Applying Nyquist’s Stability Analysis to Bode Plots With Wrapped Phase Behavior

This brief describes the stability conditions of Nyquist in terms of the data related to a Bode plot of the loop transmission function <inline-formula> <tex-math notation="LaTeX">$\mathbf {A\beta }(j\boldsymbol{\omega }\mathbf {)}$ </tex-math></inline-formula>. Specifically, it will be shown that the number of encirclements of <inline-formula> <tex-math notation="LaTeX">$\mathbf {A\beta }(j\boldsymbol{\omega }\mathbf {)}$ </tex-math></inline-formula> around the critical point–1+j0 in the complex plane <inline-formula> <tex-math notation="LaTeX">$N_{e}$ </tex-math></inline-formula> can be deduced directly from twice the directional sum of the number ±360°-phase jumps that appear in the wrapped phase portion of the Bode plot, together with the phase behavior at DC and infinity, provided the magnitude at each phase jump is greater than unity. An example will be provided to demonstrate the simplicity of the proposed theory.


Applying Nyquist's Stability Analysis to Bode
Plots With Wrapped Phase Behavior Saad Yousaf and Gordon W. Roberts , Fellow, IEEE Abstract-This brief describes the stability conditions of Nyquist in terms of the data related to a Bode plot of the loop transmission function Aβ(jω).Specifically, it will be shown that the number of encirclements of Aβ(jω) around the critical point -1+j0 in the complex plane N e can be deduced directly from twice the directional sum of the number ±360 • -phase jumps that appear in the wrapped phase portion of the Bode plot, together with the phase behavior at DC and infinity, provided the magnitude at each phase jump is greater than unity.An example will be provided to demonstrate the simplicity of the proposed theory.

I. INTRODUCTION
C ONVENTIONAL Bode stability analysis relies on phase and gain margin test metrics of open-loop behavior to determine closed-loop stability.Unfortunately, as noted in [1], this approach is fraught with inconsistencies and is simply downright unreliable.To the authors, the message here is clear -a stability analysis must be performed using the necessary and sufficient conditions described by Nyquist and no other.This simply comes down to counting the number of times the loop transmission function Aβ(s) encircles the critical point −1 + j0 in the complex plane and the direction that each circle takes.Nonetheless, this approach has its own problems as Aβ(s) can take on many twists and turns, having loops with both very small radii and ones with very large ones, that cannot be seen together on a linear scale.In the end, the user can easily miss a loop, or its direction, leaving one in doubt about the stability of the system.It appears one is no further ahead with either the Nyquist or Bode method.
One advantage of a Bode analysis is the fact that it uses a logarithmic scale to display the magnitude and phase behavior as a function of frequency.In addition, the magnitude behavior is plotted on a decibel scale, providing greater dynamic range of display.Another method that has a similar advantage to the Bode plot is the Nicolas chart [3].It too uses a decibel scale for displaying magnitude behavior and can also incorporate a logarithmic frequency scale.The theory presented here can be extended to the Nichols chart as well.
The Nyquist stability criterion provides both necessary and sufficient conditions for stability.It is therefore the objective of this brief to map the Nyquist stability conditions into equivalent conditions on a Bode plot.The development presented here follows closely the work of others, in particular [6], then [7], [8], [9], [10], but does so by extracting the encirclement number in terms of the number of ±360-degree phase jump discontinuities found from a phase wrapped Bode plot.Such Bode plots are the common output of many CAD tools, i.e., LTspice, eliminating the need for further post-processing.The proposed method also identifies the contribution to the encirclement number N e for direct current (DC) signals.The formulation depends only the phase-wrapped behavior at DC.This is quite different from the formulation of the DC contribution made in the theory of Generalized Bode Criterion (GBC) [9] where the DC contribution is system dependent (i.e., depends on the number of integrators present in the system, which one generally does not know).The proposed method also identifies the contribution to the encirclement number N e at infinity.This is necessary to form a one-tocorrespondence between the theory of Nyquist and that derived from a Bode plot.The GBC theory [9] did not consider this as that theory was limited to lowpass functions described by strictly proper transfer functions.
This brief is organized as follows.Section II of this brief describes the theory of Nyquist as it relates to single-loop negative feedback systems.Section III describes the equivalence of loop encirclement to intersections of Aβ(s) and a line along the negative real axis.In Section IV, this behavior is converted into the wrapped phase behavior of a one-sided Bode plot from which the directional sum of the ±360 • phase jump discontinuities can be counted.Section V addresses the general concerns of loop transmission functions that poles that lie on the jω-axis and how to evaluate them with existing Bode plot tools.Section VI provides one interesting example that previously proposed Bode-related stability theories would fail.Finally, conclusions are drawn in Section VII.

