soln13 - ECE320 Solutions Notes to HW 13 Cornell University...

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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) + t 0 Kc ( τ ) dτ, c ( t ) = -∞ h ( t - τ ) sin( θ s - θ v ) dτ. Putting these two equations together yields e ( t ) = θ s - θ v (0) - K t 0 -∞ 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 t 0 τ -∞ sin( e ( r )) dr. Differentiation yields ˙ e ( t ) = ˙ θ s ( t ) - K 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 0 = t, y 1 = e, y 2 = ˙ y 1 to achieve ˙ y 0 ˙ y 1 ˙ y 2 = 1 y 2 ¨ θ s ( y 0 ) - K sin( y 1 ) . 1
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(d) Identify equilibria points for the error e , if there are conditions under which they exist. To identify equilibria, we set ˙ y = 0 . However, due to the nonautonomous form of the ode , we cannot find an equilibrium unless ¨ θ s ( t ) is assumed to be constant at, say, a . In this case, the equilibrium condition becomes y * 1 = e * = arcsin( a K ) , and this is meaningful only so long as | K | ≥ | a | .
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