Respective isolines, indexed by the average current gain, are also depicted in this figure. They present progressively varying shapes, with curvature increasing towards the coordinate system origin. Observe that the average current gain increases in a non-linear fashion throughout the Early space, being characterized by steeper variations near the top lefthand side of the represented space.

Figure 11 shows the absolute variation of the current gain along the considered portion of the Early space for 0 ≤ I B ≤ 100µA and R L = 670Ω. The obtained pattern is similar to that exhibited by the average current gain, but it is less symmetrical along the horizontal and vertical orientations. The respective current gain variation isolines are also shown in this figure .   FIG. 11: The variation of current gain in the interval 0 ≤ I B ≤ 100µA for the considered circuit and parameter configurations. Respective isolines are also shown. This surface presents similar structure as that obtained for the average current gain in Figure 10 Enhanced linearity is achieved for the (V a , s) configurations near the V a or s axes. Observe that the variation values are high because the variation is normalized by the I B interval, which is equal to 100µA. Another way to quantify non-linearity as related to the current gain variations is by considering relative instead of absolute values. This type of quantification is adopted henceforth in this work. Figure 12 depicts the relative current gain variations, obtained by dividing the previous variations by the respective average current gain. This remarkable result, already hinted by previous total harmonic distortion considerations in a previous work [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF], indicates that the linearity of the transistor in the adopted circuit depends only of the s values, being completely independent of the V a values. Therefore, horizontal isolines are obtained in the figure. Observe that the isolines spacing decreases progressively with s. The most non-linear amplification is obtained for large values of s, irrespectively of V a . Interestingly, a (welcomed) substantially wide band of low variation is observed near the V a axis.

The results revealed by the average current gain and its variations can now be neatly combined into a same diagram, henceforth called the prototypic Early space, to provide an overall representation of the trade-off between current gain and linearity in the considered circuit. Figure 13 shows the so obtained prototypic space, as well as Mahalanobis ellipses obtained experimentally for several NPN and NPN silicon [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF] and germanium [START_REF] Da | Characterizing germanium junction transistors[END_REF] junction transistors. The isolines for β = 130 (dashed line) as well as for β = 240 (dotted line) are also shown in the figure, passing very near to the centers of mass (averages) of the considered groups of transistors.

FIG. 13: The Early space prototypic space, illustrating the high gain and high linearity regions, as well as prototype groups of NPN and PNP silicon (Si) and germanium (Ge) junction transistors. Though a trade-off results regarding the choice of high gain and high linearity, the region R provides a good compromise between these two goals. Also, the fact that the higher gain isolines approach the Va axis for larger values of V A , therefore leading to respectively smaller values of s, also implies that transistors with large values of Va provide a good combination of gain and linearity.

This prototypic space summarizes several key information about transistor operation, providing valuable subsidies for choice, applications, and design of transistors. If priority is placed on gain, devices near the red region should be chosen. Contrariwise, in case linearity is the main objective, devices near the green band should be considered. Now, it becomes evident that the high gain and high linearity tend to follow contrary pathways along the prototypic space, implying a trade-off between these two often sought characteristics. However, the different geometry of the average current gain and variation surfaces imply some regions of special interest when trying to optimize both gain and linearity. Though, in principle, the linearity does not depend of V a , if a large magnitude value of V a is chosen, such as in the yellow band R in the figure, higher gain values can be obtained even when choosing transistors with smaller values of s. In this way, the yellow region stands as having particular interest when aiming simultaneously at hight gain and good linearity. Also, observe the higher gain isolines tend to approach the V a axis for large magnitude values of that parameter, the implied funneling effect also implies lower values of s and improved linearity. Thus, devices with very large V a are particularly interesting for providing a good combination of high gain and low distortion.

Interestingly, real-world NPN and PNP transistors, at least for the cases experimentally characterized [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF], tend to occupy an intermediate position in the Early parameter space. The silicon NPN devices are those nearer to the yellow band, representing good candidates when aiming at both high gain and low distortion levels.

V. STABILITY FACTOR ANALYSIS

An important property of transistor-based amplifying circuits regards their stability (e.g. [START_REF] Tinnell | Electronic Amplifiers[END_REF]) with respect to the current or voltage supply oscillations. Let's obtain, by using the Early equivalent circuit, the current and voltage stability for V CC oscillations of the circuit considered in the previous sections. This can be done as follows with respect to the collector current and voltage instability with respect to V CC oscillations:

S i = ∂I C ∂V CC (I B ) = 1 + βR o I B R o + R L (32)
S v = ∂V L ∂V CC (I B ) = R L 1 + βR o I B R o + R L (33)
It follows that:

S i = ∂I C ∂V CC (I B ) = (1 -V a )tan(sI B ) R L tan(sI B ) + 1 (34)
S v = ∂V L ∂V CC (I B ) = R L (1 -V a )tan(sI B ) R L tan(sI B ) + 1 (35)
The voltage instability S v is directly related to the current stability S v , so we focus on the former index. A substantial discrepancy can be observed between the instability curves obtained by using the Early (dashed lines) and more traditional approach (solid line), as these diverge markedly, specially for larger values of I B . Interestingly, the instability obtained by using the more accurate Early equivalent circuit is smaller than that yielded by using the more traditional model. This is a probable consequence of the fact that the varying slopes of the I B -indexed isolines, which is represented in the Early equivalent circuit but not the more traditional approach, compensate for the voltage supply oscillations through a kind of saturating effect. Thus, real-world devices may turn out to have potentially increased stability than that revealed by more traditional approaches based on the (β, R o ) parameters.

This simple application of the Early equivalent circuit to the analysis of an important practical parameter in linear circuit design and application, namely the variation of the amplification as a consequence of power supply oscillations, revealed that substantial discrepancies in the quantified electronic behavior of the analyzed circuits can be produce when overlooking the more realistic variation of the I B -indexed isolines in real-world devices.

VI. PARALLEL TRANSISTOR CONFIGURATIONS

A single transistor is capable of delivering a limited amount of current to the load, being subjected to max-imum absolute ratings. A possible means for trying to increase the power provided by the output port of transistors is to combine them in parallel as illustrated in Figure 15.

FIG. 15: Parallel combination of two transistors that is sometimes employed to deliver more current to the load and to reduce the output resistance. This is not a practical circuit as it lacks some important considerations.

