Experimental and Theoretical Modeling of Moving Coil Meter

Transcription

1 Experiental and Theoretical Modeling of Moving Coil Meter Prof. R.G. Longoria Updated Suer 010

2 Syste: Moving Coil Meter FRONT VIEW Electrical circuit odel Mechanical odel Meter oveent REAR VIEW needle Series resistor

3 Moving Coil Meter Moveent This D Arsonval eter oveent is a basic EM device that responds to electrical voltage or current signals. Fro Figliola and Beasley, Theory and Design for Mechanical Measureents, John Wiley and Sons, In the particular eter being used, the coil pivots such that the conductors are always perpendicular to the agnetic field generated by a peranent agnet, as shown here.

4 Physical Modeling of Syste We focus on a nd order odel that neglects inductance. The appendix shows a 3 rd order odel with inductance, and a frequency response coparison of the nd order and 3 rd order odel shows that the siplification is reasonable. Methods for deriving transfer functions are reviewed. Frequency response derivations are suarized and exaples provided in Matlab and LabVIEW. These tools should allow coparison of odel and experiental data directly on the sae or siilar plots.

5 Meter Moveent Bond Graph A bond graph of the eter is shown below. The coil has resistance, R, and inductance, L. The needle has oent of inertia, J, and there is soe daping, B, as well. The spring has stiffness, K. These are paraeters for linear constitutive relations for each of the eleents shown in this odel. Note, the eter also has an external series resistor that is not shown here, but the value of that resistance can be added to R. We seek a atheatical odel that relates needle position, θ, to input voltage, v. Ignore inductance (very sall) Causality assignent shows this is a nd order syste. This odel can be derived fro the bond graph, or by application of Newton s Laws (echanical side) and KVL (circuit side). See appendix for syste odel with inductance included.

6 Siplified Model Equations The atheatical odel for the eter, neglecting inductance, States: State equations: EM gyrator relations: T = ri v = rω where, ( v v ) Rc + h = Jω = angular oentu θ = angular position of needle/spring h ɺ = T Ksθ Bω ɺ θ = ω i = ( R ) s If you know how to derive the equations fro a bond graph, you see how current is here deterined by the voltage drop.

7 State Equations nd order In state space for: State equations: hɺ ɺ θ r 1 B + K r R h J = + R v 1 θ 0 0 J B A Output equation: y h = θ = + θ C [ 0 1] [ 0 ] D v

8 Transfer Functions (TF) To derive a transfer function (TF), you have several options: 1. Use G(s)=C(sI-A) -1 B+D (state-space to TF). Change the state space equations into an ODE of nth order, and use Laplace transfor 3. Use a bond graph, and apply ipedance ethods It is likely that you ay have learned one or two of these approaches for deriving the TF. Method #1 is available in LabVIEW and Matlab, but only if nuerical paraeters are available.

9 nd order SS Model to TF In a siilar anner as in the previous case, the G(s) function is derived here sybolically. Keep in ind, this G(s) function has been defined by the A,B,C,D syste as: G( s) = y u θ So G( s) = v G( s) ( 0 1) G( s) G( s) R s B R s J nu r s + B + R 1 r R J J 1 1 J K s 1 + R K + r s + R s J r R r 1 r + R B + r s + R K den s r + B + R 1 s + J K J

10 TF to Frequency Response Given G(s), you can deterine the frequency response function (FRF), by setting s = jω, and deterining the aplitude and phase functions of G(s). j Use: G( jω) = G( jω) e φ Exaple (1 st order syste): G( s) = 1 G( jω) = 1 τ s + 1 jτω + 1 s= jω Aplitude function Phase function G( jω) = 1 ( τω ) + 1 I( ( den ) Re( ) ) Both of these functions are plotted versus frequency, ω. φ = φ φ = = τω = φ ω 0 tan 1 tan 1 nu den den ( ) ( ) These functions are also referred to as Bode plots, and are coonly found functions in Matlab (Control Toolbox) and LabVIEW (Control Design Toolkit). If these packages are not available, you ust derive the functions analytically so you can plot in Excel, Matlab, or LabVIEW.

11 Meter: TF to FRF ( nd order) j G( jω) = G( jω) e φ r r R J R J G( s) = G( jω) = r 1 K s= jω r 1 K s + B + s + ( jω ) + B + jω + R J J R J J ( ) jω = ω Note: 1 Aplitude function G( jω) = r R J r ω + B + K ω J R J B + 1 I( den) R 0 tan ( ) 1 J nu den Re( den) tan ( ) K φ = φ φ = = = φ ω ω J r Phase function ω

12 Meter: dc gain The dc gain is the value of the TF or FRF when s or ω go to zero, respectively. So, fro the aplitude function, r r R J R J r 1 G( jω = j 0) = = = K R K K r B + J J R J The dc gain for the oving coil eter is, θ r 1 G( jω = j 0) = = v ω= 0 R K

13 Plotting the FRF The FRFs are coonly plotted as aplitude and phase functions (soeties called Bode plots, although strictly speaking a Bode plot is an approxiated sketch of the FRF plots). LabVIEW and Matlab ay have build-in packages: In Matlab, if Control Toolbox is available, use: bode(). In LabVIEW, the Control Design Toolkit has CD Bode.vi. Use the online help if you have or want to use these tools. For this course, you should generate these plots without these tools, since it is instructive to develop the code for coputing the aplitude and phase functions. It is iportant to notice that the aplitude is often plotted in ters of decibels (db). In this context, the decibel is defined, db = 0log G( jω) Also, ake note of the frequency axes used (rad, deg, etc.). 10 See Matlab exaple on next slide.

