Signal Processing and Linear Systems-B.P.Lathi copy

# W ith o perational n otation t his e quation c an b e

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Unformatted text preview: eration £, w e h ave a ngular a cceleration ii. T he e quation o f m otion for r otational s ystems is T = J(j (in place o f F =Mx). A w ide variety of electromechanical systems convert electrical signals into mechanical m otion ( mechanical energy) a nd v ice versa. Here we consider a r ather s imple e xample o f 92 an armature-controlled de motor (with a constant field current i f) driven by a current source f (t), as depicted in Fig. 1.33a. Let lI(t) be the angular position of the rotor. The torque 7 ( t) generated in the rotor is proportional to the armature current f ( t). Therefore 7 (t) = K r f (t) JU(t) = 7 (t) - BiJ(t) Thus (JD2 + B D) lI(t) D (D + a)lI(t) = K J!(t) where a = B IJ a nd K l = K rIJ. 1.8-1 j •+ 20 10 y (t) ( t) f + 20 20 Internal and External Descriptions o f a System 10 (1.64) ( a) Fig. 1.34 A system t hat cannot be described by external measurements. (1.65) • W ith a knowledge of t he i nternal s tructure of a system, we c an write system equations yielding a n i nternal d escription of t he system. In contrast, t he system description seen from t he s ystem's i nput a nd o utput t erminals is t he s ystem's e xternal d escription. To understand an external description, suppose t hat a system is enclosed in a "black box" w ith only its input(s) a nd o utput(s) t erminals accessible. In o rder t o describe or characterize such a system, we m ust perform some measurements a t these terminals. For example, we might apply a known input, such as a u nit impulse o r a u nit step, a nd t hen measure t he s ystem's o utput. T he d escription provided by such a measurement is a n e xternal description of t he system. Suppose t he c ircuit in Fig. 1.34a with t he i nput f (t) a nd t he o utput y(t) is enclosed inside a "black box" w ith only t he i nput a nd o utput t erminals accessible. Under these conditions t he only way t o describe or specify t he s ystem is w ith external measurements. We can, for example, apply a known voltage f (t) a t t he i nput t erminals a nd m easure t he resulting o utput voltage y(t). From this information we can describe o r characterize t he system. This is t he external description. Assuming zero initial capacitor voltage, t he i nput voltage f (t) p roduces a current i (Fig. 1.34a), which divides equally between t he two branches because of t he b alanced n ature of t he circuit. Thus, t he voltage across t he c apacitor continues to remain zero. Therefore, for t he p urpose of computing the current i , t he c apacitor may be removed or replaced by a short. T he resulting circuit is equivalent t o t hat shown in Fig. 1.34b. I t is clear from Fig. 1.34b t hat f (t) sees a n et resistance of 5 0, a nd T he equivalent system as seen from t he s ystem's external terminals is d epicted in Fig. 1.34b. Clearly, for t he e xternal description, t he c apacitor does not exist. For most systems, t he e xternal a nd i nternal descriptions are identical, b ut t here a re a few exceptions, as in the present case, where t he e xternal description gives a n inadequate picture of t he systems. This happens...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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