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4.4: STABILITY BASED ON FREQUENCY RESPONSE

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    61384
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    The Routh criterion and root-locus methods provide information con­cerning the stability of a feedback system starting with either the charac­teristic equation or the loop-transmission singularities of the system. Thus both of these techniques require that the system loop transmission be expressible as a ratio of polynomials in \(s\). There are two possible difficulties. The system may include elements with transfer functions that cannot be expressed as a ratio of finite polynomials. A familiar example of this type of element is the pure time delay of \(\tau\) seconds with a transfer function \(e^{-s\tau}\). A second possibility is that the available information about the system con­sists of an experimentally determined frequency response. Approximating the measured data in a form suitable for Routh or root-locus analysis may not be practical.

    The methods described in this section evaluate the stability of a feedback system starting from its loop transmission as a function of frequency. The only required data are the magnitude and angle of this transmission, and it is not necessary that these data be presented as analytic expressions. As a result, stability can be determined directly from experimental results.

    Nyquist Criterion

    It is necessary to develop a method for determining absolute and relative stability information for feedback systems based on the variation of their loop transmissions with frequency. The topology of Figure 4.1 is assumed. If there is some frequency \(\omega\) at which

    \[a(j \omega) f(j \omega) = -1 \nonumber \]

    the loop transmission is + 1 at this frequency. It is evident that the system can then oscillate at the frequency co, since it can in effect supply its own driving signal without an externally applied input. This kind of intuitive argument fails in many cases of practical interest. For example, a system with a loop transmission of +10 at some frequency may or may not be

    stable depending on the loop-transmission values at other frequencies. The Nyquist criterion can be used to resolve this and other stability questions. The test determines if there are any values of s with positive real parts for which \(a(s)f(s) = -1\). If this condition is satisfied, the characteristic equa­tion of the system has a right-half-plane zero implying instability. In order to use the Nyquist criterion, the function \(a(s)f(s)\) is evaluated as s takes on values along the contour shown in the \(s\)-plane plot of Figure 4.16. The contour includes a segment of the imaginary axis and is closed with a large semi­ circle of radius \(R\) that lies in the right half of the \(s\) plane. The values of \(a(s)f(s)\) as \(s\) varies along the indicated contour are plotted in gain-phase form in an af plane. A possible \(af\)-plane plot is shown in Figure 4.17. The symmetry about the \(0^{\circ}\) line in the \(af\) plane is characteristic of all such plots since \(\text{Im} [a(j\omega )f(j\omega )] = - \text{Im} [a(-j\omega )f(-j\omega)]\).

    截屏2021-08-08 下午10.24.31.png
    Figure 4.16 Contour Used to evaluate \(a(s) f(s)\).
    截屏2021-08-08 下午10.26.10.png
    Figure 4.17 Plot of \(a(s) f(s)\) as \(s\) varies along contour of Figure 4.16.

    Our objective is to determine if there are any values of \(s\) that lie in the shaded region of Figure 4.16 for which \(a(s)f(s) = - 1\). This determination is simplified by recognizing that the transformation involved maps closed contours in the \(s\) plane into closed contours in the \(af\) plane. Furthermore,

    all values of \(s\) that lie on one side of a contour in the \(s\) plane must map to values of \(af\) that lie on one side of the corresponding contour in the \(af\) plane. The - 1 points are clearly indicated in the \(af\)-plane plot. Thus the only remaining task is to determine if the shaded region in Figure 4.16 maps to the inside or to the outside of the contour in Figure 4.17. If it maps to the inside, there are two values of \(s\) in the right-half plane for which \(a(s)f(s) = -1\), and the system is unstable.

    The form of the \(af\)-plane plot and corresponding regions of the two plots are easily determined from \(a(s)f(s)\) as illustrated in the following examples. Figure 4.18 indicates the general shape of the \(s\)-plane and \(af\)-plane plots for

    \[a(s) f(s) = \dfrac{10^3}{(s + 1)(0.1s + 1)(0.01s + 1)} \nonumber \]

    截屏2021-08-08 下午10.30.41.png
    Figure 4.18 Nyquist test for \(a(s) f(s) = 10^3/[(s + 1)(0.1 s + 1)(0.01s + 1)]\). (\(a\)) \(s\)-plane plot. (\(b\)) \(af\)-plane plot.

