## 2 Methodology  
### Cooling Fan System Dynamics  
An industrial cooling fan system typically comprises three essential components: a driving circuit, a rotary mechanism, and fan blades. Figure 1 briefly illustrates a cooling fan system driven by a DC motor. The driving circuit of the cooling fan system can be briefly represented by an electric circuit with a resistor and an inductor, and its dynamics can be described as  
\[L\frac{di}{dt}+iR+K_{e}\omega=V_{in} \tag{1}\]  
where \(L\) is the inductance, \(R\) is the resistance, \(K_{e}\) is the back emf constant, \(V_{in}\) is the input voltage, \(i\) is the current, and \(\omega\) is the rotor speed. For the motor mechanism, the dynamics can be written as  
\[J_{m}\frac{d\omega}{dt}=K_{t}i-B_{m}\omega-T_{L}. \tag{2}\]  
where \(J_{m}\) is the motor inertia, \(K_{t}\) is the torque constant, \(B_{m}\) is the viscous constant, and \(T_{L}\) is the external load. Based on the law of energy conservation, where electrical power is equal to mechanical power, it can be expressed as  
\[(K_{e}\omega)\cdot i=(K_{t}i)\cdot\omega \tag{3}\]  
Equation (3) shows it is true only when the constants \(K_{e}\) and \(K_{t}\) have the same value, which can be defined as  
\[K_{e}=K_{t}\equiv K \tag{4}\]  
Substituting (4) to (1) and (2), the governing equation of the cooling fan system can be represented as  
\[\left\{\begin{array}{c}L\frac{di}{dt}=-Ri-K\omega+V_{in}\\ J_{m}\frac{d\omega}{dt}=Ki-B_{m}\omega-T_{L}\end{array}\right. \tag{5}\]  
The dynamics of the cooling fan system can be represented as a 2 nd-order differential equation. Given that the response of the electrical dynamics of a cooling fan is much faster than its mechanical dynamics, the transient behavior of the electrical dynamics can be neglected. Therefore, the governing equation in (1) can be rewritten as  
\[\begin{array}{c}iR+K\omega\approx V_{in}\\ \Rightarrow i\approx\frac{V_{in}-K\omega}{R}\end{array} \tag{6}\]  
Furthermore, according to aerodynamics, the external force on the fan blade mainly comes from the drag force, which is proportional to the square of the rotor speed. Therefore, the external load can be approximated as  
\[T_{L}\approx C_{d}\omega^{2} \tag{7}\]  
Figure 1: Illustration of a cooling fan driving system [13].  
where \(C_{d}\) represents the lumped aerodynamic drag coefficient. The details of the derivation can be seen in [5]. By substituting the result of (6) and (7) in the governing Equation (5), it can be expressed as  
\[\Rightarrow\frac{J_{m}R}{K}\frac{d\omega}{dt}=V_{in}-\left(K+\frac{B_{m}R}{K} \right)\omega-\frac{C_{d}R}{K}\omega^{2} \tag{8}\]  
The equivalent coefficients can be defined as  
\[J=\frac{J_{m}R}{K},\alpha=K+\frac{B_{m}R}{K},C_{D}=\frac{C_{d}R}{K} \tag{9}\]  
Equation (8) can be expressed as  
\[J\dot{\omega}(t)+\alpha\omega(t)+C_{D}\omega^{2}(t)=\tau(t) \tag{10}\]  
After rearrangement, Equation (5) can be formed as a 1 st-order dynamical differential equation. The symbol \(\tau\) is denoted as the applied torque generated by the pulse-width modulator (PWM), which is proportional to the input voltage of the driving circuit; the equivalent coefficients \(J\), \(\alpha\), \(C_{D}\) can be regarded as the lumped moment of inertia, viscous coefficient, and drag coefficient, respectively. After a few steps of simplification, the coefficients of the physical model (10) can be derived by measuring the rotor speed and applied torque, both of which are observable.