Luciano Da

F Costa

On the Effects of Resistive and Reactive Loads on Signal Amplification

The effects of reactive loads into amplification is studied. A simplified common emitter circuit configuration was adopted and respective time-independent and time-dependent voltage and current equations were obtained. As phasor analysis cannot be used because of the non-linearity, the voltage at the capacitor was represented in terms of the respective integral, implying a numerical approach. The effect of purely resistive loads was investigated first, and it was shown that the fanned structure of the transistor isolines can severely distort the amplification, especially for Va small and s large. The total harmonic distortion was found not to depend on Va, being determined by s and the load resistance R. An expression was obtained for the current gain in terms of the base current and it was shown that it decreases in an almost perfectly linearly fashion with IB. Remarkably, no gain variation, and hence perfectly linear amplification, is obtained when R = 0, provided maximum power dissipation limits are not exceeded. Capacitive loads imply the detachment of the circuit trajectory from a straight line to an "ellipsoidal"-like loop. This implies a gain asymmetry along upper or lower arcs of this loop. By using the time-dependent circuit equations, it was possible to show numerically and by an analytical approximation that, at least for the adopted circuit and parameter values, the asymmetry induced by capacitive loads is not substantial. However, capacitive loads will imply lag between the output voltage and current and, hence, low-pass filtering. It was shown that smaller Va and larger s can substantially reduce the phase lag, but at the cost of severe distortion.

"There is geometry in the humming of the strings."

Pythagoras

I. INTRODUCTION

The dynamics of the natural world is underlain by myriad signals covering all scales of magnitude. From the light of the faintest star to the sound of the loudest thunder (not to mention quarks and supernovae), signals permeate and integrate the whole universe. From Pitagora's time, it has been known that natural signals (in that case, sound) can be understood in terms of basic harmonic functions, each with a respective magnitude. Interestingly, the way in which signals are changed by environment, such as the propagation of a bird songs throughout a forest, can be understood as changes, in specific ways, in the magnitudes of each of the signal harmonic components. For simplicity's sake, these signal magnitude alterations are henceforth called amplification, even when the resulting signal has its amplitude reduced. So, it can be said that a good deal of signal processing resumes to magnitude alterations. It is consequently hardly surprising that the development of science and technology has relied so much on the acquisition, * luciano@ifsc.usp.br amplification and analysis of signals. Indeed, all instruments critical for measuring nature, from microscopes to telescope arrays, require accurate, reliable and linear amplification of signals. A very much similar situation is found regarding the vast majority of human activity: linear signal amplification is required in telecommunications, health/leisure, and transportation, among many other critical areas. Interestingly, sometimes humans also aim for non-linear amplification, as is the case with music instrument synthesis.

Analog (or linear ) electronics (e.g. [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]), more recently accompanied by digital amplification (e.g. [START_REF] Colli-Menchi | Design Techniques for Integrated CMOS Class-D Audio Amplifiers[END_REF]), is the main scientific-technologic area focusing on the amplification of natural signals. It has provided the foundation for much of instrumentation used in science and technology. The main goal here often is to achieve as much linear operation as possible, which is a rather challenging issue given the intrinsically non-linear nature of real-world components, such as transistors. Interestingly, our world is predominantly non-linear, being rather difficult to think of natural phenomena that are perfectly linear. At the same time, most of our approaches to science, at least until recently, relied on mapping non-linear real-world phenomena into approximated linear representations. This also happened with transistor studies, as often (but not always) these devices have been mapped into linear respective representations.

Indeed, because of its many important and critical applications, much effort has been invested in deriving effective approaches to transistor characterization, represen-tation and modeling (e.g. [START_REF] Horowitz | The Art of Electronics[END_REF][START_REF] Jaeger | Microelectronic Circuit Design[END_REF][START_REF] Sedra | Microelectronic circuits[END_REF][START_REF] Stewart | Circuit Theory and Design[END_REF][START_REF] Streetman | Solid State Electronic Device[END_REF]). Though in the beginning, especially from the 50's to the 70's, these efforts tended to focus on graphic and analytical approaches -with extensive use of characteristic curves, surfaces, and transfer functions (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]), the progressive extension to larger and more complex integrated devices was largely assisted by more systematic application of numeric computational simulations. Yet, continuing research into more analytical approaches to transistor characterization, representation and modeling remains an important issue as it can not only extend our intuitive understanding of transistor operation, but also contribute do more effective and accurate simulation approaches and tools. Recently, a simple transistor model, as well as its respective equivalent circuit, was developed [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] that is founded 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]. Despite its great simplicity, consisting of a Thevenin configuration with a fixed voltage source V a in series with a variable resistance R o (I B ) = 1/tan(sI B ) (I B is the current base, and s a proportionality parameters), this model exhibits two interesting and important features: (i) it allows the representation of the transistor varying output resistance in terms of the modulating base current (corresponding to the fanned isolines that are so characteristic real-world transistor graphic representations); and (ii) the two involved parameters, namely V a and s do not vary with the transistor operation as quantified by the collector current I and voltage V C . Recall that both the parameters β and R o adopted in many traditional transistor modeling approaches are explicit functions of I and V C , i.e. β = β(I, V C ) and R o = R o (I, V C ). This parametric dependence implied that averages, or other functionals, of these two parameters had to be used for the characterization of transistor properties [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF], implying in specific or arbitrary choices about in which domain to integrate the parameters β = β(I, V C ) and

R o = R o (I, V C ) into respective averages β and R o .
Despite its recent introduction, the Early modeling approach has already led to interesting results, such as the characterization of typical properties of NPN and PNP bipolar junction transistors (BJTs) [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF], the study of germanium alloy junction devices [START_REF] Da | Characterizing germanium junction transistors[END_REF], the derivation of a prototypic parameter space characterizing trade-offs between gain and linearity [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], the study of stability of circuit operation under voltage supply oscillations [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], as well as the derivation of the Early equivalent model of parallel transistor configurations [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF]. The aforementioned study of circuit stability involved a comparison with a more traditional simplified modeling approach, which showed that largely divergent results can be obtained when considering or not the fanned structure of the base currentindexed isolines, and justifying the use of models such as the Early approach. Indeed, the development of an equivalent circuit for the Early model [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF] paved the way to applying this modeling approach to a large number of relevant problems regarding transistor characterization, modeling, simulation, and design in discrete and integrated electronics.

The present work addresses the issue of transistor amplification in circuits involving reactive loads, a situation very commonly found in practice. While purely resistive loads imply a straight line in the transistor characteristic surface, capacitive or inductive loads will define "ellipsoidal"-like pathways in these parametric spaces, also implying phase shifts between the current and voltage at the load. While these alterations would still lead to linear amplification were the transistor a linear device (i.e. had parallel, equispaced current modulated-isolines), the fanned structure actually observed in real-world devices implies that the achieved amplification will become non-linear not only because the load line will no longer be straight, but also because different paths will be followed for the positive and negative portions of the amplified signal. The fact that the induced phase shifts depend on the frequency of the input signal complicates further the situation. The availability of the Early model and respective equivalent circuit provides a viable and yet accurate way to approach this interesting and important problem of non-linear amplification considering reactive loads, and this constitutes the main motivation of the current work.