    Note that the magnitude of \(af\) in this example is \(10^3\) and its angle is zero at \(s = 0\). As \(s\) takes on values approaching +\(jR\), the angle of \(af\) changes from 0' toward \(-270^{\circ}\), and its magnitude decreases. These relationships are readily obtained from the usual vector manipulations in the \(s\) plane. For a sufficiently large value of \(R\), the magnitude of af is arbitrarily small, and its angle is nearly \(-270^{\circ}\). As s assumes values in the right-half plane along a semicircle of radius \(R\), the magnitude of \(af\) remains constant (for \(R\) much greater than the distance of any singularities of af from the origin), and its angle changes from \(-270^{\circ}\) to \(0^{\circ}\) as \(s\) goes from +\(jR\) to +\(R\). The remainder of the \(af\)-plane plot must be symmetric about the \(0^{\circ}\) line.

    In order to show that the two shaded regions correspond to each other, a small detour from the contour in the \(s\) plane is made at \(s = 0\) as indi­cated in Figure 4.18\(a\). As s assumes real positive values, the magnitude of \(a(s)f(s)\) decreases, since the distance from the point on the test detour to each of the poles increases. Thus the detour produces values in the \(af\) plane that lie in the shaded region. While we shall normally use a test detour to deter­mine corresponding regions in the two planes, the angular relationships indicated in this example are general ones. Because of the way axes are chosen in the two planes, right-hand turns in one plane map to left-hand turns in the other. A consequence of this reversal is illustrated in Figure 4.18. Note that if we follow the contour in the \(s\) plane in the direction of the arrows, the shaded region is to our right. The angle reversal places the corresponding region in the af plane to the left when its boundary is fol­lowed in the direction of the arrows.

    Since the two - 1 points lie in the shaded region of the af plane, there are two values of \(s\) in the right-half plane for which \(a(s)f(s) = - 1\) and the system is unstable. Note that if \(a_0f_0\) is reduced, the contour in the af plane slides downward and for sufficiently small values of \(a_0f_0\) the system is stable. A geometric development or the Routh criterion shows that the system is stable for positive values of \(a_0f_0\) smaller than 122.21.

    Contours with the general shape shown in Figure 4.19 result if a zero is added at the origin changing \(a(s)f(s)\) to

    \[a(s) f(s) = \dfrac{10^3 s}{(s + 1)(0.1s + 1)(0.01s + 1)} \nonumber \]

    In order to avoid angle and magnitude uncertainties that result if the s-plane contour passes through a singularity, a small-radius circular arc is used to avoid the zero. Two test detours on the s-plane contour are shown. As the first is followed, the magnitude of af increases since the dominant effect is that of leaving the zero. As the second test detour is followed, the magni­tude of \(af\) increases since this detour approaches three poles and only one zero. The location of the shaded region in the \(af\) plane indicates that the - 1 points remain outside this region for all positive values of \(a_0\) and, therefore, the system is stable for any amount of negative feedback.

    截屏2021-08-08 下午10.37.08.png
    Figure 4.19 Nyquist test for \(a(s) f(s) = 10^3 s/[(s + 1)(0.1s + 1)(0.01s + 1)]\). (\(a\)) \(s\)-plane plot. (\(b\)) \(af\)-plane plot.

    The Nyquist test can also be used for systems that have one or more loop- transmission poles in the right-half plane and thus are unstable without feedback. An example of this type of system results for

    \[a(s) f(s) = \dfrac{a_0}{s - 1} \nonumber \]

    with \(s\)-plane and \(af\)-plane plots shown in Figs. 4.20\(a\) and 4.20\(b\). The line indicated by + marks in the \(af\)-plane plot is an attempt to show that for this system the angle must be continuous as \(s\) changes from \(j0^-\) to \(j0^+\). In order to preserve this necessary continuity, we must realize that \(+180^{\circ}\) and \(-180^{\circ}\) are identical angles, and conceive of the af plane as a cylinder joined at the \(\pm 180^{\circ}\) lines. This concept is made somewhat less disturbing by using polar coordinates for the \(af\)-plane plot as shown in Figure 4.20\(c\). Here the -1 point appears only once. The use of the test detour shows that values of \(s\) in the right-half plane map outside of a circle that extends from 0 to \(-a_0\) as shown in Figure 4.20\(c\).The location of the - 1 point in either af­ plane plot shows that the system is stable only for \(a_0 > 1\).

