Geliştirilmiş Ebers-moll Modelinin Tranzistorlu Gerilim Kuvvetlendiricilerinde Minimum Distorsiyon Şartına Uygulanması

Abstract

Bu çalışmada dirençle yüklü tranzistorlu kuvvetlendiricilerde oluşan harmonik distorsiyonunun azaltılması için yapılan çalışmaların yetersiz kalan yanları araştırılmış, tek katlı kuvvetlendiricilerde baskın bileşen olan ikinci harmonik distorsiyonunun minimum yapılması nı sağlayan optimum R sürücü kaynak direncini hesaplamak üzere, akım kazancının akıma bağımlılığı da gözönüne alınarak ve Early olayının fiziksel olarak modellenmesinden hareket edilerek farklı çalışma böl geleri için yeni analitik bağımtılar çıkartılmıştır. İlk olarak lineer olmama distorsiyonunun tanımı yapılmış ve tranzistorun giriş ve çıkış karakteristiklerinin lineer olmamasının sebepleri incelenmiştir. Önce Leblebici modifikasyonundan yararlanı larak çıkartılan kolektör akımı ile baz akımı arasındaki bağıntı kul lanılmış daha sonra da kaynak gerilimi ile giriş akımı arasındaki ba ğıntı elde edilmiştir. Ortak emetörlü devrede toplam distorsiyon anlatılarak, kolektör akımının işaret kaynağı gerilimine bağlı olarak değişimini veren bağın tı, geliştirilmiş Ebers-Moll modelinde Early olayını temsil eden M ve N fiziksel parametreleri cinsinden elde edilmiştir. Bu bağıntıdan yarar lanılarak, ikinci harmonik distorsiyonunun minimum olma şartı yine aynı fiziksel parametreler cinsinden elde edilmiştir. Akım kazancı g" in kolektör akımına bağımlılığının, çıkartılan bağıntıya ne şekilde katılabileceği araştırılmış ve bu özelliğin ba ğıntı kapsamına alınmasının, EM3 modelinde aynı olayları temsil eden parametrelerin burada da kullanılmasıyla mümkün olabileceği ortaya ko nulmuştur. Değişim, önce burada belirgin olduğundan orta ve küçük akım lar bölgesinde incelenmiş, bu bölge için farklı model parametresinin kullanıldığı bir bağıntı çıkartılmış ve önerilmiştir. Orta ve küçük akımlar bölgesi için tanımlanan C_ Sah-Noyce-Schockley katsayısının elemanın davranışını belirlemek açısından önem taşıdığı gösterilmiş ve bu bölgede kolektör akımı ile baz akımı arasındaki bağıntı tam olarak verilerek, ikinci harmonik distorsiyonunun minimum olma şartını veren bağıntı geliştirilmiştir. Tranzistorlu devreye ilişkin deneysel gerçekleme yapılmış, deney sel sonuçlarla daha önce elde edilen nümerik analiz sonuçları ile teo rik sonuçlar birbirleriyle karşılaştırılarak sonuçların uygunluğu ince lenmiştir. Büyük akımlarda etkili olan Webster olayının etkisini dikka te almak üzere EM3 modelinin 6 parametresinden yararlanılmış, çıkarı lan analitik ifade kapsamına alınabilmesi amacıyla baz akımı ile kolek tör akımı arasındaki bağıntı yeniden düzenlenmiştir.