II. NYQUIST SYSTEM THEORY
A closed loop system consisting of two paths, one involving a forward path modelled with transfer function A(s) and another that models the feedback path with transfer function  β(s) as shown in Fig. 1, has the following input-output transfer function: Systems are often arranged in this manner on account of their performance properties; however, such systems may also suffer instability.Nyquist stated that stability of these system can be drawn from the behavior of the loop transmission function Aβ(s) evaluated along a contour that fully encloses the right-half plane (RHP) of the s-plane as shown in Fig. 2(a).
Evaluating the loop transmission function Aβ(s) at any point along this contour , say s = jω o , results in Aβ(jω o ).The fact that the contour is closed in the s-plane, the resulting Nyquist plot must also be closed but may experience multiple 360 • phase changes (or loops) as evident from Fig. 2(b).

A. Nyquist Statement of Mathematical Fact
According to Nyquist, the difference between the number of poles in the RHP of the closed-loop system described by the transfer function A CL (s) and the number of poles in the RHP of the loop transmission Aβ(s) is equal to the number of encirclements N e of the Nyquist plot around the critical point −1 + j0.This can be summarized in mathematical form as This equation will be referred to as Nyquist's Statement of Mathematical Fact.

B. Nyquist Stability Criterion
For a closed-loop system to be stable, it cannot have poles in the RHP, i.e., #RHP poles{A CL (s)} = 0, thus, from Eqn. (2), the following condition must be true: This expression is known as the Nyquist Stability Criterion and suggests that the number of encirclements N e of the critical points must be equal to the number of RHP poles of the loop transmission function, Aβ(s), provided the degree of the numerator of Aβ(s) is equal or less than the degree of its denominator (i.e., proper).The sign of N e refers to the direction of the Nyquist plot relative to the critical point −1+j0.A contour traveling in a clockwise direction around the critical point is considered positive whereas a contour traveling in a counterclockwise direction is considered negative.

C. Counting Encirclements by Directional Ray Crossing
In most cases, unfortunately, counting the number of encirclements N e around the critical point is fraught with many numerical difficulties.The Nyquist plot may take many complex paths around this point, with loops having small radii to other loops with extremely large radii, all plotted on one diagram.Thus, making the counting process very difficult.For example, the plot shown in Fig. 3 provides an example of the loops hidden in the region around the critical point of −1+j0.In this case, the critical point is not encircled, but great care must be exercised to determine this fact.
Alternatively, as suggested first in [6] and later in [7] and [8], counting the number of encirclements N e of the critical point can be done by counting the number of times the Nyquist plot Aβ(s) crosses the real axis from −∞ to −1 in the complex plane on the Nyquist plot.This line will be referred to as the 180 • -phase ray R, and a crossing of this line will be referred to as a ray crossing [8].

III. RAY CROSSINGS DERIVED FROM TWO-SIDED BODE PLOT
As most electronic engineers make use of Bode plots to investigate circuit behavior as opposed to Nyquist plots or Nicolas charts, it is the objective of this section is to quantify the number of encirclements N e of the loop transmission function Aβ(s) directly from the information found in a Bode plot.To identify the number of encirclements N e that the loop transmission function Aβ(s) takes around the critical point in a two-sided Bode plot, the location of the 180 • -phase ray must be mapped from the Nyquist plot to the Bode plot.When mapped onto a Bode plot, the ray from the Nyquist plot has a principal value of 180 • with periodic multiplies separated by 360 • .This is captured in Fig. 4 as shown by the horizontal lines separated by 360 • .
Superimposing the 180 • -phase ray R over the loop transmission behavior Aβ(jω) on a two-sided Bode plot allows one to identify the intersection points of the Nyquist plot and the 180 • -phase ray.These are identified as C i where i is the ith phase ray crossing [7] other than DC and at infinity, starting from the left-hand side.The ray crossings can be identified as with the newly added terms and where the operator ∧ represents the logical AND operation.Consequently, the number of encirclements N e is simply the sum of all directional ray crosses as follows: where CT represents the number of visible phase crossings with a gain greater than unity.The phase crossings at DC and infinity are assigned separate terms as these need to be tracked under different test conditions.It is these two terms that are new to the stability literature.These two terms account for the fact that both the magnitude and phase contours displayed on a Bode plot must be closed.

