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Unformatted text preview: Start Up Dynamics of DC Motor
Introduction
M. D. Fox 10 ê 25 ê 04 We have largely focussed on steady state behavior of motors and generators. Here we will examine the dynamic performance, from both a theoretical and experimental perspective. Motor dynamic response was measured by two different techniques. First, the motor was attached to the generator and the output voltage of the generator as a function of time was used to estimate the speed as a function of time. Since the coupling of shafts implies twice the moment of inertia and half the damping factor, we expect a time constant at least 4 times that of the single motor. To measure the time constant of the speed of the motor alone, we monitor the RPE of the motor under step excitation, and use the number of cycles observed in a unit time to estimate instantaneous frequency. Ultimately the results from both methods were in reasonable agreement. The damping factor was estimated at low speed, since the low speed damping factor seemed to give the best correspondence with the observed results. Below we will review the theory of motor dynamics, present the experimental data, and provide discussion of the results. Theory
R L T ω + VA + EA =Kω  J= ½ M R2 D ω +J dω/dt Electrical
Figure 1
Illustrating electrical and mechanical elements of an electric motor/generator. D is mechanical damping factor, and J is moment of inertia of the cylindrical rotor. Mechanical 2 MotorStartupTransient1025051.nb R L T Motor Jm Gen Jg r + VA + EA =Kω  ω J= ½ M r2 D ω +J dω/dt Electrical
Figure 2
Illustrating situation where motor is coupled to generator. Here J = Jm + Jg , D = DgDm. Mechanical Note the analogy between 1 and 2. If the driving Voltage in 2 is a step of amplitude Vm, @neglecting initially the K w termD you get out i = t Vm LR ÅÅÅÅÅÅÅÅÅÅÅÅ I1  E ÅÅÅÅÅêÅÅÅÅÅ M. Thus using analogous reasoning for 1, if T is a step you get Å R
t Tm JÅ w = ÅÅÅÅÅÅÅÅÅÅÅ I1  E ÅÅÅÅêÅDÅÅ M Å D dw J ÅÅÅÅÅÅÅÅÅÅ + D w = T dt ° L i + i R = VA  K w = Vm [email protected] (1) (2) (3) MotorStartupTransient1025051.nb 3 N + Rm T u[t] Jm Jg ω Rg  Electrical Equivalent to Mechanical Motor/Gen
Figure 3
1 1 Electrical equivalent to the mechanical portion Fig. 2. Here Rm = ÅÅÅÅÅÅÅÅÅÅÅÅ , Rg = ÅÅÅÅÅÅÅÅÅÅ . Å Dm Dg Take the KCL at node N to obtain dw dw w T [email protected] = Jm ÅÅÅÅÅÅÅÅÅÅ + Jg ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅ ÅÅÅÅÅÅÅÅ dt dt Rm + Rg w dw T [email protected] = J ÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅ R dt (4) If J = Jm+Jg and R = Rm+Rg, we obtain, (5) which is the same form as Eq. 1. Thus the addition of a generator to the motor results in the same first order system, but with a different time constant. By inspection it is clear that with unit step input we are charging a parallel capacitor resistor combination with an ideal current source, resulting in a transient response of the form:
J ÅÅÅ [email protected] = T R I1  E ÅÅÅÅÅR M t (6) à Simplify Analysis If Rg = Rm and Jg = Jm, as in our case of matched motor generator pairs, then Jg + Jm = 2 Jm and Rg + Rm = 2 Rm we have J R = 4 Jm Rm, i.e. the time constant of the combination motor ê generator should be four times that of the motor alone. Since there are time dependent elements in both the electrical and mechanical portions of the electromechanical circuit we could solve for the dynamic solution by simultaneously solving Eqs. 1 and 2. While this would give us a very general solution, it might not provide the best insight in what is actually happening in our specific system. Below we will do some preliminary calculations which show that we can simplify our analysis by ignoring the electric transient since it has a much shorter time constant that the mechanical portion in this case. 4 MotorStartupTransient1025051.nb ü Time constant for electrical part of circuit: Frfom direct measurement, we determine that L = 6 Milli Henry, Ra = 12.5 Ohm. TRhen the electrical time constant te is: L .006 Henry te = ÅÅÅÅÅÅÅÅÅ > ConvertA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ , Micro SecondE ÅÅÅÅÅÅÅÅ Ra 12.5 Ohm L ÅÅÅÅÅÅÅÅÅ Ø 480. Micro Second Ra ü Time constant for mechanical part of circuit From direct measurement of the rotor of our motor, we find: 1 J > ÅÅÅÅÅ M R2 ê. 8M > .104 Kilo Gram, R > .0125 Meter< 2 (9) (10) (7) (8) J Ø 8.125 µ 106 Gram Kilo Meter2 We estimate D from the steady state mechanical dissipation at a low voltage input to motor designated FOXTROT1 [2.01 Volt, see Table 1 below] as follows:
2.01 Volt .082 Ampere  H.082 AmpereL2 12.5712 Ohm ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄ ê. ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄ D > ConvertA ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ w2 2 p 930 Second1 w > ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄ , Micro Second2 WattE ÄÄÄÄÄÄÄÄ Ä 500 D Ø 587.872 Micro Second2 Watt J tm Ø ÅÅÅÅÅÅ Ø 13821. Micro Second (12) D J L We see that ÅÅÅÅÅÅ >> ÅÅÅÅÅÅ , D R so we can neglect the electrical portion of the motor circuit in this case because of its very short time constant. J 8.125 106 Gram Kilo Meter2 tm > ÅÅÅÅÅÅ > ConvertA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ , Micro SecondE ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ D 587.872 Micro Second2 Watt (11) MotorStartupTransient1025051.nb 5 Results
Three experiments were carried out. The first experiment measured electrical power into a motor designated FOXTROT1 to estimate the damping factor for the theoretical analysis above. The data is summarized in Table 1 and Figure 5 below. In the second experiment, [result shown in Fig. 6 below] we connected the motor to the generator, applied a step input, and observed the time constant of the voltage output by the generator to estimate the time course of the transient response. In the final experiment we applied a step input to the motor alone, and observed the rotary pulse encoder output [Fig. 7 below]. The number of RPE cycles in a time interval then provides an estimate of the shaft angular velocity as a function of time. The RPE can be examined by transferring the data from the scope to a computer, and then examining in detail portions of the waveform, digitized to 10000 samples of a .2 Second interval. Vin,V 0 2.01 4.14 6.2 8.06 10.17 12.24 14.04 16.28 18.23 18.9 Iin,A 0 0.082 0.075 0.076 0.078 0.08 0.087 0.088 0.09 0.09 0.09 Table 1
Speed/power data for motor designated FOXTROT1. 1 2 3 4 5 6 7 8 9 10 11 f ,kHz 0 0.93 2.972 4.837 6.542 8.467 10.3 11.98 14.04 15.83 16.47 6 MotorStartupTransient1025051.nb PM ,W 1.5 1.25 1 0.75 0.5 0.25 2.5 Mech. Diss. Vs. Speed 5 7.5 10 Figure 5 12.5 15 f,kHz Green is electrical dissipation, red is total dissipation, and black is mechanical disspation. For this device the mechanical dissipation went up linearly with speed. This differs from previous results, where the current draw, and thus electrical load increased with increasing speed. MotorStartupTransient1025051.nb 7 Figure 6
Scope trace comparing applied voltage to output voltage from motor/generator pair. The data from this trace was used to construct Fig. 3 below. 8 MotorStartupTransient1025051.nb Figure 7
Scope trace of start up transient of Minertia 8007833 motor. This data can be used to determine the time constant of a single device. MotorStartupTransient1025051.nb 9 5 4 3 2 1 0.002 0.004 Figure 8 0.006 0.008 Response of motor RPE to square wave input, showing frequency increasing fowwowing step voltage input. à Determination of best Fit for Motor/Generator experiment
Figure 8 below shows the normalized motor/generator transient response. Using regression analysis to a t solution of the form K I1  E ÅÅtÅÅ M, we obtain K = 1.0282 and t = 44.5843 Milli Second. The fit is superimposed on the experimental dat with the red line in Fig. 7 showing a good fit to the theoretical relationship. 10
BestFitParameters ê. NonlinearRegressAs3,
t MotorStartupTransient1025051.nb 8K Ø 1.0282, t Ø 0.0445843< t K I1  E ÄÄÄÄÄ M [email protected], t, 8K, t<, RegressionReport Æ BestFitParametersE Speed, Norm. 1 0.8 0.6 0.4 0.2 0.1 0.1 Figure 9 0.2 0.3 t,S Startup transient obtained from Motor/Generator hookup [time constant = 40 mS]. If the J's add the new J'=2J while the Ds are in series, resulting in new d'=D/2 so t'=4t, meaning that we estimate the time constant for a single stage to be 10 mS, close to the 8.95 mS value calculated from the motor experiment. There may be some additional drag introduced by the coupling. à Determination of best Fit for Motor alone experiment
