positive feedback tends to sharpen the frequency-response curve and to decrease the range of uniform response. This permits an increased gain and selectivity. Positive feedback in an amplifier is critical of adjustment. Too much regenerative feedback in any system may result in oscillation. Ordinarily, negative feedback is more common than positive, feedback in amplifiers.

The action of a feed-back path depends upon the frequency of operation. That is, the feedback may remain regenerative or degenerative

throughout the range of operation of the circuit, although the magnitude and phase angle of the feed-back signal may vary with frequency. It is also possible for the feedback to be positive over a certain range of frequencies and negative over another range.

The principle of feedback is illustrated in the schematic diagram of Fig. 5-16. For simplicity, series injection is shown at the input, but other forms of network coupling may be employed. In the diagram shown, a voltage Eg is applied to the input terminals of the amplifier, with the polarity as shown. Suppose that the resultant voltage at the output terminals is Epk. Suppose that a fraction ¡3 of this output is fed back in series with the input signal in such a way that the resultant signal that appears between the grid-cathode terminals has the form

But since the nominal gain of the amplifier is, by definition,

~ input potential between grid and cathode E„k then

Epk = KEgk

By combining this with Eq. (5-31), there results

EPk = KEg + KpEpk from which it follows that

But the resultant gain of the amplifier including the effects of feedback is defined as g _ output potential _ Evk r — input-signal potential Eg

Therefore it follows that

This equation expresses the resultant gain of the amplifier with feedback in terms of the nominal gain K of the amplifier without feedback, and the feed-back fraction p. It should be noted that the three quantities Kr, K, and /?, which appear in this equation, may be complex quantities.

Suppose that the feedback is negative and that the feed-back factor K(i is made large compared with unity. The resultant gain equation (5-34) becomes

This means that when | K/3 | » 1, the actual amplification with negative feedback is a function of the characteristics of the feed-back network only. In particular, if ¡3 is independent of frequency, then the over-all gain will be independent of the frequency. This permits a substantial reduction of the frequency and phase distortion of the amplifier. In fact, by the proper choice of feed-back network, it is possible to achieve almost any desired frequency characteristic.

Note that if K/3 » 1, then Kr = K/K(i « K, so that the over-all gain of the amplifier with inverse feedback is less than the nominal gain without feedback. This is the price that must be paid to secure the advantages of negative feedback. This is not a serious price to pay, since the loss in gain can be overcome by the use of additional tubes.

Clearly, if Kfi is greater than unity, then Eq. (5-35) shows that the over-all gain will not change with tube replacements or with variations in battery potentials, since /? is independent of the tube. Even if Eq. (5-35) is not completely valid, a substantial improvement results in general stability. This follows from the fact that a change in the nominal gain dK for whatever reason results in a change dKr in the resultant gain by an amount __ ______

where | 1 — K/3 | represents the magnitude of the quantity 1 — Kfi. This equation is the logarithmic derivative of Eq. (5-34). In this expression, dKr/Kr gives the fractional change in Kr, and dK/K gives the

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