A simplified common emitter configuration is adopted in which there is no emitter resistance (no negative feedback) and the load is attached between the transistor collector and the external voltage supply V CC . By using the Early equivalent circuit [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], it was possible to obtain the time-independent and time-dependent equations for the currents and voltages in the adopted circuit. The case of purely resistive loads is studied first by using the respective time-independent circuit equations, which allowed the total harmonic distortion of the amplification to be estimated for each of the possible configurations in a region of the Early space parameter, leading to remarkable results. The case of capacitive loads is studied next, and its effect on linearity and phase lag (low-pass filtering) are studied by using numeric as well as analytical approximations, also with surprising results. The work concludes by identifying prospects for future related research.

II. A SIMPLIFIED COMMON EMITTER CONFIGURATION AND ITS EARLY MODEL REPRESENTATION

Figure 1 illustrates the simplified common emitter configuration considered in this work, where Z * stands for a generic load with resistive and reactive properties. The adopted circuit configuration is said to be simplified because the base is does not incorporate some features typically found in the common emitter configuration, i.e.: (i) the base is not biased by a voltage divider, and the input signal is driven through a base resistance R B ; (ii) there is no resistance R E between the emitter and the ground, which is often used to achieve negative feedback; and (iii) the load is attached directly between the collector and the external voltage supply V CC , not attached to the collector through an additional decoupling capacitor.

The adopted simplifications, however, do not change much the properties of the circuit with respect to the more traditional common emitter configuration. For instance, collector bias can be incorporated as part of the input current I B so that, mathematically, the same effect can be achieved. The avoidance of the emitter resistance is compatible with the objective of the present work, which focus on the characterization of the circuit linearity under reactive loads, so that it is more interesting to consider the situation with more distortion. The situation with R E will be considered in a forthcoming work focusing on the study of negative feedback by using the Early formulation. Regarding simplification (iii), this does not affect the characterization of the linearity of the transistor and the so-developed analysis can be immediately extended to other situations more commonly found in practice, such as attaching the load to the collector via a decoupling capacitor.

FIG. 1: The simplified common emitter configuration adopted in the present work. The simplifications do not significantly impact the respective circuit analysis, which can be immediately extended to other circuit configurations.

The Early model equivalent circuit [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF] can be easily em-ployed to obtain the mathematical equations underlying the behavior of the circuit in Figure 1. First, we replace the transistor by the Early model equivalent circuit [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], yielding the circuit shown in Figure 2. The input section is modeled in terms of a diode, as usual, so that we have:

I B (I B ) = V i -V r R B + R i (1)
V B (I B ) = V R + V i -V r R B + R i R i (2)
Recall that the Early representation of the output port involves two parameters: the Early voltage V a and a proportionality parameter s. One of the distinguishing features of the Early model is that both these parameters do not depend on I or V C . In addition, these parameters have been formally related to averages of the more frequently adopted current gain and output resistance. Interesting, as a consequence of scaling properties of the fanning geometrical structure underlying the Early model, the effective current gain results, in an approximate sense, directly proportional to the product of V a magnitude and s, i.e.

β ≈ s |V a |. This can be immediately appreciated by considering that: (a) if V a is doubled, the effective current gain will be halved; (b) if s is doubled, this same gain is doubled. So, the net product is not to change the effective current gain. We also have that R o ∝ |V a |. It is interesting to perform the circuit analyses and discussions considering distinct parameter configurations leading to the same current gain.

In the respective equivalent circuit [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], the parameter s defines the output resistance R o (I B ) = 1/tan(sI B ) that is a function of the base current I B given a fixed value of s. Observe also that V a takes negative values. Now we can apply electric circuit analysis, in particular applying Kirchhoff's and Ohm's law, to derive equations respectively describing the adopted simplified common emitter circuit. For simplicity's sake, we first consider a purely resistive load R, which yields the follwing timeindependent equations:

I (I B ) = V CC -V a R + Ro(I B ) (3) V p (I B ) = R V CC -V a R + Ro(I B ) (4) V C (I B ) = V CC -R V CC -V a R + Ro(I B ) (5)
The obtained equations are similar in complexity [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF] to those obtained for more traditional models based on the current gain β and output resistance R o = 1/tan(sI B ).

However, a slightly more sophisticate situation arises when we considered reactive loads. Now, because the transistor is non-linear, so that the voltage and current values will vary in a non-linear way as functions of f , we cannot use the customary steady state analysis and phasor methods. Instead, we need to substitute the reactive components by their integral or differential formulation, so as to define respective frequency-dependent voltage or current terms in the respective time-dependent circuit equations.

We illustrate this procedure with respect to a capacitive load involving a capacitor C in parallel with a resistance R, as illustrated in Figure 3. For this configuration it is convenient to consider the voltage formulation for the load, so that we have the following, relatively simple, time-dependent equations:

I (I B ) = V CC -V a -V p (I B ) Ro(I B ) (6) V C (I B ) = V a + R o (I B ) V CC -V a -V p (I B ) Ro(I B ) (7)
V p (I B ) = V CC -V C (8)
At the mesh defined by the load, Kirchhoff's voltage law yields:

V R I R = V C (9)
At that same mesh, Kirchhoff's current law implies that I = I R + I C , so that we obtain:

I C = I -V C /R L (10)
Now we can derive the voltage across the capacitor as:

V p = 1 C t 0 I c (t)dt (11)
The so obtained time-independent and time-dependent equations provide all the necessary resources for analyzing the amplification properties of the circuit in Figure 1 while driving purely resistive and resisitive-reactive loads. However, first we provide some considerations regarding the numerical solution of the above circuit.

III. A SIMPLE NUMERIC APPROACH

Because steady state phasor methods cannot be used to perform AC studies on non-linear circuits with reactive loads, it is necessary to consider other methods. In this work, capacitors or inductors are substituted, as implied by each specific circuit topology, by their respective integro and differential equations, i.e.:

v C (t) = 1 C t f ti i C (t)dt (12) i C (t) = C dv C (t) dt (13) v L (t) = L di L (t) dt ( 14
) i L (t) = 1 L t f ti v L (t)dt ( 15
)
These equations can then be substituted for respective voltages or currents in the circuit, whenever implied by the circuit topology and focus of study. The integrals or differentials are then interactively estimated during the circuit analysis . Let's illustrate this approach with respect to the circuit in Figure 3. A voltage-based analysis of this circuit yielded Equations 6 to 8. Assuming that the capacitor is initially discharged (i.e. v C (t = 0) = 0V , we can derive the following interactive loop, aimed at obtaining the circuit solution from t i = 0 to t, with time resolution dt = t/(N t -1):

Algorithm 1 Numeric solution with capacitive load. Vp ← 0 3: Observe that this algorithm uses a simple trapezoidal integration scheme, so that more demanding situations may require more precise/sophisticated numeric integration methods (e.g. [START_REF] Press | Numerical Recipes: The Art of Scientific Computing[END_REF]). In the present work we have used N t > 100000, but this choice needs to take into account the frequency limit and reactance and resistance maximum values, so that larger values o N t may be required in other parametric configurations.

for it = 1:Nt do 4: Ro ← 1/tan(s * Ib[it]) 5: I[it] ← (V CC -V a -V p)/Ro 6: V C[it] ← V CC -V p 7: Ic ← IC[it] -V p/R 8: V p ← V p + Ic/C * dt

IV. AMPLIFICATION ONTO PURELY RESISTIVE LOADS

Now we proceed to investigate the characteristics of amplification of the circuit in Figure 1 by considering the case of purely resistive loads. The behavior of the circuit for this configuration is given by the time-independent Equations 3 to 5.