    Note that the - 1 points in the \(af\) plane corresponding to angles of \(\pm 180^{\circ}\) collapse to one point when the af cylinder necessary for the Nyquist con­struction for this example is formed. This feature and the nature of the af contour show that when \(a_0\) is less than one, there is only one value of \(s\) for which \(a(s)f(s) = - 1\). Thus this system has a single closed-loop pole on the positive real axis for values of \(a_0\) that result in instability.

    This system indicates another type of difficulty that can be encountered with systems that have right-half-plane loop-transmission singularities. The angle of \(a(j\omega )f(j\omega )\) is \(180^{\circ}\) at low frequencies, implying that the system actually has positive feedback at these frequencies. (Recall the additional inversion included at the summation point in our standard representation.) The s-plane representation (Figure 4.20\(a\)) is consistent since it indicates an angle of \(180^{\circ}\) for \(s = 0\). Thus no procedural modification of the type de­ scribed in Section 4.3.3 is necessary in this case.

    Interpretation of Bode Plots

    A Bode plot does not contain the information concerning values of af as the contour in the \(s\) plane is closed, which is necessary to apply the Nyquist test. Experience shows that the easiest way to determine stability

    from a Bode plot of an arbitrary loop transmission is to roughly sketch a complete \(af\)-plane plot and apply the Nyquist test as described in Section 4.4.1. For many systems of practical interest, however, it is possible to circumvent this step and use the Bode information directly.

    The following two rules evolve from the Nyquist test for systems that have negative feedback at low or mid frequencies and that have no right-half-plane singularities in their loop transmission.

    截屏2021-08-08 下午10.46.48.png
    Figure 4.20 Nyquist test for \(a(s) f(s) = a_0/(s - 1)\). (\(a\)) \(s\)-plane plot. (\(b\)) \(af\)-plane plot. (\(c\)) \(af\)-plane plot (polar coordinates).
    1. If the magnitude of af is 1 at only one frequency, the system is stable if the angle of \(af\) is between \(+180^{\circ}\) and \(-180^{\circ}\) at the unity-gain frequency.
    2. If the angle of af passes through \(+180^{\circ}\) or \(-180^{\circ}\) at only one fre­quency, the system is stable if the magnitude of af is less than 1 at this frequency.

    Information concerning the relative stability of a feedback system can also be determined from a Bode plot for the following reason. The values of \(s\) for which \(af = - 1\) are the closed-loop pole locations of a feedback system. The Nyquist test exploits this relationship in order to determine the absolute stability of a system. If the system is stable, but a pair of -1's of \(af\) occur for values of s close to the imaginary axis, the system must have a pair of closed-loop poles with a small damping ratio.

    截屏2021-08-08 下午10.50.32.png
    Figure 4.21 Loop-transmission quantities.

    The quantities shown in Figure 4.21 provide a useful estimation of the proximity of -1's of \(af\) to the imaginary axis and thus indicate relative stability. The phase margin is the difference between the angle of af and \(-180^{\circ}\) at the frequency where the magnitude of af is 1. A phase margin of \(0^{\circ}\) indicates closed-loop poles on the imaignary axis, and therefore the phase margin is a measure of the additional negative phase shift at the unity-magnitude frequency that will cause instability. Similarly, the gain margin is the amount of gain increase required to make the magnitude of \(af\) unity at the frequency where the angle of \(af\) is \(-180^{\circ}\), and represents the

    amount of increase in \(a_0f_0\) required to cause instability. The frequency at which the magnitude of \(af\) is unity is called the unity-gain frequency or the crossover frequency. This parameter characterizes the relative frequency re­sponse or speed of the time response of the system.

