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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 θ s ( t ) and the tracking or demodulated signal provided by the PLL is the angle θ v ( t ). The overall system S input is θ s and the system output is θ v . The goal is to have θ v ( t ) closely track θ s ( t ). (a) Write a system equation relating the error term e = θ s- θ v to the input θ s , to e itself, and to and the system elements. (This will not be an ode .) θ v ( t ) = θ v (0) + Z t Kc ( τ ) dτ, c ( t ) = Z ∞-∞ h ( t- τ ) sin( θ s- θ v ) dτ. Putting these two equations together yields e ( t ) = θ s- θ v (0)- K Z t dτ Z ∞-∞ h ( τ- 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 ) = θ s- θ v (0)- K Z t dτ Z τ-∞ sin( e ( r )) dr. Differentiation yields ˙ e ( t ) = ˙ θ s ( t )- K Z t-∞ sin( e ( r )) dr, ¨ e ( t ) = ¨ θ s ( t )- K sin( e ( t )) . The ode for e is second order, first degree, and nonautonomous due to the presence of θ 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 ¨ θ 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
- Graph Theory, Sin, Cornell University, differential equation form, vector differential equation