THE ORIGIN OF AUDIBLE DIFFERENCIES IN THE QUALITY OF RADIOTRANSMISSION OF THE AMPLIFIERS OF SOUND FREQUENCIES
All-state scientific institute of radio and acoustics named after A.S. Popov
The entire history of the development of transistors used for amplification of sound can be divided into three periods. During the first period the sound produced through the transistor amplifiers of sound frequency (ASF) seemed attractive, bright, and free of distortions. Unlike lamp amplifiers, the transistor ASF showed unbounded amplitude-frequency characteristics (AFC). The popularity of a “transistor sound” lasted until mid 60-ies. After that, the transistor sound gradually became a synonym for somewhat unpleasant in terms of subjective sensations, though the objective parameters of transistor ASF formally kept on passing the requirements of undistorted audio transmission. The phenomenon of unobserved audible distortions turned out to be the reason for the appearance of the term “mystical concepts”  in technical literature, where transistor was considered to be the factor responsible for the deterioration of sound. And it has just happened recently when this point of view shifted towards the very much different one, where the uncontrolled subjective distortions were considered the true reason for unpleasant subjective sensations.
The search for distortions most sensible to ear was initiated in the early 70-ies by a number of companies – manufacturers of ASF [1,2]. In particular, it was established  that among a large number of descriptive definitions, used by experts for the evaluation of quality of audio transmission of ASF, one can identify two groups: the first one may be described by the ”opacity,” “stiffness,” “metalicity,” “annoyance” of sound; the second one comprises the dullness of bass, frequency and time fuzziness, the loss of sharpness.
The first group is controlled by the effects of perception of non-linear distortions and their resultants of higher order, beats in particular. These distortions form the so-called “left” with respect to the signal, and therefore, not masked distortion spectrum. The second group turned out to be a mystery concerning what sensations should be attributed to what type of objective distortions. While performing subjective expertise of electrical recording devices, Moller  observed the sensations, which were the most similar to those mentioned above. He presumed that those sensations might be correlated to the Bass Intermodulation (BIM). Moller defined BIM as a parasite modulation of the useful signal by its distortions in the spectral range between 2 and 60 Hz. Let’s define the signal modulation by the amplitude as BIMA and by frequency as BIMF.
Why should BIM be considered as the most important parameter that characterizes the distortions of the audio recording equipment? First, the resultants of BIM coincide with the components of the useful signal that reflect the emotional context of voice and music . Second, the spectrum of BIM distortions lies within the range of critical audibility band, and, therefore, falls within the range of maximum sensitivity to the difference between the frequency and amplitude modulation of signal .
In spite of the significance of BIM distortions for the quality of audio transmission, they have not been controlled in ASF. This is a result of the absence of the physical concept, which could explain the formation of BIM. Note, that BIM distortions (known as the detonation of sound) are easily explained by oscillations of linear velocity of the sound recording material  and have been controlled in the recording devices for quite a long time.
We assumed that transistor ASF produces audible “sound detonation.” In order to prove this supposition, the detonation coefficient of several hi-fi transistor ASF was measured in the regime of signal addition of two sine signals on 20 and 3,150 Hz with the amplitude ratio 4:1 at the input of these ASF. The results unexpectedly gave us the value of the detonation coefficient equal to 0.6% when measured with the standard flutter meter, provided that the ASF has a nominal output voltage. This value of the detonation coefficient 10 times  exceeds the noticeability threshold for this type of distortions, and 2 times exceeds the tolerance value for these distortions in the C-class tape recorders . Here we set forward the physical concept for appearance of BIM distortions in ASF built on bipolar transistors.