    A particularly valuable feature of analysis based on the loop-transmission characteristics of a system is that the gain margin and the phase margin, quantities that are quickly and easily determined using Bode techniques, give surprisingly good indications of the relative stability of a feedback system. It is generally found that gain margins of three or more combined with phase margins between 30 and \(60^{\circ}\) result in desirable trade-offs be­tween bandwidth or rise time and relative stability. The smaller values for gain and phase margin correspond to lower relative stability and are avoided if small overshoot in response to a step or small frequency-response peaking is necessary or if there is the possibility of severe changes in parameter values.

    The closed-loop bandwidth and rise time are almost directly related to the unity-gain frequency for systems with equal gain and phase margins. Thus any changes that increase the unity-gain frequency while maintaining constant values for gain and phase margins tend to increase closed-loop bandwidth and decrease closed-loop rise time.

    Certain relationships between these three quantities and the correspond­ing closed-loop performance are given in the following section. Prior to presenting these relationships, it is emphasized that the simplicity and excellence of results associated with frequency-response analysis makes this method a frequently used one, particularly during the initial design phase. Once a tentative design based on these concepts is determined, more de­tailed information, such as the exact location of closed-loop singularities or the transient response of the system may be investigated, frequently with the aid of machine computation.

    Closed-Loop Performance in Terms of Loop-Transmission Parameters

    The quantity \(a(j\omega )f(j\omega )\) can generally be quickly and accurately obtained in Bode-plot form. The effects of system-parameter changes on the loop transmission are also easily determined. Thus approximate relationships between the loop transmission and closed-loop performance provide a useful and powerful basis for feedback-system design.

    The input-output relationship for a system of the type illustrated in Figure 4.10\(a\) is

    \[A(s) = \dfrac{V_o (s)}{V_i (s)} = \dfrac{a(s)}{1 + a(s) f(s)} \nonumber \]

    If the system is stable, the closed-loop transfer function of the system can be approximated for limiting values of loop transmission as

    \[A(j \omega ) \simeq \dfrac{1}{f(j \omega )} \ \ \ |a (j \omega ) f (j \omega ) | \gg 1\label{eq4.4.6} \]

    \[A(j \omega ) \simeq a(j \omega ) \ \ \ |a (j \omega ) f (j \omega ) | \ll 1 \label{eq4.4.7} \]

    One objective in the design of feedback systems is to insure that the approximation of Equation \(\ref{eq4.4.6}\) is valid at all frequencies of interest, so that the system closed-loop gain is controlled by the feedback element. The approximation of Equation \(\ref{eq4.4.7}\) is relatively unimportant, since the system is effective operating without feedback in this case. While we normally do not expect to have the system provide precisely controlled closed-loop gain at frequencies where the magnitude of the loop transmission is close to one, the discussion of Section 4.4.2 shows that the relative stability of a system is largely determined by its performance in this frequency range.

    截屏2021-08-09 上午11.12.51.png
    Figure 4.22 Nichols chart.

    The Nichols chart shown in Figure 4.22 provides a convenient method of evaluating the closed-loop gain of a feedback system from its loop transmission, and is particularly valuable when neither of the limiting approxi­mations of Equation \(\ref{eq4.4.6}\) and \(\ref{eq4.4.7}\) is valid. This chart relates \(G/(1 + G)\) to \(G\) where \(G\) is any complex number. In order to use the chart, the value of \(G\) is located on the rectangular gain-phase coordinates. The angle and magnitude of \(G/(1 + G)\) are than read directly from the curved coordinates that intersect the value of \(G\) selected.

    The gain-phase coordinates shown in Figure 4.22 cover the complete \(0^{\circ}\) to \(-360^{\circ}\) range in angle and a ratio of \(10^6\) in magnitude. This magnitude range is unnecessary, since the approximations of Equation \(\ref{eq4.4.6}\) and \(\ref{eq4.4.7}\) are usually valid when the loop-transmission magnitude exceeds 10 or is less than 0.1. Similarly, the range of angles of greatest interest is that which surrounds the \(-180^{\circ}\) value and which includes anticipated phase margins. The Nichols chart shown in Figure 4.23 is expanded to provide greater resolution in the region where it will normally be used.