We adopt a sinusoidal signal I b (t) with frequency 1kHz and amplitude 60µA. Figure 4 shows the trajectories defined by the signals I(t) and V C (t) in the circuit operation space (V C , I) for purely resistive loads with values R = 30, 60and150Ω. The adopted parameters defining the transistor operation in these cases were V a = -50V and s = 10V -1 . These Early parameters values can be found in PNP BJT transistors [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF]. Any trajectory for purely resistive loads R will be, necessarily, fixed and straight. This is a consequence of the fact that, for this type of load, the circuit operation is restricted to the straight line extending from (V C = 0, I = V CC /R) to (V C = V CC , I = 0). Nevertheless, the obtained trajectories will have inclination equal to 1/R, as in Figure 4. So, larger values of R will imply smaller transfer function slopes. The total harmonic distortion (THD) can be employed to effectively summarize the degree of distortion implied by the amplification non-linearities, which incorporate new harmonics into the amplified signal. This index is defined as:

T HD = V 2 2 + V 2 3 + . . . V 1 ( 16
)
where V i is the RMS value of the i-th harmonic component, and V 1 is the RMS value of the fundamental which, in our case, corresponds to the pure sinusoidal input I b (t).

The THD, considering only harmonic components with relative intensity larger than 1e -9 , obtained for V a = -50V and s = 10 was found to be equal to T HD(-50, 10) = 0.2999995. The THD obtained for the Early parameter configuration V a = -200V and s = 2.5 was T HD(-200, 2.5) = 0.1499999, yielding the ratio T HD(-50, 10)/T HD(-200, 2.5) ≈ 2, which is substantial. This result corroborates the fact that the fanned geometric structure underlying real-world transistors can have a major non-linear effect, implying substantial distortion on the amplification. Indeed, this could have been already guessed even by visually comparing the transfer functions obtained for the two considered parameter configurations, as that for V a = -50V and s = 10 (Fig. 5) is much more bent than that for V a = -200V and s = 2.5 (Fig. 6).

Such a strong effect of the transistor Early parameters V a and s on amplification linearity and distortion motivates a more systematic analysis. Figure 8 shows the THD values obtained for R = 150Ω, V CC = 10V , and amplitude of I B (t) equal to 60µA. These values were estimated by using the derived time-varying equations to obtain the output signals, followed by respective fast Fourier numerical analysis (e.g. [START_REF] Brigham | Fast Fourier Transform and its Applications[END_REF][START_REF] Costa | Shape Classification and Analysis: Theory and Practice[END_REF]). Remarkably, the distortion implied by the transistor inherent non-linearity depends only of the transistor Early parameter s, not of V a . This result is in full agreement with previous Early model investigations of the considered simplified common emitter circuit configuration [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF]. The smaller the value of s of a given transistor, the smaller the THD will be, and this is the only effect determining distortion as far as the fanning geometrical organization of the transistor isolines is concerned. However, considering nearly constant current gains β, if s is reduced, V a tends to increase. So, linearity can be optimized under the current assumptions by choosing s small and V a large, which is a characteristic of silicon NPN BJTs (see [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF]). Recall that no distortions at all would have been otherwise observed in case a more traditional model based on current gain β and output resistance R o , with parallel (even though inclined) characteristic isolines had been used.

It can be verified that smaller values of R will yield THD surfaces with similar shapes, but with proportionally smaller values. This is a consequence of the fact that, for smaller values of R, the load lines will approach the vertical, where the current gains implied by the radiating I B -indexed isolines are more uniform. This interesting phenomenon is further investigated in the following section.

V. AMPLIFICATION ONTO CAPACITIVE LOADS

The so far obtained results for purely resistive loads corroborate the fact that the non-linearities implied by the fanning structure of the transistor isolines can impinge large distortions on the respective amplification, depending on the Early parameters of the chosen transistors and the load resistivity. Now, we proceed to investigate how this influence takes place when the load incorporates a capacitive element, as in Figure 3.

First, let's consider the effect of a capacitor C = 250nF placed in parallel with a resistive load R = 150Ω. Figure 9 illustrates the obtained trajectories defined by V C (t) and I(t) in the circuit operation space (V C , I) for a pure sinusoidal inputs I B (t) with amplitude 60µA and frequencies f = 20Hz (a), f = 70Hz (b), f = 300Hz (c), and f = 1kHz (d). This figure also shows (in blue), as a reference, the load line obtained for a purely resistive load R = 150Ω. The adopted transistor parameters are V a = -50V and s = 10.

The transient dynamics can be observed, proceeding from the initial condition to the steady state regime. As the input frequency increases, the circuit operation progressively detaches from the straight line constraint implied by a pure resistive load, defining "ellipsoidal"-like trajectories. This is a direct consequence of the fact that, in loads defined by resistance in parallel with capacitance, the voltage will be "behind" (or "lag") the current. However, the non-linearities implied by the fanned transistor isolines preclude the application of phasor analysis typically adopted in linear steady state conditions (e.g. [START_REF] Hyat | Engineering Circuit Analysis[END_REF][START_REF] Jaeger | Microelectronic Circuit Design[END_REF]).

Observe that this detachment of the trajectories implies the distortions experienced along the upper trajec-tory cycle arc to be different from those along the lower arc. Now, it is a particularly intriguing question to investigate how this asymmetry can further impact the amplification non-linearity. Figure 10(a)-(d) shows the input I B (t) and output signals I(t) obtained for each of the four cases in Figure 9 (V a = -50V and s = 10). Observe that I B (t) has been suitably scaled up so that its shape can be directly compared to that of I C (t).

Interestingly, the distortions implied by this circuit configuration are very similar to those observed in Figure 5(b). Indeed, THD comparisons showed that the trajectory asymmetry implied by the capacitive load does not affect significantly -at least for the considered circuit, parameter and variable configurations -the linearity of the amplification. This is an interesting phenomenon that can be further investigated analytically by using two parallel purely resistive load lines, as in Figure 11 to approximate the gain variation as the trajectory excursions along the upper and lower arcs of the loop. Observe that the displacement of these load lines is defined by the supply voltage V CC .

