## 2 Modeling a Brushed DC Motor
### Brushed DC Motor Modeling Considering Back EMF Vibration  
As previously mentioned, EMF is one of the factors that can cause noise and vibration in electrical machines. This study attempted to predict the noise and vibration using the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element Name** & **Symbol** & **Unit** \\ \hline Input voltage & \(V_{in}(t)\) & [V] \\ Armature resistance & \(R_{a}\) & [\(\Omega\)] \\ Armature inductance & \(L_{a}\) & [H] \\ Armature current & \(i_{a}\) & [A] \\ Back EMF & \(e_{a}(t)\) & [V] \\ Motor armature inertia & \(J\) & [kg\(\cdot\)m\({}^{2}\)] \\ Viscous damping, friction coefficient & \(B\) & [N\(\cdot\)m/rpm ] \\ Motor torque & \(\tau\) & [N\(\cdot\)m ] \\ Motor angular position & \(\theta\) & [rad] \\ Motor angular velocity & \(\omega\) & [rad/s] \\ Motor torque constant & \(k_{\tau}\) & [N\(\cdot\)m/A] \\ Motor back EMF constant & \(k_{b}\) & [V\(\cdot\)s/ rad] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Names and symbols of electrical and mechanical elements.  
Figure 1: Schematic of a simplified equivalent representation of the brushed DC motor’s electromechanical components.  
back EMF of a brushed DC motor. The back EMF refers to the induced voltage that prevents current flow while the motor rotates, which causes voltage to be generated in the direction opposite to the voltage applied from the power source, and the voltage applied to the motor drops. The back EMF can be measured using the voltage generated by the motor when the motor rotates with no load without a power supply. Without load, the back EMF can represent the motor characteristics itself with certain DC and AC components without affecting by input current. The back EMF expressed in Equation (3) is typically modeled as a single value for simplicity but cannot be a constant in real terms. In a brushed DC motor, the stator produces a magnetic flux density \(B\) between the permanent magnets, and the rotor consists of a coil wound \(N\) times around the cross-sectional area \(S\). In a magnetic field with uniform magnetic flux density \(B\), the coil is wound \(N\) times based on the cross-sectional area \(S\), and when the coil rotates at a speed of \(\omega\), a back EMF is generated, as shown below:  
\[e_{a}(t)=NBS\omega sin(\omega t) \tag{5}\]  
In brushed DC motors, the cross-sectional area of the rotor and the number of turns of the coil are constant, and the magnetic flux density of the stator is also constant because it is generated by permanent magnets. Therefore, the back EMF can be summarized as Equation (6) by expressing \(N\cdot B\cdot S\) as a single constant \(k_{b}\).  
\[e_{a}\ (t)=k_{b}\omega sin(\omega t) \tag{6}\]  
In other words, even when rotating at the same speed, the back EMF value changes to harmonics depending on the time \(t\). The sinusoidal characteristics of this back EMF affect the voltage applied to the motor and therefore the current, torque, and angular velocity of the motor as well. This effect causes motor rotational vibration, which leads to motor noise. Using Equation (6), the back EMF, including the vibration component, can be expressed in a block diagram, as shown in Figure 2.