formula-sheet

# formula-sheet - φ p = 180 ◦-X i 6 p i X j 6 z j φ z =...

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ECE 382 Some Useful Formulae Spring 2010 Laplace Transform L ± " d dt f ( t ) # = sF ( s ) - f (0 ± ); L ± " d 2 dt 2 f ( t ) # = s 2 F ( s ) - sf (0 ± ) - ˙ f (0 ± ) L h e - at i = 1 ( s + a ) ; L h t n e - at i = n ! ( s + a ) n +1 L [sin ωt ] = ω s 2 + ω 2 ; L [cos ωt ] = s s 2 + ω 2 Second-order Systems C ( s ) R ( s ) = T ( s ) = ω 2 n s 2 + 2 ζω n s + ω 2 n ; solution s 1 ,s 2 = - ζω n ± n q 1 - ζ 2 Rise time, t r = π - θ ω n 1 - ζ 2 ; Peak time, t p = π ω n 1 - ζ 2 Maximum overshoot, M p = exp - πζ 1 - ζ 2 ! ; Settling time, t s = 4 ζω n Analogous Systems Table 1. Quantities in F-V and F-I Analogous Systems. Mechanical Electrical Systems System/Components Force-Voltage Force-Current Force, F Voltage, e Current, i Mass, M inductance, L capacitance, C Dashpot, b or f resistance, R conductance, 1 R Spring, K reciprocal of reciprocal of capacitance, 1 C inductance, 1 L Displacement, x charge, q magnetic ﬂux linkage, ψ Velocity, ˙ x current, i voltage, e 1

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Mason’s Gain Formula T ij ( s ) = C i ( s ) R j ( s ) = k P k Δ k Δ where Δ is the determinant of the graph. Δ = 1 - X i L i + X 2 Y j L j - X 3 Y j L j + ··· + ( - 1) k X k Y j L j Root Locus (Angle of departure or arrival)
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Unformatted text preview: φ p = 180 ◦-X i 6 p i + X j 6 z j ; φ z = 180 ◦ + X i 6 p i-X j 6 z j • Bode Plot Resonant frequency, ω r = ω n q 1-2 ζ 2 for ≤ ζ ≤ . 707 Maximum peak at resonant frequency, M r = 1 2 ζ √ 1-ζ 2 • Nyquist Plot Gain Margin, G m G m = 1 | G ( jω ) | ω = ω π , ω π = phase crossover frequency. Phase Margin, φ PM . φ PM = 180 ◦ + 6 G ( jω ) | ω = ω c , ω c = gain crossover frequency. • Compensator Design zero location = ω n cos( θ + φ 2 ) cos( θ-φ 2 ) , pole location = ω n cos( θ-φ 2 ) cos( θ + φ 2 ) . α = 1-sin φ 1 + sin φ ; sin φ = 1-α 1 + α 2...
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## This note was uploaded on 02/04/2012 for the course ECE 382 taught by Professor Staff during the Spring '08 term at Purdue.

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formula-sheet - φ p = 180 ◦-X i 6 p i X j 6 z j φ z =...

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