We have from the Early modeling of the considered circuit with a purely resistive load R has its output current I(I B ) expressed as:

I(I B ) = V CC -V a R + R o (I B ) = (V CC -V a ) tan(sI B ) R tan(sI B ) + 1 (17)
It follows that the current gain β(I B ) along the load line, in terms of I B is given as follows:

β(I B ) = ∂I(I B ) ∂I B = s (V CC -V a ) csc 2 (s I B ) (cot(s I B ) + R) 2 (18)
Observe that this current gain is a function of the transistor Early parameters V a and s as well as the circuit parameters R, and V CC , as well as I B . Now, we take the output current I α (I B ) along a load line parallel (same R) to the previous one. This new load line has it position along the horizontal axis determined by ξV CC , where ξ is a proportionality coefficient (e.g. ξ = 1.1):

I ξ (I B ) = (ξV CC -V a ) tan(sI B ) R tan(sI B ) + 1 ( 19
)
The current gain β α (I B ) along this load line, in terms of I B , can be derived as:

β ξ (I B ) = s (ξV CC -V a ) csc 2 (s I B ) (cot(s I B ) + R) 2 (20)
Now we can obtain the proportion ρ between the current gain between the second and first load lines as:

ρ = β ξ (I B ) β(I B ) = ξV CC -V a V CC -V a (21)
Remarkably, ρ turns out to be independent of I B , s and R, varying only with V CC , V a and ξ. This is shown in Figure 12 for 5 ≤ V CC ≤ 50, where successive constant gains are obtained for V CC steps of 50V .

Figure 13 illustrates the current gain β(I B ) in terms of 0 ≤ I B ≤ 15µA, for 50Ω ≤ R ≤ 5kΩ. As a consequence of the I B -indexed fanned characteristic of the isolines underlying the transistor operation, the current gain varies in a virtually linearly decreasing way with I B for all considered values of R. Remarkably, as less gain variation implies in enhanced linearity, almost perfect linear amplification is achieved for loads with very smal R. Indeed, perfect linear operation is achieved in the limit R = 0Ω, provided I B (t) is controlled so as to have I C (t) not exceeding the voltage supply maximum current and the transistor maximum power dissipation. Now, substituting for V a = -200V , V CC = 10V and ξ = 1.1, we have that ρ = 1.004762. For a good average gain (NPN) β = sV a = 2.5| -200| = 500, this implies in a relative gain variation of ((500)1.004762) + 500)/500 = 0.004762, which is negligible given the typically obtained values of gains, parameters and variables for the considered configurations.

So, for the considered hypothesis and configurations, the capacitive component of the load does not strongly influences the linearity of the amplification despite the respectively implied asymmetry along the upper and lower arcs of the loop. This happens particularly for large values of V a , which is known [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] to be directly related to the transistor average output resistance R o (I B ) , irrespectively of s. So, transistors with larger values of V a not only will have better linearity for a same gain, but will also (and for the same reasons) imply better resilience to capacitive load effects. However, large output resistance also implies bigger lag between the output voltage and current, and hence typically unwanted low-pass filtering, which is another important aspect deserving further consideration.

Figures 10(e)-(h) show the collector voltage V C (t) superimposed onto the input current I B (t), with respect to f = 20Hz (e), f = 70Hz (f), f = 300Hz (g), and f = 1kHz (h), assuming V a = -50V and s = 10. Observe that V C had to have its signal inverted, and then I B (t) was scaled up so as to allow the comparison between the shape of these two signals. Unlike what was verified between the input and output currents, which is always in phase, there is a very perceptible lag between the output voltage and current that increases with f , even though this is not a particularly high frequency.

So, the choice of large values of V a implies larger average output resistance R o that tends to worsen the lag, implying more intense low-pass filtering. So, a trade-off is revealed between linearity and low-pass filtering implied by capacitive loads. This tends to become particularly critical for loads with large resistive or capacitive components (i.e. large τ = RC).

Because filtering represents a distortion of the original signal, it is important to try to devise means for avoiding the effects of capacitive loads. An immediate possibility is to use transistors with smaller V a , leading to smaller average output resistance R o . Figures 10(i)-(l) depict the output voltage V C (t) and input current I B (t) for V a = -10V and s = 50 (same gain as before, but much smaller output resistance), with respect to f = 20Hz (e), f = 70Hz (f), f = 300Hz (g), and f = 1kHz (h). Though a large distortion has been implied to the voltage signal, the smaller output resistance allowed by V a = -10V allowed an almost perfect phase match between the input current I B (t) and the output voltage V C (t). However, the severity of the impinged distortion renders this solution unviable for most applications, except as a possible means of dealing with synchronization problems.

Another possibility to avoid the complications implied by capacitive loads is to add an inductor in series with the load, with reactance perfectly matched so as to cancel the load capacitance (in a manner remindful of load correction), yielding a purely resistive load as a result. Interestingly, the compensation required for any given signal frequency will also work for other frequencies. This possible technique is allowed as a consequence of the transistor non-linearities having been found in this work, at least for the considered hypotheses, not to influence too much the asymmetry of the amplification along the two arcs of reactive loops, otherwise the compensation could be undermined. This possible technique also requires the capacitance to be canceled not to be a function of other parameters (e.g. voltage, electromagnetic field or temperature) than the signal frequency.

VI. CONCLUDING REMARKS

Linear amplification remains a critically important issue for a vast number of electronics applications required by science/technology and as assistance to humans. However, the non-linearity intrinsic to transistors has substantially constrained more systematic and complete studies of transistor-based amplification. The recent development of a simple and yet accurate transistor model [START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF][START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF][START_REF] Da | An Early model of transistors and circuits[END_REF] capable of incorporating the key nonlinearities characterizing the operation of these devices, paved the way to allowing more accurate studies of transistor amplification. One particularly relevant and challenging aspect related to the non-linearity of transistor amplification concerns the influence of loads with reactive components on the circuit operation. This issue has been particularly complicated to be approached because of the lack of simple and accurate transistor models capable of reflecting the fanned structure of the characteristic surface of real-world transistors. This structure implies non-linearities precluding the complex phasor approach commonly used in AC circuit analysis to be used. To address this problem in a more systematic way provided the main motivation for the present work.

A simplified common emitter circuit configuration was adopted and represented in terms of its Early model, which involves only two parameters, V a and s, that are invariant to the circuit operation as defined by the base current I B (t), collector current I(t) and collector voltage V C (t). The Early equivalent circuit of a transistor was then applied to derive simple, and yet accurate time-independent and time-dependent voltage and current equations describing the circuit behavior for purely resistive loads as well as for reactive loads. The approach in this work focused capacitive loads because the inductive counterpart can be treated in a completely analogue way except for the sign of relative lag between the output voltage and current. As the non-linearities implied by the fanned characteristic of real-world transistors preclude the application of well-established phasor AC analysis, we resorted to a numeric-interactive solution in which the integral expression for the voltage in a capacitor is integrated by using a simple numerical method (trapezoid method), which was presented in detail.

