By Farid Golnaraghi, Benjamin C. Kuo
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Extra info for Automatic Control Systems
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.