It is known [9,10] that the temperature increase at the p-n junction of bipolar transistor is accompanied by the increase of the current gain from 0.3%/K to 0.5%/K and the decrease of the base-emitter voltage at 2.2mV/K. The thermal sensitivity of these parameters has been accounted for earlier from the position of providing the thermal stability for the operating point of the amplification stage . Paul  pointed out that the temperature oscillations at the p-n junction, produced by the oscillation of the dissipated power on the collector, affect the transforming properties of transistors. He gave the name Mitlaufeffekt (non-isothermal effect) to this effect. However, the formation of these distortions, caused by the thermoelectric mechanism of signal transformation in ASF, was for the first time addressed by Opperman . This communication didn’t raise much interest among professionals, since it lacked the quantitative estimations for the observed distortions. Later, Fishtain  noticed the end-to-end phase shifts in the infrasound region at the differential stage built on bipolar transistors. Those shifts lied within the frequency region where the inertia of electrical processes shouldn’t play any role. The infrasound region distortions were explained by the influence of the AC component of p-n junction temperature on the thermosensitive parameters of bipolar transistors. However, the connection between BIM distortion in ASF and temperature-caused distortions was not demonstrated in any of the publications, to the best knowledge of the author. Due to the obscurity of the reasons for BIM appearance, we have studied the physical processes responsible for the formation of linear and non-linear distortions in typical 3-stage transistor ASF. As a result, it was proved  that thermal distortions are being formed not only in separate stages, but practically in all transistor sub-circuits in typical ASF. The most significant of the obtained results was the establishment of the connection between thermal distortions and formation of BIM distortions.
Fig.1. The principle electrical circuit of a typical ASF built on bipolar transistors.
The study of the conditions for the appearance of thermal distortions was conducted in ASF, according to Fig.1, where the output stage (OS) (transistors VT6 – VT11) comprises three Darlington-type class B emitter-followers. The driver stage (DR) (transistor VT4) is built as a common emitter, where the collector circuit has a dynamic load (DL) (transistor VT5). The differential stage (DS) (transistors VT1, VT2) is completed according to the asymmetrical circuit with a dynamical load (transistor VT3) in the emitter circuit.
Thermal distortions in the output stage.
Any change in the amplitude of the voice or musical signal is accompanied by the temporal change of the p-n junction temperature of OS transistors. This is clear if one recognizes that for B-class OS stage the average power dissipated on the collector of the output stage transistors during one period of the amplified signal is a function of the amplitude of that signal :
where Um3 is the amplitude of the signal at the output of ASF in Volts, E is the source voltage of the OS leg in Volts, and RH3 is the OS load resistance in Ohms.
Though the rate of change of p-n junction temperature of OS transistors is limited by the inertia of thermoelectric transformation, the value of the thermal time constant junction-case is not sufficient for smoothing out the gradually changing instantaneous power on the collector of OS transistors. When the dissipated power at the collector of OS transistor has a maximum value, the instantaneous p-n junction temperature could possibly reach its maximum allowed value 150°C, and during the intervals of the absence of signal the temperature can decrease to the temperature of the transistor case .
The maximum operating temperature of the OS transistor case is 60-70°C, which is preset during the fabrication of ASF . This value tends to be a compromise between the effective heat transfer away from the output transistors and maintaining of the strict requirements to the standards of massive ASF. It can be concluded that the optimally designed ASF can reveal 80-90°C difference in the values of the instantaneous p-n junction temperature of OS transistors of B-class ASF.
In order to quantitatively estimate the influence of AC component of the p-n junction temperature on distortions, we will have to find some relative increase in current gain of the transistors. Let’s use the value of the temperature jump across the p-n junction and the dependence of current gain on temperature  that was determined earlier. It can be shown that the change of gain can be as high as 50% for each transistor in OS, and as high as 300% for the circuit of 3 transistors in OS. Taking into account all said above, the voltage gain for OS is:
m3 = RH3 / (RH3 + R3); R3 = R2 / (β1 + 1)( β 2 + 1)( β 3 + 1) (1)
