4.4 Global Negative Feedback in a vacuum tube amplifier

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

A technique to reduce distortions, across all amplifier stages, is the global negative feedback. The effect of the global negative feedback can be explained intuitively as follows.

Global negative feedback consists in using a negative feedback loop, which subtracts the output signal, appropriately attenuated, from the input signal. If the input and the output signals have identical shapes, the subtraction has the only effect of attenuating the input signal and consequently the produced output signal.

Similarly, when a distorted output signal is fed back, and subtracted from the input signal, distortion is also attenuated. To better understand this, first note that distortion level produced by an amplifier stage depends on the output level and does not depend on its input level. Suppose we compare two output signals having the same level: one produced with global negative feedback and the other without[3]. Given that the distortion depends on the output level, in both cases, the distortion internally introduced is the same. However, the outputs signal produced with negative feedback has less distortion, given that the produced distortion itself has been subtracted and 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 R₁ and R₂, from the output signal. Resistor R₁ is generally called the feedback resistor. The feedback signal V_fb, 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 V_fb from the input signal V_in. 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 V_fb is obtained using the voltage divider equation:

.

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 R₁, to form a 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:

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The quantity Aβ 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_d, where V_d is the harmonic distortion introduced by the amplifier itself. The harmonic distortion percentage 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 to 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_out (without negative feedback) or  (with negative feedback). Given that we set  equal to the output signal without feedback V_out, the harmonic distortion V_d, internally introduced by the amplifier, is the same in both cases. However, the second of the above equations says that when using negative feedback, V_d is also fed back and attenuated to . Simplifying, as before, we obtain that

.

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_dB of 20 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 f₁₈₀ is the frequency where the 180° phase-shift occurs, and A₁₈₀ is the amplifier gain at this frequency. If the loop gain A₁₈₀β is smaller than 1 (that is ), positive feedback introduces just a gain peak at f₁₈₀. However, if A₁₈₀β 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 A₁₈₀β becomes smaller than 1. Low pass filters, obtained using grid stopper resistors, and high pass filters, produced by inter-stage coupling capacitors, can accomplish to this task.

However, not always low-pass and high-pass filters are able to effectively eliminate the conditions for oscillation and instability. In these cases, a step-network can be used to move the 180° phase-shift where A₁₈₀β 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_sn and resistor R₁.

Example 15: Step network for negative feedback stability Consider, for instance, an amplifier with an open loop gain A=500 (54dB). Suppose that the solid and dashed red plot in Figure 32 respectively give the gain and phase-shift of the amplifier, with respect to the frequency. We can see that the amplifier introduces a phase-shift of -180°, around 240K Hz, where the gain A240KHz is still 25 (28dB). Suppose we apply a feedback factor β of 0.1, corresponding to a loop gain β∙A=0.1∙355=35.5 (fbdB=31.2dB), by setting R1 to 10K Ohm and R2 to 1K Ohm. At 240K Hz we have that the loop gain is β∙A240KHz=0.1∙25=2.5 (fbdB=10.9dB). Since the loop gain at 240K Hz is greater than 1 (fbdB>6dB) and the phase shift is -180°, the amplifier will oscillate. To avoid oscillation, we have to move the -180° phase-shift at a frequency f where the loop gain βAf is smaller than 1 (fbdB<6dB). To have βAf<1, with β=0.1, we need f such that the amplifier gain Afat frequency f is Af < 1/β=10 (20dB). The frequency where we have a 20dB gain is f=400K Hz. At this frequency, the phase-shift is -220°. To go below -180° we have to compensate at least 40°. Using a capacitor Csn of 80 pF, in parallel with R1, we obtain the frequency compensation shown by the red dashed line in Figure 33. We can see that the phase compensation at 240K Hz is 45° and at 400 Hz is 54°. The solid and dashed blue line in Figure 32 represents the loop gain and its phase-shift, with this compensation. At 240K Hz the phase-shift is now -135°, at 400K Hz it is -166°. The -180° phase-shift now occurs at 500K Hz, where the amplifier gain A500KHz is 5 (14 dB), thus below the threshold 10, which we identified before. The amplifier now is stable, in fact, βA500=0.1∙5=0.5, and fbdB=3.5dB.