The circle’s radius is now found from the Pythagorean theorem:

Note that as the gain, Kp β, is increased, the closed-loop system poles become more and more damped, until they become real, one approaching the zero at s = –a and the other going to −•.

The damping of a complex-conjugate (CC) pole pair in the s-plane can be determined quantitatively by drawing a line from the origin to the upperhalf plane pole. The damping factor, ζ, associated with the CC poles can be shown to be the cosine of the angle that the line from the origin to the CC pole makes with the negative real axis, i.e., ξ = cos(φ). If the poles lie close to the jω axis, φ 90∞ and ξ 0. The length of the line from the origin to one of the CC poles is the undamped natural frequency, ωn, of that CC pole pair. Recall that the CC pole pair is the result of factoring the quadratic term, [s2 + s(2ξωn) + ωn2]. By way of example, the damping of the open-loop CC pole pair in Figure 5.18(B) is ξ = cos[tan−1(γ/α)] = α/ωn.

Many other interesting examples of root-locus plots are to be found in the control systems texts by Kuo (1982) and Ogata (1970) and in the electronic circuits text by Gray and Meyer (1984). Circle root-locus plots are easy to construct graphically by hand. More complex root-locus plots should be done by computer. In closing, it should be stressed that root-locus plots show the closed-loop systems poles. Closed-loop zeros can be found algebraically; they are generally fixed (not functions of gain) and do affect system transient response. Merely locating the closed-loop poles in an apparently good position does not necessarily guarantee a step response without an objectionable overshoot.

Figure 5.19(A) through Figure 5.19(L) illustrate typical root-locus diagrams for linear SISO feedback systems. Locus branches for loop gains with NFB (left column) and for PFB (right column) are shown. The unstable systems oscillate when complex-conjugate closed-loop pole pairs move into the righthalf s-plane; they saturate when one closed-loop pole moves into the right-half s-plane.

5.5Use of Root-Locus in the Design of “Linear” Oscillators

5.5.1Introduction

“Linear” oscillators are used to generate sinusoidal signals for many applications; probably the most important is testing the steady-state sinusoidal

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FIGURE 5.19

(A) through (L): Representative root-locus plots for six loop gain configurations for NFB and PFB conditions. PFB root-locus plots are on the right.

frequency response of amplifiers. They are also used in measurements as the signal source for ac bridges used to measure the values of circuit components R, L, and C, as well as circuits Z(jω) and Y(jω). Still another oscillator application is a signal source in acoustic measurements. In biomedicine, oscillators are used to power ac current sources in impedance pneumography and plethysmography (Northrop, 2002).

Linear oscillators are called “linear” because the criteria for oscillation are based on linear circuit theory and root locus, which is applied to linear systems. In reality, linear oscillators are designed to have a closed-loop, complex-conjugate pole pair in the right-half s-plane so that oscillations,

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FIGURE 5.19 (continued)

when they start, will grow exponentially. Oscillations start because circuit noise voltages or turn-on transients excite the unstable system. To limit and stabilize the amplitude of the output oscillations, a nonlinear mechanism must be used to effectively lower the oscillator’s loop gain to a value that will sustain stable oscillations at some design output amplitude.

Linear oscillators can be designed to use negative or positive voltage feedback. The first to be analyzed is the phase-shift oscillator, which uses NVF.

5.5.2The Phase-Shift Oscillator

A schematic for a phase-shift oscillator (PSO) is shown in Figure 5.20. To simplify analysis, the op amps are treated as ideal. The PSO uses negative feedback applied through an RC filter with transfer function, Vi/Vo = β(s). The second op amp provides an automatic gain control that reduces the oscillator’s loop gain as a function of its output voltage, thus stabilizing the output amplitude. A small tungsten lamp is used as voltage-dependent resistance. As the voltage across the lamp, Vb, increases, its filament heats up (becomes brighter). Metals such as tungsten have positive temperature coefficients, i.e., the filament resistance increases with temperature. Filament temperature is approximately proportional to the lamp’s electrical power input, so the curve of Rb(Vb) is approximately square-law. As the lamp’s voltage increases, so does its resistance, and the gain of the inverting op amp stage decreases as shown in the figure. The second op amp’s gain is:

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−R2 /Rbo

FIGURE 5.20

Top: schematic of an R–C phase-shift oscillator that uses NFB. The lamp is used as a nonlinear resistance to limit oscillation amplitude. Bottom left: gain of the right-hand op amp stage as a function of the RMS Vb across the bulb. Bottom right: resistance of the lamp as a function of the RMS Vb. The resistance increase is due to the tungsten filament heating.

R2 is made equal to the desired Rb = RbQ at the desired Vb = VbQ, so Av2 = −1 when Vb = VbQ . If Vb > VbQ , then Av2 < 1. The gain of the first (noninverting) op amp stage is simply Vb /Vi = (1 + RF/R1) = Av1.

The gain of the feedback circuit is found by writing the three node equations for the circuit:

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The three roots of β(s) are real and negative, i.e., they lie on the negative real axis in the s-plane. Their vector values are found with Matlab’s ROOTS utility: s1 = −5.0489/(RC), s2 = −0.6431/(RC) and s3 = −0.3080/(RC). The PS oscillator’s loop gain as a function of frequency is, in vector form,

Alternately, AL can be written as an unfactored frequency response polynomial:

The Barkhausen criterion can be used to find the frequency of oscillation, ωo, and the critical gain required for a pair of the oscillator’s closed-loop, complex-conjugate poles to lie on the jω axis in the s-plane. To apply the complex algebraic Barkhausen criterion, set AL(jωo) = 1–0∞ and note that, for AL(jωo) to be real, the real part of its denominator must equal zero. From this condition,

and solving for (1 + RF/R1)(R2/Rb). Thus:

For oscillations to start, Vb 0, and (1 + RF/R1)(R2/Rbo) must be >29, so the system’s poles will be in the right-half s-plane and the oscillations (and Vb ) will grow exponentially. When Vb reaches VbQ, R2/Rb = R2/RbQ = 1, (1 + RF /R1)(R2/Rb) = 1, and the PS oscillator’s poles are at s = ±jωo, giving stable oscillations with Vo = VbQ.

© 2004 by CRC Press LLC