where R2, R3 are the output resistance of DS and OS, respectively, in Ohms; β 1, β 2, β 3 are the current gains of OS transistors of the emitter-follower connected in series. The output resistance of the emitter-follower circuit is variable, due to change of the current gain of the OS transistors, whereas the relative change of the value of R3 can be as high as 3. According to (1), the change in the value of R3 is the reason for the instability of the current gain μ3. This shows itself as a parasite amplitude modulation of the useful signal at OS output by the envelope of the same signal. The experts from “Sansui”  named this type of distortions as “envelope distortions,” although those distortions rather belong to the BIMA type due to their spectral characteristics. These distortions are observed only for non-stationary signal at the input of ASF, and therefore cannot be detected by the traditional methods used to measure intermodulation distortions. One of the effective methods for the “envelope distortion” control is the response of OS to the amplitude-modulated signal or a pitch burst . In the latter case the relative change of the amplitude of the pitch burst at the output of OS (during the first 100 ms from its initiation) can be considered as a measure of the envelope distortion. The magnitude of the envelope distortion for OS as an emitter-follower (Fig.1) can be written as:
where we have used the voltage gain of OS, defined according to (1) at the maximum and the minimum temperature of p-n junction. In case of the typical values of R2 > RH3(β 1 + 1)( β 2 + 1)( β 3 + 1) for OS control from the current source, these envelope distortions can be as high as 300%. If the opposite inequality is being held (OS is controlled by the voltage source), the thermal sensitivity of the current gain can be neglected. The driver stage DR (see Fig.1) built on the bipolar transistor VT4 operates in the regime A as a common-collector amplifier. The thermal distortions in DR are formed when the instantaneous temperature of p-n junction of VT4 changes due to oscillations of the dissipated power on its collector. This power dissipation can be calculated for operating points of VT4, where the AC component of this dissipated power can be represented as:
where UÏ2 is the source voltage for DS, UO2 is the collector-emitter voltage when the signal with the instantaneous value of uk is not present, is the load resistance in the collector leg.
Since DS operates in the regime of powering the input from the voltage source, it turns out that the thermal sensitivity of p-n junction base-emitter voltage for VT4 becomes quite significant. This voltage causes the creation of the unaccounted feedback coupling between collector and base-emitter circuit . The feedback coupling (FC) circuit includes the transformation of AC voltage U2 into instantaneously dissipated power Pk(t), followed by the change in p-n junction temperature and then by the instantaneous change of the base-emitter voltage with the coefficient – 2.2 mV/K. This thermoelectric feedback coupling (TEFC) in transistor VT4 is very much similar to the electromechanical one. The expression for the transfer characteristic of TEFC circuit of DR can be written in the operational form as :
where RT2 and tÏ-Ê are the thermal resistance, in Kelvin/Watt, and the thermal time constant of the junction-case of transistor VT4, in sec, respectively; Pk(s) – is the AC component of the instantaneous power dissipated on the collector of that transistor written in the operational form, in Watts; U2(s) is the AC component of the voltage at the output of DR in the operational form, in Volts. This characteristic exhibits inertial non-linearity of the second order.
The transfer characteristic of the closed system with the inertial non-linear FC can be represented in a simplified form  as a product of the frequency-dependent function of the linear part H2(s) and non-linear transfer function F2 (u2) with the resistive character. As consequence, we get :
where is the resistance in the emitter leg of VT4, in Ohms; is the time constant that corresponds to the pole frequency of the transfer function of DR, in sec. This function can be determined using the following condition:
where - is the voltage gain of DR without TEFC.
From the expression (3):
For the values of the frequency, i.e. at, the dependencies of AFC and non-linear distortions of DR (see Fig.2) has been obtained . It is seen from the figure that the influence of TEFC ceases above the frequency , the AFC of the DR stage becomes flat, and the non-linear distortions die with the rate of 6 dB/oct. If in the equation (2), and if Um2 < Un2 ~ Uo2 (Um2 is the amplitude of the AC component of the voltage at the output of DR), then the value of the coefficient of the second harmonic of DR is given by the formula :
One important feature of TEFC in DR is the instability of the pole frequency ïîëþñà at u2>0. In other words, this frequency has a deviation with the relative magnitude:
This means that the useful signal (with the frequency ) at the output of DR is phase-modulated by the interfering signal with the frequency if the two of them are simultaneously present at the input of DR, where the useful signal has frequency , and the interfering signal (infrasound) has a frequency and the amplitude larger than that of the useful signal. The phase deviation in this case equals:
According to the known relationship :
the phase deviation represents itself as a detonation of the useful signal, i.e. the BIM distortions appear at the frequency close to the pole frequency as result of infrasound oscillations at the input of ASF in DR.
Fig.2. Dependence of distortions (AFC and non-linear) on frequency in the driver stage (DR) of ASF.
Calculation of distortions in DR.