In order to provide a reference for the analysis of capacitive loads, we investigated the purely resistive case first. This was done by using the simpler time-independent circuit equations directly derived from the Early approach. Several interesting results were obtained. In particular, it was shown that the Early parameters V a and s have major effect on the linearity of the transistor amplification. This effect is revealed mostly as a saturation of the current gain as I B increases, as a consequence of the transistor radiating isoline characteristics. This implies a gain asymmetry between the positive and negative cycles of the AC version of the input current. Transistors with larger V a magnitude and smaller s tend to yield improved linearity for the same gain of transistors with smaller V a magnitude and larger s capable of delivering the same current gain. Power spectra obtained for such different parameter choices revealed a linearly decaying magnitude along the lower harmonics, an effect that is much more pronounced in the case of large s and smaller V a . This different spectral composition motivated the considera-tion of total harmonic distortion (THD) measurements. These measurements were easily obtained by using the time-dependent version of the circuit equations, followed by fast Fourier transform, allowing the systematic identification of the THD for each possible parameters configuration in wide region of the Early parameter space (V a , s). The obtained, more accurate, results confirmed previous related studies using gain approximation equations [START_REF] Da | Towards a simple, and yet accurate, transistor equivalent circuit and its application to the analysis and design of discrete and integrated electronic circuits[END_REF], with the THD depending only of s in an almost linear way, while being completely independent of V a . The THD for the considered circuit and parameter configurations was also found to decrease with smaller values R of the resistive load, with the remarkable result that perfectly linear amplification is achieved for R → 0Ω, provided the transistor and circuit maximum dissipations are observed.

The reactive load situation was addressed next, with focus on capacitance. This case is particularly interesting because the lag between voltage and current, which increases for larger signal frequencies, implies the trajectory defined by the circuit operation in the (V c , I) space to detach from the straight line characterizing purely resistive loads and for "ellipsoidal"-like loops. So, as the circuit goes along the upper or lower arcs of this loop, as I B varies, it experiences different non-linearities as a consequence of the fanned geometry of transistor amplification. This could, in principle, complicate the intrinsic non-linearities already observed for purely resistive loads. In the case of typical Early parameter values configurations, and under the considered choices and parameter configurations, it was found numerically and theoretically that the transistor linearities implied by the loop-induce gain asymmetry are relatively minor in the sense that most of the impinged distortions are a consequence of the transistor fanned structure solely on the resistive load. The analytical approach to this result involved the construction of an approximation to the loop-induced gain asymmetry by using two parallel straight load lines. It was remarkably shown that under these circumstances the ratio of the current gains along the two parallel load lines does not depend on I B or R. As a matter of fact, the current gain was found to vary in almost perfectly linearly decreasing fashion with I B , which accounts for the gain independence of I B .

As the transistor non-linearities were found not to significantly contribute additional distortions as a consequence of the gain asymmetries along the upper and lower arcs of the loop, choosing transistors with larger V a will improve the linearity of the amplification. However, this also increases the average output resistance R o , implying in larger lags between voltage and current. A trade-off is therefore established between linearity and phase lag (low-pass filtering) by choice of larger V a . The possibility to try reducing the lag by using smaller V a (and therefore smaller R o resulted in almost null lag, but also in severe signal distortion. As an alternative for avoiding the unwanted capacitive effects on transistor amplification, a series inductor can be incorporated to the load which, when properly matched, can eliminate the capacitive behavior. This requires both capacitance and inductance not to depend on other parameters such as temperature or voltage.

The reported methodology and results pave the way to many future investigations. In particular, it would be interesting to consider other amplifier configurations, such as more conventional common emitter circuits. In such cases, it would be also worth studying the contribution of negative feedback on enhancing linearity, as it has been experimentally shown that negative feedback may not be able to completely eliminate parameter variations that are intrinsic to real-world transistors. It would also be interesting to consider other parameter configurations, such as higher frequencies and V C C, as well as more substantial capacitance and resistor values. The remarkable lag elimination provided by transistors with small V a and large s can also be eventually applied in electronic synchronization problems. Another research worth of future investigation is the case of controllable loads, such as amplification taking place over transistors. FIG. 11: The geometrical construction that can be adopted in an approximate study of the asymmetry of gains in the upper and lower arcs of capacitive loops in load lines. The two arcs can be approximated by two parallel straight load lines, one for V CC (lower arc of the loop) and the other for an incremental increase ξV CC (upper arc). Remarkably, as derived in the text, the ratio between the current gains along any such two parallel (same R) load lines, even when further apart, does not depend on s or R, being solely a function of V CC , Va and ξ.

FIG. 2 :

 2 FIG. 2: The equivalent circuit obtained by replacing the transistor in Figures 1 by the respective Early model equivalent circuit.

FIG. 3 :

 3 FIG.3:The output mesh of the circuit configuration in Figure2, with the load Z * corresponding to a capacitor C in parallel with a resistor R.

1 :

 1 procedure SolveCap(Va,s,f,VCC,R,C,Nt,Ib) 2:

FIG. 4 :

 4 FIG. 4: The trajectories of the signals I(t) and V C (t) defined in the (V C , I) space for purely resistive loads R = 30Ω (a), 60Ω (b) and 150Ω (c) are perfectly straight. These results consider Va = -50V and s = 10, a parameter configuration that can be found in silicon PNP BJTs [4].

Figure 5 (

 5 a) shows the transfer function[START_REF] Silva | Seeking maximum linearity of transfer functions[END_REF] defined by the transistor while going from I b (t) to I(t) (i.e. I(t) = g(I B (t)), and Figure5(b) shows the superimposition of I(t) and I B (t), the latter having been scaled to match the amplitude of I(t). It is evident from this figure that the fanned structure of real-world transistors can strongly deform the original signal, as reflected in the bent transfer function and the mismatch between the shapes of the input and output currents. An overall current gain β ≈ s |V a | = 500 was obtained.

FIG. 5 :

 5 FIG. 5: The transfer function (a) defined by the considered circuit for Va = -50V and s = 10. This curve exhibits a marked curvature, which implies the amplified current I(t) (b, solid line) to be substantially deformed as compared to a scaled version of the pure sinusoidal input I B (t) (b, dotted line).

Figure 7

 7 Figure 7 depicts the power spectrum of the original signal I B (t) (a) and of the amplified signal I(t) (b).The distortion implied by the transistor geometry on the lower harmonics is evident. Observe that the residual power spectrum values smaller than -50dB are a consequence of numerical round-off noise, and can be disregarded. Now, we repeat the study above, but considering V a = -200 and s = 2.5, which yields approximately the same current amplification. This parametric configuration can be found in silicon NPN BJTs[START_REF] Da | Characterizing complementary bipolar junction transistors by early modeling, image analysis, and pattern recognition[END_REF]. Figure6depicts the respectively obtained transfer function (a) and superimposition of the input and output currents (b). The obtained transfer function resulted substantially less bent for for V a = -200V and s = 2.5, implying in much smaller amplification distortion. This confirms that the Early transistor parameters can have great influence on the amplification linearity. The power spectrum of the output current I(t) obtained is shown in Figure 7(c).The total harmonic distortion (THD) can be employed to effectively summarize the degree of distortion implied by the amplification non-linearities, which incorporate

FIG. 6 :

 6 FIG. 6: The transfer function (a) defined by the considered circuit for Va = -200V and s = 2.5. This curve exhibits a less marked curvature, and hence smaller distortion, than that in Figure5.

