By Farid Golnaraghi, Benjamin C. Kuo

Automated keep an eye on platforms offers engineers with a clean new controls e-book that locations specific emphasis on mechatronics. It follows a progressive process via really together with a actual lab. additionally, readers will locate authoritative insurance of contemporary layout instruments and examples. present mechatronics functions construct motivation to profit the fabric. broad use of digital lab software program is usually built-in during the chapters. Engineers will achieve a robust comprehend of regulate structures with assistance from glossy examples and routines.

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**Example text**

Right-hand rule convention: positive phase is in counter clockwise direction. v2 (2-3) c^tan-1v Introducing Eq. (2-2) into Eq. (2-1), we get z = RcosO + jRsmO (2-4) Upon comparison of Taylor series of the terms involved, it is easy to confirm eJe -Rcos0 + sin9 (2-5) Eq. (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.

It is useful to remember that the error is 3 dB at the corner frequency, and it is 1 dB at 1 octave above (co = 2/T) and 1 octave below (co — 1/27") the corner frequency. 3 dB. Based on these facts, the procedure of drawing |1 + jcoT\dB is as follows: 1. Locate the corner frequency co = l/T on the frequency axis. 2. Draw the 20-dB/decade (or 6-dB/octave) line and the horizontal line at 0 dB, with the two lines intersecting at co = l / T . 3. If necessary, the actual magnitude curve is obtained by adding the errors to the asymptotic plot at the strategic frequencies.

V)-plane is described as single-valued (Fig. 2-3). If the mapping from the G(tf)-plane to the s-plane is also singlevalued, the mapping is called one-to-one. However, there are many functions for which the mapping from the function plane to the complex-variable plane is not single-valued. s, = (7, +jo)] s-plane cox *. v)-plane. 20 Chapter 2. Mathematical Foundation it is apparent that, for each value of s, there is only one unique corresponding value for G(s). y-plane. 2-1-4 Analytic Function A function G(s) of the complex variable s is called an analytic functionin a region of the splane if the function and all its derivatives exist in the region.