# Feedback: negative and positive (part 1)

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 Introduction 'Feedback' is the term which is given to the introduction of a signal voltage, derived from the output of what is usually an amplifying stage, into the input of that stage. This feedback can be present as a deliberate feature of the circuit design, but may also occur inadvertently, either because of shortcomings in the amplifying devices or other circuit components which are used, or because of oversights in the design of the circuitry in which they are employed. Whatever the cause, the presence of feedback can significantly alter the performance of the circuit, so, although this can be a rather complex aspect of circuit theory, it’s necessary for its effects to be broadly understood by any engineer seeking to try his hand at circuit design. The basic effects of feedback If the polarity of the signal which is fed back is of the same sign as that input signal which generated it, then the feedback signal is said to be 'positive'. If its polarity is opposite to that of the input signal, it’s said to be 'negative'. (Because negative feedback is such a common feature of circuit design, it’s often just referred to as NFB). In the case of positive feedback (which I shall call PFB), because the signal which is fed back is of the same polarity as the input signal, it will act to increase the size of this signal, which will, of course, then increase the size of the output signal, which will, in turn, increase the size of the voltage which is fed back. Clearly, unless there is some loss of signal somewhere in the system, or some constraint on the possible input or output voltage swing, this process would cause the gain block to produce an infinitely large output. In a DC system, this effect will usually cause the system to 'latch up' -- a condition in which the output voltage is driven fully towards one or other of the voltage limits of which the output is capable. In an AC coupled system, the result will usually be a continuous and uncontrolled oscillation. Both of these effects of PFB can be utilized in circuit design. FIG1 The use of positive feedback to provide bistable or mono-stable logic elements. For example, in the DC systems shown in FIG1, the circuit layout can be arranged to provide either a 'bi-stable' or a 'mono-stable' function block. In the first of these forms, shown in FIG1a, the output voltage can be made to jump rapidly from one stable output voltage state to another, where it will rest until a suitable input signal restores the original output voltage condition of the circuit. In the second case, shown in FIG1b, the output voltage can be made to change from its normal rest condition to a different voltage level, where it will remain for a period of time, determined in this case by the values chosen for Cfb and f ?fb, before reverting to its original state. Both of these types of function are widely used in logic elements. Positive feedback, in an AC circuit, can be used to produce a continuous oscillatory voltage output, which could be used as a signal source, although for this purpose some means of controlling both the frequency of oscillation and the amount of feedback -- which will determine the output signal voltage -- is necessary. As an example of this, in the simple 'Wien bridge' oscillator circuit, shown in FIG2, the system is made to oscillate continuously by the use of positive feedback, which is applied from the output to the non-inverting input of a gain block (Ai), via the RC Wien network -- R\Cu RlCi -- (in which R1= R2, and C1=C2). This network has the interesting characteristic that, at a frequency at which the impedance of R2+C2 is twice that of R1 in parallel with C\ (for which the shorthand notation R\\\C\ is often used), the out put, at point ?\ is in phase with the input, at point ? \ and if the gain of the amplifier is equal to, or slightly exceeds 3, the circuit will give a sinusoidal output signal at a frequency defined by the equation: F=1/(2 pi CR). In practical oscillator circuit designs of this type, negative feedback, by way of a signal applied to the inverting input of the gain block, is then used to control the gain of the amplifier, so that is high enough to cause the circuit to oscillate, but not so high that the output sinewave signal is distorted by 'clipping', due to the limits on the output voltage swing imposed by the supply voltage rails. This can be done quite conveniently, by the use of an output voltage-dependent component, such as the low-power thermistor (TH1), shown in FIG2, whose resistance value decreases as the output voltage increases, and which will thereby control the amount of the negative feedback signal applied to the inverting input of the gain block. If the value of 7?f b is chosen correctly, this NFB input will reduce the gain so that it’s exactly equal to 3 at the desired AC output voltage swing. FIG2 A 'Wien bridge' oscillator using both +ve and -ve feedback. It’s convenient, for the purposes of explanation of circuit behavior, to postulate an ideal amplifying gain block, having inverting (-in) and non-inverting (+in) inputs, an infinitely high value of internal gain, and a zero impedance output point, a concept which is called an 