Push-pull stages are able to reduce harmonic distortions, specifically second order distortions, produced in the stage itself. The input stage has a single-ended configuration. A single-ended stage, operating in non-linear areas, produces distortions. If the distorted pre-amplified signal is given to a push-pull stage, distortions are amplified as well.

A technique to reduce distortions, across all amplifier stages, is the *global negative feedback*. Global negative feedback consists in using a *negative feedback loop*, which subtracts the output signal, appropriately attenuated, from the input signal.

The effect of the global negative feedback can be explained intuitively as follows. If the input signal and the output signal have identical shapes, the subtraction has the only effect of attenuating the input signal to be amplified and consequently the produced output signal. Similarly, when distortion signal is fed back, and subtracted from the input signal, distortion is also reduced. Suppose now we compare two output signals having the same level, one produced with global negative feedback and the other without[3]. Note that distortion level depends on the output level, and does not depend on input level. This implies that, in both cases, the distortion internally introduced is the same. However, the outputs signal produced with negative feedback has less distortion, given that distortion itself has been attenuated through the feedback loop.

Figure 31 shows the basic schema of a global negative feedback loop. The feedback signal is produced by the voltage divider, composed of resistors *R _{1}* and

*R*, from the output signal. Resistor

_{2}*R*is generally called the

_{1 }*feedback resistor*. The feedback signal

*V*, provided by the voltage divider, is applied to the cathode of the input stage vacuum tube. When the output signal and the input signal have the same phase, this schema has the effect of subtracting the feedback signal

_{fb}*V*from the input signal

_{fb}*V*. In fact, in this case, the feedback signal shifts the cathode voltage in the same direction of the input signal, reducing the grid to cathode voltage. In this way the signal at the grid of the input stage is . The feedback signal

_{in}*V*is obtained using the voltage divider equation:

_{fb}.

The factor β

* *

is generally referred as the *feedback factor*. The amount of feedback can be set by choosing β, by way of the voltage divider resistors.

Note that, if the phase of the output signal were inverted, with respect the input signal, the circuit would produce a positive feedback, which increases distortions and produces oscillations. When phase shifts are produced through the amplifier’s stages and affect the correct operation of the feedback loop, a capacitance *C _{sn}* can be used, together with resistor

*R*, to form a

_{1}*step network*, which has the purpose of maintaining a correct phase of the feedback signal at all relevant frequencies, guaranteeing stability of the amplifier. This is better discussed in Section 4.4.3.

### 4.4.1 Gain with negative feedback

The gain of the amplifier without negative feedback is referred as the *open loop gain* of the amplifier. The* *open loop gain *A* is the ratio between the output and the input signals:

.

We said that, when there is global negative feedback, the signal seen by the grid is . Therefore, the output signal is

.

Simplifying we obtain

.

The *closed loop gain* *A ^{fb}* of the amplifier, which is the gain of the amplifier when global negative feedback is used, can be obtained as the ratio between the output signal, with negative feedback loop, and the input signal:

.

The quantity *A*β is generally referred as the *loop gain*, which is the gain seen *in* the feedback loop.

It is generally useful expressing the amount of negative feedback applied as the reduction of the gain in the amplifier. This can be easily obtained by expressing the gain in dB and computing the amount of feedback *fb _{dB}* as:

.

For instance, if the gain of the amplifier without negative feedback is 20dB and the gain with global negative feedback is 14dB, we say that we apply *fb _{dB}*=6dB of feedback.

### 4.4.2 Benefits of negative feedback

The use of negative feedbacks has several advantages. It stabilizes the gain of the amplifier, decreases output impedance, increases input impedance, increases bandwidth, and reduces distortions.

In the following, as an example, we discuss how negative feedback reduces harmonic distortions[4] by a factor , at the same output level of the amplifier without negative feedback.

Suppose there is no negative feedback loop. In correspondence of the input signal *V _{in}*, the amplifier produces the output signal

*V*+

_{out}*V*, where

_{d}*V*is the harmonic distortion introduced by the amplifier itself. The harmonic distortion percentage

_{d}*HD*is measured as the ratio between the harmonic distortion signal and the output signal:

Suppose now negative feedback loop is used and the input signal is increased from *V _{in}* to , to compensate the gain loss and obtain an output signal equal to the output signal obtained without negative feedback. In other words, the increased input signal is such that . Now, the output of the amplifier, considering also distortions, is . We can treat separately these two components using the equations discussed before:

.

The harmonic distortion *V _{d}* depends only on the output signal

*V*(without negative feedback) or (with negative feedback). Given that we set equal to the output signal without feedback

_{out}*V*, the harmonic distortion

_{out}*V*, internally introduced by the amplifier, is the same in both cases. However, the second of the above equations says that when using negative feedback,

_{d}*V*is also feed back and attenuated to . Simplifying, as before, we obtain that

_{d}.

With negative feedback, the produced harmonic distortion is attenuated, with respect to the harmonic distortion *V _{d}* generated without negative feedback, at the same output level. For instance, a negative feedback

*f*of 20

_{dB}*dB*implies a closed loop gain 10 times lower than the open loop gain. However, harmonic distortion, at the same output level, will be ten times lower as well.

### 4.4.3 Stability of negative feedback

The schema for negative feedback, given in Figure 31, requires that the input signal and the negative feedback signal, applied to the cathode, have the same phase. However, when the input signal goes through the amplifier stages, its phase might shift significantly, reaching in some cases a 180° phase-shift. With a 180° phase-shift, the feedback signal has an opposite phase with respect to the input signal and the feedback circuit becomes a positive feedback circuit. Positive feedback is dangerous because it might introduce instability and oscillations.

Consider that both low-pass and high-pass filters produce a 45° phase-shift at their cut-off frequency and that phase-shifts are accumulated in a sequence of filters. Therefore, a 180° phase-shift becomes probable at frequencies near the borders of the bandwidth of the amplifier.

Suppose *f _{180}* is the frequency where the 180° phase-shift occurs, and

*A*is the amplifier gain at this frequency. If the loop gain

_{180}*A*β is smaller than 1 (that is ), positive feedback introduces just a gain peak at

_{180}*f*. However, if

_{180}*A*β is greater or equal to 1 (that is ), the amplifier will oscillate. In order to avoid this, the amplifier has to be designed so that the frequency, where the 180° phase-shifts occurs, is where

_{180}*A*β becomes smaller than 1. Low pass filters, obtained using grid stopper resistors, and high pass filters, due to inter-stage coupling capacitors, can accomplish to this task.

_{180}However, not always low-pass and high-pass filters are able to effectively eliminating the conditions for oscillation and instability. In these cases, a *step-network* can be used to move the 180° phase-shift where *A _{180}*β is smaller than 1. A step network is obtained by using a capacitor in parallel with the feedback resistor, as shown in Figure 31, with capacitor

*C*and resistor

_{sn}*R*.

_{1}