# formulae - _—————-——-—-—————-—...

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Unformatted text preview: ____—————-——-—-—————-— EEE3€57 ' Summary of Formulae ___—____———-—-———- Some Laplace transforms 6(t) 1 10) l S 1 no) — \$2 it"’1(t) 1 l m! Sm+ exp(—at)l(t) S + a sin(a)t)l(t) 2 ‘0 2 S + a) cos(wt)l(t) 2 s 2 s M) f(t — T) <—> (“17(5) e"””f(t) <—) F(s+a) Laplace transform theorems If f(!) <-) F(s) , then ‘ f(t) * go) H mm» if,le sF(s)—f<o—) 2 u d if” e s2F(s)~sf(0—)-f(0-) F (s) s :[f(t)dt e) 0- WT f(0+) = lim sF(s) s—no FVT lim f(1) = lingsF(s) provided all poles I—DG) of sF(s) have negative real part. Simpliﬁed form of Mason’s gain rule 2 path TF3 1 — 2 loop TFs For complex poles at s = ~0i jwd , (on =,/0'2+a)f,, g=0’/w,, 2 a=gwrn (0d:er lug G(s) = where a)” is the undamped natural frequency and g is the damping ratio. The associated quadratic factor is s2 + Zgwns + a): For the second order system with transfer function (02 H(s) : .——-’L— \$2 + 2gro,,s + mi ’ tr 3 ﬂ’ﬂ sec, where ﬂ = tan—'(wd/O') V wd t, = 72/0,, Mp — {Gr/cod)” '(g/ 1'52)” 4 4 ts = — = 0' gwn A necessary condition for all roots of the polynomial a(s) = aos" + 013"“ + (12'.t""'2 +...+an, 00' > 0 to have negative real part is a, > 0, all i . A necessary and suﬂicient condition is given by the Routh array: - ' n S I do a; a4 a5 ..... s“' a, a3 a5 a7 s“ b, b2 b3 s"’3 c. c; (:3 s 1 * where = alaz — 0003 b2 = 01(14 — 0005 etc 9 \$ 01 01 _ bla3 — 01172 c 9135 ‘ “lbs etc Ci " wb‘ , 2 — b1 ’ The number of sign changes in the ﬁrst column is equal to the number of roots with real part _ positive. Special cases (1) When the ﬁrst entry in a row is zero (and the entire row is not zero), replace the zero with e = 0 + and (2) When an entire row is zero, replace it with the derivative of the row above. A unity feedback system with feedforward ~ transfer function G(s) is ‘type k’ if G(s) has k poles at the origin. The static position, velocity and acceleration error constants are KP = lim G(s), Kv =1ingsG(s), K“ = ling szG(s). S—DO For a unit step, unit ramp and unit parabolic input, the steady state error .955 is equal to 1 l 1 Type 0, type 1 and type 2 systems have a ﬁnite as, in response to step, ramp and parabolic inputs respectively. For the characteristic equation 1+ G(s)H(s) = 1+ K A“) = o , B(s) where G(s)H(s) has n poles and m zeroes (n 2 m ), the root locus is the locus of roots for positive K. A point 3 = so is on the locus if and only if 40(s0)H(so) = 180° i [360° . The steps in drawing a root locus are: 1. Mark poles and zeroes of G(s)H(s) . 2. Draw the locus on the real axis to the left of an odd number of real axis poles plus zeroes. 3. Draw the asymptotes. The centroid is given by . ca = 2 Pi “ Z 2i n — m ' and the angles of the asymptotes by 0 0 ¢a =M’1 = 0, 12,." n — m 4. Find breakaway and break-in points by solving B(s)A'(s) = A(s)B'(s) 5. Calculate departure angles from poles and arrival angles at zeroes by considering a point on the' root locus inﬁnitesimally close to the pole or zero. 6. Calculate imaginary axis crossings using the characteristic equation and substituting s = ja) or using the Routh criterion. The value ofK at a point s = so on the locus can be found from [C(so)H(so)| =1 . Lead and lag compensators have transfer functions ofthe form T3+1=K 7' aTs+l c 1 3+— aT with 0 < a <1 for a lead compensator and a >1 for a lag compensator. - For a system with open loop transfer function , G(s)H(s) the Nyquist stability criterion is Z=N+P where Z = number of zeroes of H G(s)H (s) in the RHP, N = number of clockwise encirclements ofthe (-l,0) point, and P = number of poles of G(s)H(s) in the RHP. Suppose 0(3) is the open loop transfer function of a unity feedback system. The gain margin is the inverse of lG(jw)| at the phase crossover frequency (when 1000) = —1 80° ). The phase margin is the angle by which 4000;) exceeds —1 80° at the gain crossover frequency (when lKGQ‘w)‘ =1 or 0 dB). ' For the lead (lag) compensator Ts+l s =_K DU aTs+1 Kca ,a<l a) The max (min) phase ¢m occurs at frequency where mu’ 1 a) =—— max shuns—"0‘, a ———1‘ST“¢W 1+a 1+sm¢max The state space equations for a LTI system are x = Ax + Bu y = Cx + Du and the transfer function is . G(s) = C(sI — A)" B + D The eigenvalues of A are the characteristic roots (poles) of the system and are found by solving |sI—A|=0. ...
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