IV. RAY CROSSINGS DERIVED FROM ONE-SIDED BODE PLOT WITH WRAPPED PHASE BEHAVIOR
For compatibility with various circuit-related CAD tools, such as LTspice, PSpice, etc., the phase of a complex function is plotted as a "wrapped" phase function rather than an "unwrapped" function.Such a phase function is bounded between −180-degrees and +180-degress, which may result in additional ±360 • phase jump discontinuities not seen before.Such a wrapped phase behavior is illustrated with the two-sided Bode plot shown in Fig. 5.This two-sided Bode plot Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
is equivalent to the Bode plot of Fig. 4 except the phase has been limited to a −180-degree to +180-degree phase range.
Here one can see the first jump discontinuity labeled as J 1 jump upwards from −180 degrees to +180 degrees.This jump signifies an upward ray crossing.The frequency of this jump corresponds to the same frequency as the first 180 • phase ray crossing frequency, ω C 1 .Likewise, the next phase jump discontinuity labelled as J 2 jumps upward from −180 degrees to +180 degrees resulting in an upward ray crossing.The frequency of this jump corresponds to the same frequency as the second 180 • phase ray crossing frequency, ω C 2 .Following this, the jump at DC labeled as J DC jumps in the opposite direction from +180 degrees to −180 degree resulting in a downward ray crossing.The remaining jumps labeled J 3 and J 4 follow the same behavior as J 1 and J 2 .These jumps occur at frequencies ω C 3 and ω C 4 .As the magnitude of Aβ(jω) goes to zero as ω → ∞, any phase jump at infinity does not need to be considered for this case.
If one defines an upward jump discontinuity as +1 and a downward jump discontinuity as −1, one can capture what was just said and write: where represents the algebraic difference operation and @ means defined at a point operation.Likewise, the ±360 • phase jump discontinuities at DC denoted as J DC , can be quantified as and ±360 • phase jump discontinuities at infinity, denoted as J ∞ , is stated as As these ±360 • phase jump discontinuities have a oneto-one correspondence with the phase of Aβ(s) crossing the principal angle of +180 • and its ±360 • phase multiplies, the number of encirclements, N e , of the loop transmission function, can be redefined as where JT is defined as the total number of ±360 • phase jump discontinuities that correspond to a magnitude of Aβ(s) that is greater than unity (0 dB) other than at DC or infinity.One can go further and subdivide the visible ±360 • phase jump discontinuities between the positive and negative frequencies and write: Further, as the phase behavior of Aβ(jω) exhibits odd symmetry about the DC axis, one can state: This statement is immediately obvious from the two-sided Bode plot of Fig. 5. Thus, N e can be written exclusively in terms of the positive frequency behavior of the Bode plot as This is most convenient as most CAD tools provide only one-sided Bode plots consisting of only positive frequencies.

