By Farid Golnaraghi, Benjamin C. Kuo
Automated regulate platforms offers engineers with a clean new controls e-book that areas certain emphasis on mechatronics. It follows a progressive method by way of truly together with a actual lab. furthermore, readers will locate authoritative assurance of contemporary layout instruments and examples. present mechatronics purposes construct motivation to benefit the cloth. large use of digital lab software program can also be built-in in the course of the chapters. Engineers will achieve a powerful comprehend of keep watch over platforms with the aid of smooth examples and routines.
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2-5) is also known as the Euler formula. As a result, Eq. (2-1) may also be represented in polar form as z = Rej0 = R16 (2-6) We define the conjugate of the complex number z in Eq. (2-1) as z* =x- jy Or, alternatively, z* = Rcos9- jRsm9 = Re~j0 (2-7) (2-8) Note: (2-9) Table 2-1 shows basic mathematical properties of complex numbers. 18 Chapter 2. tl + X2) - /(>'! V1) \22 = / ? i«>tf| [ Z2 = -¾ + j}'2 \ -2 = / ? 2 _ Z\ Z2 _ (X\X2 + V1V2) + j(x\V2 +X2V1 ) zi z2 _»,-- f£/u fe '-^ -w -*-(*)*<*-*> x$ + y2 EXAMPLE 2-1-1 F i n d / a n d / .
2-76) to Eq. (2-77), which gives to = l/T (2-78) This frequency is also the intersect of the high-frequency approximate plot and the lowfrequency approximate plot, which is the 0-dB axis. The frequency given in Eq. (2-78) is also known as the corner frequency of the Bode plot of Eq. (2-74), because the asymptotic plot forms the shape of a corner at this frequency, as shown in Fig. 2-15. The actual \G( jco) | dB plot of Eq. (2-74) is a smooth curve and deviates only slightly from the straightline approximation.
The magnitude in dB and phase of the Bode plot of Eq. (2-80) are shown in Fig. 2-15. The data in Table 2-3 are still useful for the simple-pole case if appropriate sign changes are made to the numbers. For instance, the numbers in |1 + jcoT\dB, the straight-line approximation of |1 + jcoT\dB, the error (dB), and the /(1 + jcoT) columns should all be negative. At the corner frequency, the error between the straight-line approximation and the actual magnitude curve is — 3 dB. 2-2-8 Quadratic Poles and Zeros Now consider the second-order transfer function G(s) = cot s2 + 2$a)„s + co2 1 + (2;/con)s + (l/co2n)s2 (2-83) We are interested only in the case when £ < 1, because otherwise G(s) would have two unequal real poles, and the Bode plot can be obtained by considering G(s) as the product of two transfer functions with simple poles.