THE THEORY of FEEDBACK. A PROLONGATION

The same formula (7.4), as well as formula obtained is obtained by a solution of a set of equations (7.1). Using submission of the formula (7.4) as an ascending power series (7.3) it is possible to make two variants of the equivalent schemes of amplifiers with  feedback    (Fig. 46,47).    

The sign in a denominator   (7.4) depends on that,   positive or   the negative connection envelops   the amplifier. It is known [19] that     transfer   function     of a circuit,     in particular of outline of feedback, can be represented as:   (7.5), that is vector. Making a replacement in a Fig. 47 (7.3) agrees (7.5), the transmission factor of the amplifier can be presented as the sum of vectors (Fig. 48) 

From a Fig. 48 it is visible, that for want of |K | < 1, irrespective of phase shift, sum of vectors has limit as vector, from which, for want of passing of a signal through a direct circuit, the voltage output is formed, whence it is possible to make a conclusion, that a stable mode here is shown. In a Fig. 49 is shown, that for want of |K | > 1, the sum of vectors of a limit has no, more point, it the limit is determined by the boundaries of a linearity of a circuit. Such mode is unstablis. The addition is made on a complex plane.

                It is possible to reach such conclusion and analytically. The row (7.3) is majorized by a row, which for want of there is no limit. As the addition is made in the complex form, the time is eliminated, that is correlation between vectors only causal, but not temporary.
                     The addition of vectors happens instantly, therefore for want of vector of the sum   rotates with a velocity of equal infinity, that is the frequency of a vector is equal to infinity.

  Naturally,   that such signal can not pass through an actual circuit, that allows to make a conclusion about stability of the amplifier, if   (for want of n = 0,1,2,…¥ ). Such conclusion meets to a criterion of Nikewist. However if the signal, passing on a circuit of an outline feedback, acquires temporary delay Dt, the frequency of a rotated vector will be final also amplifier can become unstablis, even if the hodograph of a vector of a transmission factor of a breaked circuit feedback does not envelop a point with coordinates (-1; 0). In more detail problem on applicability of a criterion of Nikewist will be considered below. 
                The stability of the scheme (Fig. 45) does not depend on, whether the semi-infinite line composed from links ,has by convective or absolute instability [24,26].
                5. Let's consider now problem on influence of transfer function to a sign of feedback.  The necessity it is caused by that the definition of a sign feedback with use of concept of inversion was above given and was mentioned, that it allows to consider a sign feedback independent from frequency characteristics of a circuit. The reason it is explained below. Any transfer function of the valid and positive circuit can be, represented as continuous fraction, that is, the valid and positive circuit can be presented as the scheme shown on a Fig. 50 [19]:

Each from pairs (Zn; Yn) represents an elementary link, which transfer function can be designated -. Then the transfer function of all circuit can be presented as [19,27]:. Each from links has generally complex transmission factor. In some, small enough, initial period of time of the transient being a response on single horse racings sent on input(entrance) of a circuit, sent the on can be approximated by expression:; where t - constant of time of a circuit. The equivalent scheme of an elementary link (Zn; Yn) can be represented as a gang from three elements - active resistance (R), inductance (L) and capacity (С), which can represent both separate design elements, and electrical performances of a construction of elements and editing. 
            The transition function of a link can be expressed by the formula [19]:

  ( ), were:  - Pair of the conjugate radicals,

   equal: .

For want of to small magnitude Dt all links can be considered as aperiodic, inertial links, as the initial part of a transition function can be approximated by function:. The derivative  if t = 0 is positive for this function. It is known [19], that for want of to transfer on an input of a circuit representing elementary link, voltage   u (t) = 1 for want of t < 0 and u(t) =0 for want of t > 0, the transition function   will accept a kind. If the link has a strengthening active element,. For want of junction of such links in a circuit, the transfer function will look like:  , or, if all links are identical:. The derivative of such function for want of t=0 has the same sign, as the derivative single horse racing sent on an input of a circuit, which is equal to infinity for want of t=0.

If on an inputof the amplifier the voltage   equal 0 for want of moves, on an output of an outline feedback there will be a response also equal to zero for want of , and the derivative will have it, for want of t=0, same sign, as derivative of source function. It signifies, that from t=0 before some time the polarity of the response on source effect will coincide polarity of source effect, that is the sign feedback will be determined only by availability, or absence of inversion. From above-stated follows, that a parity of performances of source function and it of the response for want of t > 0 and for want of t > ¥ so differ, that the conclusions obtained for want of the analysis of a stabile mode, can be opposite to conclusions obtained for want of the analysis of a transient regime. 

