PETER BAXANDALL - Audio Amplifier Design - 2 WORK IN PROGRESS - WORK IN PROGRESS
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Secondo articolo della serie scritta da Peter Baxandall per Wireless World nel corso del 1978. La prima parte la trovate qui .
"Il miglior risultato ottenibile con la matematica
è essere in grado di farne a meno"
(Oliver Heaviside)
AUDIO POWER AMPLIFIER DESIGN - 2
di Peter Baxandall - Wireless World , March 1978, pp. 41-46
I principi della retroazione negativa
In the January 1978 we talked about the slew-rate limit and its possible consequences with regard to his case, deriving from the potential inability of the input stage of an amplifier to provide the current required by the compensation capacitor placed between the base and collector of the second stage that follows it (VAS). And we have also seen that, with proper design practices, the problem can be rendered irrelevant. This time, before you submit in the discussion of specific circuits that we will face in the coming articles, we will discuss the basic concepts related to feedback negative and transfer functions.
Feedback: definition of terms
Figure 1 shows the typical case of an amplifier with overall feedback. The signs "+" and "-" next to the symbols of the tensions, indicating their polarity when their instantaneous values \u200b\u200bare considered positive. Vout / Vin is the gain of the circuit in the presence of feedback, often just closed-loop gain. From scheme is clear that:
(1) Vout = A * (Vin + ßVout)
(except at the center frequencies of the bandwidth of the circuit, the "+" of addition is to be understood as a transaction that occurs between variables to be taken into account their phase differences) .
From (1) is obtained from:
(1A) Avin = (1-Aß) * Vout
(1B) Vout / Vin = A / (1-Aß)
This formula can rightly be considered the "feedback and general formula is valid for applications in positive feedback (multiplier factor of merit, some types of active filters and oscillators also), and for those in negative feedback as the If we're talking about amplifiers.
to the midrange, where it is assumed that there are significant mismatches on the signal, the quantity "A" should be considered in an inverting amplifier, no more a negative number that multiplies the result obtained by subtracting the input circuit of the amount by Vin Vout * ß, or feedback signal from the exit. This means that these mid frequency amplifier in a negative feedback, Aß is a negative number, which will be compared to the single quantity A, attenuation of the gain otherwise available ( which does not prevent him to remain in each If a gain for all intents and purposes, ndt ).
Sometimes the denominator of (1B) is given in the form:
(1C) Vout / Vin = A/(1+Aß)
In questo caso di "A" e "ß" vanno inserite solo le grandezze senza il segno mentre l'intera formula va considerata una forma specifica per la sola retroazione negativa, mentre la sua equivalente per la retroazione positiva sarà scritta come:
(1D) Vout/Vin = A/1-Aß
Questa si può certainly be considered a complication is not necessary and indeed may induce confusone especially in applications where it is not always clear whether a feedback should be considered positive or negative.
The loop gain is just the gain in the ring feedback, shown in Figure 1 by Aß, a concept that in the context of idealized illustration is relatively simple. However in many practical situations it is necessary to pay attention to detail when you want to calculate or measure the loop gain of a circuit. For example, how do you calculate the loop gain of Figure 2?
If the ring is opened by removing the output capacitor and a voltage test is applied between the output and ground, then this will produce, at the junction between R2 and R3 is connected to the emitter of Q1, a voltage equal to Vi * ß. This voltage applied to the emitter of Q1 will find himself through a series resistance equal to R2 / / R3 (the result of the parallel between R2 and R3) which in turn goes into reverse series of 1/gm1, thereby reducing the transconductance and the gain of that stage. But we can also proceed in another way, by calculating the equivalent resistance resulting from the parallel 1/gm1 and R2 and using the result instead of R2, and find, by applying the Thevenin theorem, the effective feedback to the emitter voltage of Q1 at the time of ' injection on the output voltage VI. In both cases, to obtain from the corresponding circuit output voltage that we obtained closed-loop, we must provide, when tested, the addition of an output load replacement of the feedback network R2 and R3 we disconnected precisely to act on the test itself ( Baxandall's speech is incomplete at this point a part: it is clear that the two ways of calculating loop Open the effects of the test voltage Vi, which lead to different results, in the original is not given the reason. In fact, the basic problem of the feedback in this type of circuit, in whatever form they come together, is that it is at once both a GENERAL FEEDBACK entire circuit is only one element of a local feedback Q1, where the point of application - the emitter of Q1 - is the same for both - ndt ).
illustrated in Figure 3 are the most common types feedback and specific name (voltage, current, series and parallel). As discussed below, the definition of feedback "serious" or "parallel" is on the way in which the feedback itself is inserted into the input circuit (in series or in parallel precisely the source of the input signal) while the feedback voltage or current related to the size (voltage or current) to which the feedback signal demoted from the network "ß" is rendered extinct proportional. A feedback voltage to the circuit tends to assume a characteristic behavior of the voltage with a lower output impedance than your circuit without feedback, while the opposite happens with the current feedback.
