## 3 Theoretical Analysis of the CTA Circuit  
A constant temperature anemometer is defined by its general differential equation. Therefore, the energy stored in a wire of length \(l\) and diameter \(d\) can be written as:  
\[\frac{dQ_{s}}{dt}=\frac{dQ_{c}}{dt} \tag{1}\]  
where \(Q_{s}\) is the energy stored in the hot wire, \(Q_{c}=R_{w}l^{2}\) is the energy provided by the Joule effect and \(Q_{fc}\) is the energy transferred from the wire to the outside. By injecting a current \(l\), the wire is brought to a temperature higher than the fluid temperature. The main source of dissipation in the hot wire is the convection force where \(h\) is the heat transfer coefficient. It is related to the Nusselt number, which can be written as:  
\[N_{h}=\frac{hd}{k_{f}} \tag{2}\]  
where \(k_{f}\) is the thermal conductivity of the surrounding fluid. By replacing \(h\) by its expression as a function of the Nusselt number and by considering a wire of finite length \(l\), the formula of the heat transfer by convection force can be written as:  
\[Q_{fc}=\pi lk_{f}(T_{w}-T_{0})Nu \tag{3}\]  
The Nusselt number \(Nu\) expresses the efficiency of heat transfer by convection. The linear dependence between the resistance of the wire and its temperature can be written as:  
\[R_{w}l^{2}=\pi lk_{f}(T_{w}-T_{0})Nu \tag{4}\]  
where \(R_{w}\) is the operating resistance of the hot wire. The temperature variation of the CTA system can be written as:  
\[T_{w}-T_{0}=\frac{R_{w}-R_{0}}{\alpha_{0}R_{0}} \tag{5}\]  
where \(R_{0}\) is the resistance of the wire at room temperature and \(\alpha_{0}\) is the temperature coefficient of the material used at room temperature. Adding Equations (4) and (5) we will have  
\[R_{w}l^{2}=\pi lk_{f}\bigg{(}\frac{R_{w}-R_{0}}{\alpha_{0}R_{0}}\bigg{)}Nu \tag{6}\]  
Figure 2: SEM microphotograph of the fabricated sensor with a magnified image of the sensitive section. From [6], it is demonstrated that the Nussel number \(Nu\) can be written as:  
\[Nu=0.42\text{Pr}^{0.2}+0.57\text{Pr}^{0.33}\text{Re}^{0.5} \tag{7}\]  
where Pr is the Prandtl number and Re is the Reynolds number. Using Equation (7) in Equation (6), the linear dependence between the resistance of the wire and its temperature can be written as:  
\[\frac{R_{w}l^{2}}{(R_{w}-R_{0})}=A+BV^{0.5} \tag{8}\]  
with  
\[A=0.42\frac{\pi k_{f}l}{\alpha_{0}R_{0}}\text{Pr}^{0.2} \tag{9}\]  
and  
\[N_{h}=\frac{hd}{k_{f}} \tag{2}\]0  
where \(\rho\) is the density of the fluid, \(\mu\) is the dynamic viscosity of the fluid and \(V\) is the average speed of the fluid. Therefore, Equation (8) represents the fundamental relationship of the constant temperature hot-wire anemometer. The CTA circuit uses a regulation loop. It is composed of two opamps as shown in \[N_{h}=\frac{hd}{k_{f}} \tag{2}\]1 The first opamp has a time constant of M1 and the second opamp has a time constant of M2. Therefore, this technique allows having a second order system. As a result, the transfer function of the complete system can be written as:  
\[N_{h}=\frac{hd}{k_{f}} \tag{2}\]2  
where \(M^{\prime}\) is the first order time constant of the differential equation of the control loop with \(M^{\prime}=M1+M2\), \(M^{\prime}\) is the second order time constant of the differential equation of the control loop with \(M^{\prime\prime}=M1\times M2\) and \(G\) is the total gain of the regulation loop. The third order time constant of the constant temperature hot-wire anemometer equation is an invariable intrinsic parameter which depends on the flow velocity, the superheat coefficient, and the time constant of the regulation loop [34]. This parameter defines the general cutoff frequency of the anemometer. The system response to a square wave has been analyzed [35]. To obtain a better cut-off frequency, the anemometer must be adjusted so as to have an index response with 15% overshoot.