The exact treatment of nonlinear harmonic distortion for obtain ing analytical expression presents mathematical difficulties. One of the approximate methods for characterization of the nonlinear distor tion in low frequency amplifier is to use the Taylor's series expansion of the d.c. transfer characteristic. The others are: a. Numerical analysis by introducing nonlinear elements in the equivalent circuit of the active devices, b. By using approximate expressions in the system d.c. charac terization. In this dissertation, nonlinear harmonic distortion in low frequency common emitter amplifier has been studied and exact analy tical expressions have been obtained between source resistance and second harmonic distortion. A good agreement was observed between the results of the derived formulas with the results of both full numerical computation and experi mental measurements. As it is well-known that nonlinear distortion is a fundamental parameter in electronic system design, because it limits the accuracy of signal processing. This. statement particularly applys to active filters, oscillators and amplifier design. A single-stage amplifier is a fundamental element in both integ rated system design and in multi-stage amplifiers. A transistor is a nonlinear three-terminal electronic device which is used in amplifier design. Because it is a nonlinear device harmonics of the excitation signal appear in the output. In the first publications on the subject by Strut t [l] and Meyer [2], it is pointed out to the important fact that the second harmonic distortion becomes zero for a certain value of the signal source rezistance. Subsequent work has confirmed the above result and some analy tical expressions of.the said distortion have been obtained. However, because of their complexity they have not found a wide-use. On the other hand experimental results show the important role of second -rx- harmonic distortion for single-stage audio amplifier. Another means of reducing nonlinear distortion is the applica tion of negative feedback. Various studies on such circuits have been made by Hönicke [5] and in 1968 Lötsch [6 J summarized the up-to-date results. It is. possible to reduce the total harmonic distortion at the output of amplifiers by applying negative feed back. However the usable amount of feed back is limited by stability problems, so that the reduction of total harmonic distortion is also limited. There fore it proves more practical to first design a stage with the lowest possible harmonic distortion and then reduce it further by the appli cation of negative feed back. In 1975 Leblebici [4] has derived a method for the reduction of the second harmonic distortion which is the biggest term in the total harmonic distortion and established that it is theoretically possible to reduce second. harmonic to zero. In the above work it has been assumed that the output curves of a transistor for constant base current have a fixed slope. Since. this is not perfectly true the expression derived with. this assumption are only approximate and there fore must be improved upon. On the other hand the expressions used for the output conductance was also approximate and contributed to some of the errors in the final result. Following the above explanations and after reminding some de finitions, the mathematical condition.for annulation of the second harmonic has been investigated. It has been also shown that the sig nal source resistance, satisfying the above condition can be precisely calculated for any type of transistor, using its equivalent circuit parameters as derived from its relevant physical parameters. A new transistor model had to be used for the derivation of the new expressions. With the present nonlinear transistor models in which the last portion of the constant base-current curves are assumed to be linear, a sufficient agreement between the experimental and theoretical results can not be achieved. To obtain this agreement a new modified Ebers-Moll model, as proposed by Leblebici L8J has been used after a brief description of the same which takes into account the Early effect in a more adequate fashion as compared to the previous model [4]. To improve the optimization of the signal source resistance, its expression had to be rearranged to take account not only of col lector current (I ) and collector-emitter voltage (VT), but also of the EM3 model parameters such as C"? n"T, 6 and the leakage resistance R_R of the collector base junction. [l9], [2lJ. In section-2 the source of nonlinearity in any bipolar tran sistor has been explained. It was pointed out that if i" vs iR is not linear, the relation between ip and i^ can be expressed By power series as shown by eq.(2.2). If the excitation signal consists of one frequen cy and some mathematical transformations are made, the eq.(2.3) is obtained in which the harmonics can be seen in addition to fundamental and d.c. components. This type of distortion is called "harmonic distortion or amplitude distortion". If the signal consists of two -x* or more frequencies, besides the terms having frequencies which are multiples of the signal frequencies, other terms also having the sums and differences of the signal frequencies. This type of distortion is called "intermodulation distortion". Also given in this section, with eq.