'operational amplifier'. Devices such as the integrated circuit wide band gain blocks, such as the '741' type device, and particularly the later and improved FET input versions of this kind of circuit, such as an 'LF351' or the TL051' come very close to satisfying the operational amplifier specification, in that they have a very high gain (100,000 or greater), a low output impedance (less than 200 ohms), and, especially in the case of the FET-input versions, very high input impedance, and are indeed described as opamps.' in the makers' catalogs. If an IC of this kind is used as the gain block, shown symbolically as Ai in FIG2, operated from +/ 15V supply rails, the output voltage swing is set, by the choice of Rfb, to lie between 1V and 3V RMS, the total harmonic distortion of the circuit would be of the order of 0.01-0.03%, at 1kHz. In the notation used in FIG3 (and which I have also used in FIGs 1 and.2), the symbols '+' and '-' within the triangular opamp. diagram, are taken to represent the 'non-inverting' and 'inverting' inputs of the amplifier. The + and -- symbols outside the triangle are taken to represent connections to some external DC power supply (typically +/-15 V), though these connections are often omitted from the circuit diagram unless their connections are, in some way, significant in respect of the operation of the circuit. FIG3 Conventional notation for 'operational amplifier' type of gain block Positive feedback can also be used, with care, to increase the AC voltage gain, or output swing obtain able from an amplifier circuit. A very common example of this is the use of a 'bootstrapped' load resistor, as in the circuit of FIG4 (the term is said to derive from the fanciful idea of lifting oneself up by ones own bootstraps'), which is commonly used to increase the effectiveness of the driver stage in simple audio amplifier designs. In this circuit, a volt age signal derived from the outputs of the two push pull emitter-followers (Q2/Q3) (point R), is coupled back to the +ve end of RA (point S), by way of O, so that if the output voltage at R, swings up or down, the potential at point S will follow this voltage excursion. This greatly increases the stage gain due to Q1, by virtue of the positive feedback signal, because it makes the dynamic impedance of RA much higher than its static value. However, except in circuit layouts which are deliberately designed to function as 'free running' oscillators -- a topic which is explored at greater length in Sections 10 and 13 -- positive feedback is generally avoided, and precautions are frequently taken to avoid its inadvertent occurrence, since it can lead to undesirable effects on the circuit performance. FIG4 The use of a bootstrapped load resistor (R3) to increase available output voltage swing FIG5 Schematic gain block with feedback loop The effect of feedback on gain The basic characteristics of a simple series feedback circuit (i.e. one in which the feedback signal is applied in series with the input signal) can be explained by reference to the block diagram shown in FIG5, in which the square symbol is used to represent a circuit element with a gain of A, and which has two independent and isolated input and output circuits, m -- n and p -- q, arranged so that an input signal voltage e, applied between m and n will generate an output voltage of Ae between p and q, without sign inversion. If we define the signal actually applied between the input points m and n as e\% and the signal appearing at the output points p and q as £0, then the stage gain of the amplifying block, A, is: D A=^ which is known as the 'open-loop' gain -- i.e. the gain before any feedback is applied. However, if there is an electrical network between the output and the input of this gain block which causes a proportion ß of the output signal to be returned to the input circuit, in series with the applied input signal Eu then the actual input signal fed to the gain block will be altered so that e\ will actually be E\ + ß£0, or … From these equations, we can determine the effect of feedback on the stage gain, bearing in mind that, in the absence of a feedback signal, E\ = e\. The closed-loop stage gain, i.e. that when feedback is applied, I have called A', and is defined by the equation … There are several conventions for the names used in feedback systems, but the one I prefer is that in which the term 'ß?', the product of the 'open-loop' gain and the proportion of the signal reintroduced to the input by the feedback network, is referred to as the "feed back factor". It must be remembered that, in this equation, the sign of ß may be either positive or negative depending on the configuration of the feedback network. In the case of a NFB system, ß is negative, so that the gain becomes A/(l + ß?) [A confusing usage of the term feedback factor, which I think should be avoided, is when it’s employed, in NFB systems, to describe the amount of negative feedback used in the design. This is usually expressed in dBs, and is normally defined as the extent to which the open-loop gain is reduced when negative feedback is applied {i.e. A'/A or 1/(1 + ß?)