# Time Domain Investigation of Hybrid Intelligent Controllers Fed Five-Phase PMBLDC Motor Drive
## 2 Modeling of Five-Phase PMBLDC Motor Drive SYSTEM  
There are different methods for modeling a PMBLDC motor, but when it is needed to study time domain characteristics, only time domain modeling is preferred, which has been carried out in this paper for the proposed multiphase machine.  
The modeling of the five-phase Permanent Magnet BLDC motor as based on the electrical equivalent circuit diagram shown in Figure 2b. Figure 2a,b depicts both the entire drive system configuration of the proposed motor drive and its mathematical motor model. The pulse width modulation (PWM)-based inverter has a ten-switch, five-leg voltage source configuration energized with a fixed dc-link voltage (V\({}_{\rm d}\)).  
The following list of assumptions were made for simplifying the analysis.  
* The stator field winding is unsaturated;
* Stator per phase resistance in all the windings is the same;
* Mutual and self-inductances are fixed;
* Ideal power semiconductor switches in the inverter;
* Negligible iron losses.  
A delta-connected five-phase PMBLDC motor incorporating the above assumptions is represented in phase variable model as,  
\[\begin{bmatrix}\rm V_{ab}\\ \rm V_{bc}\\ \rm V_{cd}\\ \rm V_{de}\\ \rm V_{ea}\end{bmatrix}=\rm R_{S}\begin{bmatrix}\rm i_{a}-\rm i_{b}\\ \rm i_{b}-\rm i_{c}\\ \rm i_{c}-\rm i_{d}\\ \rm i_{d}-\rm i_{e}\\ \rm i_{e}-\rm i_{a}\end{bmatrix}+\rm L_{S}\frac{d}{dt}\begin{bmatrix}\rm i_{a}- \rm i_{b}\\ \rm i_{b}-\rm i_{c}\\ \rm i_{c}-\rm i_{d}\\ \rm i_{d}-\rm i_{e}\\ \rm i_{e}-\rm i_{a}\end{bmatrix}+\begin{bmatrix}\rm E_{a}-\rm E_{b}\\ \rm E_{b}-\rm E_{c}\\ \rm E_{c}-\rm E_{d}\\ \rm E_{d}-\rm E_{e}\\ \rm E_{e}-\rm E_{a}\end{bmatrix} \tag{1}\]  
where, V, L, M, i, and E, represent the line-to-line voltage, self-inductance, mutual inductance, current, and back emf of the corresponding five phases a, b, c, d, and e, respectively. Considering the phase resistances of each phase are balanced and equal, the effective line-to-line resistance Rs is expressed as Rs = 2R in ohm.  
\[\rm R_{a}=\rm R_{b}=\rm R_{c}=\rm R_{d}=\rm R_{e}=\rm R\;;\rm L_{a}=\rm L_{b}= \rm L_{c}=\rm L_{d}=\rm L_{e}=\rm L\;;\]  
\[\rm M_{ab}=\rm M_{bc}=\rm M_{cd}=\rm M_{de}=\rm M_{ea}=\rm M\;;\]  
\[\rm i_{a}=\rm i_{b}=\rm i_{c}=\rm i_{d}=\rm i_{e}\;;M_{\rm i_{b}}+\rm M_{ic}+M_ {\rm i_{d}}+\rm M_{ie}=-\rm M_{\rm i_{a}} \tag{2}\]  
Figure 1: Technical drawing of the proposed work.  
Figure 2: Five-phase PMBLDC motor drive and its motor’s electrical model. (**a**) Five-leg MOSFET inverter-fed BLDC motor drive. (**b**) Equivalent electrical model of the motor.  
Incorporating Equation (2) into Equation (1), and after simplifying, we obtain the equation in a state space form as (3), and implement it in a MATLAB Simulink environment,  
\[\frac{\mathrm{d}}{\mathrm{dt}}\begin{bmatrix}\mathrm{i_{a}-i_{b}}\\ \mathrm{i_{b}-i_{c}}\\ \mathrm{i_{c}-i_{d}}\\ \mathrm{i_{d}-i_{e}}\\ \mathrm{i_{e}-i_{a}}\end{bmatrix}=\frac{1}{\mathrm{L_{s}}}\begin{bmatrix} \begin{bmatrix}\mathrm{V_{ab}}\\ \mathrm{V_{bc}}\\ \mathrm{V_{cd}}\\ \mathrm{V_{de}}\\ \mathrm{V_{ea}}\end{bmatrix}-\mathrm{R_{S}}\begin{bmatrix}\mathrm{i_{a}-i_{b}} \\ \mathrm{i_{b}-i_{c}}\\ \mathrm{i_{c}-i_{d}}\\ \mathrm{i_{c}-i_{d}}\\ \mathrm{i_{d}-i_{e}}\\ \mathrm{i_{e}-i_{a}}\end{bmatrix}&\begin{bmatrix}\mathrm{E_{a}-E_{b}}\\ \mathrm{E_{b}-E_{c}}\\ \mathrm{E_{c}-E_{d}}\\ \mathrm{E_{d}-E_{e}}\\ \mathrm{E_{e}-E_{a}}\end{bmatrix}\end{bmatrix} \tag{3}\]  
The mechanical part of the motor is the process of electromagnetic torque production, and the equation governing this process is given by,  
\[\mathrm{T_{e}}=\mathrm{T_{L}}+\mathrm{J}\frac{\mathrm{d}\omega}{\mathrm{dt}}+ \mathrm{B}\omega \tag{4}\]  
where TL denotes load torque (Nm), J denotes moment of inertia (Kgm\({}^{2}\)), and B denotes damping coefficient of friction in (Nms/rad). The equations are used to construct the Simulink model of the five-phase PMBLDC motor drive in the MATLAB environment.