Feedback Amplifiers part5

Equivalence of Circuits from a Feedback-Loop Point of View

From the study of circuit theory we know that the poles of a circuit are independent of the external excitation. In fact the poles, or the natural modes (which is a more appropriate name), are determined by setting the external excitation to zero. It follows that the poles of a feedback amplifier depend only on the feedback loop. Thus, a given feedback loop may be used to generate a number of circuits having the same poles but different transmission zeros. The closed-loop gain and the transmission zeros depend on how and where the input signal is injected into the loop.

As an exam nsider the feedback loop of Fig. 5.14(a). This loop can be used to generate the nverting op-amp circuit by feeding the input voltage signal to the terminal of R is connected to ground; that is, we lift this terminal off ground and connect it to he same feedback loop can be used to generate the inverting op amp circuit by feeding the input voltage signal to the terminal of hat is connected to ground.

Recognition of the fact that two or more circuits are equivalent from a feedback- loop point of view is very useful because stability is a function of the loop. Thus one needs to perform the stability analysis only once for a given loop.

THE STABILITY PROBLEM

Transfer Function of the Feedback Amplifier

In a feedback amplifier such as that represented by the general structure of Fig. 5.1, the open-loop gain A is generally a function of frequency, and it should therefore be more accurately called the open-loop transfer function, A(s). Also, we have been assuming for the most part that the feedback network is resistive and hence that the feedback factor is constant, but this need not be always the case. We shall therefore assume that in the general cas ack transfer function is β(s). It follows that the closed-loop transfer functio v

It is the manner in which the loop gain varies with frequency that determines the stability or instability of the feedback amplifier. To appreciate this fact, consider the frequency at which the phase angle ( ) becomes 180°. At this frequency, , the loop gain ( ( ) ( ) will be a real number with a negative sign. Thus at this frequency the feedback will become positive. If at = , the magnitude of the loop gain is less than unity, then from Eq. (5.27) we see that the closed-loop gain ( ) will be greater than the open-loop gain ( ) since the denominator of Eq. (5.27) will be smaller than unity. Nevertheless, the feedback amplifier will be stable. On the other hand, if at the frequency the magnitude of the loop gain is equal to unity, it follows from Eq. (5.27) that ( ) will be infinite. This means that the amplifier will have an output for zero input; this is by definition an oscillator. To visualize how this feedback loop may oscillate, consider the general loop of Fig. 5.1 with the external input set to zero. Any disturbance in the circuit, such as the closure of the power-supply switch, will generate a signal xi(t) at the input to the amplifier. Such a noise signal usually contains a wide range of frequencies, and we shall now concentrate on the component with frequency = that is, the signal sin( ). This input signal will result in a feedback signal given by

Since is further multiplied by -1 in the summer block at the input, we see that the feedback causes the signal Xi, at the amplifier input to be sustained. That is, from this point on, there will be sinusoidal signals at the amplifier input and output of frequency .Thus the amplifier is said to oscillate at the frequency . If at , the magnitude of the loop gain is greater than unity, the circuit will oscillate, and the oscillations will grow in amplitude until some nonlinearity (which is always present in some form) reduces the magnitude of the loop gain to exactly unity, at which point sustained oscillations will be obtained. This mechanism for starting oscillations by using positive feedback with a loop gain greater than unity, and then using a nonlinearity to reduce the loop gain to unity at the desired amplitude, will be exploited in the design of sinusoidal oscillators. Our objective here is just the opposite: Now that we know how oscillations could occur in a negative-feedback amplifier, we wish to find methods to prevent their occurrence

The Nyquist Plot

The Nyquist plot is a formalized approach for testing for stability based on the discussion above. It is simply a polar plot of loop gain with frequency used as a parameter. Figure 5.15 shows such a plot. Note that the radial distance is | | and the angle is the phase angle . The solid-line plot is for positive frequencies. Since the loop gain - and for that matter any gain function of a physical network—has a magnitude that is an even function of frequency and a phase that is an odd function of frequency, the plot for negative frequencies (shown in Fig. 5.15 as a broken line) can be drawn as a mirror image through the Re axis. The Nyquist plot intersects the negative real axis at the frequency .

i. Thus, if this intersection occurs to the left of the point (-1, 0), we know that the magnitude of loop gain at this frequency is greater than unity and the amplifier will be unstable.

ii. On the other hand, if the intersection occurs to the right of the point (-1, 0) the amplifier will be stable.

iii. It follows that if the Nyquist plot encircles the point (-1, 0) then the amplifier will be unstable.

EFFECT OF FEEDBACK ON THE AMPLIFIER POLES

The amplifier frequency response and stability are determined directly by its poles. We shall therefore investigate the effect of feedback on the poles of the amplifier.

Stability and Pole Location

For an amplifier or any other system to be stable, its poles should lie in the left half of the s- plane. A pair of complex-conjugate poles on the axis gives rise to sustained sinusoidal oscillations. Poles in the right half of the s plane give rise to growing oscillations. To verify the statement above, consider an amplifier with a pole pair at = ± . If this amplifier is subjected to a disturbance, such as that caused by closure of the power-supply switch, its transient response will contain terms of the form

This is a sinusoidal signal with an envelope . Now if the poles are in the left half of the s-plane, then will be negative and the oscillations will decay exponentially toward zero, as .shown in Fig. 5.16(a), indicating that the system is stable. If, on the other hand, the poles are in the right half-plane, then will be positive, and the oscillations will grow exponentially (until some nonlinearity limits their growth), as shown in Fig. 5.16(b). Finally, if the poles are on the axis, then will be zero and the oscillations will be sustained, as shown in Fig. 5.16(c).

Poles of the Feedback Amplifier

From the closed-loop transfer function in Eq. (5.26), we see that the poles of the feedback amplifier are the zeros of 1 + ( ) ( ). That is, the feedback-amplifier poles are obtained by solving the equation

which is called the characteristic equation of the feedback loop. It should therefore be apparent that applying feedback to an amplifier changes its poles.

In the following, we shall consider how feedback affects the amplifier poles. For this purpose we shall assume that the open-loop amplifier has real poles and no finite zeros (i.e.,all the zeros are at = ∞ . This will simplify the analysis and enable us to focus our attention on the fundamental concepts involved. We shall also assume that the feedback factor is independent of frequency

Amplifier with a Single-Pole Response

This process is illustrated in Fig. 5.16(a). Figure 5.16(b) shows Bode plots for | | and |Af|. Note that while at low frequencies the difference between the two plots is 20 log(l + Aoβ), the two curves coincide at high frequencies. One can show that this indeed is the case by approximating Eq. (5.33) for frequencies

Physically speaking, at such high frequencies the loop gain is much smaller than unity and the feedback is ineffective. Figure 5.16(b) clearly illustrates the fact that applying negative feedback to an amplifier results in extending its bandwidth at the expense of a reduction in gain. Since the pole of the closed-loop amplifier never enters the right half of the s plane, the single-pole amplifier is stable for any value of β. Thus this amplifier is said to be unconditionally stable. This result, however, is hardly surprising, since the phase lag associated with a single-pole response can never be greater than 90°. Thus the loop gain never achieves the 180° phase shift required for the feedback to become positive.