    One effective way to view the Nichols chart is as a three-dimensional surface, with the height of the surface proportional to the magnitude of the closed-loop transfer function corresponding to the loop-transmission parameters that define the point of interest. This visualization shows a "mountain" (with a peak of infinite height) where the loop transmission is +1.

    The Nichols chart can be used directly for any unity-gain feedback sys­tem. The transformation indicated in Figure 4.10\(b\) shows that the chart can be used for arbitrary single-loop systems by observing that

    \[A(j \omega) = \dfrac{a(j\omega)}{1 + a(j\omega) f(j \omega)} = \left [\dfrac{a(j\omega) f(j \omega)}{1 + a(j\omega) f(j \omega)} \right ]\left [\dfrac{1}{f(j \omega)} \right ] \nonumber \]

    The closed-loop frequency response is determined by multiplying the factor \(a(j\omega) f(j \omega)/[1 + a(j\omega) f(j \omega)]\) obtained via the Nichols chart by \(1/f(j\omega)\) using Bode techniques.

    One quantity of interest for feedback systems with frequency-independent feedback paths is the peak magnitude \(M_p\) equal to the ratio of the maxi­ mum magnitude of \(A(j\omega)\) to its low-frequency magnitude (see Section 3.5). A large value for \(M_p\) indicates a relatively less stable system, since it shows that there is some frequency for which the characteristic equation approaches zero and thus that there is a pair of closed-loop poles near the imaginary axis at approximately the peaking frequency. Feedback amplifiers are frequently designed to have \(M_p\)'s between 1.1 and 1.5. Lower values for \(M_p\) imply greater relative stability, while higher values indicate that stability has been compromised in order to obtain a larger low-frequency loop transmission and a higher crossover frequency.

    截屏2021-08-09 上午11.19.18.png
    Figure 4.23 Expanded Nichols charts.

    The value of \(M_p\) for a particular system can be easily determined from the Nichols chart. Furthermore, the chart can be used to evaluate the effects of variations in loop transmission on \(M_p\). One frequently used manipulation determines the relationship between \(M_p\) and \(a_0f_0\) for a system with fixed loop-transmission singularities. The quantity \(a(j\omega )f(j\omega )/a_0f_0\) is first plotted on gain-phase coordinates using the same scale as the Nichols chart. If this plot is made on tracing paper, it can be aligned with the Nichols chart and slid up or down to illustrate the effects of different values of \(a_0f_0\). The closed-loop transfer function is obtained directly from the Nichols chart by evaluating \(A(j\omega )\) at various frequencies, while the highest magnitude curve of the Nichols chart touched by \(a(j\omega )f(j\omega )\) for a particular value of \(a_0f_0\) indicates the corresponding \(M_p\).

    Figure 4.24 shows this construction for a system with \(f = 1\) and

    \[a(s) = \dfrac{a_0}{(s + 1)(0.1s + 1)} \nonumber \]

    The values of \(a_0\) for the three loop transmissions are 8.5, 22, and 50. The corresponding \(M_p\)'s are 1, 1.4, and 2, respectively.

    While the Nichols chart is normally used to determine the closed-loop function from the loop transmission, it is possible to use it to go the other way; that is, to determine \(a(j\omega )f(j\omega )\) from \(A(j\omega )\). This transformation is occasionally useful for the analysis of systems for which only closed-loop measurements are practical. The transformation yields good results when the magnitude of \(a(j\omega )f(j\omega )\) is close to one. Furthermore,the approximation of Equation \(\ref{eq4.4.7}\) shows that \(A(j\omega ) \simeq a(j\omega )\) when the magnitude of the loop transmission is small. However, Equation \(\ref{eq4.4.6}\) indicates that \(A(j\omega )\) is essen­tially independent of the loop transmission when the loop-transmission magnitude is large. Examination of the Nichols chart confirms this result since it shows that very small changes in the closed-loop magnitude or angle translate to very large changes in the loop transmission for large loop- transmission magnitudes. Thus even small errors in the measurement of \(A(j\omega )\) preclude estimation of large values for \(a(j\omega )f(j\omega )\) with any accuracy.