As illustrated in Figure 16, the Early equivalent circuit can be easily employed to analyze the parallel configuration of two transistors (the result can be immediately extended to more transistors). The so obtained circuit can now be analyzed by using Kirchhoff's and Ohm's laws, as well as the Thevenin equivalent theorem, to obtain the following equations of the resulting effective output resistance R o,eq (I B ) and Early voltage V a (I B ), both of which potentially dependent of I B :

R o,eq (I B ) = R o1 (I B )R o2 (I B ) R o1 (I B ) + R o2 (I B ) (36)
V a,eq (I

B ) = V a2 + R o2 (I B ) V a1 -V a2 R o1 (I B ) + R o2 (I B ) (37)
For simplicity's sake, and without loss of generality, we can make s 2 = γs 1 and V a2 = λV a1 . First, we have that:

R o2 (I B ) R o1 (I B ) = tan(s 1 I B ) tan(γs 1 I B ) (38)
and, consequently, it follows that:

R o,eq (I B ) = 1 tan(s 1 I B ) + tan(γs 1 I B ) (39)
V a,eq (I B ) = λV a1 tan(s

1 I B ) + V a tan(s 1 I B ) tan(s 1 I B ) + tan(γs 1 I B ) (40)
Interestingly, V a,eq turns out to be almost perfectly constant with I B for the parameters and variables typically found in the considered circuit configurations to the point that the variation often results smaller than double floating point precision. So, the equivalent transistor effectively has, effectively, a constant Early voltage determined only by s 1 , s 2 , V a1 and V a2 . This nearly constant value of V a,eq can be conveniently obtained by applying L'Hospital theorem to Equation 40, which yields:

V a,eq (I B ) = λ + 1 γ + 1 V a (41)
The dependence of V a,eq with γ and λ is depicted in Figure 17. It is assumed that Ro1 = 1 and that Va1 = 1.

So, increasing λ promotes linear larger values of V a,eq , and this increase is more accentuated for smaller values of γ. However, the effective V a,eq can never exceed the maximum between the two original Early voltages V a1 and V a2 .

Another remarkable property of the parallel transistor combination is that the proportionality ratio R o1 /R o,eq and R o2 /R o,eq is also virtually invariant with I B . This means that the pairwise combination of transistors will have an equivalent output resistance R o,eq that will not interfere with the degree of linearity of the resulting device. As a consequence, parallel combinations of transistors cannot imply in new types of characteristic surfaces deviating from the Early model geometry. Similarly, by applying L'Hospital theorem:

R o,eq = γ γ + 1 R o2 (42)
Figure 18 shows the range of R o,eq that can be obtained by varying γ. Observe that the equivalent resistance tends to saturate at 1 for very large values of γ.

FIG. 18: The equivalent output resistance Ro,eq in terms of γ = s2/s1. Ro,eq increases with γ, but tends to saturate soon. It is assumed that Ro1 = 1.

As the equivalent V a and s initially seemed to depend on I B , this could lead to novel qualitative behavior of the parallel combination of transistors, such as eventual displacement of the point V a along the V C axis during normal circuit operation, or non-linear dependencies of θ with I B implied by s not being constant with I B . However, the interesting obtained results imply that the parallel combination of transistors, at least for the considered configuration and parameters and variables ranges, will effectively result in a new transistor described by the same qualitative behavior as dictated by the Early equivalent circuit (see Figure 19). So, parallel combinations can be used to "design" new devices with parameters that are intermediate between the two original transistors. For instance, a transistor with smaller V a,eq can be obtained by combining two transistors with larger V a values. Observe that the resulting V a,eq and R o,eq values will never be larger than the respective values of the original combined transistors.

VII. CONCLUDING REMARKS

Transistors have been around since the mid 40's, playing a decisive role for the development and popularization of modern electronics. Yet, their seemingly complex operation, which constrains the linearity of applications, has motivated much effort aimed at better understanding, modeling, applying and designing improved devices and circuits. There are two main vectors motivating the continued interest in transistors: (i) they are used in a vast range of applications underlying most of human activities, some of them critically; and (ii) oftentimes, transistors are required to operate with as much linearity as possible, which implies several challenges given the practical variability of transistor parameters allied to their relatively complex behavior. Originally, the focus of interest in transistor research was directed toward discrete devices and circuits, and great attention tended to be given to graphical representation and modeling approaches. With the introduction of integrated technology, interest shifted to transistors as the basic elements in integrated circuits, often studied by using numericcomputational simulations. However, several applications still justify, or even require (as in power electronics), the application of discrete devices. Be that as it may, discrete and integrated transistors follow the same physical constrains, so that they can be treated by the same unified approaches.

As a consequence of the continuing interest in transistor electronics, a vast quantity of works have been reported aimed at characterizing, modeling and design these devices. One of the main approaches consists in deriving, experimentally or from basic physical principles, the functional response of the devices (often done in a graphic way, with the help of transfer functions and other types of diagram), and then obtaining respective equivalent circuits that can be incorporated into circuit analysis performed analytically or through simulations. Several of the traditional transistor models are based on two parameters, the current gain β and the (collector) output resistance R o . While these two parameters are particularly intuitive, as they are closely related to transistor electronic operation in circuits, they have an intrinsic limitation in the sense that they vary, often significantly, with normal circuit operation, taking different values for each current and voltage values taken by the transistor output.

A simple, and yet accurate, transistor model was developed recently [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] founded on the Early effect, characterized by an interesting geometry of operation. In addition to the Early voltage V a sometimes used in transistor modeling, a second proportionality parameter s is involved that governs how the angle θ of the base current-indexed transistor isolines vary with that current I B . Interestingly, this parameter s has been experimentally found [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Characterizing germanium junction transistors[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] to underly the simple linear relationship θ = sI B , at least for hundreds of small signal silicon and germanium BJTs, to vary in nearly full linear fashion. Therefore, both parameters V a and s are fixed and constant for each given transistor, defining to a great extent its electronic properties. In addition, the mapping of transistors into the Early parametric space, instead of in more traditional (β, R o ) spaces, has the advantage of allowing a better characterization and separation of different types of transistors [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Characterizing germanium junction transistors[END_REF], possibly as a consequence of the fact that the Early approach captures directly the geometry of operation of real-world devices in the so-called "linear" operation region.

Despite having been introduced quite recently, the Early model has already been applied with encouraging success to several problems in electronics, including the estimation of transfer functions [START_REF] Da | An Early model of transistors and circuits[END_REF], the characterization of complementary BJTs including the derivation of a prototypic space [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF], as well as the study of the electronic properties of alloy junction germanium transistors [START_REF] Da | Characterizing germanium junction transistors[END_REF].