FIG. 8 :

 8 FIG. 8: The THD values estimated for each of the Early parameter configurations in the rectangular domain 0 ≤ s ≤ 10 and -200V ≤ Va < 0. The THD is found to vary only with s, and not with Va, increasing in an almost linear way with s. The THD is also a function of R, assuming similar shapes but smaller absolute values for smaller values of R.

FIG. 9 :FIG. 10 :

 910 FIG. 9: The trajectories in the circuit operation space (V C , I) defined by the input I B (t) and output I(t) currents for R = 150Ω, C = 250nF and frequencies f = 20Hz (a), f = 70Hz (b), f = 300Hz (c), and f = 1kHz (d). Observe the transient converging (for this range of frequencies) swiftly to an "ellipsoidal"-like loop. The detachment of the trajectory from the resistive load line increases with f to the point (not shown in this figure) as to reduce the V C (t) voltage gain. The load lines for R = 150Ω are also shown (in blue) for reference, and the considered transistor is characterized by Va = -50V and s = 10.

FIG. 12 :

 12 FIG. 12: β is constant with I B for 5 ≤ V CC ≤ 50, increases with equispaced steps as the Va magnitude is drecreased. Results obtained for R = 150Ω, Va = -200 and s = 2.5 with 50V steps.

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


Mohamed Tahar

The Pilot Project for the Library Mohamed Tahar in Timbuktu

On the fringe of the Saharan desert, nestled in the elbow of the Niger River, Timbuktu is one of Africa's most fabled cities. A myth envelopes the city of Timbuktu. In his historical work Tārīkh al-Sūdān, a local scholar in the 17 th century, al-Sa'dī, speaks of an "exquisite city, pure, delicious and illustrious". In the Middle Ages, for Arabs and then for Europeans, Timbuktu was a fascinating city. The Arabs knew the richness of the Songhai Empire (1464-1492) and the Askiya Dynasty (1493-1595). Europeans spoke of the treasures transported in caravans toward the city in the midst of the sands. Timbutch and the name of the emperor Mansa Musa were inscribed on a Catalan atlas made in the 14 th century for King Charles V of France.

The Muslim Africans have a completely different image of Timbuktu. They speak first of a center for the transmission of knowledge. In the 16 th century there was in the city a "virtual industry for the copying of works coming from the North". 1 It was also a center of diffusion for Muslim mysticism. Timbuktu was the city of 333 saints, 333 being a symbolic and magical number.

At the beginning of Islam, from the 8 th to the 11 th century, sub-Saharan Africa was first considered as a territory with a many riches: gold, ivory, precious species of wood, and slaves. Ibadite Berbers from the Maghreb developed commerce along the caravan routes, but few traces remain of the Islamization at that time. 2 In the 13 th century Timbuktu became a way-station at the heart of Saharan commerce.

In 1352, when Ibn Baṭṭūta undertook his voyage in the Bilād al-Sūdān, he described the city of Timbuktu. 3 He knew the story about the emperor of Mali, Mansa Musa, whose caravan passed through Cairo in 1324 on the pilgrimage route to Mecca.4 This is the emperor who built the main mosque and the University of Djingareyber and made Timbuktu a center of cultural influence, both Islamic and economic. The historian Ibn Khallikān tells us that the emperor brought back many manuscripts on the return trip. Thus, as of the 14 th century, tradesmen and scholars who visited Fez, Cairo, and the Hejaz in Arabia took with them manuscripts that were used for teaching in Timbuktu and were also recopied there.

According to Leo the African (d. 1552), "it was here that various manuscripts and written books were brought from the Barbary Coast, and were sold at a higher price than any other merchandise". 5 When the city fell into the hands of the pasha Jouder of Morocco in 1591, Timbuktu was a major center of Islamic learning. Ahmed Baba al-Timbukti (d. 1627), the most highly regarded scholar in the city,6 tells us that there were more than 20,000 students. They attended universities attached to three mosques in Timbuktu, Sankoré, Sidi Yahya and Djingareyber. 7 It is at this time that Moulay Ibrahim, the ancestor of Abdoulwahid Haidara, collected the first manuscripts for the future Mohamed Tahar Library.

The history of the library "Mohamed Tahar"

The Mohamed Tahar library is a reflection of the history of the region of Timbuktu since the end of the 15 th century. Today there still exist thirty-six private libraries in the city.

The Ahmed Baba public library is the most important and is presently in Bamako, because the manuscripts were moved there in 2012 during the occupation of the Islamic fundamentalists. 8 Abdoulwahid Haidara has been the owner of the Mohamed Tahar library since 2003.

The library is situated in the Sankoré quarter, which was formerly one of the three teaching centers in Timbuktu. At present it is being rehabilitated by UNESCO.

According to the family tree found in one of the manuscripts, 9 the ancestor of the family, Moulay Ibrahim, came from Yanbū' and settled in Arawan in the 16 th century.

Situated six days by camel to the north of Timbuktu, Arawan was founded in 1385. Already in the 14 th century it was a crossroads for caravans transporting salt and was also the birthplace of many famous scholars, among whom was Ahmed Baba.

Moulay Ibrahim acquired the first manuscripts now found in the library. At that time, in addition to the Koran, scholars owned manuscripts on grammar, theology, Islamic law (fiqh) and sometimes on medicine and astrology. In the library of Mohamed Tahar, as in other private libraries in Timbuktu, the oldest books date from the 15 th century. The oldest manuscript in the city is a Koran dating from the late 12 th century, copied in al-Andalus and kept in the Fondo Kati library.

Another eminent member of the family, known by the name of Baba Chirfi, lived in the 17 th century and the library proper was founded by one of his sons. Since that time, it has been entrusted to the member of the family who knows Arabic best and who is committed to 8 The 38,000 manuscripts and most of the material in the library were exfiltrated by members of the Institute for Higher Learning and Islamic Research (IHERI-AB). 9 An entry concerning the family of Abdoulwahid Haidara is found in a local chronicle entitled Kitāb alturjumān fī tārīkh al-saḥrāʾ wa-l-sūdān wa-balad timbuktu wa-shinjīṭ wa-arawān (ms. 762 in the library of the IHERI-AB). The author is Muḥammad Maḥmūd b. al-shaykh al-Arawānī. See the family tree at the end of this article.

preserving the family patrimony. Today the library bears the name of Mohamed Tahar, a family scholar who left Arawan in the 19 th century and settled in Timbuktu. He was a wellknown jurist and one of the best calligraphers in the region. He taught grammar and religious science in Timbuktu.

A few remarks concerning the collection and particularities of the manuscripts

The Mohamed Tahar library contains about 2,000 manuscripts kept in iron trunks. As in Chinguetti in Mauritania, the private libraries in Timbuktu hold a large number of manuscripts: between 1000 books for the smallest, and up to 8,000 in the Fondo Kati Library.

