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Crc Press Mechatronics Handbook 2002 By Laxxuss Episode 1 Part 8

Tham khảo tài liệu 'crc press mechatronics handbook 2002 by laxxuss episode 1 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | or equivalently by observing that the current flowing in the series circuit is related to the capacitor voltage by i t CđvC đt and that Eq. 11.55 can be rewritten as dvC d vC t RC T7 LC------2--- vC t - vs t n.57 dt dt2 Note that although different variables appear in the preceding differential equations both Eqs. 11.55 and 11.57 can be rearranged to appear in the same general form as follows d y t dy t a2 - 2 a 1 d a0 y t F t 11.58 where the general variable y t represents either the series current of the circuit of Fig. 11.49 or the capacitor voltage. By analogy with Eq. 11.54 we call Eq. 11.58 a second-order ordinary differential equation with constant coefficients. As the number of energy-storage elements in a circuit increases one can therefore expect that higher-order differential equations will result. Phasors and Impedance In this section we introduce an efficient notation to make it possible to represent sinusoidal signals as complex numbers and to eliminate the need for solving differential equations. Phasors Let us recall that it is possible to express a generalized sinusoid as the real part of a complex vector whose argument or angle is given by mt Ộ and whose length or magnitude is equal to the peak amplitude of the sinusoid. The complex phasor corresponding to the sinusoidal signal A cos mt Ộ is therefore defined to be the complex number Aej Aej complex phasor notation for A cos wt f 11.59 1. Any sinusoidal signal may be mathematically represented in one of two ways a time-domain form v t A cos wt f and a frequency-domain or phasor form V j w Aejf 2. A phasor is a complex number expressed in polar form consisting of a magnitude equal to the peak amplitude of the sinusoidal signal and a phase angle equal to the phase shift of the sinusoidal signal referenced to a cosine signal. 3. When using phasor notation it is important to make a note of the specific frequency m of the sinusoidal signal since this is not explicitly apparent in the phasor expression.