14 % Basic script for plotting aplitude and phase plots % Exaple: 1st order syste. RGL, % Plot at selected frequencies f = [ ]; % w = *pi*f; N = length(w); R=81.36e3; C=0.005e-6; tau = R*C; j = sqrt(-1); for i=1:n, gsys(i) = 1/(j*w(i)*R*C+1); agsys(i) = abs(gsys(i)); Note, you can handle the coplex function in Matlab directly. db(i) = 0*log10(agsys(i)); angsys(i) = atan(iag(gsys(i))/real(gsys(i))); end % subplot(,1,1), seilogx(f,db,'o') subplot(,1,), seilogx(f,angsys*180/pi,'o') Exaple Script

15 Appendix Details on the physical odeling and on the 3 rd order odel

16 B-field N + S current flow direction copper rod Basic Electroechanics This slide suarizes the basic force-current relation in each conductor. In a bond graph, this can be odeled by a gyrator, which gives a net relation between torque and current. q V F F = q V B B Peranent agnet supplies the B-field The differential force on a differential eleent of charge, dq, is given by: where B is the agnetic field density, and i the current (oving charge). It can be shown that the net effect of all charges in the conductor allow us to write: where dl is an eleental length. For a straight conductor of length l in a unifor agnetic field, you can integrate to find the total force: F = il B With angle α between the vectors, you can arrive at the desired relation: v i G F x We find this odulation as: r = Bl sinα V ɺ x df = dqv B df = idl B ( sinα ) F = Bl i gyrator odulus F v = r i = r V

17 Meter Moveent Bond Graph A bond graph of the eter is shown below. The coil has resistance, R, and inductance, L. The needle has oent of inertia, J, and there is soe daping, B, as well. The spring has stiffness, K. These are paraeters for linear constitutive relations for each of the eleents shown in this odel. Note, the eter also has an external series resistor that is not shown here, but the value of that resistance can be added to R. We seek a atheatical odel that relates needle position, θ, to input voltage, v. Causality assignent shows this is a 3 rd order syste. This odel can be derived fro the bond graph, or by application of Newton s Laws (echanical side) and KVL (circuit side).

18 Full Model Equations The atheatical odel for the eter, including all the effects described is, h = Jω = angular oentu States: λ = Li = flux linkage θ = angular position of needle/spring hɺ = T Ksθ Bω State ɺ equations: λ = v ( Rc + Rs ) i v ɺ θ = ω EM gyrator relations: T v = = r i r ω Note: the needle and the spring have the sae velocity.

19 Siplified Fors and Relations We often choose to ake use of siplified forulations; the state space equations ay not be suited for answering questions we have. Note how the state variables are related to other useful variables. The state equations are related to both 1 st and nd order fors that we ight use. h = Jω = J ɺ θ λ = Li = flux linkage θ = angular position of needle/spring d hɺ θ = J ɺɺ θ = J T = Ksθ B dt di ɺ λ = L = v ( R + R ) i v dt ɺ θ = ω c s dθ dt

20 State Equations In state space for: R r 0 L J ɺ λ λ 1 r B hɺ K h 0 State equations: = + v J J ɺ θ θ B J A λ y θ [ ] h Output equation: = = + [ 0] v C θ D

21 Full Model SS to TF Using the G(s) forula, apply directly to the derive the for shown here to the right. With a sybolic processor, this is easily accoplished (e.g., Matlab, MathCAD, or Matheatica). G( s) ( 0 0 1) G( s) G( s) nu C s 3 L J s 3 L J r L J R s + L r L 0 r J B s + J 1 J (si-a) -1 0 K s r 0 + s L B + s L K R s + J + R s B + R K + r s ( ) s r B + R J + L B + R B + L K + r s + R K D den s 3 R + L + B J s + R B L J + K J + r L J s + R K L J

22 clear all % exaples paraeters for oving coil % this exaple plots both the 3rd and nd order syste global R r J K B L Exaple using bode() R = 15085; r = 0.003; J = e-7; K = 10e-6; B = 9e-7; L = 0.05; % 3 rd order case A1 = [-R/L -r/j 0;r/L -B/J -K;0 1/J 0]; B1 = [1;0;0]; C1 = [0 0 1]; D1 = [0]; sys1 = ss(a1,b1,c1,d1); [nu1,den1]=sstf(a1,b1,c1,d1) % nd order case A = [-(B+r*r/R)/J -K;1/J 0]; B = [r/r;0]; C = [0 1]; D = [0]; sys = ss(a,b,c,d); [nu,den]=sstf(a,b,c,d); bode(sys1,sys) Magnitude (db) Phase (deg) Bode Diagra Frequency (rad/sec) nd order syste Note how the two odels only deviate at very high frequency we will never excite the eter at this frequency range!! The nd order syste is clearly applicable for all cases of interest. 3 rd order syste