    The relative stability of a feedback system and many other important characteristics of its closed-loop response are largely determined by the behavior of its loop transmission at frequencies where the magnitude of this quantity is close to unity. The approximations presented below relate closed-loop quantities defined in Section 3.5 to the loop-transmission properties defined in Section 4.4.2. These approximations are useful for predicting closed-loop response, comparing the performance of various systems, and estimating the effects of changes in loop transmission on closed-loop performance.

    截屏2021-08-09 上午11.25.30.png
    Figure 4.24 Determination of closed-loop transfer function for \(a(s) = a_0/[(s + 1)(0.1s + 1)]\), \(f = 1\).

    The assumptions used in Section 3.5, in particular that \(f\) is one at all frequencies, that \(a_0\) is large, and that the lowest frequency singularity of \(a(s)\) is a pole, are assumed here. Under these conditions,

    \[M_p \simeq \dfrac{1}{\sin \phi_m}\label{eq4.4.10} \]

    截屏2021-08-09 上午11.31.13.png
    Figure 4.25 \(M_p\) for several system with \(45^{\circ}\) of phase margin.

    where \(\phi_m\) is the phase margin. The considerations that lead to this approxi­mation are illustrated in Figure 4.25. This figure shows several closed-loop­-magnitude curves in the vicinity of \(M_p = 1.4\) and assumes that the system phase margin is \(45^{\circ}\). Since the point \(|G| = 1, \measuredangle G = -135^{\circ}\) must exist for a system with a \(45^{\circ}\) phase margin, there is no possible way that \(M_p\) can be less than approximately 1.3, and the loop-transmission gain-phase curve must be quite specifically constrained for \(M_p\) just to equal this value. If it is assumed that the magnitude and angle of \(G\) are linearly related, the linear constructions included in Figure 4.25 show that \(M_p\) cannot exceed approximately 1.5 unless the gain margin is very small. Well-behaved sys­tems are actually most likely to have a gain-phase curve that provides an extended region of approximate tangency to the \(M_p = 1.4\) curve for a phase margin of \(45^{\circ}\). Similar arguments hold for other values of phase margin, and the approximation of Equation \(\ref{eq4.4.10}\) represents a good fit to the relationship between phase margin and corresponding \(M_p\).

    Two other approximations relate the system transient response to its crossover frequency \(\omega_c\).

    \[\dfrac{0.6}{\omega_c} < t_r < \dfrac{2.2}{\omega_c} \nonumber \]

    The shorter values of rise time correspond to lower values of phase margin.

    \[t_s > \dfrac{4}{\omega_c} \nonumber \]

    The limit is approached only for systems with large phase margins.

    We shall see that the open-loop transfer function of many operational amplifiers includes one pole at low frequencies and a second pole in the vicinity of the unity-gain frequency of the amplifier. If the system dynamics are dominated by these two poles, the damping ratio and natural frequency of a second-order system that approximates the actual closed-loop system can be obtained from Bode-plot parameters of a system with a frequency-independent feedback path using the curves shown in Figure 4.26\(a\). The curves shown in Figure 4.26\(b\) relate peak overshoot and \(M_p\) for a second-order system

    to damping ratio and are derived using Equations 4.3.24 and 4.3.28. While the relationships of Figure 4.26\(a\) are strictly valid only for a system with two widely spaced poles in its loop transmission, they provide an accurate approxima­tion providing two conditions are satisfied.

    1. The system loop-transmission magnitude falls off as \(1/\omega\) at frequencies between one decade below crossover and the next higher frequency singu­larity.
    2. Additional negative phase shift is provided in the vicinity of the cross­over frequency by other components of the loop transmission.
    截屏2021-08-09 下午12.26.16.png
    Figure 4.26\(a\) Closed-loop quantities from loop-transmission parameters for system with two widely spaced poles. Damping ratio and natural frequency as a function of phase margin and crossover frequency.

