3.1. Equation of state
Equation (14) shows the mathematical model that is proposed in this work, which represents an equation of state that relates the variables that are considered in this study of a BDCM system. With this mathematical model, it is possible to determine the contribution of each element that has been considered within the circuit, both for the electrical power \(P\) (Equation 15) and for the work \(W\) (Equation 16).
\(V=\alpha T{R}_{0}{I}_{a}+\frac{1}{2q}{L}_{a}{I}_{a}^{2}+{\frac{1}{q}T}_{m}\theta\)
|
(14)
|
\({P}_{Elec}=V{I}_{a}=\alpha {I}_{a}^{2}{R}_{0}T+{I}_{a}{L}_{a}\frac{d\left({I}_{a}\right)}{dt}+{T}_{m}\dot{\theta }\)
|
\(\left(15\right)\)
|
\({W}_{Elec}=Vq=\alpha T{R}_{0}{I}_{a}q+\frac{1}{2}{L}_{a}{I}_{a}^{2}+{T}_{m}\theta\)
|
\(\left(16\right)\)
|
Figure 4 shows the value of the electrical power calculated using Equation (15). It can be observed (Fig. 4a) the three contributions and the total calculated power which presents a value of 1.7649 W, while the electrical, magnetic, and mechanical contributions show values of 0.2981 W, 0.0 W, and 1.4669 W. The mechanical contribution is the largest followed by the electrical supply, and finally, the magnetic is null in steady state. Likewise, it is possible to obtain the electrical power values by operating the PWM at different percentages of its nominal capacity. Fig. 4b shows the power transients at five different PWM% values, and it is observed that the power presents a first-order dependence to the PWM%.
Figure 5 shows the calculated work value using Equation (16), and Fig. 5a shows the three contributions during operation at 100% of the PWM rated capacity. The total calculated work is 1,582.7 kW, while the thermal, magnetic, and mechanical contributions show values of 267.79 W, 29.2 µW, and 1,315 kW, during the 100% operation of the PWM. The mechanical contribution is the largest, followed by the electrical and finally the magnetic. Likewise, it is possible to obtain the electrical work values when operating the PWM at different percentages of its nominal capacity. Fig. 5b shows the work transitory, and it can be observed that the work presents a first-order dependence to the PWM%.
3.2. Estimation of thermo-electromechanical variables of the BDCM.
Given that the state function represents the dependence of the electric potential \(\left(V\right)\)on changes in electric current\(\left({I}_{a}\right)\), temperature \(\left(T\right)\) and torque \(\left({T}_{m}\right)\). It is established that \(V=V\left({I}_{a},T,{T}_{m}\right)\) which gives the total differential (Equation 17). In the same way, \({I}_{a}={I}_{a}\left(V,T,{T}_{m}\right)\) (Equation 18), \(T=T\left(V,{I}_{a},{T}_{m}\right)\) (Equation 19) and \({T}_{m}={T}_{m}\left(V,{I}_{a},T\right)\) (Equation 20) can be expressed.
\(dV={\left(\frac{\partial V}{\partial {I}_{a}}\right)}_{T,{T}_{m}}d{I}_{a}+{\left(\frac{\partial V}{\partial T}\right)}_{{I}_{a},{T}_{m}}dT+{\left(\frac{\partial V}{\partial {T}_{m}}\right)}_{T,{I}_{a}}d{T}_{m}\)
|
\(\left(17\right)\)
|
\(d{I}_{a}={\left(\frac{\partial {I}_{a}}{\partial V}\right)}_{T,{T}_{m}}dV+{\left(\frac{\partial {I}_{a}}{\partial T}\right)}_{V,{T}_{m}}dT+{\left(\frac{\partial {I}_{a}}{\partial {T}_{m}}\right)}_{T,V}d{T}_{m}\)
|
\(\left(18\right)\)
|
\(dT={\left(\frac{\partial T}{\partial V}\right)}_{{I}_{a},{T}_{m}}dV+{\left(\frac{\partial T}{\partial {I}_{a}}\right)}_{V,{T}_{m}}d{I}_{a}+{\left(\frac{\partial T}{\partial {T}_{m}}\right)}_{V,{I}_{a}}d{T}_{m}\)
|
\(\left(19\right)\)
|
\(d{T}_{m}={\left(\frac{\partial {T}_{m}}{\partial V}\right)}_{{I}_{a},T}dV+{\left(\frac{\partial {T}_{m}}{\partial {I}_{a}}\right)}_{V,T}d{I}_{a}+{\left(\frac{\partial {T}_{m}}{\partial T}\right)}_{V,{I}_{a}}dT\)
|
\(\left(20\right)\)
|
To obtain the corresponding values of the electrical material properties from the partial derivatives. The derivatives for each variable are obtained using the equation of state (see Equation 14), and subsequently using Maxwell's equations, the values of the partial derivatives of Equations (18), (19), and (20) are obtained.
With the differential equations solutions is possible to establish the values of each of the variables, as shown in Fig. 6, where the values were obtained using the adiabatic BDCM system, and the value calculated using the mathematical model of the variables: Voltage (3a), Current (3b), Temperature (3c) and Torque (3d), during the operation at 20, 40, 60, 80 and 100% of the nominal capacity of the PWM.
Figure 7 shows the percentage accuracy of the mathematical model for all the parameters evaluated. The voltage presents a higher variability during the first moments due to the motor start impulse and then stabilizes, reaching a 0.2% error in the measured signal. Current, temperature, and torque, on the other hand, present a better response with an average value of 100±0.1%.
The equations to obtain the values of the thermo-electromechanical variables are shown below.