}. For example, if the closed-loop gain of an amplifier, with NFB, was reduced to one hundredth of its open-loop value, (-40dB), this would be described as 40dB of negative feedback.] Considering the relationships shown in equation (4), in the particular case where the network is non-inverting, and the feedback signal is returned to the input in the same phase (positive feedback), so that {A'/A = 1/(1- ß?)}, the situation arises in which the larger the feedback factor Aß, becomes, the smaller the denominator in this equation will be, and the higher the effective gain of the system -- up to the point where Aß = 1, when the denominator in the equation would become zero, and the closed-loop gain, A, would become infinite. This is the condition which will lead to 'latch-up' in a DC coupled circuit, or continuous oscillation in an AC coupled one, as, for example, in the Wien network oscillator described above, which will oscillate when E0 = 3Ei, and ß = 1/3, which leads to the condition that ß£0 = E\. If the feedback network is a phase inverting one, so that ß is negative, the gain equation will become ... In this case, the closed-loop gain will decrease as the amount of feedback is increased, up to the point at which ßA is so much greater than 1 that the equation approximates to ... This is particularly appropriate to those cases in which the open-loop gain (i.e. the gain before the application of feedback) is very high; as would nearly always be the case with operational amplifier integrated circuits at low to medium signal frequencies. In these circum stances, the gain is determined almost entirely by the attenuation ratio of the feedback network, so that if ß is 1/10, then the gain will be 10, and if ß is 1/100, then the circuit gain will be 100. This is a very useful effect, in that it makes the gain of the circuit arrangement almost completely independent of the gain of the amplifier block itself, so long as this is high in relation to ß. A specific advantage which arises in this case is that, if the feedback net work has an attenuation ratio which is constant, and independent of frequency, then, so long as the gain of the amplifier block remains high in relation to ß, then the system will have a stage gain of 1/ß and this will also be independent of operating frequency. The use of NFB to reduce distortion In addition to the improvement which NFB can make to the flatness of the frequency response of an amplifier, it will also, within limits, reduce harmonic and waveform distortion, as well as any 'hum' and noise generated within the amplifying circuit, which can indeed be considered as just another aspect of signal distortion. If the instantaneous relationship, between the input signal applied to a circuit block and the output voltage from that block, is considered to be its amplification factor, then any 'distortion' in the output signal can be regarded simply as an indication that the effective amplification factor of the system has changed during the transmission of the signal. This change might be the result of a variation in output voltage with time, within the duration of a nominally constant signal, perhaps because of the intrusion of mains hum or noise. Alternatively, it could be due to the characteristics of the system itself -- as shown in the case of the square-wave signal illustrated in FIG6. Again, it could be an unwanted change in the signal wave form, in the case of a continuously varying input signal voltage, because the gain of the system is influenced by the instantaneous magnitude of the input signal, due to nonlinearity in the input/output transfer characteristic, shown in FIG7. FIG6 Distortion of step-waveform due to circuit characteristics FIG7 Distortion due to nonlinear amplifier transfer function However, if the amplifier gain, A, is sufficiently high that the closed-loop gain (AO, becomes simply 1/ß, and if the factors which determine the value of ß are themselves noise and hum free and distortionless, then the output signal must also be distortionless, noise and hum free, simply because variations in the open-loop amplifier gain A, as a function of input voltage or time, are no longer factors which influence the output volt age. This is only strictly true for infinitely high values of A, but even for lower values of A the distortion will still be reduced by the feedback factor (1 + ß?), as shown by the following analysis, due to Langford Smith (Radio Designers Handbook, 4th Edition, Section 7). Let a sinusoidal input voltage, E1, of an amplifier employing negative feedback be defined by the term E'un cos ??, where ? = Inf. Then, if the amplifier introduces harmonic distortion, the output will be …where w om, w 2m and w 3m are the peak values, respectively, of the fundamental, second and third harmonic frequencies, etc. The feedback voltage (ß w 0) will therefore be … The voltage applied to the input of the amplifier will be … Since E0 = Aeu the output voltage will therefore be …where Ho and H3 are the ratios of the second and third harmonic voltages to the voltage of the fundamental in the amplifier without feedback. But the output voltage has already been defined by equation (6), so, by equating the fundamental components of equations (6) and (7) Equating the second harmonic components in equations (6) and (7) ... Inserting the value of e_im from (8)… Therefore,
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