    The value of these curves is that they provide a way to determine an approximating second-order system from either phase margin, \(M_p\), or peak overshoot of a complex system. The validity of this approach stems from the fact that most systems must be dominated by one or two poles in the vicinity of the crossover frequency in order to yield acceptable performance. Examples illustrating the use of these approximations are included in later sections. We shall see that transient responses based on the approximation are virtually indistinguishable from those of the actual system in many cases of interest.

    截屏2021-08-09 下午12.27.41.png
    Figure 4.26\(b\) \(P_0\) and \(M_p\) versus damping ratio for second-order system.

    The first significant error coefficient for a system with unity feedback can also be determined directly from its Bode plot. If the loop transmission includes a wide range of frequencies below the crossover frequency where its magnitude is equal to \(k/\omega^n\), the error coefficients \(e_0\) through \(e_{n - 1}\) are negligible and \(e_n\) equals \(1/k\).

    PROBLEMS

    Exercise \(\PageIndex{1}\)

    Find the number of right-half-plane zeros of the polynomial

    \[P(s) = s^5 + s^4 + 3s^3 + 4s^2 + s + 2\nonumber \]

    Exercise \(\PageIndex{2}\)

    A phase-shift oscillator is constructed with a loop transmission

    \[L(s) = -\dfrac{a_0}{(\tau s + 1)^4} \nonumber \]

    Use the Routh condition to determine the value of ao that places a pair of closed-loop poles on the imaginary axis. Also determine the location of the poles. Use this information to factor the characteristic equation of the system, thus finding the location of all four closed-loop poles for the critical value of \(a_0\).

    Exercise \(\PageIndex{3}\)

    Describe how the Routh test can be modified to determine the real parts of all singularities in a polynomial. Also explain why this modification is usually of little value as a computational aid to factoring the polynomial.

    Exercise \(\PageIndex{4}\)

    Prove the root-locus construction rule that establishes the angle and intersection of branch asymptotes with the real axis.

    Exercise \(\PageIndex{5}\)

    Sketch root-locus diagrams for the loop-transmission singularity pattern shown in Figure 4.27. Evaluate part \(c\) for moderate values of \(a_0f_0\), and part \(d\) for both moderate and very large values of \(a_0f_0\).

    截屏2021-08-09 下午12.40.04.png
    Figure 4.27 Loop-transmission singularity patterns.

    Exercise \(\PageIndex{6}\)

    Consider two systems, both with \(f = 1\). One of these systems has a forward-path transfer function

    \[a(s) = \dfrac{a_0 (0.5s + 1)}{(s + 1)(0.01s + 1)(0.51 s + 1)} \nonumber \]

    while the second system has

    \[a'(s) = \dfrac{a_0 (0.51s + 1)}{(s + 1)(0.01s + 1)(0.5 s + 1)} \nonumber \]

    Common sense dictates that the closed-loop transfer functions of these systems should be very nearly identical and, furthermore, that both should be similar to a system with

    \[a''(s) = \dfrac{a_0}{(s + 1)(0.01s + 1)} \nonumber \]

    [The closely spaced pole-zero doublets in \(a(s)\) and \(a'(s)\) should effectively cancel out.] Use root-locus diagrams to show that the closed-loop responses are, in fact, similar.

    Exercise \(\PageIndex{7}\)

    An operational amplifier has an open-loop transfer function

    \[a(s) = \dfrac{10^6}{(0.1s + 1)(10^{-6} s + 1)^2}\nonumber \]

    This amplifier is combined with two resistors in a noninverting-amplifier configuration. Neglecting loading, determine the value of closed-loop gain that results when the damping ratio of the complex closed-loop pole pair is 0.5.

    Exercise \(\PageIndex{8}\)

    An operational amplifier has an open-loop transfer function

    \[a(s) = \dfrac{10^5}{(\tau s + 1)(10^{-6} s + 1)}\nonumber \]

    The quantity \(\tau\) can be adjusted by changing the amplifier compensation. Use root-contour techniques to determine a value of \(\tau\) that results in a closed-loop damping ratio of 0.707 when the amplifier is connected as a unity-gain inverter.