Is valid, if the sinusoidal function gived on an input of a circuit for want of t > ¥, has on a comparison with it by the response on an output a phase lag equal j, for want of t > 0 such shifts tend to zero. That is, it is possible to consider, that in area t > 0 source sinusoidal effects in phase with the response on an output, while for want of t > ¥ these functions can be oppositphased.

It is convenient to consider the analysis of processes happening in the amplifier with feedback, on an example of the amplifier, in which, for want of of breaked circuit of an outline feedback, response on source effect as the single jump  will be also jump changed only on amplitudes and have delay t. The conclusions obtained for such amplifier, can be distributed and on the amplifier with the limited passband, that will be shown below.

6.As was shown above, transition function of a circuit of an outline of the amplifier with feedback, as it is possible to approximate a response on single positive jump, for want of small enough Dt from t=0, function:  , or   . For a signal bypassing an outline n of time:  ,         where  .

  ;                             if  А > 1.

Therefore, for want of amplification factor А> 1, irrespective of a constant of time t, the derivative of a transition function tends to infinity. The amplitude also tends to infinity, as a Limit of such transition function it is possible to consider a step-function with amplitude of equal infinity. This implies, that the linear dispersing circuit, live in an outline feedback, provides, on the one hand, limited passband, on the other hand, results in process of summation similar to summation in the impulse amplifier, which will be considered below.

7. As impulse amplifiers now have received broad distribution and for deriving a broad passband, for want of with which the form of impulse can qualitative enough be transmitted, the feedback is widely applied, is of interest to consider work of such amplifier. Besides  negative feedback is applied and to shaping more broad passband in broadband amplifiers. As was shown above, the analysis of the dispersing amplifier can be shown to the analysis of the impulse amplifier.

For the analysis of process, in this case, it is supposed, that there is an amplifier, with    and with temporary delay t, enveloped negative feedback. On an input of the amplifier is sent of single jump. In a fig. 52 the process in time is shown, where     are voltages. 
                From the schedules it is visible, that the sum of voltages  represents oscillatory process with amplitude increasing on an exhibitor for want of .

For want of it the amplifier will be stable. However for want of   final (if in a circuit of an outline feedback there are delay) amplifier with negative feedback can generate oscillations. Such process can be in an emitter follower [28]. Generally speaking, the actual system should have final frequency, even owing to a final velocity of distribution of source perturbation and final sizes of the system. Therefore any system can be considered as stable for want of to magnitude of oscillation with frequency so small, that it can be neglected. It is possible to name such operational mode of the amplifier with negative feedback, for example, cvasystabile. The problem on oscillation frequency in the present work is not considered, as it is determined by a number of the factors going out for it of a framework.

In any case, a sufficient stability criterion of the amplifier with negative feedback is:. Concerning impulse amplifiers such criterion is accepted in works [29,27,28,26].

8. Now we shall consider application of a method of an ascending power series to the analysis of processes in the oscillator. Oscillator, in this case, is meant as the amplifier with positive feedback with the module of a transmission factor on an outline feedback greater that one, with linear frequency dependent  by a passive part of a circuit of an outline feedback. It is considered, that in an outline feedback of delay, elements with not  minimum phase AFC and the negative resistances are not present.

The addition of elements of a row can be considered as addition of impulses, with amplitude tending to infinity, that in case with the amplifier with  positive feedback results in avalanche increase of voltage in the party of revolting effect. Such increase happens until there will come saturation of an active element.

For want of saturation, submission on an input of the amplifier of revolting effect of the same sign, as well as the sign of the previous revolting effect, can not cause a response on an output of an outline feedback. That is, for want of saturation the feedback is teared, the status of stability occurs, from which the amplifier can be introduced by revolting effect of an opposite sign and sufficient magnitude [29]. For want of it avalanche process again happens which is transferred  the amplifier in other status of saturation. For a realization of constants of auto-oscillations it is necessary that in the amplifier with feedback there was a source of with alternating signs revolting effects.

The sources of with alternating signs revolting effects can be realized by various methods, but one from most widespread is the oscillatory outline, which free oscillations are sinusoidal oscillations have positive and negative halfcycles. For want of it the frequency of transfers of the amplifier from one stable (saturated) status in other will be determined by frequency of own oscillations of an outline. For want of such submission of process happening in the oscillator, it is possible to show, that:

- for want of in the circuit of the amplifier of a source of with alternating signs source revolting effects, the amplifier with positive feedback is unstablis and tends to proceed in one from extreme statuses, saturation or lock-out. If to take measures, that these statuses were stable enough, such device is possible to receive, as the trigger, if there will be one status stable, it is possible to receive an one-vibrator;

- the maintenance of balance of phases is not necessary for work of the oscillator, it is necessary only that the conditions "swing" of an oscillatory element feedback were observed.