As an example, let us calculate the range of frequencies and amplification roll-off, as well as the second harmonic coefficient and BIM distortions caused by TEFC in DR. For this example we will take the values relevant for the transistor KT602Á, which has RT=45 K/W  and = 6 ms. The transistor mode in DR is the following:
Uo= 20 V, Un= 125 V, Rn= 15 kOhm, Ie = 7 mA, Re= 4.3 Ohm (at the average p-n junction temperature), Um= 15 V. Using these parameters (equations (2), (3), (5), and (6)-(8)), the calculation yields the following values :
· Relative amplification roll-off at low frequency d=3.24 (10dB);
· The frequency where the amplification decay starts (at the level –3 dB) fp = 86 Hz;
· The frequency where the amplification decay stops (at the level +3 dB) = 26.3 Hz;
· Harmonic distortions at = 11.7%;
· The relative deviation of the pole frequency fp equals 10.3%; the deviation of the phase of the signal for the frequency in the vicinity of the pole frequency is 0.053 rad, which corresponds to the 1.3% of the frequency detonation.
Thermal distortions in the differential stage.
The observation of the thermal distortions in DR (see Fig.1) motivated us to study TEFC in the transistors VT1, VT2 of the differential stage DS , although the low output voltage of DS and insignificant power dissipated on the collector of its transistors do not create any conditions for sufficient impact of TEFC. The attributes of TEFC in DS are the following:
· The voltage at the output of DS is negligibly small (approximately 1000 times smaller than at the output of DR), therefore, the transfer characteristic of TEFC in DS can be assumed linear;
· TEFC in DS has a sign opposite to that of TEFC in DR, since the source voltage of DS and collector-emitter voltage of transistors in DS are approximately equal;
· The signals at the output of TEFC in VT1 and VT2 are added together, since they are connected between the emitters of those transistors in series and in phase.
Based on the statements above, the transfer function of DS can be written as :
where is the voltage transfer coefficient of DS (without the influence of TEFC);
01(s) is the transfer characteristic of TEFC in DS; is the thermal time constant of junction-case of the transistors VT1, VT2, in sec; is the time constant corresponding to the pole frequency of the transfer function of DS with TEFC (is defined similar as, see the equation (4)), in sec; are the resistance of the DS load and the emitter of VT1, VT2, respectively (see Fig.1), in Ohms; RT1 is the thermal resistance of the junction-case of the transistors VT1, VT2, in K/W; U01 is the collector-emitter voltage of transistor VT1, in Volts.
In practical embodiments of DS circuits the following inequality holds: . Hence, TEFC is positive. Given that , it means that the instability of the DS regime takes place at the frequency close to zero. This instability can be excluded if the condition or 2.2 is satisfied, where R is the thermal resistance of p-n junction-case of transistors VT1 and VT2 of DS, in K/W.
Thereby, substantial linear distortions appear in DS in the infrasound frequency region. They show themselves as a leading shift of the signal phase, and also in the boost of AFC. This boost is limited, since the operational point also shifts into the region of TEFC conditional stability. Similar results were obtained in .
Thermal distortions in the dynamical load.
The collector circuit of transistor VT4 of DR and emitter circuit of transistors VT1, VT2 of DS (see Fig.1) comprises the dynamical load (DL) built on the bipolar transistor. The external circuit of the base of that bipolar transistor has a relatively small resistance. As a result of it, the influence of TEFC is considerable in DL (like in DS and DR). Since DL is a two-terminal device, the TEFC appears as a change in the total dynamic resistance of DL in the low-frequency sound region. In this case TEFC can be calculated as being connected in series by voltage.
The dependence of the total dynamic resistance of DL on the parameters of the sub-circuit can be written as:
where is the dynamical resistance of DL (without the influence of TEFC), in Ohms; * is the transfer characteristic of TEFC in DL; RK is the resistance of all circuits parallel to DL (including self) without any influence of TEFC, in Ohms; R*Ý is the resistance in the emitter circuit of DL, in Ohms.
When DL is included in DS, i.e. in the legs of the transistors VT1, VT2, DS behaves similar to a linear two-terminal device. This can be explained by the insufficiency of in-phase part of the signal applied to DL. Although DL is not a source of the non-linear distortions in DS, the change in the value of the dynamical resistance of the former contributes to the in-phase part of the signal at the output of ASF. Consequently, it will signify the appearance of other types of distortions  when ASF operates in the negative feedback regime.
