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The Loop Gain - Nyquist Diagram

Author: N.H. Crowhurst

 The angle between two voltages - phase

We are fortunate that this time difference between the voltage fluctuation at different points in an amplifier at various frequencies can be designated as a phase angle. This means that we can draw a "picture" of the voltage with what is called a vector. It is a line drawn at an angle to correspond with the phase angle from a starting point representing zero.

The loop gain A(3 is the important thing to consider. We can draw a series of lines from a starting point called O (for Origin). Each line represents both the magnitude and the phase angle of Ap for a different frequency. Joining the outer ends of these lines produces a curve that represents all possible positions for the tip of the A(S vector and shows how Ap varies in phase and magnitude due to frequency changes. This curve is called a Nyquist diagram.

 Basic starting point for locus curve at frequency where there is no phase shift

In the case of negative feedback, the feedback factor is (1 4- Ap). First we draw a line in one direction from O, representing Afi. Assuming that there is no phase shift, we can mark off 1 in the opposite direction from O and the total length of the line is then the feedback factor (1 + Ap). In the case of positive feedback, both A(3 and the distance will be measured in the same direction from O and the disance between the end of the line representing Ap and 1 will represent the feedback factor (1 - Af5).

Either way, the feedback factor is given by the distance between the A(3 locus and the point measured off 1 unit from O. The convenience of this method is that the distance between any other point on the locus and this position, measured 1 unit from O, also represents both the magnitude and phase angle of the feedback factor.

Thus the feedback factor can be called (1 4- AP); bear in mind that Ap is not just a simple number now, but that it includes a phase angle. When this phase reaches 180°, Ap has become negative instead of positive, and represents fully positive feedback.

A rigorous proof of this would involve mathematics beyond the scope of this book, and an exact explanation would call for a knowledge of complex numbers. Looking at it as a simple geometrical diagram, however, can give us a good picture of what happens without knowing all the mathematics.

 Loop gain (magnitude and phase)
 A loop gain diagram

Now let's look a little more at the geometry relative to this diagram. The formula for the gain of an amplifier with feedback is Af = A/(l + AfJ).

In most amplifiers, the feedback fraction p is constant; it does not change either in magnitude or phase as we change frequency. The internal amplifier gain A is the part that changes with frequency and produces the phase shift. We could multiply the top and bottom of the fraction giving the amplification with feedback by p and still have the same results. We obtain a part of the formula for amplification with feedback that does not change with frequency 1/p, and a part that does change with frequency A|3/(l + AP).

In the locus diagram, we have a curve representing the locus of a point whose distance and angle from the point O represent the value of Ap in magnitude and phase. Also, the distance and angle from the point measured off 1 unit from O represents the magnitude and phase of (1 + AP). So the ratio of the distances of any point on the curve from these two points gives us the frequency-varying part of the formula for amplification with feedback.

It is a fact in geometry that all the points whose distances from two fixed points are in a fixed ratio form a circle. If we draw a family of circles representing different ratios of gain variation Ap/(l + AP), we have a background that will help us interpret this locus diagram curve. In the background circles shown here, the vertical straight line joins all points where Ap = 1 4- Ap. From there, curves are drawn at 0.5-db differences in ratio up to 3 db either way [Ap = 1.414 X (1 + Ap) or 1 + Ap = (1.414 X Ap)}. From there to 10 db, the circles are at 1-db intervals, and from 10 to 20 db, at 2-db.

 Family of circles representing different ratios of gain variations
 Constant gain diagram

The locus vector curve itself is called a Nyquist diagram. If the curve representing the locus of Ap follows one of the circles which represents a constant ratio of distances from O and 1, the gain of the amplifier with feedback would be constant, although there would be phase change in both A(5 and (14- Ap), as well as a transition from positive to negative value of Ap.

This particular response is impossible with any practical amplifier. At some frequency, A must fall to zero. Usually this happens at a very low frequency and at a very high frequency, due to ultimate loss in the coupling capacitors at the low-frequency end and stray capacitance between stages at the high-frequency end. Either way, when Ap falls right down toward zero, the curve must turn in and go to the point O. None of the circles representing constant ratio goes through the point O; all pass between the points O and 1 and out beyond them on opposite sides, one side or the other, except for the line that represents Ap = 1 4- Ap which is a straight line perpendicular between the two points. As a practical amplifier has frequency limitations and eventually loses gain completely at extreme frequencies, it cannot follow any of these constant-gain lines (circles) all the way.

We can use this diagram, however, to predict the overall response of the amplifier (with feedback) by the way the locus curve criss-crosses the circles drawn to represent different values of constant ratio. A complete Nyquist diagram starts from O and finishes at" O, representing frequencies zero and infinity. For simplicity, we have shown half, starting from a mid-range frequency where there is no phase shift either way.

 Overall frequency response

 Left: Gain and phase margins; right: measuring gain margin

Now we can see how this same diagram shows how much margin of safety we have between the way we are working and the beginning of oscillation. Increasing the value of Ap at all frequencies uniformly multiplies up the whole size of the curve in proportion. If the curve goes round more than 180° before it turns into the source or origin point O, increasing its size eventually causes it to go through the 1 point, which means oscillation occurs. We can see how much margin there is between the point where the curve passes through the positive direction and the 1 point. As the whole curve multiplies up in proportion as the gain or feedback of the amplifier is changed, the ratio between this distance and 1 gives the amount by which the loop gain AfJ can be increased before oscillation commences.

This ratio, expressed in db, is called the gain margin, because an increase by this much gain starts oscillation. This margin can actually be measured by increasing the amount of feedback until oscillation commences and then calculating the difference in loop gain between the condition at which the amplifier actually works and the amount needed to make it oscillate.

The other criterion of stability, as these margins are called, is the phase margin. This can be shown on the Nyquist diagram, but it is not at all easy to measure. Hence, the gain margin is probably of the most value in assessing the performance of an amplifier. On the diagram, the phase margin is the angle of the vector Ap at the point where the curve passes through a radius of 1 from point O, and point 1 from which distances (1 + Ap) are measured. It means that increasing the loop phase shift in the amplifier (or anywhere in A or P) by this angle would cause the amplifier to oscillate.

This statement assumes a change of the phase shift without a change of the amplification characteristic in any other way, which, in practice, is not possible, This is another reason why the phase margin is not a very practical criterion: it has no real significance, but now we have the tools.

Last Update: 2010-11-03