formula-sheet

formula-sheet - p = 180 -X i 6 p i + X j 6 z j ; z = 180 +...

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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: p = 180 -X i 6 p i + X j 6 z j ; z = 180 + X i 6 p i-X j 6 z j Bode Plot Resonant frequency, r = n q 1-2 2 for . 707 Maximum peak at resonant frequency, M r = 1 2 1- 2 Nyquist Plot Gain Margin, G m G m = 1 | G ( j ) | = , = phase crossover frequency. Phase Margin, PM . PM = 180 + 6 G ( j ) | = c , c = gain crossover frequency. Compensator Design zero location = n cos( + 2 ) cos( - 2 ) , pole location = n cos( - 2 ) cos( + 2 ) . = 1-sin 1 + sin ; sin = 1- 1 + 2...
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formula-sheet - p = 180 -X i 6 p i + X j 6 z j ; z = 180 +...

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