The great majority of the texts are in Arabic, but in Timbuktu there are many ethnic groups and languages. One finds Peuls, Touaregs, Songhai, and Haoussa. Some manuscripts are written in Hasaniya Arabic, a dialect found in the Sahara, and in Songhay, and more rarely in Hausa dialect. 10 Sometimes these different languages are still written in Arabic letters.

Certain manuscripts are voluminous more than three hundred folios, but many others contain only a few leaves. This is the case of contracts, short treatises on Islamic law, Sufi prayers, poems and magic recipes, for example. The manuscripts that arrived by caravan from the Orient or the Maghreb could be very expensive. Such is the case of the Korans, for example. A copy of the famous Shifā' by Qāḍī 'Iyāḍ (d. 1149) was sold for 45 mithqals of gold in 1486. 12 But when teaching developed in the 16 th century at the Sankoré University, Timbuktu became the most important center in Africa for copying manuscripts. The number of local scribes increased and the price varied according to the reputation of the scribe, the quality of the paper, and the type of transcription.

Most of the manuscripts in the Mohamed Tahar Library were copied in Timbuktu or in the region during the 17 th and 18 th centuries, but sometimes they are the only witnesses to medieval texts, as is the case of the treatises by Ibn Maymūn.

Additions often found in the manuscripts are a source of precise information on textual transmission. The first or last leaf of the manuscripts in Timbuktu occasionally contains the contract made between the scribe and his client with a mention of the price of the copy and the signature of two witnesses, as required by Muslim law. 13 The Library kept a volume concerning the life of the Prophet Mohammed, al-Sīrat al-Kalā'iyya. It was written by Sulaymān b. Mūsā al-Kalā'ī (d. 1237), a 13 th century Andalusian scholar who lived at the time of Averroes. This is the only work by that author found in manuscript. The others are lost. Perhaps we will find others in Timbuktu. At the end of the last leaf, an addition in a hand 11 The other copies are in Cairo, Rabat, and Berlin. The esoteric commentary on grammar is an important literary genre in Arabic, which was initiated by Qushayrī in the 10 th century with his treatise Naḥw al-qulūb: the grammar of the hearts. See. F. Chiabotti, "Naḥw al-qulūb al-ṣaġīr: La 'grammaire des coeurs' de ʿAbd al-Karīm al-Qušayrī", BEO, LVIII (2009): 385-402. 12 A mithqal equals about 3 ½ grams. This is reported by al-Bakrī. This work on the merits of the prophet continued to be read and taught in Sankoré Mosque in the 16 th century according to al-Sa'dī (Tarikh es-Soudan) 13 The price can also be stated in bartered merchandise or animals. other than that of the scribe tells us that the transcription was bought for 3 ryal on behalf of Ahmed Baba from the Sufi brotherhood of Aḥmad al-Bakkāy ibn Yaḥyā and that the sale was made according to the rules. The copy was thus made at the latest at the beginning of the 17 th century. Many copies contain numerous additions in the blank spaces, grouped around beautiful central decorations that are typical of manuscripts from Timbuktu. Others was reused in order to copy completely unrelated texts.

Codicological elements

Most of the manuscripts in the Mohammed Tahar Library are copied on paper with a watermark, usually composed of three crescents. This paper began to be used in the 16 th century. Made in Venice and Genoa in Italy, it progressively replaced the papers with watermarks representing the cross or angels. Until the 15 th century, it was at Fez that one found the largest number of paper-makers in the western Maghreb. So-called Arabic paper has no watermark. Thereafter non-watermarked paper was made in Cairo and arrived in Timbuktu with the caravans, but was not as popular as Italian paper. 14 The manuscripts of Timbuktu are not made of quires, but of loose leaves, often unnumbered and without catchwords, which complicates the re-composition of the texts when the leaves are in disorder. This is unfortunately frequently the case.

Conservation conditions in the Mohammed Tahar Library are precarious. The manuscripts are kept in iron trunks that are often moved about and sometimes buried in the sand during conflicts. They were hidden for a year during the last occupation of the city in 2012. The bindings when they exist are generally made of goat or sheep skin. The leather also arrived by caravan, but there was also an important local production in the city of Gao. The 14 J.M. Blood, "Paper in Sudanic Africa", in The Meaning of Timbuktu, ed. Shamil Jeppie et Suleyman Bachir Diagne (Cape Town: HSRC Press in association with CODESRIA, 2008), 45-55. In the 17 th century, goods were transported via Tripoli. bindings may or may not have a flap, and sometimes there are leather ties for keeping the book closed. Many books have no binding. In Timbuktu, the books suffer from dryness, but also rainstorms because homes are not watertight and flooding is frequent. The paper is often fragile and brittle, and the insects that nibble the leaves are voracious.

The different types of writing in the Saharan manuscripts have been little studied. The calligraphy was originally adopted from that used in Muslin Andalusia and the medieval Maghreb, with the same particularities for the writing of certain letters. According to Malians, a type of writing called sūdānī was created in Timbuktu in the 16 th century before spreading to the Sahara. This writing then evolved with different ethnic groups: hausawī (from the name of the Hausa people), al-takrūnī (adopted by the Toucouleurs), or al-sūqī (from the name attributed to the tamasheq Kel as-suq ethnic group in the region of Adrar, and finally the Sahrawi writing (in the Sahara). One book that is frequently read in Abdoulwahid Haydaira's family contains a poem by Ibn Wāhib, an author from the region of Timbuktu, written in khaṭṭ al-sūqī. The people of Timbuktu recite the poem for several days before the feast of the birth of the Prophet.

Representative manuscripts

All the disciplines taught during the golden age of Timbuktu are present in the Mohamed Tahar Library: religion, grammar, history, literature, science. Certain texts are found in all the libraries in the Muslim world, but others are unusual. The authors of the oldest manuscripts are essentially from the western Maghreb, and from the region of Timbuktu from the 17 th century onward. They also attest to the notoriety of certain texts in the Saharan world.

Religion (dīn) is understandably the first to be represented with five Korans and a large number of fragments. A manuscript, Naẓm ṣughrā al-Sanūsī, is written by a local author, little known, for example a beautiful manuscript containing a commentary by a local author, 'Alī al-Ḥarīshī, on the foundation text of Malikism: Irshādāt al-sālik fī sharḥ Muwaṭṭa' Mālik. The Maghrebin jurists of the 14 th , 15 th and 16 th centuries, authors of works that are found in the library, are for the most part Sufis. Muslim mysticism is another major strength of the library.

Taṣawwuf

The library contains texts by Shadhili authors, including works by 'Alī Ibn Maymūn.