9. In the literature on the theory of radiocircuits, theory of automatic control and engineering cybernetics for want of consideration of the theory of stability of systems with feedback a stability Nikevist criterion , as a rule, is resulted [17,30,18,31,29 etc.]. This criterion, due to the mathematical obviousness and convenience in practical application, has received broad distribution. However, for want of it those conditions and restrictions are not always strictly taken into account which are resulted in works Nikevist and Baud [18,32].

Conditions superimposed on a system with feedback, by which we apply a Nikevist criterion , the following [18,32]:       
                - an analyticity of a transmission facttor   with teared ОС;       
                - a linearity;    &   
                - absence  of not minimal phase  circuits. 
                Besides the conclusion about stability or instability is made then, when the transients cease, that is for want of t ®¥.
            If to assume, that, the process in the amplifier with positive feedbac for want of to submission on an input of sinusoidal voltage, can be represented, according to the formula (7.3), as the sum of an infinite amount of sinusoids, and, if the phase lag j between sinusoids is equal to zero or is multiple 2p, this sum will represent a sinusoid with amplitude of equal infinity, that meets to a mode of instability (Fig. 53а):

If the phase lag is not equal to zero and is not multiple 2p, in the given instant the amount of addends of sinusoids, for want of as n=¥, both positive, and negative polarity, will be uniformly and equally infinity. And, the magnitudes of voltages  in this instant will be distributed under the sinusoidal law. The sum of all voltages at any moment will be equal to zero (Fig. 53б), as shows justice and sense of a criterion Nikevict.

From such submission of a criterion Nikevist  the significance of conditions superimposed on a system with feedback, for want of conclusion of a criterion is visible. It is clear, that default of above-stated conditions in any degree result in unauthenticity of a criterion. In the literature [18,33] the reason of an inapplicability of a criterion Nikevist to not minimal phase circuits is resulted, explicitly request of an analyticity of transfer function of a system also is explained. The influence of nonlinearity of a system to a degree of reliability of a criterion in the literature, as a rule, is not considered. Really, the criterion was justified by the author for an abstract linear system, which behaviour is described by the linear differential equations. To nonlinear systems, which all actual systems are, it, strictly speaking, is not aplicable.

However, because of of experience of  application of the linear analysis to nonlinear systems without feedback, it is accepted to consider, that the outcomes obtained for want of the analysis of linear systems, can be applied, though and with some errors, and to actual, nonlinear systems. Is of interest as far as it fairly in application to systems with feedback.

As was above shown, the phase relations of a source sinusoidal signal and target differ depending on, whether we consider them for want of t®0 or for want of t®¥. For want of t®0 the source and target sinusoidal voltages can be considered as sinphased, whereas for want of t®¥, when the process is considered stabile, between source   and target sinusoidal signals the certain phase shift is installed. Therefore conclusion about stability of the amplifier, if the hodograph it of transfer function will not envelop a point with coordinates - (-1,0), obtained Nikevist for want of  t®¥ , for want of  t®0 can be not valid.

Really, irrespective of phase shifts in circuits for want of t = ¥ for want of to transfer on an input of the amplifier from voltage, the process of increase of amplitude of voltage output tends to infinity. However increase it limits by nonlinearity of the actual scheme, and the feedback for want of output of the amplifier on nonlinear operational mode is teared. 

Sometimes in the literature there is a submission, that the instability is characterized by undamped oscillations. However it not always so. Sometimes instability is exhibited that under an operation of external perturbation the system avalanchely passes in one from limiting operational modes - saturation, or lock-out, in which remains so long as the next effect will not throw it in other limiting operational mode. The balance of phases is necessary for taking into account only in application to oscillators with an oscillatory outline toensure "swing" of an outline, but also the significant range of shifts of phases is in this case supposed. 

Thus, the  Nikevist criterion, (as well as Mihailovs, and Raus-Hurwitz), despite of it mathematical faultlessness, can not be valid in application to hardware systems with feedback, as their behaviour considerably differs from a behaviour of a mathematical model accepted by Nikevest for want of conclusion of the criterion.

10. The material stated in the present work, does not reach all possibilities of an offered method. In it the essence, principles of its application to various kinds of systems with feedback, and also attitudeof a circumscribed method to accepted theoretical  sights on feedback is circumscribed, in main, it. A number of problems concerning a method of an ascending power series, still require theoretical study.

Is of interest to develop settlement techniques for the analysis and synthesis of concrete actual systems with feedback and to introduce a circumscribed method in practice of computer design of the radiocircuits, radio systems, electronic devices, systems of automatic control and management etc.  
                The Method of an ascending power series has obviousness, that allows to use it not only as by a means of the analysis, but also as by a convenient means of the description of processes happening in the devises or system with feedback.

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