V. ACCOUNTING FOR POLES ON THE Jω AXIS
If there are poles associated with Aβ(s) on the jω-axis, then the contour that encloses the RHP must be modified so that it does not pass through these points.Otherwise, the value of Aβ(s) at these points would be undefined.A common method to avoid these points is to detour around them from the righthand side.Here the path of the arc is described as εe jϕ where ε is the radius of the arc and φ is the phase function.φ would range from -90 • to +90 • and ε would tend towards 0. While adopting such an approach is fine for hand analysis, standard Bode plot routines do not offer such adaptations, as they simply assume that the function to be plotted has no poles on the jω-axis.To circumvent this limitation, one can simply evaluate Aβ(s) along a new contour that runs in parallel to the jω-axis at a very small distance ε away it in the RHP, i.e., It is important to note, however, that the application of the Nyquist statement of mathematical fact given by Eqn.(2) applies to the closed-loop system with loop function Aβ(s + ε) and not Aβ(s).Take for instance, Aβ(s) = 1/s 2 evaluated along the path defined by s = ε + jω.Here the Nyquist plot will not enclose the critical point.Thus, N e =0 and there are no RHP closed-loop poles, as there are no RHP poles associated with Aβ(s + ε).This is consistent with solving directly for the closed-loop poles from

VI. EXAMPLE
This section will demonstrate how a one-sided Bode plot can be used to extract the same information as a Nyquist plot without the use of a Nyquist diagram.The example provided is an unstable closed-loop system A CL (s) having four RHP poles with a stable loop transmission function Aβ(s).
To test the relation between Nyquist and Bode plots defined in this brief, consider a unity-gain negative feedback configuration with the following feedforward path described as A(s) = −5 (s + 1000) 14   (s + 100) 9 (s + 10000) The above list reveals that the closed-loop system has four poles in the RHP, thus would be unstable.
To convince the reader that a Bode analysis of the loop transmission function Aβ(jω) can reveal these same facts, consider the Bode plot shown in Fig. 6.Here one can clearly see in the phase response having one upward phase jump around 125 rad/s, one downward phase jump at 500 rad/s and another downward phase jump at 1900 rad/s.However, we only consider the one upward phase jump around 125 rad/s since for the other two phase jumps the magnitude is less than 0 dB.Additionally, the phase as one approaches DC can be seen to be equal to +180 • .Thus, according to Eqn. (10), there is an upward 360 • phase jump discontinuity at DC. Similarly, as one approaches infinity, the phase angle is +180 • while the magnitude is greater than unity, thus a downward phase jump discontinuity occurs at infinity.This leads one to conclude J DC = +1, J i | ω>0 = 1 and J ∞ = +1.From Eqn. (15) the number of encirclements can be equated to This agrees with the closed-loop RHP poles found directly from the closed-loop transfer function seen listed above.It is interesting to note that the gain and phase margin from the plots shown in Fig. 6 is 59.7 dB and −46.5 deg, respectively, suggesting the closed-loop system is unstable.While the Bode stability criterion provides leads to the same conclusion as the proposed Bode analysis, in general, as pointed out by Lumbreras et al. [9], the Bode stability criterion is limited to transfer functions with no RHP poles or zeros.If the coincident zeros of the transfer function of Eqn.(17) are changed from −1000 to +1000, one would find the gain and phase margin is 19.8 dB and 143 degrees, respectively, suggesting the closed-loop system is unstable but is indeed stable.However, the proposed Bode method, along with Nyquist, suggests the correct stability behavior.

VII. CONCLUSION
The stability principle established by Nyquist relating the RHP poles of the closed-loop system to the frequency behavior of the loop transmission function Aβ(jω) and its number of RHP poles is an essential tool for any engineer.At the heart of the method is the number of times Aβ(jω) encircles the critical point, -1+j0 on a Nyquist plot, denoted as N e .In this brief, the mathematical conditions have been provided that allow one to extract N e directly from a phase wrapped Bode plot.This brief recognizes that N e is simply the directional sum of the total number of visible ±360 • jump discontinuities, including those at DC and infinity, provided |Aβ(jω)| > 1 at the frequency of each phase jump.

Fig. 3 .
Fig. 3. Illustrating the often-hidden behavior of a Nyquist plot in the region around the critical point of −1+j0.

Fig. 4 .
Fig. 4. A two-sided Bode plot of an arbitrary loop transmission function Aβ(s) with the 180 • -phase ray R superimposed.Highlighting the phase region where ray crossings occur and don't occur.

Fig. 5 .
Fig. 5. Magnitude and phase wrapped Bode plots for all frequencies of an arbitrary loop transmission function Aβ showing the ray crossings as phase jump discontinuities.
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