# 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  
Pulse Shaping  
The transfer function of the CR-(RC)n shaping amplifier is given as follows, assuming that the time constants of the differentiation and the \(n\) integrations are all the same value \(\tau\):  
\[H(s)=\frac{s\tau}{\left(1+s\tau\right)^{n+1}}. \tag{1}\]  
The impulse response function of the CR-(RC)n shaping amplifier, \(h(t)\) is derived from Equation (1) as:  
\[h(t)=\frac{d}{dt}\bigg{[}\frac{1}{n!}\bigg{(}\frac{t}{\tau}\bigg{)}^{n}e^{- \frac{t}{\tau}}\bigg{]}. \tag{2}\]  
Figure 2: Block diagram of the proposed digital logarithmic power signal generation. When the input signals are assumed with a sampling time interval of \(T\), the \(z\)-transform of the impulse response function of the CR-(RC)\({}^{n}\) shaping amplifier, \(H(z)\) will be  
\[H(z)=\sum_{m=0}^{\infty}h(mT)z^{-m}. \tag{3}\]  
and, the \(z\)-transform of CR-(RC)\({}^{4}\) shaping amplifier with four stages of integration, \(H_{4}(z)\) is approximated as:  
\[H_{4}(z)=\frac{\left\{ \begin{array}{l}z^{4}\alpha T^{3}\Big{(}4-\frac{T}{ \tau}\Big{)}+z^{3}\alpha^{2}T^{4}\Big{(}12-\frac{11T}{\tau}\Big{)}\\ +z^{2}\alpha^{3}T^{3}\Big{(}-12-\frac{11T}{\tau}\Big{)}+z\alpha^{4}T^{3}\Big{(} -4-\frac{T}{\tau}\Big{)}\end{array} \right\}}{24(z-\alpha)^{5}} \tag{4}\]  
where \(\alpha=e^{-\frac{T}{\tau}}\). The \(z\)-transforms for input and output signals of CR-(RC)\({}^{4}\) shaping amplifier, \(X(z)\) and \(Y(z)\), have the relationship:  
\[Y(z)=H_{4}(z)X(z). \tag{5}\]  
Calculating the corresponding difference by taking the inverse \(z\)-transform of Equation (5) [12; 13]:  
\[ \begin{array}{l}y(n)=5\alpha y(n-1)-10\alpha^{2}y(n-2)+10\alpha^{3}y(n-3) \\ -5\alpha^{4}y(n-4)+\alpha^{5}y(n-5)+\frac{\alpha}{24}\Big{(}-\frac{T^{4}}{ \tau}+4T^{3}\Big{)}x(n-1)\\ +\frac{\alpha^{2}}{24}\Big{(}-\frac{11T^{4}}{\tau}+12T^{3}\Big{)}x(n-2)-\frac{ \alpha^{2}}{24}\Big{(}\frac{11T^{4}}{\tau}+12T^{3}\Big{)}x(n-3)\\ -\frac{\alpha^{4}}{24}\Big{(}\frac{T^{4}}{\tau}+4T^{3}\Big{)}x(n-4).\end{array}  \tag{6}\]  
The transfer function of the pole-zero cancellation is:  
\[H_{\text{CR}}(s)=-\frac{R\Big{(}1+sR_{pz}C_{d}\Big{)}}{R_{pz}(sRC+1)} \tag{7}\]  
While the transfer function of the CR-(RC)\({}^{n}\) shaping amplifier with the pole-zero cancellation is  
\[H_{pz}(s)=-\frac{R\Big{(}1+sR_{pz}C_{d}\Big{)}}{R_{pz}(sRC+1)^{n+1}}. \tag{8}\]  
The \(z\)-transforms of input and output signals of the first CR section of the shaping amplifier with the pole-zero cancellation is approximated as:  
\[H_{\text{CR}}(z)=-\beta\frac{1+\frac{2\tau_{\text{pr}}}{T}\frac{z-1}{z+1}}{1+ \frac{2\tau_{\text{RC}}}{T}\frac{z-1}{z+1}}=-\beta\frac{\Big{(}T-2\tau_{pz} \Big{)}+\Big{(}T+2\tau_{pz}\Big{)}z}{(T-2\tau_{RC})+(T+2\tau_{RC})z}. \tag{9}\]  
Calculating the corresponding difference by taking the inverse z-transform of Equation (9) gives  
\[y_{1}(n)=-Ay_{1}(n-1)-Bx(n)-Cx(n-1) \tag{10}\]  
where \(A=\frac{T-2 RC}{T+2 RC},B=\frac{T+2R_{pz}C_{d}}{T+2 RC},C=\frac{T-2R_{pz}C_{d}}{T +2 RC}\). Signal to Noise Discrimination  
There are several digital signal processing techniques to discriminate pulse signals induced by neutrons from those by gamma rays and electric noises, including pulse height discrimination, pulse shape discrimination and discrete Fourier transform method. The pulse height discrimination is set to a discrimination threshold and to generate an output pulse when the amplitude of an input pulse exceeds the threshold. The pulse shape discrimination originates from the different properties of neutron and gamma ray pulse shapes. A ratio of two areas obtained by integration of a pulse in various time intervals is used to discriminate the neutron interaction from the gamma ray interaction. The ratio \(R\) is defined as [14]:  
\[R=A_{2}/A_{1} \tag{11}\]  
where \(A_{1}\) is the total area of the pulse and \(A_{2}\) is the area of the tail part of the pulse. The ratio of the neutron pulse is generally larger than that of the gamma ray pulse.