In this work, an equivalent circuit is developed for the Early model in order to pave the way to its direct application in analysis and design of discrete and integrated circuits. We started by discussing the geometrical characteristics of real-world transistors, and then revised one of the simplified traditional models based on β and R o , that have been largely used and applied in electronics. Next, the equivalent circuit of the Early model was derived, starting from characteristic surfaces with isolines radiating from a common point V a , corresponding to the Early voltage. The obtained circuit resulted substantially simple, with its output port including only a resistance and a voltage source connected in Thevenin series. How-ever, this resistance turns out to be dependent on the base current I B , as well as on the Early parameters V a and s, i.e. R o = R o (I B |V a , s).

To illustrate the potential of this model, it was applied to the characterization of a simplified common emitter circuit configuration in which the load is directly attached between the transistor collector and the external voltage supply V CC . The collector current, as well as the voltage across the load, were easy and conveniently derived by taking into account the Early equivalent circuit. Interestingly, these equations turned out to be no more complex than those obtained for the more traditional approach based on β and R o , despite the fact that that the more traditional model considers a much more simplified situation in which the current gain does not vary during the transistor operation as a consequence of the equispaced parallel (though inclined) isolines. Then, equations for the DC and AC current, voltage and power gains were derived and, again, resulted not to be more elaborated than the respective counterparts obtained for the simpler, more traditional model.

The current and voltage gain equations (the power gain can be directly obtained from these to gains) were found to exhibit an interesting mathematical structure involving ratios between tangents. Two kernel functions g() and h() were derived, respectively, from these two gain equations. It is believed that these two equations are behind the intricacies of transistor operation. As illustrated graphically, the current and voltage gains vary in an almost perfectly linear fashion with I B , which suggests the derivation of the average and variation of the gains by taking into account only the extremities of the excursion of the gains as they vary with the same range of I B . This yielded two very simple expressions for quantifying the average current and voltage gains, as well as their variations, given the Early parameters V a and s, the transistor input port resistance R o , as well as the circuit configuration (i.e. R L ). The gain variations are particularly interesting because they are directly related to the linearity of the amplification. For instance, null gain variation necessarily implies perfect linearity.

The development of equations for the average and variation of the current and voltage gains paves the way to a series of interesting applications and analysis of discrete and integrated circuits. This potential was preliminary illustrated in the current work with respect to: (a) the identification of trends, along the Early space, underlying gain and linearity; (b) the derivation of a prototypic space that can assist transistor characterization, choice, design, and applications; (c) the quantitative study of circuit stability analysis; and (d) the characterization of parallel configurations of transistors.

Regarding the gain and linearity of transistors, as re-vealed by the Early approach, we found that they follow a trade-off, with higher gain typically implying reduced linearity. However, it was possible to identify a region in the Early space in which a good compromise can be achieved between these two requirements. In addition, we also found that larger values of V a magnitude tend to be particularly interesting, as larger gains can be obtained for a fixed s when V a is increased while relatively small values of s can be achieved. The obtained prototypic Early space for several real-world NPN and PNP silicon and germanium transistors revealed that the silicon NPN BJTs are the devices characterizing a particularly good combination of gain and linearity, at least as far as the considered configuration and parameters and variable settings are concerned. The obtained prototypic Early space provides valuable means for characterizing, analyzing, designing and applying discrete and integrated transistors.

The application of the Early equivalent circuit to the stability study of effects of power supply oscillations on the load voltage revealed that the stability of real-world devices in the considered circuit type and configuration may be substantially better than that revealed by the more traditional modeling approach based on β and R o . This fact suggests that deviations between these two models can also be expected regarding other important electronic properties of several types of circuits. Indeed, the Early equivalent circuit allowed the variation of the slopes I B -indexed isolines to be taken into account, and this has been somewhat provided some compensation for the power supply oscillations.

Parallel transistor configurations were also approached by using the proposed Early circuit. Here, we found that the resulting device follows almost perfectly the Early representation, as the resulting Early voltage does not depend on I B and the resulting equivalent output resistance depends on that current according to the Early voltage hypothesis, namely R o,eq = s eq I B . Such combinations provide an interesting possibility for engineering "new" equivalent devices with intermediate parameters as the two original devices. These results can be immediately extended to parallel combinations of 3 or more transistors.

The reported concepts, developments, and results pave the way to a large number of immediate future developments. These include the investigation of more sophisticated circuit configurations, as voltage followers, current mirrors, push pull stages, phase splitters, as well as the more complete common emitter configuration. Specially promising is the possibility to use the Early equivalent circuit for developing new, improved circuits. It would also be interesting to compare the performance of several types of transistors in these circuits by using the Early approach, so as to identify their respective main advantages. The Early equivalent model can also be used to investigate the effect of several types of noise and environment variations on the circuit performance. A particularly promising would be to characterize transistor amplification when applied to reactive loads. All these possibilities can be used in both discrete and integrated device and circuit design.

Appendix: Series Expansion of Early Model Current and Voltage Gains

For typical values of I B in small signal transistors (approximately in the order of tenths or hundreds of microamperes), the Early-based equations obtained for the circuit considered in Section IV can be simplified with very small error by respective series expansion at I B = 0: a i (I B ) ≈ -V a s 1 -sR L I B + s 2 3 (3R 2 L + 1)I 2 B (A.1)

a v (I B ) ≈ - V a R i sR L 1 -sR L I B + s 2 3 (3R 2 L + 1)I 2 B (A.2)
Often, a good fit can be obtained even by leaving the last term out.

FIG. 4 :

 4 FIG.4:The circuit configuration considered in this work. The load resistance R L , together with a series voltage supply V CC , are attached to the transistor collector (here playing the role of output). The inclined parallel isolines in the adopted characteristic surface are modeled in terms of the variable current source Is and the respective output resistance Ro.

FIG. 5 :

 5 FIG. 5: The geometrical construction underlying the Early model of transistors. The set of I B -indexed isolines converge at V C = Va as a consequence of the Early effect. The load line defined by a resistive load R L attached to the collector in series with the voltage supply V CC is also shown. The circuit operation therefore remains limited along this line. Observe that the slope of the load line is determined by R L , while the inclination of the isolines correspond to 1/Ro(I B ), which is intrinsically compatible with the geometric structure observed for the real-world transistor in Figure 1. Only a second hypothesis underlies the Early model: the linear relationship between the angle θ of the isolines and the base current I B , i.e. θ = sI B , implying a second parameter, the proportionality constant s, to the Early model.

