Unformatted text preview: i Method To Estimate RA Contents M. D. Fox 3/20/05 Estimating RA ................................................................................................................. 1 Abstract ............................................................................................................... Introduction ........................................................................................................ Theory .................................................................................................................. 1 1 2 2 2 General ........................................................................................................... Determine RA from experimental data ........................................................... 3 Methods .............................................................................................................. Results ................................................................................................................. 4 Result 1 .......................................................................................................... Framework for taking raw data. ............................................................... Framework for assigning data to the variable d1. ......................................... Framework for fitting and plotting data. ..................................................... Determination of Ra. ................................................................................ Testing the Result ........................................................................................... Plot experimental points and theory line in one step ...................................... 9 4 4 4 5 6 7 8 Discussion and Conclusions ..................................................................... ©19882005 FOX Research All rights reserved. Estimating RA
Martin Fox 3/16/05 Abstract
The armature resistance RA must be deduced to be able to calculate the droop in speed with increasing torque in DC motors, and to estimate electrical losses and separate them from mechanical losses. The basic VA iA RA Å Å equation for a DC motor is f = ÅÅÅÅÅÅÅÅÅ  ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ which relates the output frequency of the rotary pulse encoder C C [RPE] to the measured armature voltage and current. To deduce the unknown constants C and RA from a set of measurements of f versus VA and IA it is necessary to do a multivariate regression. Performing such a regression using Mathematica indicated that for the particular DC motor studied in this experiment, the armature resistance was estimated to be 15.1219 Ohm. Introduction
For a DC motor, we know that the speed varies linearly with the internal or generator voltage EA . For noload conditions, we can usually get a straight line relationship between the applied external voltage VA and the motor speed, since current draw is small. Such a relationship will exhibit an offset from the origin that can be used to deduce the armature resistance if the current draw of the motor is known. In this note we will provide an example of how to do this calculation using the multvariate regression function in Mathematica. By providing the constants that give the best fit for a number of experimental points, this provides an approach to estimation of the effective armature resistance under dynamic conditions. In the following pages, we will provide a brief theory, and then work out a detailed example of the determination of armature resistance from typical motor speed versus input voltage and current measurements. 2 MethodToEstimateRA0317051.nb Theory
à General
T RA + VA iA + EA Figure 1 Steady state permanent magnet DC motor model. The steady state model of a DC motor is as shown in Fig. 1. The basic equations for the electrical and mechanical sides are: VA = iA RA + EA and T = TM  wM D By conservation of energy EA iA = TM wM Also, the speed of the motor will be directly proportional to the internal voltage EA , so EA = K wM Putting 3 into 4, we obtain EA iA TM EA = K ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ; and iA = ÅÅÅÅÅÅÅÅÅÅÅÅ or iA = Kp TM . Å Å TM K 1 where Kp is ÅÅÅÅÅÅ . Thus the armature current is directly proportional to the mechanical torque. K (5) (4) (3) (2) (1) TM D à Determine RA from experimental data
From (1) and (4) we have, VA = iA RA + K wM (6) MethodToEstimateRA0317051.nb 3 which can be rearranged to express the speed wM as a function of the independent variables VA and IA as follows : VA  iA RA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ . ÅÅÅÅ wM = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ K If we use cyclical rather than radial frequency this becomes, VA  iA RA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ ÅÅÅÅ fM = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ C where C = 2 p K. (8) (7) Methods Figure 2 Experimental setup for measurement of speed as a function of input voltage and current of a DC motor. The experimental setup was as illustrated in Fig. 2. Input voltage could be read directly from the digital output of the GW Dual Tracking DC supply, current was measured using the RiteTech Multimeter in current mode on the mA range, and RPE output frequency was measured using the Tektronix digital scope. 4 MethodToEstimateRA0317051.nb Results
à Result 1
ü Framework for taking raw data.
TableForm@ [email protected] i, i, i<, 8i, 1, 10<D , TableHeadings > 8Automatic, 8"Vin,V", "Iin,mA", "f,kHz"<<D 1 2 3 4 5 6 7 8 9 10 Vin,V 2 4 6 8 10 12 14 16 18 20 Iin,mA 1 2 3 4 5 6 7 8 9 10 f ,kHz 1 2 3 4 5 6 7 8 9 10 "" 1 2 3 4 5 6 7 8 9 10 "Vin,V" 2.10 4.04 6.19 8.04 10.47 12 14 16 18 20 "Iin,mA" 59.8 66.3 71.8 75.4 78.7 6 7 8 9 10 "f,kHz" 1.082 2.677 4.577 6.150 8.279 6 7 8 9 10 ü Framework for assigning data to the variable d1.