Figure 4 shows how a combination of voltage and current feedback can be used to obtain an amplifier whose output impedance can be defined as well required, for example, sending signals over a telephone line. This technique uses more efficiently the output power available than does a normal voltage or current amplifier followed by a passive impedance adapter.
If we examine Figure 4th and let us assume to use an amplifier ideal we see that the input voltage is given by:
(2A) Vin = ß Vuscita + R4 * Iusco
you can also write:
(2B) Vin = ß * (* Rload Iusco ) + R4 * I output
which show:
(2C) Iusco = Vin / (R4 + ß * Rload)
write as:
(2D) Iusco = (Vin / ß) / (R4 / ß + Rload)
The above shows that the circuit of Figure 4a is the equivalent circuit of Figure 4b. Then rearranging the voltage drop across R4 to provide a positive rather than negative feedback, you can also otteenere a negative output resistance.
amplifier is said often that x dB of negative feedback at a given frequency, a claim that is open to more a possible interpretation. Sometimes it is understood that:
20log 10 *! Loop gain the open-loop situation.
If you are reassessing now for a moment, the equation (1), it becomes apparent that the two definitions just given in quantifying the rate of negative feedback are not exactly equivalent, and indeed differ significantly when rate feedback is included. With the definition usually preferred, the negative feedback is considered at a given frequency if it reduces the gain of the circuit and vice versa is considered positive if it increases.
In real amplifiers subjected to negative feedback is often a peak in its frequency response immediately a ridosso della frequenza alla quale il guadagno di anello si riduce all'unità; questo fa sì che, in tale regione, il guadagno del circuito sia in effetti più alto in presenza di retroazione che non in sua assenza ed è qui pertanto ritenuta una retroazione positiva anziché negativa come è per il resto della banda passante degli amplificatori qui considerati.
Altre volte la retroazione è ritenuta negativa se la componente reale della tensione di retroazione
ßV
out
è in controfase rispetto alla tensione di ingresso V' in (Figure 1, with V ' in real), and is considered positive if the real component of SSV out is in phase with V 'in . This concenzione, although popular, however, is wrong and - as will become evident during the course of this series of articles - inconsistent with the distinction between positive and negative feedback data poco fa.
Considerazioni sulla stabilità
L'argomento della stabilità negli amplificatori retroazione è quanto mai vasto e su di esso sono stati scritti molti testi specialistici e matematicamente complessi. I più famosi tra questi sono probabilmente quelli di Hendryik Nyqvist ed Hendryik Bode, entrambi impiegati ai Bell Telephone Laboratories negli anni trenta e cinquanta del secolo scorso. Per quanto stagionati questi testi trattano gli aspetti fondamentali the problem extensively and in-depth and are still considered valid and current. Many electrical engineers my colleagues, particularly those ignorant of any formal training in the theory of feedback, you probably feel intimidated by the amount and complexity of the technical literature available, and concepts such as complex frequency, poles and zeros, convolution operators Heaviside, Laplace transforms and flow charts of the signals appear to many to be insurmountable obstacles.
For my part I think the key point, however, that the matter be referred to gain a theoretical understanding of the topic sufficiently adequate to understand the reasons for various phenomena that verificano nei circuiti e soprattutto le possibilità disponibili nella modifica dei circuiti per ricavarne le prestazioni migliori possibili. In effetti l'ammontare di preparazone teorica necessaria per raggiungere lo scopo è in realtà sorprendetemente ridotto, nonostante questo verrà sicuramente negato dai più entusiasti patiti della matematica!
Vi sono parecchi motivi per ritenere non necessario per un buon progettista di amplificatori tutta la conoscenza della teoria della retroazione che viene altrimenti supposta necessaria. In primo luogo la maggor parte dell'analisi compiuta in origine aveva lo scopo di INDIVIDUARE i criteri di stabilità connessi ai circuiti retroazionati e a esprimerli in un formato conveniente e pratico da utilizzare per gli ingegneri. Una volta che ciò sia stato computo e sta stato tutto definito senza equivoci, gli ingegneri possono limitarsi a utilizzarne i risultati senza per questo dover essere in grado di dimostrarne la validità. In secondo luogo, assodata l'esistenza di una adeguata comprensione qualitativa dei problemi, i valori ottimi di alcuni componenti da utilizzare nella realtà sono più praticamente ed effiicientemente stabiliti per via sperimentale. Ciò è in gran parte dovuto al fatto che alle frequenze più elevate a cui deve lavorare un circuito - che possono arrivare anche a parecchi MHz - un qualche grado di approssimazione sulla conoscenza del comportamento reale dei transistori è inevitabile, anche when you can benefit from assistance provided by information technology (
and this remains valid even today, more than thirty years after the writing of this article: even today, despite the now wide availability of electronic simulation programs seemingly "perfect "- SPICE in the first place - the proper evaluation and prediction of actual behavior of a circuit not only can not be separated from the experience gained by the designer, but very different from what we like to believe many fans and amateurs who rely on this rather uncritically type of media, including the simulations more "accurate" can not really provide a representation of the actual behavior of a circuit that go beyond what is, at best, it can be considered nothing more than a good estimate of what really happens.