FIG. 6 :

 6 FIG. 6: The Early model equivalent circuit. While the input port is the same as before, the output port now incorporates a variable output resistance Ro(I B ) = 1/tan(sI B ) and a fixed voltage source Va,where Va is the Early voltage. The two diodes are included for generality and can be omitted in typical amplifying circuit configurations.

FIG. 7 :

 7 FIG. 7: Application of the Early model to study an simplified common-emitter configuration with resistive load R L and voltage source V CC . Recall that Va has negative value and that the diodes can be omitted for the assumed biases. While Va does not vary with either I B , I C or V C , the proportionality parameter s depends of I B , implying the output resistance Ro also to be a function of I B .

FIG. 8 :

 8 FIG. 8: The function g(I B |s, R L ) in terms of 0 ≤ I B ≤ 100µA characterizing the current amplification (except by a factor -V a)properties of a transistor for s = 2 and 50Ω ≤ R L ≤ 1kΩ.

  B,min ) + h(I B,max ) 2 (29)

FIG. 12 :

 12 FIG. 12: The function h(I B |s, R L ) in terms of 0 ≤ I B ≤ 100µA characterizing the voltage amplification (except by a factor -Va/Ri) properties of a transistor for s = 2 and 50Ω ≤ R L ≤ 1kΩ.

Figure 14

 14 Figure 14 illustrates the behavior of S i in terms of 0 ≤ I B ≤ 100µA assuming V a = -70V and R L = 670Ω.

FIG. 16 :

 16 FIG.16:The circuit obtained by substituting the two transistor in the parallel combination by their respective Early equivalents.

FIG. 17 :

 17 FIG. 17: The equivalent Early voltage Va in terms of γ = s2/s1 and λ = Va2/Va1. Va,eq increases linearly with λ for any fixed value of γ.It is assumed that Ro1 = 1 and that Va1 = 1.

FIG. 19 :

 19 FIG. 19: The resulting Early equivalent circuit for parallel combination of two transistors. The resulting device has been found, at least for the considered configurations, to follow almost perfectly the Early model in the sense that the resulting equivalent Early voltage is constant and the obtained equivalent output resistance depends on I B as dictated by the Early approach.

  Towards a Simple, and Yet Accurate, Transistor Model and Its Application to the Analysis and Design of Discrete and Integrated Electronic Circuits Luciano da F. Costa * São Carlos Institute of Physics, IFSC-USP, São Carlos, SP, Brazil (Dated: 05 February 2018)

Acknowledgments.

Luciano da F. Costa thanks CNPq (grant no. 307333/2013-2) for sponsorship. This work has been supported also by FAPESP grants 11/50761-2 and 2015/22308-2.


Amine Boumaza
email: amine.boumaza@loria.fr

Bruno Scherrer
email: bruno.scherrer@loria.fr

Optimal control subsumes harmonic control

We consider trajectory planning within the frameworks of optimal control and harmonic control. We present a formal evidence, in the continuous domain and in a standard discretization, that harmonic control is the limit case of some optimal control problem in which we make the noise level tend to infinity. In other words we show that optimal control subsumes harmonic control. We discuss properties of both paradigms and present simulations that illustrate this relationship.

I. INTRODUCTION

The use of harmonic functions (solutions to the Laplace equation) as potential fields for trajectory planning was proposed by Connolly et al. [START_REF] Connolly | Path planning using laplace's equation[END_REF] and Akishita et al. [START_REF] Akishita | Laplace potential for moving obstacle avoidance and approach of a mobile robot[END_REF] in 1990. In such an approach, obstacles in the configuration space correspond to maxima of the potential, while goals correspond to minima; control algorithms then reduce to locally descend the potential until they reach the global minimum. The harmonic approach for control (we will now on write "harmonic control") has had some impact on the robotics community [START_REF] Masoud | Motion planning in the presence of directional and obstacle avoidance constraints using nonlinear anisotropic, harmonic potential fields: A physical metaphor[END_REF]- [START_REF] Wang | A new potential field method for robot path planning[END_REF]. However, the use of potential functions derived from partial differential equations (PDE), does not date from these seminal works on harmonic functions. Optimal control and reinforcement learning [START_REF] Sutton | Reinforcement Learning : An Introduction[END_REF] are based on the computation of a potential: the value function. In a continuous domain, the theory of dynamic programming, which was pioneered in the 1950s by Bellman [START_REF] Bellman | Dynamic Programming[END_REF], implies that the value function satisfies a PDE, the Hamilton-Jacobi-Bellman equation. Similarly to the harmonic case, the globally optimal control is locally derived from the value function.

The motivation of this paper is to emphasize a strong relationship between harmonic and optimal control. We argue that the former is a special case of the latter: we formally show, in the continuous domain as well as in a standard finite difference discretization counterpart, that harmonic control is the limit case of trajectory planning with optimal control in a space with isotropic noise 1 , when the noise level tends to infinity. Our aim is not to propose some new control algorithms but to explain that many practitioners of harmonic control do optimal control without necessarily knowing it.

To our knowledge, the articles that discuss both harmonic and optimal control do not highlight such a specific-togeneral relationship. Some articles consider both approaches as complementary alternatives [START_REF] Singh | Robust reinforcement learning in motion planning[END_REF]- [START_REF] Perkins | Lyapunov-constrained action sets for reinforcement learning[END_REF], whereas in others the connection between harmonic control and optimal control is close but not stated by the authors. Connolly et al. [START_REF] Connolly | Nonholonomic path planning using harmonic functions[END_REF] incorporate non-holonomic constraints in harmonic control by using Neumann conditions that constrain the system to satisfy its real degrees of freedom. Though this is what is done in optimal control through the state dynamics' function (see section II), no link with optimal control is made explicit. Masoud et al. [START_REF] Masoud | Motion planning in the presence of directional and obstacle avoidance constraints using nonlinear anisotropic, harmonic potential fields: A physical metaphor[END_REF] suggest a way to enforce directional constraints in some parts of the state space by introducing an anisotropic harmonic potential. This is similar to having a constant drift (in the dynamics function) and a non trivial diffusion matrix in optimal control.