The decrease of the total dynamic resistance of DL, built on transistor VT5, has a different effect on the operation of DR. As it has been shown in , the low-signal transfer function of DR (with DL) can be written as:
where is the thermal time constant junction-case of the transistor VT5, in K/W; is the time constant corresponding to the pole frequency of the fall of amplification of DR due to TEFC in DL (is determined similar as , see equation (4)), in sec; RK2 is the resistance of all legs connected in parallel to the output of DR, including the dynamic resistance of collector of VT4 (without TEFC), in Ohms; is the thermal resistance of the junction-case of the transistor VT5, in K/W; are the voltage source of DL (extrapolated value) and the collector-emitter voltage of the transistor VT5 (without a signal), respectively, in Volts; is the resistance in the emitter circuit of VT5, in Ohms.
It is important to notice that the employment of DL in DR ensures the partial linearization of the stage because of mutual compensation of the higher-order terms, due to TEFC in both VT4 and VT5.
Thermal distortions in the amplifier with the common negative feedback coupling.
The discussed thermal distortions in DS, DR and OS are being observed as various non-linear distortions of the signal at the output of ASF, connected in a common negative feedback coupling.
Harmonic distortions. The non-linearity of the transfer characteristic of DR according to (5) and the decrease of the loop gain of ASF on the signal frequencies are the main cause of the noticeable growth of harmonic distortions in the infrasound region at the output of ASF. This growth of harmonic distortions was observed earlier in transistor-based ASF ; however, this phenomenon has not been satisfactorily explained until now.
Infrasound intermodulation distortions. In the course of operation of the household ASF one can observe the electrical oscillations with the frequency 3 - 10 Hz at its input. These oscillations are can be produced, for example, when a slightly warped vinyl disk is being played. As a result of the increase in a low-frequency gain at the output of ASF caused by the frequency correction of the recording, these oscillations can have the amplitude equal to one half of the maximum amplitude of the recording . Under these conditions, the error signal is formed between the two inputs of DS; this error signal is comprised of both the useful signal and the infrasound oscillation component with the amplitude larger than that of the useful signal. The observed exchange of amplitudes can be explained by the reduction of the loop gain of the common negative feedback coupling in the infrasound region that is caused by the decay of amplification of DR below the frequency . If thereby the amplitude of the signal doesn’t leave the linear part of the transfer characteristic of DS, then the infrasound intermodulation distortions of BIMA type  are being formed in DS. The mechanism of the formation of these distortions is similar to the mechanism of the indermodulation distortions TIM .
As it has been shown in , the coefficient of the infrasound intermodulation distortions in the ASF with the common negative feedback coupling can be determined according to the following formula:
where is the oscillation amplitude of the infrasound frequency at the output of ASF, in Volts; UR1 is the constant voltage at the load DS, in Volts; is the modulus of the total dynamic resistance of the load of DR at the infrasound frequency, in Ohms; is the sensitivity function that connects the amplification of the ASF with the common negative feedback coupling to the change in amplification of its forward transfer at the frequency of the useful signal. Since the loop gain of the common negative feedback coupling of ASF turns by phase ±p/2 between the frequency of the dominant pole of the common negative feedback coupling and the frequency , then the parasite amplitude modulation becomes the frequency modulation as a result of the amplitude-phase conversion (APC) , i.e. one can observe the distortions of BIMF type at both low and high sound frequencies at the output of ASF with the common negative feedback coupling.
Distortions of the envelope function. Distortions of the envelope function in the experimental ASF (see Fig.1) also appear in DS because of non-linear coupling of the transfer characteristic of the useful signal, which has a form of the amplitude-modulated oscillations, with the signal representing the envelope function of these oscillations. The latter is formed as a result of non-linear conversion of the amplitude-modulated oscillations on the transfer characteristic of DR (see Fig.2) at the frequencies .
If we recognize the fact that the envelope function of the voice or musical signal has a finite frequency spectrum, then the amplitude of the modulated oscillations can as well be less than the amplitude of the signal representing the envelope function (taking into account the loop gain of ASF dies at the frequency ).