3.2.1 Voltage calculation
From the solution of Equation 17, we obtain Equation 21, which represents the variation of the electric potential as a function of changes in current, temperature, and torque.
\(\varDelta V=\frac{{L}_{a}}{2q}\left[{{{I}_{a}}_{2}}^{2}-{{{I}_{a}}_{1}}^{2}\right]+\alpha {R}_{0}T\left[{{I}_{a}}_{2}-{{I}_{a}}_{1}\right]+\alpha {R}_{0}T\left[{T}_{2}-{T}_{1}\right]+\frac{\theta }{q}\left[{{T}_{m}}_{2}-{{T}_{m}}_{1}\right]\)
|
(21)
|
The BDCM system voltage operates as expected with a first-order linear increase between PWM% and Voltage (Figure 6a), presenting a rate of change of 91.62 mV/PWM% and a correlation (r2) of 0.9922. The values obtained by the model fit the measured voltage values, with an average coefficient of variation of 0.9855 (Fig. 7).
Figure 8 shows the voltage value estimated using Equation (14), showing the three contributions, during operation at 100% of the PWM rated capacity. The total calculated voltage is 9.7876 V, while the electrical, magnetic, and mechanical contributions show values of 1.656 V, 48.67 µV, and 8.132 V. The mechanical contribution is the highest followed by the electrical and finally the magnetic contribution.
3.2.2 Current calculation
From the solution of Equation (18), we obtain Equation (22), which represents the variation of current as a function of changes in voltage, temperature, and torque.
\(\varDelta {I}_{a}=\frac{q}{{I}_{a}{L}_{a}+\alpha {R}_{0}T}\left[{V}_{2}-{V}_{1}\right]-{I}_{a} ln\left[\frac{{I}_{a}{L}_{a}+\alpha {R}_{0}{T}_{2}}{{I}_{a}{L}_{a}+\alpha {R}_{0}{T}_{1}}\right]-\frac{\theta }{{I}_{a}{L}_{a}+\alpha {R}_{0}T}\left[{{T}_{m}}_{2}-{{T}_{m}}_{1}\right]\)
|
(22)
|
The current of the BDCM system operating as expected with a first-order linear increase between PWM% and current (Fig. 6b), presenting a rate of change of 1.04 mA/PWM% and a correlation (r2) of 0.9991. The values obtained by the model fit the measured current values, with an average coefficient of variation of 1.0002 (Fig. 7 and 9).
3.2.3 Temperature calculation
The solution of Equation 19 gives Equation 23, which represents the variation of temperature as a function of changes in voltage, current, and torque.
\(\varDelta T=\frac{\left[{V}_{2}-{V}_{1}\right]}{\alpha {R}_{0}{I}_{a}}-\frac{{L}_{a}}{\alpha {R}_{0}q}\left[{{I}_{a}}_{2}-{{I}_{a}}_{1}\right]-Tln\left[\frac{{{I}_{a}}_{2}}{{{I}_{a}}_{1}}\right]-\frac{\theta }{\alpha {R}_{0}{I}_{a}q}\left[{{T}_{m}}_{2}-{{T}_{m}}_{1}\right]\)
|
(23)
|
Figure 6c shows the comparison between the temperature records using the adiabatic BDCM system and the value calculated using the mathematical model of equation 23. It is observed that the values obtained by the model fit the measured temperature values, with an average variability coefficient of 0.99913 between the measured value and the modeled one. However, they do not present a first-order dependence concerning the PWM%. Nevertheless, it is possible to observe two regions of average temperature.
It is possible to observe in Fig. 10, the BDCM system operates in two conditions, in low conditions for values lower than 40 of PWM%, with a recorded temperature of 31.85±0.29°C and an average calculated temperature of 31.90±0.21°C. While at values above 60 of PWM% a temperature of 37.74±0.90 and a calculated temperature of 37.75±0.91°C are recorded.
Figure 11 shows the value of the temperature calculated using Equation (23). Although, it would be possible to show the contributions to the temperature since the mathematical expression allows it. It has no physical representation, so the temperature cannot be considered in this way, being an intensive property that reaches quasi-stationary states, and the different heat capacities and heat transfer rate of the materials.
On the other hand, the heat given depends on each contribution of the given amount of heat, but that analysis will be left pending for a later part of this series of articles.
3.2.4 Torque calculation
Solving Equation (20) results in Equation (24), which represents the variation of torque as a function of changes in voltage, temperature, and current.
\(\varDelta {T}_{m}=\frac{q}{\theta }\left[{V}_{2}-{V}_{1}\right]-\frac{{L}_{a}}{2\theta }\left[{{{I}_{a}}_{2}}^{2}-{{{I}_{a}}_{1}}^{2}\right]-\frac{\alpha {R}_{0}Tq}{\theta }\left[{{I}_{a}}_{2}-{{I}_{a}}_{1}\right]-\frac{\alpha {R}_{0}{I}_{a}q}{\theta }\left[{T}_{2}-{T}_{1}\right]\)
|
(24)
|
Figure 12 shows the comparison between the torque records using the adiabatic BDCM system and the value calculated using the mathematical model of Equation (24). It can be observed that the values obtained by the model fit the measured torque values, with an average variability coefficient of 1.00021 between the measured value and the modeled one. Furthermore, it can be observed that the current of the CD Brushed system operates as expected with a first-order linear increase, presenting a rate of change of 1.04 µNm/PWM% and a correlation (r2) of 0.9579.
Finally, this work does not consider a load connected to the motor shaft. Although, the model considers that the systems are in an equilibrium system under static conditions. Therefore, when a load is connected to the brushed DC motor system, the other variables must change to keep the required torque constant.