The work presented in this article closely relates to [START_REF] Connolly | Harmonic functions and collision probabilities[END_REF] where Connolly provides an analysis of harmonic functions in terms of collision probabilities. He shows that, on a grid, the discrete harmonic function at one point is related to the probability of colliding with an obstacle given that the process follows a random walk from that point. He explains that a reinforcement learning algorithm, TD(0), can calculate such a harmonic function (in this case, the harmonic function is indeed equal to the value function of the random uniform policy). Though close, the link with optimal control is not explicitly stated: 1) This analysis does not clarify to which extent "descending the gradient of the value function of the random uniform policy" makes sense. 2) The relaxation technique for computing the harmonic function is presented as a new rapid technique for computing the value function even though it is equivalent to the Value Iteration algorithm. 3) Last but not least, only a discrete version of the problem is studied and the continuous setting, from which the harmonic function comes, is not addressed. The work we present in this paper addresses these issues: we show that a harmonic controller is the optimal controller of some navigation problem on a (finite difference) discretized version, as well as on the (original) continuous domain.

The organization of the paper is as follows. Section II gives a brief introduction to the optimal control framework. Section III shows how to perform trajectory planning with isotropic noise within such a framework. Section IV describes our main result: control with harmonic function corresponds to the case where, in this planning approach, we make the noise level tend to infinity. Finally, section V provides a discussion that clarifies some properties that are shared by the specific (harmonic) case and the general (optimal control) case and others that are not (interesting particular properties of the specific case over the general case).

II. OPTIMAL CONTROL: A BRIEF INTRODUCTION

We begin by a brief introduction to optimal control. One considers a system defined at time t by its state x (t) ∈ Ω (the state space) where Ω ⊂ IR n is the closure of an open set Ω and ∂Ω is its boundary ( Ω = Ω ∪ ∂Ω). This system is controlled by u (t) ∈ U where U is a compact set (the control space). The dynamics of such a system is governed by a stochastic differential equation (SDE):

dx = f (x (t) , u (t)) dt + σ (x (t) , u (t)) dw, (1)
where w corresponds to a m-dimensional Wiener process or Brownian motion and σ is a n × m diffusion matrix. We consider the case of infinite time horizon. For any initial state x 0 , any control law u (.) and trajectory x (.), we note τ the exit time of x (.) from Ω, with the convention that τ = ∞ when the trajectory stays infinitely within Ω. We define the discounted cost functional J as the expected cost over all possible trajectories:

J (x, u (•)) = E »Z τ 0 γ t c (x (t) , u (t)) dt + γ τ C (x (τ )) - , (2)
where c is the instantaneous cost in state x ∈ Ω, C(x) is the terminal cost at x ∈ ∂Ω and γ ∈ (0, 1) is a discount factor which guarantees that J remains bounded. We define the optimal value function J * , function of the initial state x, as the minimum of the cost functional J for all control laws u (.):

J * (x) = min u(.) J (x, u (•)).
We let a = σ • σ T and note ∇J * the gradient of J * . Under reasonable assumptions 2 , it can be proved [START_REF] Krylov | Controlled Diffusion Processes[END_REF] that J * is C 2 and satisfies the following PDE known as the Hamilton-Jacobi-Bellman (HJB) equation:

J * (x) ln (γ) + min u∈U    c (x, u) + ∇J * (x) • f (x, u) + 1 2 n i=1 n j=1 a ij ∂ 2 J * ∂x i ∂x j (x)    = 0, (3)
for x ∈ Ω, with boundary conditions ∀x ∈ ∂Ω, J(x) = C (x). The optimal value function is particularly interesting since it enables to compute a deterministic optimal controller. For every state x the optimal control u * (x) is the argument u for which the min in eq. 3 is attained, which is also known as the greedy controller with respect to J * :

u * (x) = arg min u∈U    c (x, u) + ∇J * (x) • f (x, u) + 1 2 n i=1 n j=1 a ij ∂ 2 J * ∂x i ∂x j (x)    . (4)
2 Sufficient conditions are: 1) the matrix a = σ • σ T satisfies a "uniform parabolicity assumption": there exists a c > 0 such that ∀x ∈ Ω, ∀u ∈ U and ∀y ∈ IR n , P n i=1 P n j=1 a ij (x, u) y i y j ≥ c y 2 ; 2) f , σ and the boundary ∂Ω are of class C 2 , c and C are Lipschitzian.

III. APPLYING OPTIMAL CONTROL TO TRAJECTORY

PLANNING

In order to make the connection with harmonic control, we now focus on a simple instance of optimal control. Consider that the state space Ω is an environment in which an agent navigates. For simplicity of exposition, we consider the 2dimensional case Ω ⊂ IR 2 ; note however that our result generalizes directly to any dimension n. The boundary of the environment is decomposed into 2 sets ∂Ω = O ∪ G: O is the set of obstacles and G the set of goals. At each time t > 0, the agent is characterized by its coordinates x(t) = (x 1 (t), x 2 (t)) and its dynamics is related to a command direction θ(t) according to the following SDE:

dx = - → u (θ(t)) dt + σ dw.
where

- → u (θ) is a unit vector in the direction θ ∈ [0, 2π],
and σ is a positive constant. In other words, we consider an optimal control problem with a unit speed control (u = θ and f (x, u) = -→ u (θ)) and a constant isotropic noise (σ ∈ IR). In this case it amounts to replace the matrix a by a real number σ 2 . Furthermore, we consider that there are no intermediate costs (∀x ∈ Ω, c (x) = 0), only terminal costs C on the boundary: C(x) = 1 on obstacle boundaries (∀x ∈ O) and C(x) = -1 on goal boundaries (∀x ∈ G). Note ∆J * the Laplacian of J * . In this specific case, the HJB equation (eq.