(2.7) is the definition of the second harmonic distortion, in terms of power coefficients. In section-3 the input and output characteristics of a transis tor which is used as an electronic device has been showen to be nonli near and reasons have been explained. By means of these characteris tics the transfer characteristic of a transistor has been obtained and it was shown that both the static and dynamic transfer characteris tics are nonlinear. One of the reasons for this is the Early effect which has been well defined in Modified Ebers-Moll model [8]. For many applications this model represents the bipolar transistor accu rately, especially for distortion analysis, where nonlinearity of the transistor characteristics are very important. Instead of the geometrical representation with the Early voltage V. by Lindholm and Hamilton [23], Leblebici described the Early effect in a different way where the following assumptions are made : a. Emitter efficiency is very high b. BF»l c. The. width of the emitter-base space charge layer can be neglected compared with the width of the base region. d. The inverse Early effect can be neglected in the forward active operation region. Leblebici has defined two new model parameters, called respec tively M and N, for a more accurate representation of the Early effect [8], [12]. Ebers-Moll expressions are given by the eqs.(3.1) and (3.2), with some approximations these expressions can be simplified as given in expressions (3.3) and (3.4). Both are functions of the collector- emitter voltage VT. Using the V", voltage swing about the quiescent point and the power series expansion, it can be seen that all harmo nics occurs. Since Ij. is an exponantial function of VR" the input characte ristic is also nonlinear. The eq.(3.7) which expresses the transfer characteristic of a transistor in terms of the modified Ebers-Moll model parameters, is derived from the eqs.(3.3) and (3.4). The voltage transfer curve can also be obtained by means of the other characteris tics of a transistor and this transfer curve is also nonlinear as shown in fig. 3.5. Eq.(3.11) shows that the dynamic current transfer characteristic is obtained only in terms of Modified Ebers-Moll para meters M and N. This is a very important expression, because its Taylor's series expansion enables calculation of all the coefficients in terms of the M,N parameters and the quiescent point coordinates. -xirr The relation between excitation signal voltage and input cur rent is given by eq.(3.26). If the inverse power series is used, input current v. s excitation voltage can be obtained as given in eq.(2.27). Both these expressions include the source resistance Rg. In section-4, the total harmonic distortion in a resistance loaded single-stage amplifier has been taken up. If the plots of base current i, v. s excitation voltage Vg and collector current i v. s base current i, are given, the relation between collector current i and excitation voltage v can be given by eq.(4.1). The coefficients of c.'s in the eq.(4.1) can be obtained in terms of the coefficients of b.'s and a. 's in the eqs.(3.16) and (3.28) respectively..The second harmonic distortion which is given by eq.(4.3) can be simplified as eq.(4.4) using an adequate approximation. In this section it has been explained that the second harmonic distortion plays a great role in the total harmonic distortion of a single-stage amplifier. This can be easily seen, comparing d" with other terms in both the numerical and experimental results. The expression (4.11) which gives the optimum source resistan ce R,. in terms of Modified Ebers-Moll parameters and quiescent operating point, can. be obtained by using eqs.(4.6), (4.7), (4.8), (4.9) and (4.10). All the parameters which are required for the ex pression (4.11) can be measured. It has been proved that M and N pa rameters are easily calculated [8j. The method of measurements for the 3-p and I are also known [l9j. As an example, for the tran sistor &.39T the measurement values are 3 = 206, M = 0.03, I = 0.172 E-13 and N = 0.464. The base current I can be found from eq(4.14) but it will be more convenient to use the expression (4.16) because it's calculation is based on collector-emitter voltage Vr" at the quiescent point. Compared to eq.(4.14), eq.(4.16) gives the current gain 3-p, with a greater precision because the latter includes M and N parameters as well as C-E voltage VT at the operating' point. The expression (4.11) is very important because it gives the optimum source resistance R, > in terms of Modified. Ebers-Moll pa rameters M,N and of the quiescent point. When this expressions is compared with the. expression found in the previous study [4j the fol lowing differences are seen: 1. The previous study is unsufficient because of two reasons. Firstly, the Early Effect has been accounted for after an inexact geo metric representation. Secondly, the last partion of the constand. base- current curves has been taken as a straight line. In fact, these por tions are curved. 2. In the same study [4], the current gain 3p is found by using the plot 3-, v.s I". This method causes some error. If the current gain 3" for V__ ¦=* 0, which is easily measured, is known [19 J, the cur rent gain 6 at. any quiescent point can be calculated precisely in terms of Modified Ebers-Moll parameters M,N and of the collector-emitter voltage by using eq.(3.17). So, the optimum source resistance at ' various quiescent points can be calculated easily. -xxi«* 3. Eq.(3.7) shows the relation between collector and base cur rents 1.8 J. Since the parameters M and N belong to Modified Ebers-Moll model used in the expression (4.11), the optimum source resistance at the desired. quiescent point can be found more exactly than with the previous expression. 4. In the previous study the current gain $ must be determined at two points. This causes errors because these values must be ob tained by the plot of 3" v.s I,,. The output conductance g00 which is found in this way is therefore relatively incorrect. 5. By means of the expression (4.11) the optimum source resis tance Rg(0 pt) can be easily calculated for the various currents at the desired quiescent point. Furthermore it is convenient for compu ter calculation, which is not the case with the previous study [4]. 