    Exercise \(\PageIndex{9}\)

    A feedback system that includes a time delay has a loop transmission

    \[L(s) = -\dfrac{a_0 e^{-0.01 s}}{(s + 1)}\nonumber \]

    Use the Nyquist test to determine the maximum value of \(a_0\) for stable operation. What value of \(a_0\) should be selected to limit \(M_p\) to a factor of 1.4? (You may assume that the feedback path of the system is frequency independent.)

    Exercise \(\PageIndex{10}\)

    We have been investigating the stability of feedback systems that are generally low pass in nature, since the transfer functions of most opera­tional-amplifier connections fall in this category. However, stability prob­lems also arise in high-pass systems. For example, a-c coupled feedback amplifiers designed for use at audio frequencies sometimes display a low-frequency instability called "motor-boating." Use the Nyquist test to demonstrate the possibility of this type of instability for an amplifier with a loop transmission

    \[L(s) = -\dfrac{a_0 s^3}{(s + 1)(0.1s + 1)^2}\nonumber \]

    Also show the potentially unstable behavior using root-locus methods. For what range of values of \(a_0\) is the amplifier stable?

    Exercise \(\PageIndex{11}\)

    Develop a modification of the Nyquist test that enables you to determine if a feedback system has any closed-loop poles with a damping ratio of less than 0.707. Illustrate your test by forming the modified Nyquist diagram for a system with \(a(s) = a_0/(s + 1)^2\), \(f(s) = 1\). For what value of \(a_0\) does the damping ratio of the closed-loop pole pair equal 0.707? Verify your answer by factoring the characteristic equation for this value of \(a_0\).

    Exercise \(\PageIndex{12}\)

    The open-loop transfer function of an operational amplifier is

    \[a(s) = \dfrac{10^5}{(0.1s + 1)(10^{-6} s + 1)^2}\nonumber \]

    Determine the gain margin, phase margin, crossover frequency, and \(M_p\) for this amplifier when used in a feedback connection with \(f = 1\). Also find the value off that results in an \(M_p\) of 1.1. What are the values of phase and gain margin and crossover frequency with this value for \(f\)?

    Exercise \(\PageIndex{13}\)

    A feedback system is constructed with

    \[a(s) = \dfrac{10^6 (0.01s + 1)^2}{(s+1)^3}\nonumber \]

    and an adjustable, frequency-independent value for \(f\). As \(f\) is increased from zero, it is observed that the system is stable for very small values off, then becomes unstable, and eventually returns to stable behavior for sufficiently high values of \(f\). Explain this performance using Nyquist and root-locus analysis. Use the Routh criterion to determine the two borderline values for \(f\).

    Exercise \(\PageIndex{14}\)

    An operational amplifier with a frequency-independent feedback path exhibits 40% overshoot and 10 to 90% rise time of 0.5 ps in response to a step input. Estimate the phase margin and crossover frequency of the feed­ back connection, assuming that its performance is dominated by two widely separated loop-transmission poles.

    Exercise \(\PageIndex{15}\)

    Consider a feedback system with

    \[a(s) = \dfrac{a_0}{s[(s^2/2) + s +1]}\nonumber \]

    and \(f(s) = 1\).

    Show that by appropriate choice of ao, the closed-loop poles of the system can be placed in a third-order Butterworth pattern. Find the crossover frequency and the phase margin of the loop transmission when \(a_0\) is selected for the closed-loop Butterworth response. Use these quantities in conjunc­tion with Figure 4.26 to find the damping ratio and natural frequency of a second-order system that can be used to approximate the transient response of the third-order Butterworth filter. Compare the peak overshoot and rise time of the approximating system in response to a step with those of the Butterworth response (Figure 3.10). Note that, even though this system is con­siderably different from that used to develop Figure 4.26, the approximation predicts time-domain parameters with fair accuracy.


    This page titled 4.4: STABILITY BASED ON FREQUENCY RESPONSE is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James K. Roberge (MIT OpenCourseWare) .

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