The state space model needs 5 parameters, so it is not as convenient to use in this control study.

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1 Trasfer fuctio for of the odel G θ K ω 2 θ / v θ / v ( s) = = 2 2 vi s + 2ζωs + ω The followig slides detail a derivatio of this aalog eter odel both as state space odel ad trasfer fuctio (TF) as show above. Why use the TF odel? The static experiets provide a easure of the static gai, so oly two ore paraeters are eeded 3 total. Also, the step respose looks like a classic 2 d order syste respose. The state space odel eeds 5 paraeters, so it is ot as coveiet to use i this cotrol study. TF odel oly eeds dapig ratio ad atural frequecy, which ca be readily deteried i the laboratory. Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

2 Aalog ovig coil eter odel The eter physical odel is preseted i two differet versios: full odel ad 2 d order. The 2 d order odel eglects iductace, which is a good assuptio for this applicatio. The electroechaical (EM) torque ca be odeled usig a gyrator. Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

3 Recall the odels show before i scheatic for: Electrical circuit odel Vi i Rotatioal syste Meter oveet eedle Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

4 The EM torque iduced o the ovig coil is related to the curret a gyrator odel This slide suarizes the basic force-curret relatio i a coductor. I a bod graph, this ca be odeled by a gyrator, which gives a et relatio betwee force ad curret. For a otor or for the case of the rotatioal ovig coil, this force is resolved ito torque. B-field N + S curret flow directio copper rod q V F F = q V B B Peraet aget supplies the B-field The differetial force o a differetial eleet of charge, dq, is give by: where B is the agetic field desity, ad i the curret (ovig charge). It ca be show that the et effect of all charges i the coductor allow us to write: where dl is a eleetal legth. For a straight coductor of legth l i a uifor agetic field, you ca itegrate to fid the total force: F = i l B With agle α betwee the vectors, you ca arrive at the desired relatio: gyrator odulus Departet of Mechaical Egieerig Dyaic Systes ad Cotrols Lab v i G F x We fid this odulatio as: r = Bl siα F = r i v = r V V ɺx df = dqv B d F = id l B ( ) F = Blsiα i

5 Aalog ovig coil eter bod graph A bod graph of the eter ca take the for show below. The coil has resistace, R, ad iductace, L. The eedle has oet of iertia, J, ad there is soe dapig, B, as well. The sprig has stiffess, K s. These are paraeters for liear costitutive relatios for each of the eleets show i this odel. Note, the eter also has a exteral series resistor that is ot show here, but the value of that resistace ca be added to R. We seek a atheatical odel that relates eedle positio (equal to sprig deflectio) to iput voltage, v i. This odel ca be derived fro the bod graph, or by applicatio of Newto s Laws (echaical side) ad KVL (circuit side). E Electrical circuit odel v i i i I : L R : RT RT = R + Rs I R : J ɺλ i r ɺh ω i i v T T 1 G 1 s i ω s ɺθ vr ir ω coil + series EM coversio Rotatioal syste T B B : B C : 1 K s See Appedix B for explaatio of gyrator odel for EM trasductio. Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

6 Full 3 rd order odel, with iductace The state-space odel for the eter, icludig the iductace, is 3 rd order. h = J ω = eedle agular oetu 3 States: λ = L i = flux likage θ = agular positio of eedle/sprig ɺh = J ɺω = T K s θ Bω i = λ 3 State ɺλ = L ( di dt) = v i (R + R s )i v i equatios: ɺθ = ω Note: the eedle ad the sprig have the sae EM gyrator T = r i velocity. relatios: v = r ω Also, ca choose either the eter flux likage or curret as the state. L Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

7 3 rd order odel state-space equatios I state space for: State equatios: ɺλ ɺh ɺ θ Output equatio: = Dyaic Systes ad Cotrols Lab R T r 0 L J r B K J J s J A y = θ = C λ h θ λ h θ v i D B Departet of Mechaical Egieerig v i

8 Aalog ovig coil eter bod graph eglect iductace If we eglect the iductace, we see that the odel reduces to secod order. Note the chage i causality (if you uderstad bod graphs). This assuptio is reasoable give that we observe a step respose i the experiets that looks 2 d order, uderdaped. Now the oly states of iterest are the eedle agle (related to sprig deflectio) ad the eedle rotatioal oetu. E v i i i I : Lc I : J ɺλ i r ɺh ω i i v T T 1 G 1 s i ω s ɺθ vr ir ω T B B C : 1 K s R : RT RT = R + Rs coil + series R : B See Appedix B for explaatio of gyrator odel for EM trasductio. Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

9 2 d order odel, eglectig iductace The atheatical odel for the eter, eglectig iductace, h = J ω = agular oetu States: θ = agular positio of eedle/sprig State ɺh = T K s θ B ω equatios: ɺθ = ω EM gyrator T = r i relatios: v = r ω where, ( ) i i = i R = v v i (R + R s ) The eter curret is ow deteried by the voltage drop, ot by the iductor state. Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

10 2 d order state-space equatios I state space for: State equatios: ɺh B + r 2 ɺθ = R 1 K T J s 1 0 J A Output equatio: y = θ = 0 1 C h θ h θ r + R T 0 B + 0 v i D v i Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

11 Let s covert this ito a 2 d order ODE equatio First, cosider just the first equatio ad write it i ters of the agle: h = J ω = J ɺ θ ɺ h = J ɺɺ θ Substitute ito the oetu equatio: J ɺɺ θ + B + r 2 R T 1 J J ɺ θ + K s θ = r R T v i Reeber that the agle ad agular velocity are related, so write i ters of the agle. This gives the 2 d order ODE we wat: ɺɺ θ + B + r 2 R 1 T J 2ζω ɺ θ + K s J θ = ω 2 r J R T v i = K r s J v K s R i T 2 ω u(t) Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

12 Relate to the static gai easured i lab ɺɺ θ + B + r 2 R 1 ɺ θ T J + K s J θ = K r s J v K s R i T 2 2 ω ω u(t) 2ζω For a costat iput voltage, the steady-state agle (equilibriu) is foud by akig the derivative ters zero, ɺɺ θ + B + r 2 R 1 ɺ θ T J + K s θ J ss = K r Recall: s v J K s R i T 90 deg =0 =0 θ = 15 V r θ ss = static gai v K s R i = K θ /v v i odel T So, if you wat a certai agle, you siply apply, v = K / θ θ i v desired Works well if paraeters are kow, reai costat, ad dyaic effects are ot sigificat. v i Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

13 Now write i trasfer fuctio for Start i the stadard for: ɺɺ θ + 2ζω ɺθ + ω 2 θ = ω 2 u(t) u(t) = K θ /v v i Trasfor to s-doai, ad solve for agle-to-voltage relatio: θ K 2 θ / v = 2 2 i + 2ζω + ω v s s You ca see this odel returs the static gai relatio whe you ake s go to zero (i.e., steady-state). So, all we eed is the dapig ratio ad the atural frequecy to paraeterize this dyaic odel. These ca be foud either fro the physical paraeters or by experietally deteriig the values i the lab. We ll do the latter. ω Dyaic Systes ad Cotrols Lab Departet of Mechaical Egieerig

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