6. As it is shown in Fig. 4.4 and Fig. 4.5 the collector-emit ter voltage plays an important role in the amount of second harmonic distortion. Numerical analysis was carried out. using TIME1 and HALSEN prog rams which have been prepared for the purpose of distortion analysis [l3j, [l4], [25]. At any quiescent point, for example at Vc" = 12.5 V and for various quiescent currents the obtained numeric results are given at Table-4.1 and theoric results using eq.(4.11), in Table-4.2. When both tables are examined, some differences are noticed between optimum source resistance Rg(opt) values found for given currents: The values found in Table-4.1 are higher than those of Table-4.2 for decreasing collector current values. For high level currents there is no considerable difference between numeric and theoretical.results. But in low-level currents the differences are getting more important:. In section -5 the causes of these differences are investigated. These differences between the numeric and theoretical results stems from very low values of current gain 3" for low-currents. Although the base current has been calculated from expression (3.7), this can not represent 3-, in all current ranges. In other words, expression (3.7) does not represent accurately enough the relation between col lector current I and base current I. Generally there are three re gions of interest in the variation or 3T with current. A typical va riation of 3" with Ic is shown in Fig. 4.6. Region I is the low-cur rent region in which $" increases with !".'. Region II is the mid-cur rent region in which 3-p is constant (43T). Region III is the high- current region in which 3 drops as the current is. increased J_L9j. The decreases observed in regions I and' in the two portions of region III have different causes and are termed "Sah-Noyce-Schockley effect", "Webster effect" and Kirk effect" respectively [l5], [l6], [l7], [l8], [24] : Audio amplifiers are generally designed for low-and mid-currents, therefore region I and region II are examined first. In. the next sec tions region III is taken into consideration. For the base current eq.(4.18) can be written as eq.(4.19) which involves both regions I and II while eq.(3.20) is given only for region I. This expression invol ves two more parameters sig^ and C2 which belong to EM3, where n^ is called "the low-current, forward region emission coefficient"and its typical value is 2 and C« is called""Sah-Noyce-Schockley coefficient". -xiii- Since at the quiescent point V and I is known base-emitter voltage V can be determined by eq.(4.24). Now if the Taylor's series expan sion is applied to the eq.(4.25) the power series expr.(4.31) is ob tained which relates i^ to the excitation source signal vg. And if similar mathematical operations are made the optimum resistance R, » is obtained by eq.(4.38) in terms of the new parameters ng^ and C2- C2 was measured with parameter analyser (HP 4145) and its value was obtained as 180 for the transistor M139T. Using the expr.(4.38) the obtained results are given in Table-4.3 and when compared with the Table-4.2, it is seen that the expr.(4.3) is more exact than the expr. (4. 11), but it is not yet sufficient because there still are considerab le differences. To find the reason of the differences the current gain 3" has to be considered once again, r The drop in 3"at low-currents (region I) is caused by extra compo nents of Lp. which had been ignored before. For the normal, active re gion with V = 0 there are three extra components which are caused by a) the recombination of carriers at the surface, b) the recombination of carriers in the emitter-base space- charge layer, c) the formation of emitter-base surface channels. So, dividing eq(5.1) by eq.(5.5), eq.(5.7) will be obtained which gives the exact relation between Ip and I, in the region 1 and II. Taylor's series expansion is used for the eq.(5.7) to obtain the coefficients in terms of n", and C". If the value of b../b_ given by eq.(5.12) is substitute m expr. (4.37), expr. (5.13) is obtained which enables calcu lations of the Rg(opt) value precisely in the region I and II. After obtaining the analytical expressions, in section-6 ex periments were made with the transistor Mİ39T, The circuit diagram is shown in fig. 6.1 and the actual set-up in fig. 6. 2. The results of this experiment are given in the Table-4.5 which gives the experimental re sults, a good agreement is observed except for one point. In high- current (region III) there is difference between experiment and theoric results. This stems from the Webster and Kirk effects [l7], [24]. So, it is necessary to find a new analytical expression for the regions II and III. In section-7, it is shown that at high-currents 1^,. must be repT resented with the expr. (7.2) [17 j. 6 is the new model parameter of EM3 which is called "High-current coefficient". So, the exact relation between I and I can be obtained with the expr. (7.8). When Taylor's series expansion is used for eq(7.8) about the quiescent point, the relation is obtained between ic ve i^ by eq.(7.12). From the expr. (7.12) the ratio b.. /b? can be obtained in terms of M,N and 8 parameters and if this ratio is substituted in expr. (4.9), expr. (7.13) will be obtained, giving exactly the optimum source resistance Rg(opt) for high-currents (region-Ill). The parameters C" and 0 can be easily measured, with parameter analyser HP 4145, using direct or indirect methods [l9]. Comparing the results obtained from expr. (7.13) and -xiv- from the experience, a good agreement is observed. In fig-7.4 all re sults have been plotted. In last section it has been shown that all expressions can be simplified and that some approximations can be made for most practical cases without introducing appreciable errors. As a conclusion, new expressions have been developed for a clo ser calculation of the optimum source resistance which minimizes total harmonic distortion of a single-stage common emitter amplifier where the transistor may be used in the low-mid-and high current ranges. With a similar approach equivalent formulas can be developed for use with emitter-coupled amplifier stage.