The s-plane

In traditional ac analysis in the complex plane, the voltages and currents were complex numbers. But the frequencies were always real. However now, in an effort to include virtually arbitrary waveforms into our analysis, we have in effect created a complex frequency plane too, (a + jco). This is called s-plane, where s = a + jco. Analysis in this plane is just a more generalized form of frequency domain analysis. 

Resistance still remains just a resistance (no dependency on frequency or on s). 

Let us also see how impedances add up in the s-plane — in particular when we parallel or series-combine reactances. 

For a parallel combination, we know that the reciprocals of the impedances add up to give us the reciprocal of the effective impedance. So 

Therefore, two inductors in series have an effective impedance equal to their sum 

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Fig. 5.5 Effect of closed-loop pole position in the s-plane on system transient response.

Fig. 5.10 Roots of the characteristic equation fora second-order system shown in the s-plane.

Stability in the z-plane 7.6.1 Mapping from the s-plane into the z-plane... 

The results are exact—we do not need to make approximations as we had to with root locus or the Routh array. The magnitude plot is the same as the first order function, but the phase lag increases without bound due to the dead time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G(jco) locus starts at Kp on the real axis and then "spirals" into the origin of the s-plane. 

The roots of the characteristic equation can be very conveniently plotted in a two-dimensional figure (Fig. 6.8) called the s plane. The ordinate is the imaginary part m of the root x, and the abscissa is the real part a of the root 5. The roots of Eqs. (6.117) to (6.119) are shown in Fig. 6.8. We will use these s-plane plots extensively in Part IV. 

The system is stable if all the roots of its characteristic equation lie in the left half of the s plane. 

The dynamics of this openloop system depend on the roots of the openloop characteristic equation, i.e., on the roots of the polynomials in the denominators of the openloop transfer functions. These are the poles of the openloop transfer functions. If all the roots lie in the left half of the s plane, the system is openloop stable. For the two-heated-tank example shown in Fig. 10.16, the poles of the openloop transfer function are 5 = 1 and s = — j, so the system is openloop stable. 

The dynamic performance of a system can be deduced by merely observing the location of the roots of the system characteristic equation in the s plane. The time-domain specifications of time constants and damping coefficients for a closedloop system can be used directly in the Laplace domain. 

There is a quantitative relationship between the location of roots in the s plane and the damping coefficient. Assume we have a second-order system or, if it is of higher order, assume it is dominated by the second-order roots closest to the imaginary axis. As shown in Fig. 10,5 the two roots are Si and and they are, of course, complex conjugates. From Eq. 6.68) the two roots are... 

Notice that lines of constant damping coefficient are radial lines in the s plane. Lines of constant time constant are cirdes. 

There is one root and there will be only one curve in the s plane. Figure 10.6 gives the root locus plot. The curve starts ats = — 1/rg when X, = 0. The dosedloop root moves out along the negative real axis as K, is increased. 

Note that the effect of adding a zero or a lead is to pull the root locus toward a more stable region of the s plane. The root locus starts at the poles of the open-loop transfer function. As the gain goes to infinity the two paths of the root locus go to minus infinity and to the zero of the transfer function at s = -2. We will find that this is true in general the root locus plot ends at the zeros of the openloop transfer function. 

When a system has poles that are widely different in value, it is difficult to plot them all on a root locus plot using conventional rectangular coordinates in the s plane. U is sometimes more convenient to make the root locus plots in the log s plane. Instead of using the conventional axis Re s and Im s, an ordinate of the arg s and an abscissa of the log s arc used, since the natural logarithm of a complex number is defined ... 

A process has a positive pole located at (-1-1,0) in the s plane (with time in minutes). The process steadystate gain is 2. An addition lag of 20 seconds exists in the control loop. Sketch root locus plots and calculate controller gains which give a dosedloop damping coeHicient of0.707 when... 

We will explore this phenomenon quantitatively in the s plane. We will discuss linear systems in which instability means that the reactor temperature would theoretically go off to infinity. Actually, in any real system, reactor temperature will not go to infinity because the real system is nonlinear. The nonlinearity makes the reactor temperature climb to some high temperature at which it levels out. The concentration of reactant becomes so low that the reaction rate is hmited. 

This type of controller design has been around for many years. The pole-placcmcnt methods that are used in aerospace systems use the same basic idea the controller is designed so as to position the poles of the closedloop transfer function at the desired location in the s plane. This is exactly what we do when we specify the closedloop time constant in Eq. (11.63). 

If a complex function Fj,) has Z zeros and P poles inside a certain area of the s plane, the number N of encirclements of the origin that a mapping of a closed contour around the area makes in the F plane is equal to Z — P. 

B, APPLICATION OF THEOREM TO CLOSEDLOOP STABILITY. To check the stability of a system, we are interested in the roots or zeros of the characteristic equation. If any of them lie in the right half of the s plane, the system is unstable. For a closedloop system, the characteristic equation is... 

Let us illustrate the mapping of the contour that goes around the entire RHP of the s plane using some examples. 

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