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6.51,6.52,6.72,6.74,6.77

6.51,6.52,6.72,6.74,6.77 - EEE 480 HW 8 SOLUTIONS NOTE...

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EEE 480 HW # 8 SOLUTIONS NOTE: Ballpark computations relating bandwidth ( w BW ) and crossover frequency ( w GC ). In classical compensator designs, it is often useful to convert closed-loop bandwidth speci¯cations to crossover spec- i¯cations. The closed-loop bandwidth is the frequency where the complementary sensitivity T drops below p 0 : 5 of its initial (DC) magnitude. Recall that T ( s ) = L ( s ) 1 + L ( s ) where L ( s ) is the loop transfer function (typically L = GC , G being the plant and C being the compensator transfer function). In a ¯rst approximation, w BW ¼ w GC . This is justi¯ed by the observation that the loop transfer function magnitude decreases (usually rapidly) past the crossover frequency. While this is a good ballpark estimate, the two frequencies can di®er by approximately a factor of 2. For a more re¯ned approximation, we can use some assumptions on the rate of decay and the phase of the loop transfer function to develop a relationship between w BW and w GC . Since such a relationship is used before the compensator is designed, it is important to express our assumptions in terms of quantities that appear in the typical design speci¯cations. First, observe that at the crossover frequency j T ( jw GC ) j = ¯ ¯ ¯ ¯ L ( jw GC ) 1 + L ( jw GC ) ¯ ¯ ¯ ¯ = 1 j 1 + cos Á + j sin Á j where Á = 180 ¡ PM , in degrees. Furthermore, j T ( jw BW ) j j T ( jw GC ) j = j L ( jw BW ) j j L ( jw GC ) j j 1 + L ( jw GC ) j j 1 + L ( jw BW ) j For simplicity, we take the last factor to be approximately 1. Also, for typical designs, L ( jw ) decreases with a rate around 20db/dec around the crossover frequency. This implies that j L ( jw BW ) j j L ( jw GC ) j ¼ w GC w BW (Depending on prior knowledge, one can use the expression ( w GC =w BW ) m where m ¼ 0 : 5 ¡ 1 : 5 for a 10-30db/dec, respectively.) Substituting these expressions in the de¯nition of the bandwidth j T ( jw BW ) j = p 0 : 5 j T (0) j , we get w BW ¼ w GC p 2 p (1 + cos Á ) 2 + sin 2 Á j 1 + L (0) j j L (0) j When the plant and/or the controller contain an integrator, this expression simpli¯es to w BW ¼ w GC p 2 p (1 + cos Á ) 2 + sin 2 Á For example, for a 40 o phase margin the last expression yields w BW ¼ 2 w GC , while for a 60 o phase margin w BW ¼ p 2 w GC . As usual, these expressions are not exact and have a limited range of validity. Their best use is often to adjust the crossover frequency selection in the second design iteration.
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