Table 2 below summarizes the frequency as a function of time for the motor FOXTROT1. Regression analysis of the data [Eq. 13 and 14 below] results in a time constant of 8.94797 Milli Second, and a delay of 4.0792 Milli Second. Examination of Fig. 2 above shows that only about 1 Milli Second is true delay and that most of the delay is due to a difference between the trigger point and 0 time on the oscilloscope. Close examination of Fig. 8 shows there is a small delay time there also [note discrepancy in early points]. Fig. 9 below, which shows the experimental data and the regression line shows good agreement with theory. If we add the delay of 1 Milli Second to the time constant of 8.94 Milli Second, we get an effective time constant of 9.94 Milli Second which is roughly one quarter the time constant of 44.6 Milli Second for the motor/generator pair. The discrepancy could be due to additional mechanical resistance introduced by the coupling. t,mS 5 7 9 10.8 12.8 22.8 f ,Hz 4 9 13 17.5 22 30 1 2 3 4 5 6 MotorStartupTransient1025051.nb 11 7 8 9 42.8 62.8 103 33 33 34 Raw data for construction of response curve for single motor. BestFitParameters ê. NonlinearRegressAd6, K I1  E ÅÅÅÅtÅÅ Å M [email protected]  dD, t, 8K, t, d<, RegressionReport Ø BestFitParametersE
td (13) (14) 8K Ø 33.6369, t Ø 8.94797, d Ø 4.0792< f,Hz 30 25 20 15 10 5 20 40 60 Figure 10
Change in frequency as function of time for Minertia motor alone. From 11 above, time constant t = 8.94 mS. t of Fig. J 5 [44.5 mS] is more than four times as large, reflecting larger ÅÅÅÅÅÅ . D 80 100 t,mS 12 MotorStartupTransient1025051.nb Discussion and Conclusions
Both the motor alone and the motor/generator pair exhibited a first order rise consistent with a controlled current source [torque] driving a parallel RC network. As predicted by the theory the time constant of the t JÅ motor/generator pair was about 4 times that of the motor alone. Theory lines of the form K I1  E ÅÅÅÅÅRÅ M tracked the results well as shown in Figs. 9 and 10 above. For the particular motors employed here, Minertia DC developed for servo applications, the electrical transient could be ignored, and the major contributor to the transient response was the moment of inertia of the rotor and the Damping factor [1/mechanical resistance] of the device. Due to the low and relatively constant current draw, the electrical losses were fairly flat as speed increased for FOXTROT1, while the mechanical losses closely tracked the total loss [see Fig. 5]. This behavior implies a reduction in the steady state damping factor as a function of velocity. More study needs to be done to determine if this is idiosyncratic to FOXTROT1 or a property of the Minertia motors. Key results include a time constant of about 9 Milli Second for motor FOXTROT1 alone, and a time constant of 44.5 Milli Second for the motor generator pair. Using the steady state data to estimate the transient damping factor D, revealed that the effective damping factor appears to decrease at higher steady state speeds. The damping factor observed at low speeds appeared to be the best value for modelling the transient response of the motor and motor/generator. Overall, we have shown that in this case motor dynamic response is well modelled by a simple first order analysis of the mechanical system. The rotor acts as an energy storage device due to the inertia of a rotating cylindrical element. For design of cars for racing for example, small time constants like this mean that the vehicle comes to its steady state velocity almost immediately, and thus racing speed is best approached through optimization of the steady state model. Another interesting application of this type of data would be to introduce elements [like electrical inductors or capacitors] to slow the startup and thus reduce the transient current required for startup. This can make it possible to start motors using smaller supplies and thus reduce stresses on both mechanical and electrical elements. ...
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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
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