# Digitalization of the Ex-Core Neutron Flux Monitoring System for APR1400 Nuclear Power Plant
## 2 Design of the Digital ENFMS for APR1400
### Logarithmic Power Signal Generation
#### Algorithms for the Digital Signal Processing
The discrimination point of \(R\) to classify the neutrons and gamma ray interactions can be chosen at the cross point of the neutron and the gamma ray curve. The pulse shape discrimination can also be done by evaluating the similarity of how close the patterns of two different pulse shapes are when each of the pulses is represented by a vector. The similarity \(S(X,Y)\) is defined as [15]  
\[S(X,Y)=\frac{X\cdot Y}{|X||Y|}=\cos\theta \tag{12}\]  
where \(X\) is a pattern vector to be identified and \(Y\) is an appropriate reference or discriminant vector, \(X\cdot Y\) is the scalar product, \(|X|\) and \(|Y|\) are norms of the vectors, and \(\theta\) is the angle between the two vectors. The angle \(\theta\) is used as an index of the pulse shape discrimination. In the digital system, a pulse shape can be represented by the vector:  
\[x=(x_{1},x_{2},x_{3},\dots,x_{n}) \tag{13}\]  
where \(n\) is the number of elements of the vector \(X\) and the elements can be obtained directly from the input of the signal-to-noise discrimination function. The typical profile of the pulse shape can be expressed as  
\[I(t)=I_{0}\Big{(}e^{-t/\tau_{d}}-e^{-t/\tau_{r}}\Big{)} \tag{14}\]  
where \(\tau_{r}\) is the rise time and \(\tau_{d}\) is the decay time. While taking a pulse shape from the measured data as the discriminant vector, \(S(X,Y)\) for each pulse signal is calculated through Equation (12) and, if a neutron pulse shape is used as the discriminant vector, then the similarity index distribution can be obtained as given in Figure 3.  
The figure of merit, \((FOM)\), which is the quantity to represent the discrimination power, is defined as:  
\[FOM=\frac{T_{n-\gamma}}{W_{n}+W_{\gamma}} \tag{15}\]  
where \(T_{n-\gamma}\), \(W_{n}\) and \(W_{\gamma}\) are the distance between the neutron and gamma ray peaks, the full width at half maximum (FWHM) of the gamma ray interactions and the FWHM of neutron interactions, respectively. The discrimination threshold of \(S(X,Y)\) to classify the neutrons and gamma ray interactions can be chosen at the cross point of the neutron and the gamma ray curve, which is at around 0.5 (Figure 3). \[H_{pz}(s)=-\frac{R\Big{(}1+sR_{pz}C_{d}\Big{)}}{R_{pz}(sRC+1)^{n+1}}. \tag{8}\]0 The similarity index distribution (Sample). The rise time method and the discrete Fourier transform method can be applied as a pulse shape discrimination technique in the mixed neutron and gamma ray fields [15; 16]. Mean Squared Voltage Calculation  
The MSV signal, \(S_{MSV}\) and \(C_{norm}\) relationship is given as:  
\[H_{pz}(s)=-\frac{R\Big{(}1+sR_{pz}C_{d}\Big{)}}{R_{pz}(sRC+1)^{n+1}}. \tag{8}\]1  
Logarithmic Power Signal Generation  
The logarithmic reactor power signal, \(S_{LogPower}\), is generated by combining the pulse count signal and the MSV signal, which are then adjusted by a reactor power calibration factor \((C_{RP})\). The key parameters for the calibration of the logarithmic reactor power signal are the start and end points of the overlapping region between the pulse count signal and the MSV signals \((S_{start},S_{end})\), and \(C_{RP}\). The overlapping region of the pulse signal \(\left(S_{pulse}\right)\) and \(S_{MSV}\) is characterized by \(S_{start}\) and \(S_{end}\), and is used for signal selection and interpolation as follows [1]:  
\[S_{LP}=\left| \begin{array}{lll}S_{pulse},&for&S_{pulse}<S_{start}\\ \eta S_{pulse}+(1-\eta)S_{MSV},&for&S_{start}\leq S_{pulse}\leq S_{end}\\ S_{MSV},&for&S_{pulse}>S_{end}\\ \\ S_{LogPower}=C_{RP}\times S_{LP}\\ \end{array} \right. \tag{17}\]  
where \(\eta=\left(S_{end}-S_{pulse}\right)/\left(S_{end}-S_{start}\right)\), and \(C_{RF}\) is the calibration factor.