Therefore, the error signal of the common negative feedback coupling is being formed between the two inputs of DS; this error signal has a dominant infrasound oscillation component, which is a derivative of the envelope function of the useful signal. Thereby, if the amplitude of the signal doesn’t leave the linear part of the transfer characteristic of DS, then the infrasound intermodulation distortions of BIMA type, defined in  as envelope function distortions, are formed in DS.
Taking into consideration the common negative feedback coupling, the distortion coefficient of the envelope function of ASF can be determined according to the following formula :
where is the maximum value of the amplitude of the modulated signal at the output of ASF, in V; is the modulus of the total dynamic resistance of the load of DS at the frequency in Ohms; S(w) is the sensitivity function that connects the amplification of the ASF with the common negative feedback coupling with the change in amplification of ASF forward transfer at the frequency of the spectrum of the modulated oscillations.
As it was in prior cases, in the discussed instance one can observe the conversion of BIMA into BIMF, therefore, the distortions of the envelope function are being transformed into distortions of the instantaneous carrier frequency of the useful signal as a result of APC.
The results of the experiment.
During the experimental studies of the effects performed on the prototype device (see Fig.3), it turned out that the most convincing result was the observation of the linear thermal distortions on the transistor VT4, since there are no other possibilities for the appearance of other types of distortions in the given frequency window. We measured the AFC and the transfer reaction of the error signal in the summing junction of the amplifier. It was determined that the error signal in the summing junction of the amplifier grows from 60 to 20 when a sine wave voltage is applied at the input of that amplifier. The relative gain of the signal was 7 dB. The measurements were carried out on the middle sound frequencies, where we found another frequency window from 2.8 – 1.5 kHz with a relative gain of the error signal 5 dB. This phenomenon can be explained by the existence of the local heating-up zones in the crystal that, as it is well known, are the reason for the formation of the secondary breakdown region in transistors.
It has been revealed that practically every transistor in the amplifier is a source of the thermal distortions in the low sound frequency region. The application of the composite signal at the input of ASF in this case leads to the transformation of these distortions into infrasound intermodulation distortions, known as Bass Intermodulation (BIM) .
Thermal distortions appear when the instantaneous signal interferes with the thermosensitive parameters of the transistor – current amplification coefficient and the base-emitter voltage, due to thermoelectric conversion in transistors .
Fig.3. The principle electric circuit of the ASF prototype that has been used for the measurements of thermal distortions.
The current gain of the output stage (see Fig.1) can change by 200-300% due to the change of the dissipated power on the collector of the transistors during the transmission of the real (non-stationary) signal. This leads to the change of the voltage gain of that stage. The rate of this change is low, being limited by the thermal time constants junction-case of the transistor. However, even this slow change affects the distortion of the signal envelope function and it is significant for perception of the signal.
The most significant parameter in the driver stage and its dynamical load is the thermal sensitivity of the base-emitter voltage of the transistors. As a result of the conversion of the stage output voltage into instantaneously dissipated power on the collector of the transistor, then into the temperature of p-n junction, and then into the change of the base-emitter voltage, the non-linear thermoelectric feedback coupling is formed, which causes the reduction of the gain of that stage and the increase of the non-linear distortions in the low sound frequency region.
The differential stage, in spite of the low signal value at the output, also shows the noticeable influence of the thermoelectric feedback coupling, which is a linear one in this case, and moreover, it is positive.
Thermal distortions can be detected at the output of ASF with a common negative feedback coupling (see Fig.1). One can observe :
· Harmonic distortions in the low sound frequency region (under 100 Hz);
· Infrasound intermodulation distortions of BIMA and BIM type;
· Envelope function distortions, produced in the output and differential stages; in the latter case
· due to the transfer of the significant error voltage back to its input by the negative feedback loop.
The infrasound intermodulation distortions and envelope function distortions, converted into the parasite frequency modulation (BIMF) due to AFC, cannot be controlled in the household ASF. At the same time, it is known  that even the smallest values (approximately 0.06%) of the frequency modulation in the composite signal are audible. This subjective difference in perception of different ASF with similar traditionally controlled parameters can apparently be explained by the distinct values of the BIM type distortions.
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