2.10 4.04 d1 = 6.19 8.04 10.47 59.8 66.3 71.8 75.4 78.7 1.082 2.677 4.577 6.150 8.279 882.1, 59.8, 1.082<, 84.04, 66.3, 2.677<, 86.19, 71.8, 4.577<, 88.04, 75.4, 6.15<, 810.47, 78.7, 8.279<< MethodToEstimateRA0317051.nb 5 ü Framework for fitting and plotting data.
p1 = [email protected]@@All, 81, 3<DD, PlotStyle Æ PointSize@.02D, PlotRange > 8811,  1<, 8 1, 11<<D 10 8 6 4 2 2
Ü Graphics Ü 4 6 8 10 thy1 = [email protected]@@All, 81, 3<DD, 81, x<, xD  0.762203 + 0.861739 x [email protected], [email protected], 8x,  1, 11<D, AxesLabel > 8"VA ,V", "f,kHz"<, PlotLabel > "f Vs. VA "D 8 6 4 2 2 4 6 8 10 6 MethodToEstimateRA0317051.nb f,kHz 10 8 6 4 2 2
Ü Graphics Ü f Vs. VA 4 6 8 10 VA ,V ü Determination of Ra. If conditions were truly no load, there would be no current flow. Current flows due to mechanical damping in the motor. Energy is dissipated both in mechanical damping and in armature loss. We can rearrange the input equation (8) for the motor as follows: VA iA RA Å Å f = ÅÅÅÅÅÅÅÅÅ  ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ . C C Another useful form is: f = Cp VA  Cp iA RA 1 where Cp is ÅÅÅÅÅÅ . C (9) (10) Import the nonlinear fit package to get multidimensional regression capabilities. This is a very useful and powerful package.
<< Statistics`NonlinearFit` Now perform a fit on the data set d1.
[email protected], Cp VA  Cp iA RA , 8VA , iA <, 8RA , Cp <D  0.0134908 iA + 0.892135 VA Comparing with Eq. 2, we see that Cp RA = .0134908, thus
.0134908 RA = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄÄÄÄ .892135 0.0151219 MethodToEstimateRA0317051.nb 7 Since current was recorded in Milli Ampere, Resistance is output in Kilo Ohm so RA is 15.1219 Ohm. à Testing the Result
To test this result, we subtract out the effect of RA and see if we get a straight line passing through the origin. Here we compile a table of values d2 = { (EA = VA  iA RA L , f}. If our theory is correct, f = Cp EA , so the resultant curve should be a straight line
d2 = [email protected]@@i, 1DD  .0151219 d1@@i, 2DD, d1@@i, 3DD<, 8i, 1, [email protected]<D 881.19571, 1.082<, 83.03742, 2.677<, 85.10425, 4.577<, 86.89981, 6.15<, 89.27991, 8.279<< passing through the origin [i.e. with yaxis intercept at zero]. Plotting we obtain: p2 = [email protected], PlotRange > 880, 10<, 810, 0<<, PlotStyle > PointSize@.02DD 10 8 6 4 2 2
Ü Graphics 4 6 8 10 Regress d2 to obtain the best straight line fit:
thy2 = [email protected], 81, x<, xD 0.000263543 + 0.892096 x Combine the plots to see if the straight line passes through the origin:
[email protected], [email protected], 8x, 0, 11<D, AxesLabel > 8"EA ,V", "f,kHz"<, PlotLabel > "f Vs. EA "D 8 MethodToEstimateRA0317051.nb 10 8 6 4 2 2 f,kHz 10 8 6 4 2 4 6 8 10 f Vs. EA 2
Ü Graphics Ü 4 6 8 10 EA ,V The straight line does intersect the origin, so it appears we have successfully deduced the armature resistance and removed it as a factor. à Plot experimental points and theory line in one step
The following statement plots the theory and experimental data in a single Mathematica expression [but it is a hard one to remember!].
[email protected], 8x, 0, 11<, AxesOrigin Æ 80, 0<, PlotRange Æ 8Automatic, 80, 10<<, Epilog Æ [email protected] êû d2, [email protected] , AxesLabel > 8"EA ,V", "f,kHz"<, PlotLabel > "f Vs. EA "D; MethodToEstimateRA0317051.nb 9 f,kHz 10 8 6 4 2 f Vs. EA 2 4 6 Figure 3 8 10 EA ,V Motor speed in Hz as a function of generator or internal voltage EA in Volt. Discussion and Conclusions
We have demonstrated how to deduce motor armature resistance from a set of measurements of motor speed as a function of input voltage and current. By using all the input data in a regression we can anticipate that this will provide a more accurate estimation of RA over a range of operating conditions compared to a calculation based on a single data point. For the particular motor studied here we obtained an armature resistance of 15.1219 Ohm or approximately 15 Ohm. Other readings for similar motors have ranged from less than 10 Ohm to greater than 20 Ohm. The armature resistance depends not only on the winding resistance, which we would expect to be fairly stable, but also the condition of the carbon brushes and the commutators, which can be expected to degrade with use. In the future it would probably be a good idea to repeat this procedure several times to assess the variability of the results and thereby estimate error limits. ...
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This note was uploaded on 11/18/2010 for the course ECE 3211 taught by Professor Marks during the Spring '08 term at UConn.
 Spring '08
 Marks

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