3) reduces to:

J * (x) ln(γ) + min θ∈[0,2π] {∇J * ( x )• - → u (θ)} + σ 2 2 ∆J * (x) = 0 (5)
for x ∈ Ω with boundary conditions ∀x ∈ ∂Ω, J * (x) = C(x). Under this model, it is easy to see that the optimal control θ * (x) in state x is colinear with -∇J * (x) ∇J * (x) : indeed, in eq. 5, the only term involved in the min is

∇J * (x) • - → u (θ),
which in turn is minimal when θ is in the opposite direction of the gradient ∇J * (x). Furthermore, for every γ ∈ (0, 1), the value function of the controller θ(•) (eq. 2) reduces to:

J (x, θ (•)) = E [γ τ |x(τ ) ∈ O] P r [x(τ ) ∈ O] -E [γ τ |x(τ ) ∈ G] P r [x(τ ) ∈ G] . (6)
Note that the integral term in eq. 2 vanishes because there are no instantaneous costs c(•). Also recall that the optimal controller minimizes this value function. Let us now interpret what this means. In order to minimize J (x, θ (•)), the optimal controller tries both to minimize

E [γ τ |x(τ ) ∈ O] P r [x(τ ) ∈ O] and to maximize E [γ τ |x(τ ) ∈ G] P r [x(τ ) ∈ G].
Since τ → γ τ is a decreasing function of τ , this intuitively means that the optimal controller both tries to maximize the time of hitting an obstacle and to minimize the time of reaching the goal. In the simple deterministic case 3 when σ = 0, and for any value of γ ∈ (0, 1), the optimal controller which consists in following the gradient of the optimal value function, is the one which minimizes the path length (and the time) to reach the goal. In practice, one cannot compute exact analytical solutions to eq. 5 and a usual approach, which we follow here, consists in building a finite difference scheme. This is what we describe now. To do so, we follow the lines of [START_REF] Kushner | Numerical Methods for Stochastic Control Problems in Continuous Time[END_REF] and build a discrete-time discrete-space controlled Markov chain. Given a resolution δ > 0, we build a grid Σ δ and its border ∂Σ δ on the domain of the problem. For simplicity we assume that any point of the border of the discretized domain belongs to the border of the initial domain: ∂Σ δ ⊂ ∂Ω. Given a grid resolution δ, the function J is approximated by a function J δ defined on Σ δ ∪ ∂Σ δ . We define cos + , cos -, sin + and sin - as the positive and negative parts of cos and sin: cos ± (θ) = max(± cos θ, 0) and sin ± (θ) = max(± sin θ, 0). Notice that cos θ = (cos + θcos -θ) and | cos θ| = (cos + θ + cos -θ). Furthermore, we define the following transitions probability from grid point (x, y) to its four neighbors when going in direction of θ:

   p θ [(x, y), (x ± δ, y)] = 1 N θ σ 2 2 + δ cos ± θ p θ [(x, y), (x, y ± δ)] = 1 N θ σ 2 2 + δ sin ± θ (7)
where

N θ = δ(cos + θ +cos -θ +sin + θ +sin -θ)+4σ 2 /2 = δ(| cos θ| + | sin θ|) + 2σ 2 can
be regarded as a normalizing factor that ensures that the the transition probabilities p θ ((x, y), (x , y )) sum to 1. These transition probabilities on the discretized grid have a natural geometric interpretation: all of them have the same noise component ( σ 2 2N θ ), and two of them (the ones that are in the control direction θ) have nonzero weights (see figure 1(a)). We finally write τ which is a quantity that can be interpreted as the time needed to go from one grid point to another when following direction θ. Then, an approximation of J is obtained by computing the unknown J δ in the following system:

(θ) = δ 2 N θ , - → u (θ) α β σ 2 2Nθ α Nθ + σ 2 2Nθ σ 2 2Nθ β Nθ + σ 2 2Nθ (a) (b)
J δ (x, y) = γ τ (θ J δ x,y ) X (x ,y )∈Σ δ p θ J δ
x,y ˆ(x, y), (x , y ) ˜Jδ (x , y ), [START_REF] Kazemi | An incremental harmonic function-based probabilistic roadmap approach to robot path planning[END_REF] for (x, y) ∈ Σ δ and J δ = C on ∂Σ δ , and where θ J δ x,y is the angle that corresponds to the steepest slope direction at (x, y) when considering a piecewise linear interpolation of J δ (see figure 1(b)). The notation θ J δ

x,y may seem a bit heavy, but it is important to remember that the optimal angle depends on the coordinate (x, y) and the value function J δ . Introducing the following operator B δ on Σ δ :

B δ [W ](x, y) = γ τ (θ W x,y ) X (x ,y )∈Σ δ p θ W x,y ˆ(x, y), (x , y ) ˜W (x , y )
for (x, y) ∈ Σ δ and B δ [W ] = C on ∂Σ δ , eq. 8 becomes

J δ = B δ [J δ ]. Since γ < 1 and τ (•) > δ δ+2σ
2 > 0, the operator B δ satisfies a contraction property (with contraction factor at least γ δ δ+2σ 2 ). Therefore J δ is unique, and can be computed by relaxation, that is as the limit of the sequence J δ n+1 ← B δ [J δ n ] when n tends to infinity. Once J δ is computed, the (approximate) optimal control is the direction θ J δ

x,y that descends the gradient of (the linear interpolation of) J δ . It can be proved [START_REF] Kushner | Numerical Methods for Stochastic Control Problems in Continuous Time[END_REF] that such a finite difference scheme is convergent: J δ uniformly tends to J * and the approximate optimal controller ∇J δ tends to the optimal controller ∇J * when the discretization resolution δ tends to 0.

IV. HARMONIC CONTROL IS A LIMIT CASE OF OPTIMAL

CONTROL Now that we have introduced an instance of optimal control problem for trajectory planning, we turn back to the main claim of this study: we demonstrate that when the noise level σ tends to infinity, the optimal control of our simple model tends to harmonic control.

As we are interested in the dependence on σ, we denote J σ the value function J * of eq. 5. In order to study the behavior of J σ when σ → ∞, we regard it as a function of the Sobolev space [START_REF] Evans | Partial Differential Equations[END_REF]. When σ → ∞, we show that J σ tends to a harmonic function, that is a function Φ that satisfies Laplace equation ∆Φ = 0.

H 1 (Ω) with its norm f H 1 (Ω) = ( Ω f 2 + Ω (∇f ) 2 ) 1/2
Theorem 1: Let Φ be the unique function satisfying ∆Φ = 0 on Ω and Φ = C on ∂Ω. Then the family of solutions (J σ ) σ>0 of equation 5 satisfies

lim σ→∞ J σ -Φ H 1 (Ω) = lim σ→∞ Ω (J σ -Φ) 2 dx + Ω (∇J σ -∇Φ) 2 dx 1/2 = 0
Notice that this convergence is strong in the sense that 1) J σ tends to Φ and that 2) its gradient ∇J σ tends to ∇Φ. We can prove a similar result for the standard finite difference approximations of J σ and Φ. For a resolution δ > 0 and the grid Σ δ ∪ ∂Σ δ introduced in section III, let Φ δ be the standard 5-point finite difference approximation of Φ, that is the unique solution to:

Φ δ (x, y) = 1 4 Φ δ (x + δ, y)+ Φ δ (x -δ, y) + Φ δ (x, y + δ) + Φ δ (x, y -δ)
for all (x, y) ∈ Σ δ and Φ δ = C on ∂Σ δ . When the noise level σ tends to infinity, the approximate value function J δ σ tends to the approximate harmonic function Φ δ : Theorem 2: Fix a discretization step δ > 0. Let Φ δ be the standard 5-point finite difference approximation of the harmonic function Φ. Then the family (J δ σ ) σ>0 of discretizations of the value function defined in section III satisfies lim σ→∞ J δ σ = Φ δ uniformly on Σ δ ∪ ∂Σ δ . The proofs of theorems 1 and 2 can be found in [START_REF] Boumaza | Optimal control subsumes harmonic control[END_REF]. On one hand, the proof of theorem 2 is rather straightforward (notice that it is specific to the finite difference schemes used to discretize both problems). On the other hand, if the result of theorem 1 seems intuitive (in eq. 5, we expect that when the term σ 2 2 before ∆J * tends to ∞, the other terms get negligible), the proof we make relies on a few technical manipulations of eq. 5 and non-trivial properties of the Sobolev space H 1 (Ω). In fact, our proof of theorem 1 in [START_REF] Boumaza | Optimal control subsumes harmonic control[END_REF] is more general that what we really need here: we show that the value function J σ and its gradient ∇J σ tend to the harmonic function Φ and its gradient ∇Φ for any bounded control function f and any bounded cost c (recall the general optimal control equations 1 and 3); in our simple instance model, we take u = θ, f (x, u) = -→ u (θ) and c = 0 so that the optimal controller matches the opposite direction of the gradient ∇J σ . Overall, we can summarize our convergence results (when σ → ∞) and the standard numerical schemes convergences (when δ → 0) in the following diagram:

(J σ , ∇J σ ) σ→∞ -→ (Φ, ∇Φ) δ→0 ↑ ↑ δ→0 (J δ σ , ∇J δ σ ) σ→∞ -→ (Φ δ , ∇Φ δ )
When the noise level σ tends to infinity, the value function becomes harmonic, and the corresponding optimal control therefore consists in descending the gradient of this harmonic function. This limit case of optimal control is strictly equivalent to what Connolly [START_REF] Connolly | Path planning using laplace's equation[END_REF] and Akishita [START_REF] Akishita | Laplace potential for moving obstacle avoidance and approach of a mobile robot[END_REF] proposed in 1990: path planning is done by gradient descent of the function Φ. Furthermore, our formal analysis gives some insight on the discrete analysis of harmonic control by Connolly [START_REF] Connolly | Harmonic functions and collision probabilities[END_REF] discussed in the introduction. If harmonic control happens to be equivalent to "descending the gradient of the value function of a random uniform policy", this is because the agent's actions become negligible when the noise level gets infinitely large: at the limit, its movements are mostly due to the isotropic noise. Also, as we said that theorem 1 is general (it is true for any control function f and any cost c), the connection we make has a corollary that may be of interest to harmonic control practitionners: in the limit case, and for general f (x, u) and c(x, u), the optimal control theory prescribes to take the controller given by equation 4, which reduces here to:

u * (x) = arg min u {c(x, u) + ∇Φ(x).f (x, u)} ,
(and this is only equal in general to the gradient direction when u = θ, f (x, u) = -→ u (θ) and c = 0). Recall that f (x, u) describes the real dynamics for action u in state x while c(x, u) can be seen as a bias to prevent form taking action u in state x. Though such a general control law might generate some complications (the controller may in general get stuck 4 ), it constitutes a natural candidate for incorporating kinematical and dynamical constraints such as 4 A non-zero c can block the trajectory away from the goal (since in practice the gradient of harmonic functions can be very small). If c = 0, then a sufficient condition for always getting to a goal is that f allows to move in any direction (possibly with different speeds) in the state space.

non-holonomicity and/or a notion of cost within the harmonic control framework.

We now present some simulations in order to illustrate what happens when the noise level tends to infinity. We have , and starting at equally distributed grid points in the environment. All trajectories where computed using optimal control (with J δ σ ) except the last one which is derived from harmonic control (with Φ δ ). considered a maze environment containing two goals (G 1 and G 2 in figure 2). Dark lines represent obstacles, white represents free space and gray lines represent trajectories computed along the value function's gradient. Goal G 1 at the top left of the environment is accessible through a narrow corridor, and goal G 2 is located in a fairly uncluttered area. Figure 2 illustrates, for five increasing values of the noise level σ, the trajectories computed starting from equally spread initial positions.

In the case where the dynamics is deterministic (σ = 0), the optimal trajectories are the shortest; most of the trajectories starting in the upper part of the environment go toward goal G 1 whereas all others go toward goal G 2 . When we increase the noise level (σ = 0.4), the trajectories stray away from the walls and start to become smoother. With little more noise (σ = 0.5), the corridor leading to goal G 1 becomes hazardous and most of the trajectories go toward goal G 2 . At σ = 1, trajectories are smoother, and we begin to see some of them halting before reaching a goal (trajectories starting on the right side): paths through narrow corridors becoming more dangerous, the controller maintains the agent in safer areas away from obstacles. This behavior intensifies at σ = 50. A new phase begins at σ = 500: the noise being elevated, strategies consisting in staying in safe areas are not efficient anymore (the noise inevitably leads towards an obstacle). Then, as a kamikaze that would know he is going to die anyway, it becomes more interesting to start moving again towards a goal. This behavior is more noticeable when we go on increasing the noise level. Eventually observe that the last simulation (σ = 10 6 ) experimentally confirms our theoretical result: the trajectories are close to the trajectories computed using a harmonic control.

V. DISCUSSION

We believe that harmonic control has had a significant impact in the literature because as a paradigm, it has several nice properties: completeness, incrementality, flexibility, and parallel implementation; we discuss them in this section. Morevover, as we showed that harmonic control is a special case of optimal control, it is interesting to check which of these properties are also shared by the general framework of optimal control. Conversely, we will highlight specific advantages of harmonic control over optimal control.

A fundamental property of harmonic functions is the absence of local optima. Whatever the starting position, planned trajectories always lead to a goal: such trajectories as said to be complete. This property makes harmonic function particularly interesting compared to other local potential field navigation [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF] which might exhibit local optima. In general, the value function of an optimal control problem can have local optima. In the model we described in this paper, the underlying control law is complete in two cases: in the deterministic case (σ = 0) and, obviously, in the limit (harmonic) case (σ = ∞). In an environment with a large amount of noise, the control law may not attain a goal if this proves riskier (in the sense of the criterion defined in equation 6) than moving towards the goal.

