ABSTRACT The objective of the controls laboratory is to use multiple methods to characterize parameters of a DC motor and compare these experimental methods to theory. By determining system parameters accurately, closed loop control can be used to optimize systems not only involving DC motors but essentially any mechanical system. Gains for several controllers were calculated and the effect of adding mass to the flywheel of the motor was investigated. Initially, the dead-zone and saturation voltages were obtained for the motor, to define the operating range. To determine system parameters (namely the time constant Tm and motor gain Km) two methods were investigated: step voltage response and frequency response. Using the step response, the steady-state speed of the DC motor was recorded for incremented input voltages and plotted. The slope of the linear regression of this plot is an estimation of the motor gain. The plot can be found at the end of this abstract. Additionally, for each input voltage a characteristic speed was determined by taking the product of the steady-state speed ωss and (1-e-1). Since the time constant is defined as the time taken for the exponent to be negative one (i.e. e-1), the characteristic speed occurs at the time constant. The time constant was obtained for each input voltage and averaged. To determine these parameters using frequency response, the input voltage was incremented over a range of frequencies. For each input frequency, the amplitude of the input voltage and motor speed were determined (since they both varied sinusoidally), and the phase lag between the peaks was obtained by the relationship φ=(∆t*fin*360) for each input frequency fin where ∆t is the time between input and output peaks. The magnitude of the ratio of motor speed amplitude to input amplitude was plotted (in dB) versus the input frequency on a logarithmic scale, to produce a bode plot. The phase lag was also plotted against the same frequency scale. These plots may also be found attached at the end of this report. By our understanding of bode plots, the speed corresponding to the corner frequency of the magnitude plot can determine the time constant by the relationship ωc=1Tm. By determining where a 3 dB change occurred, the time constant can be estimated. With this same plot, the motor gain may be estimated by considering the first input frequency of 0.1 Hz, and solving for the gain by the relationship 20logKm=(plot value @ 0.1 Hz). Using the phase lag plot, the time constant may be estimated as well by again determining the speed associated with the corner frequency, which should occur at a 45° phase lag. With these parameters, the gains for P, PI, and PID controllers were estimated to use closed-loop control of the DC motor. Next the theoretical response of the motor was compared to the experimental results using the theoretical expression ωt= VinKm(1-e-tTm). Per the instructions in lab, the theory was compared against the experimental results for an input voltage of 3 Volts. Additionally, a weight of arbitrary mass was added to the flywheel of the motor and the results were recorded again. Both experimental results and the theory are displayed at the end of this report. For the DC motor used in lab, the saturation voltage was given as ± 5 V, and the dead-zone range was found to be ± 0.11 V. When using the step response method, the motor gain was determined to be Km=8.981 and the time constant was determined to be Tm=0.0998 s. Using frequency response, the time constant and motor gain according to the magnitude bode plot were Km=8.933 and Tm=0.0994 s. Using the phase lag plot, the time constant was estimated to be Tm=0.0965 s. For both bode plots, the corner frequency was determined to be ~10 Rad/sec, or fin≈1.6 Hz. Using values of Km=8.957 and Tm=0.0986 s (the average of all experimentally determined results), gains were calculated for use in closed loop control. For a P (proportional) controller, KP=0.81, for a PD (Proportional – Derivative) controller, KP=2.016 and KD=0.0645, while for a PID (Proportional – Integral – Derivative) controller, KP=3.425, KD=0.153, and KI=16.128. The experiment was rather well controlled, and the results showed good agreeance. The plot of steady-state speed versus input voltage was highly linear, and the bode plots were very similar to theoretical plots of the mathematical model of the system. Note that the theoretical 3V input response is offset slightly from the experimental response once a steady-state speed is achieved. This is likely due to minor losses electrically and mechanically in the motor, as well as possible friction or other sources of loss.

APPENDIX

Figure 1 – Determination of System Parameters by Step Response

Figure 2 – Comparison of Experimental and Theoretical Responses

Figure 3 – Determination of System Parameters by Frequency Response