ndt ) Some believe that to achieve optimal values \u200b\u200bof certain components through a series of trial and error is not a modern and decent design technique, a view with which I disagree very much. In fact, a way of seeing this empirical approach is to consider the circuit under test nothing more than a sort of "analog computer" whose response to changes in the components can be seen on the screen of an oscilloscope. Plus an analog computer that has the undeniable merit of "simulate" nothing but himself. This system, if pursued with intelligence and policy, I think a more direct technical, economic and sensitive to gain information on the operation of a circuit than otherwise obtained through the simulation models and programs that run on a computer, I realize, however, that but in the end all this is just a matter of preparation and even personal preferences of the designer.
In some instances we tend to believe that the circuit designer has to spend his time primarily between charts, formulas, and computers and leave to others the practical work, a work philosophy which I personally do not agree with a lot: the lavoro sperimentale e in sé molto stimolante e fecondo e in molti casi l'osservazione diretta di possibili effetti inattesi può condurre a migliorie circuitali a cui altrimenti non si sarebbe neppure pensato. Queste non solo possono sperimentate immediatamente ma possono dare l'avvio a riflessioni e considerazioni che, oltre a consolidare la concreta esperienza del progettista, possono generare nuove idee, anche nuovi approcci analitici che a loro volta danno l'avvio a una nuova fase sperimentale. E nella mia esperienza è proprio dall'alternanza continua tra attività teorica e sperimentale che proviene l'evoluzione di nuovi e migliori progetti. Ovviamente una conseguenza inevitabile di questo modo di procedere si ha nel fatto che ciò che inizia come a clean circuit board tends later to resemble a chaotic nest of birds but are otherwise convinced that the majority of projects amplifiers with truly original features have evolved mainly through this road before reaching the final form of a smart plate Printed circuit board.
A very real danger that run many engineers looking too absorbed in their mathematical work (
or, nowadays, the simulation work on a computer ndt
) is to end up overlooking other more down-to-earth but no less important the entire design process. A few years ago in my article (3), I found myself saying that "while it is virtuous to be able to analyze a project, may be more so to be able to see when its detailed analysis is unnecessary, or even conceive a practical circuit whose behavior can be more easily planned. "
Retracing our steps, the purpose of what I am about to write is to present the theoretical minimum needed to understand and appropriately address issues related to the stability of feedback amplifiers. They will also not be used any more complicated or complex numbers "j-notation" (4). Per chi comunque volesse avvalersi di basi teoriche più consistenti non manca di certo la letteratura tecnica in grado di accontentarli, usando anche nell'ambito della progettazione di amplificatori i concetti di frequenza complessa, poli e zeri ecc.
Al riguardo, per una introduzione elementare, raccomando la lettura dell'eccellente serie di articoli che M.G. Scroggie ("Cathode Ray") pubblicò nel 1962 su Wireless World
(6), (7), (8). Una trattazione più completa e avanzata della teoria e pratica della retroazione la si può ritrovare nell'ottimo libro di Cherry e Hooper, Amplifying Devices and Low-Pass Amplifier Design (9), testo although that does not face problems using determinants and other concepts of calculus when it sees fit, remains a true test of engineering in which there are certainly practical considerations to various design problems.
An AC-coupled amplifier can have stability problems with both extreme low and high extremes of its bandwidth. Below will be considered only the case of possible instability at high frequency, the assumption (now as then
reasonable if one considers that the use of transformers output, the main element in a feedback system that can generate instability problems at low and very low frequency, is now essentially reduced to minority and niche situations, ndt
) that the amplifiers are coupled in continuous speech, however, that follow can easily be extended to the case of instability at low frequency where required by circumstances.
First we will examine some basic notions about the functions of transfer and whose understanding will help us to better frame the whole issue of negative feedback. The transfer function of a feedback amplifier as with any other circuit is simply an equation that quantifies its output signal V out
in relation to its input signal V
in . Usually assumes for convenience that the amplifier is free of gross forms of nonlinearity, but apart from this simplification, its transfer function contains all information about its frequency response, phase, transient and its margins stability needed to characterize the behavior. The downside is that, except for simple cases, derive and simplify the function transfer of a feedback amplifier, is a tedious and exasperating for those who, unlike the writer, has a natural aptitude for this type of tasks. Nyquist and Bode diagrams we shall consider later in the solution of these problems are a far more practical approach and sent for most of the situations regarding the design of amplifiers. It is always theoretically possible to use the sole use of complex numbers to calculate the currents and voltages at any point of the amplifier circuit connected to V in
and V out and hence obtain the transfer function. purposes of discussion we will use the schema shown in the figure above (Fig. 5), which is an amplifier rather simple and idealized, but fully adequate to explain the concepts that will be shown below.