Computation of harmonic functions using relaxation methods is performed iteratively, which allows incremental updates of the environment. Newly detected obstacles may be integrated in the model as new boundary conditions during computation. This property permits to use them in dynamic environments or in environments explored on-line, as for instance in [START_REF] Zelek | Complete real-time path planning during sensor-based discovery[END_REF]- [START_REF] Feder | Real-time path planning using harmonic potentials in dynamic environments[END_REF], [START_REF] Kazemi | An incremental harmonic function-based probabilistic roadmap approach to robot path planning[END_REF]. Value functions, which can also be computed using relaxation, also permit incremental updating of the environment's model: new obstacles and goals may be integrated during computation. More generally, updating incomplete or dynamic models of the environment within the framework of optimal control is known as "indirect reinforcement learning". Moreover, research on "direct reinforcement learning" shows that incremental planning can be done even without maintaining an explicit model of the environment (see [START_REF] Sutton | Reinforcement Learning : An Introduction[END_REF] for a general introduction, and [START_REF] Munos | Reinforcement learning for continuous stochastic control problems[END_REF] for a study of the continuous time case).

Trajectories generated by harmonic control can be quite flexible. Using Dirichlet boundary conditions, one can generate safe trajectories that have the tendency to stray away from obstacles. This is due to the fact that the potential flow is orthogonal to the boundaries. Using Neumann conditions, the potential flow is tangential to the boundary which permits to have riskier trajectories. It is also possible to combine both boundary conditions [START_REF] Singh | Robust reinforcement learning in motion planning[END_REF], [START_REF] Connolly | On the applications of harmonic functions to robotics[END_REF] which allows to have different intermediate behaviors. In optimal control, trajectories may also be refined in numerous ways. Contrary to the harmonic case, we can express precisely, via the cost functions c and C, the relative severity of hitting certain obstacles compared to others, the relative importance of attaining certain goals compared to others and the relative importance of attaining goals rather than hitting obstacles. In the same way, the laws of dynamics f , and the noise parameter σ, are parameters that may further explicitly influence the nature of the produced trajectories.

Another interesting property of harmonic functions that has been advocated is their inherent distribution of computation. The relaxation methods used to solve Laplace's equation are naturally distributable; computing grid values rely only on the local information at neighboring cells. Low level hardware implementation were proposed by Trassenko and Blake [START_REF] Trassenko | Analogue computation of collisionfree paths[END_REF] implemented as resistive grid arrays. Furthermore parallelization makes this model a potential candidate to explain the computations of control undergone in the brain. For instance, Connolly and Burns [START_REF] Connolly | The basal ganglia and the planning of actions[END_REF] argue that the basal ganglia could compute harmonic function for motor control. The computation of the value function is also a natural candidate for parallelization. Iterative methods such as Gauss-Seidel or Jacobi, which solve the discrete version of the problem, may be implemented completely asynchronously using massively parallel architectures [START_REF] Bertsekas | Parallel and Distributed Computation: Numerical Methods[END_REF]. Though it is not clear whether these algorithms can be implemented in very low level architectures such as resistive grids, it is possible to implement them on parallel processors with communication delays and no synchronization [START_REF] Bertsekas | Distributed asynchronous computation of fixed points[END_REF]. Finally, as it is the case for harmonic functions, such parallel implementations motivated biological analogies with what may happen within the brain [START_REF] Scherrer | Asynchronous neurocomputing for optimal control and reinforcement learning with large state space[END_REF].

As we have just seen, most of the interesting properties of harmonic control are shared with optimal control, except the possibility of very low level implementation such as resistive grids. There is, however, another important property which is characteristic of harmonic control: Laplace equation (which characterizes the harmonic function) is a linear PDE for which one can derive analytical fundamental solutions in any dimension whereas this is generally not the case for the Hamilton-Jacobi-Bellman PDE. This does not make a real difference when one uses a relaxation technique for iteratively computing the potential. However, the linearity and the existence of such fundamental solution was exploited by Viéville et al. [START_REF] Viéville | An improved biologically plausible trajectory generator[END_REF] to tackle the curse of dimensionality: the harmonic function is decomposed in a sum of fundamental (possibly high-dimensional) harmonic functions. With such an approach, the authors can compute trajectories for a 10degrees of freedom virtual robotic arm in the presence of obstacles. Extending this idea to optimal control in general does not seem obvious and constitutes future research.

VI. CONCLUSION

This paper presented a link between harmonic control and optimal control: we argued that optimal control subsumes harmonic control. The results presented show formally that for a well chosen optimal control problem, the value function becomes harmonic when the noise level tends to infinity and that in both cases, control consists in following the gradient of these functions. The convergence is showed in the continuous domain and in a standard finite difference scheme. We believe that such an analysis provides more insight on harmonic control. We discussed different properties of both theories, and showed that many interesting properties of harmonic control are shared by optimal control. However, a specific power of harmonic controllers lies in the ability to implement them on very low level hardware as resistive arrays, and this does not seem to be the case for the Hamilton-Jacobi-Bellman equation. It would be interesting to further investigate the possibilities for such hardware implementations of optimal control. Furthermore, harmonic control relies on the computation of a simple linear PDE, which has some analytical fundamental solutions; this has recently been exploited to tackle the curse of dimensionality [START_REF] Viéville | An improved biologically plausible trajectory generator[END_REF]. The transposition of this work to the optimal control framework is currently under investigation.

Fig. 1 .

 1 Fig. 1.(a) Transition probabilities p θ [(x, y), (x , y )] in the discrete model: neighbors in direction θ have a component that is proportional to the coordinates of the unit vector -→ u (θ). (b) A discrete representation of J δ , illustrating the orientation of the gradients, and an (approximate) optimal trajectory computed by following the gradient.

Fig. 2 .

 2 Fig.2. Evolution of trajectories with respect to the noise level σ. Each figure illustrates different trajectories leading to goals G 1 and G 2 , and starting at equally distributed grid points in the environment. All trajectories where computed using optimal control (with J δ σ ) except the last one which is derived from harmonic control (with Φ δ ).

Isotropic noise means that the noise is the same in every direction.

In this case, the analysis of the HJB is a bit more complex, involving viscosity solutions (see[START_REF] Munos | A study of reinforcement learning in the continuous case by the means of viscosity solutions[END_REF]) which we do not address in details here.