The Sufi order of Sunni Islam known as Shadhili was very present in the Maghreb and Sahara in the 16 th century. There are also texts by authors of the Sufi order known as Qadiri, which established itself in Timbuktu in the 17 th century, and texts of the Tijaniyya, which developed in the Sahara in the 18 th century. Today the family of Abdoulwahid Haidara is in the Tijani order. According to him, the library attests to the harmony that exists between the Sufi confraternities in the city. Among many manuscripts, we can mention: 1/ a copy of the Dalā'il

  The oldest books in the Mohamed Tahar Library are fragments of the Koran and religious texts. There are three mystic Muslim texts (taṣawwuf) by the same Maghrebin author, 'Alī Ibn Maymūn al-Fāsī (d. 1511), which are assembled in a single manuscript. Several works by that author are still unpublished and surviving witnesses are rare. The author is a scholar from Fez who lived in Cairo, then in Damascus where he wrote ten treatises on Muslim mysticism. Only two texts have been published in recent years. Abdoulwahid Haidara agreed to photograph one of these texts, the al-Risāla l-Maymūniyya fī tawḥīd al-Ᾱjurrūmiyya, and an edition and French translation are now being made by Clara Murner as part of a doctoral thesis. It is an esoteric commentary on one of the most famous treatises on Arabic grammar, al-Ᾱjurrūmiyya. 11 It is difficult to date the phenomenon of the arrival of the first manuscripts in Arabic in western Sudan. Ibn Baṭṭūṭa speaks already of children learningArabic in Koranic schools in Timbuktu in the 14 th century. He reports that the children used wooden tablets (lawḥa) and that they made their own pens and ink, just as they do today.

  al-khayrāt by Muḥammad b. Sulaymān al-Jazūlī (d. 1465). It is a major work of Sufism which one finds throughout the Muslim world. At Timbuktu it is read on the feast days or at ceremonies organized to ward off war or drought. 2/ a copy of the famous poem in 103 verses by Sīdī Abū Madyan al-Andalusī (died at Tlemcen in 1197). This poem, al-Ra'iyya is well known to the Maghrebin Sufis and exists only in manuscript form. Many documents in the Mohamed Tahar Library complement those of the IHERI-AB documentation on the local history of the 18 th and 19 th centuries: in particular, several letters written by Aḥmad al-Bakkāy al-Kuntī. Sheikh of the Qadiriya brotherhood and political leader, Ahmad al-Bakkāy is the author of a vast correspondence kept at IHERI-AB but also in Timbuktu and in the Bibliothèque nationale de France, in Paris. It would be very useful to collect, catalogue, and digitize these letters and documents in order to broaden our knowledge

J.-L. Triaud, « La fabrication du savoir » in J.-M. Dijan, Les manuscrits de Tombouctou. Secrets, mythes et réalités (Paris: éditions Jean-Claude Lattès,

2002), 117.2 The 300 epigraphic inscriptions in Arabic and Tifinagh, deciphered in the work of Paulo Fernando de Moraes Farias, attest to the ancient Arabization and Islamization, but none of the inscriptions come from the area of Timbuktu: Arabic Medieval Inscriptions form the Republic of Mali. Epigraphy, Chronicles and Songhay-Tuareg History (London: British Academy, 2004). See also J.-L. Triaud, "L'éveil à l'écriture: un nouveau Moyen Âge sahélien" Afrique et histoire 4 (2005), 185-241.

Ibn Baṭṭūṭa recounts that in the house of an emir near Timbuktu he read a copy of the work of Ibn al-Jawzī (the famous Hanbalite from Bagdad who died in 1200) on the lexical difficulties in the Koran (Gharīb al-Qur'ān).

Many chronicles, like that of al-'Umarī (m. 1384), Masālik al-abṣār fī mamālik al-amṣār tell of his encounter with the Mameluke sultan of Egypt, al-Nāṣir Muḥammad Ibn Qalā'ūn. One also finds the story of his voyage in the works of Abū Sa'īd 'Uthmān al-Dukkālī, Ibn Khaldūn, and Ibn Baṭṭūṭa, then in the Tarikh es-Soudan of al-Sa'dī (translated by O. Houdas, Paris, Adrien-Maisonneuve, 1964).

 5 Léon l'Africain, Description de l'Afrique, transl. A. Epaulard (Paris: Adrien Maisonneuve, 1956), 468-69

.6 He is the author of fifty works. Copies of some of his texts are found in the Mohamed Tahar library and his name is often cited in the margins of

manuscripts.7 In the Tarikh el-fettach, Maḥmūd Katī recounts that there were also between 150 and 180 Koranic schools for young boys (ed. M. Delafosse, trad. O. Houdas, 1 st ed. 1913-14, reed. Paris: Adrien-Maisonneuve, 1964), 316.

This language is spoken essentially in Niger and Nigeria.

This correspondence was partially studied in two doctoral theses: Abdelkader Zebadia, The career and correspondence of Ahmad Al-Bakkay of Timbuctu: an historical study of his political and religious role from

to 1866 (London, SOAS, 1974) and Ismail Traore, Les relations épistolaires entre la famille Kunta de Tombouctou et la Dina du Macina (1818-1864) (Université de Lyon, ENS, 2012).

of 19 th -century Mali 17 . Many of them, kept in private libraries such as the Mohamed Tahar Library, are still unknown, and many texts by local authors are still unpublished. UNESCO has just finished work on restoring the building. An inventory and digitalization of the Mohamed Library is now in process in cooperation with the IRHT.

Muriel ROILAND (IRHT-CNRS, Paris) muriel.roiland@irht.cnrs.fr

Muḥammad b. Aḥmad al-Wankarī, who died in 1066/1655. Al-Wankarī comments in verse on the theological work of al-Sanūsī who lived in the 15 th century at Tlemcen and rejuvenated the Ash'arite religious doctrine. His works were taught in Timbuktu in the 16 th -century universities. [START_REF] To | Al-Sanûsî, un africain ash'rite au 15ème siècle[END_REF] The occult sciences

The treatises on magic are numerous in the library, and this is the case throughout the Muslim world. Few comparative studies exist of the texts on magic in Africa and elsewhere. [START_REF]In the Middle Ages, the opinions of jurists and scholars are often divided on this subject, between condemnation and acceptance: accepted by a famous Malikite jurist named Ibn Abī Zayd al-Qayrawānī (m. 996)[END_REF] We find many recipes for the reconciliation of the spouses, religious formulas, invocations to angels, and magic squares. One manuscript contains an astrological treatise by Muḥammad Baghayogho (d. 1593), a well-known scholar in Timbuktu in the 16 th century. Baghayogho, imam at the Sidi Yahya Mosque, was one of the rare scholars who was not deported to Morocco after the city of Timbuktu was taken and the Songhai Empire fell in 1592. The manuscript is the only copy known of this text in Timbuktu.

Islamic law (fiqh)

The library contains many tracts by Malikite authors and commentaries on Malikite treatises. Malikism is the school of law adopted in the Maghreb and Sahara. It is the largest subject represented in the Mohamed Tahar Library. Law was fundamental in university teaching and all scholars were required to study it. The large number of manuscripts is also explained by the fact that the ancestors of Abdoulwahid Haidara were jurists (fuqahā').

Consequently, besides the law treatises, we find many contracts. Certain of these texts are