# Filament X-ray Tube Current Control Method Using Indirect Filament Temperature Estimation
## 2 Principle of Operation  
### Thermionic Emission  
Filament temperature can be regulated by understanding the electrical characteristics of the filament as a function of temperature. The filament X-ray tube generates X-rays using thermal electrons emitted into a vacuum from a high-temperature metal that receives energy greater than the work function. The electrons emitted by the heating generate an electron cloud around the filament, and the number of space charges remains constant at a constant filament temperature [12]. Under a constant filament temperature condition, the magnitude of the current density flowing in the vacuum changes owing to the thermal electrons based on the anode voltage. This magnitude can be classified into the space charge-limited regime and temperature-limited regime based on the variation in current density [13]. The equations of [PERSON]'s law and [PERSON]'s law exhibit the correlation between the anode voltage and tube current at a constant filament temperature. Here,  
Figure 1: Schematic of the X-ray image acquisition.  
denotes the thermal electron acceleration distance; \(e_{0}\) indicates the free space permittivity; \(m_{e}\) represents the electron mass; \(e\) denotes the electron charge; \(V\) indicates the anode voltage; \(A_{0}\) represents the Richardson constant of tungsten, which equals 60 cm\({}^{-2}\cdot\)K\({}^{-2}\); \(\varphi\) denotes the work function of tungsten; \(k\) indicates the Boltzmann constant, which equals 1.3806 \(\times 10^{-23}\) m\({}^{2}\cdot\)kg\(\cdot\)s\({}^{-2}\cdot\)K\({}^{-1}\); and \(T\) represents the absolute temperature K.  
\[J=\frac{4e_{0}}{9}\sqrt{\frac{2e}{m_{e}}}\frac{V^{3/2}}{d^{2}} \tag{1}\]  
\[J=A_{0}T^{2}e^{-\varphi/kT} \tag{2}\]  
After determining the anode voltage and tube current values using Equations (1) and (2), the filament temperature is the only variable associated with the tube current. An increase in the temperature of the metal increases the vibrational energy of the metal atoms, which in turn increases the electrical resistance. Table 1 summarizes the electrical resistivity and the change in resistivity rate of tungsten based on the temperature [14; 15]. Herein, \(\rho\) denotes the resistivity of tungsten, and \(R/R_{300K}\) indicates the ratio of the resistivity at the current temperature to that at room temperature.  
Figure 2 illustrates the resistivity ratio of tungsten considering the temperature listed in Table 1. Based on Figure 2, the electrical resistivity can be expressed as a function of temperature using Equation (3).  
\[R_{w}=48\Big{(}1+4.8297\times 10^{-3}T+1.663\times 10^{-6}T^{2}\Big{)}[n\Omega \m  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} \(R/R_{\text{Joux}}\) \\ \(R/R_{\text{Joux}}\) \\ \end{tabular} } & \multicolumn{2}{c}{\
\begin{tabular}{c} **Temperature** \\ **(K)** \\ \end{tabular} } & \multicolumn{2}{c}{\(\rho\) (s\(\Omega\)cm)} & \multicolumn{2}{c}{\(R/R_{\text{Joux}}\)} & \multicolumn{2}{c}{
\begin{tabular}{c} **Temperature** \\ **(K)** \\ \end{tabular} } & \multicolumn{2}{c}{\(\rho\) (s\(\Omega\)cm)} & \multicolumn{2}{c}{\(R/R_{\text{Joux}}\)} & \multicolumn{2}{c}{\(\rho\) (s\(\Omega\)cm)} & \multicolumn{2}{c}{\(\rho\) (s\(\Omega\)cm)} \\ \hline
1.0 & 300 & 5.65 & 5.48 & 1200 & 30.98 & 10.63 & 2100 & 60.06 & 16.29 & 3000 & 92.04 \\
1.43 & 400 & 8.06 & 6.03 & 1300 & 34.08 & 11.24 & 2200 & 63.48 & 16.95 & 3100 & 95.76 \\
1.87 & 500 & 10.56 & 6.58 & 1400 & 37.19 & 11.84 & 2300 & 66.91 & 17.62 & 3200 & 99.54 \\
2