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Unformatted text preview: ECE320 Solutions Notes to HW 13 Spring 2006 Cornell University T.L.Fine 1. Consider the phase locked loop (PLL) system displayed in Figure 11.17 of the class notes. The information-bearing phase modulated signal is the angle s ( t ) and the tracking or demodulated signal provided by the PLL is the angle v ( t ). The overall system S input is s and the system output is v . The goal is to have v ( t ) closely track s ( t ). (a) Write a system equation relating the error term e = s- v to the input s , to e itself, and to and the system elements. (This will not be an ode .) v ( t ) = v (0) + Z t Kc ( ) d, c ( t ) = Z - h ( t- ) sin( s- v ) d. Putting these two equations together yields e ( t ) = s- v (0)- K Z t d Z - h ( - r ) sin( e ( r )) dr. (b) Now assume that h represents an ideal integrator and, after appropriate differentiation, write an ode for the error e . Classify this ode as to order, degree, and whether or not it is autonomous. With this choice of h we have e ( t ) = s- v (0)- K Z t d Z - sin( e ( r )) dr. Differentiation yields e ( t ) = s ( t )- K Z t- sin( e ( r )) dr, e ( t ) = s ( t )- K sin( e ( t )) . The ode for e is second order, first degree, and nonautonomous due to the presence of s ( t ). (c) Rewrite this ode in our standard vector differential equation form. Let y = t, y 1 = e, y 2 = y 1 to achieve y y 1 y 2 = 1 y 2 s ( y )- K sin( y 1 ) . 1 (d) Identify equilibria points for the error e , if there are conditions under which they exist....
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- Spring '06