# Research on the Autonomous Control Technology Used in the Slurry Mixing System of Cementing Units
## 2 Working Principle and Model Establishment of the Slurry Mixing System
### Establishment of the Slurry Mixing System Model  
In order to achieve precise control of cement slurry density, a set of automatic control systems that can timely and accurately adjust and control the size of the electromagnetic proportional valve have become the key components of cementing slurry mixing systems; therefore, it is necessary to establish a mathematical model for electromagnetic proportional valves.  
(1) Proportional amplifier  
The proportional amplifier is a component that provides a specific current to the solenoid valve, and it has a significant influence on the performance of the entire density control system. When the proportional amplifier is operating, it receives a weak control electrical signal and proportionally outputs the current required by the operation of the control solenoid valve. The proportional amplifier is generally a proportional element, and the transfer function is shown in Equation (1).  
\[K_{a}=\frac{I(s)}{U(s)} \tag{1}\]  
where \(I(s)\) represents the input current of the proportional amplifier, A; \(U(s)\) represents the analog voltage that the digital controller converts into, V; and \(K_{a}\) represents the gain of the proportional amplifier, A/V.  
Figure 2: Cementing mixing system density control process.  
The proportional amplifier outputs the current to act on the valve actuator; then, the valve openness is changed under the action of electromagnetic force, and the gas flow of each throttle is regulated. Namely, the input of the solenoid valve is the control current, and the output is the openness of the valve.  
If the electromagnetic actuation force of a solenoid proportional valve is \(F_{em}\), its expression is as follows:  
\[F_{em}=\frac{\left(iN\right)^{2}}{2}\times\frac{dG}{dx_{ev}} \tag{2}\]  
where \(x_{ev}\) is the effective movement of the proportional valve core, m; \(i\) is the current passing through the coil of the proportional valve, A; \(N\) is the number of coils of coil inductance; and \(G\) is the air gap permeability.  
On the control coil of the solenoid proportional valve, the relationship between the input coil voltage u and the current through the coil \(i\) is given by Equation (3):  
\[u=L\frac{di}{dt}+(R_{1}+R_{2})i+K_{si}\frac{dx_{ev}}{dt} \tag{3}\]  
where \(R_{1}\) is the internal resistance of the coil, \(\Omega\); \(R_{2}\) is the internal resistance of the electric controller, \(\Omega\); \(L\) is the coil inductance, H; and \(K_{si}\) is the coefficient of self-inductance electromotive force, H.  
The force on the valve core inside the solenoid proportional valves is balanced, and the displacement motion equation of the valve core can be obtained by using [PERSON]'s second law. The equation is as follows:  
\[m\frac{d^{2}x_{ev}(t)}{dt^{2}}+c\frac{dx_{ev}}{dt}+K_{s}x_{ev}+F_{x_{ev}}=F_{ em} \tag{4}\]  
where \(m\) is the mass of the core, kg; \(c\) is the damping coefficient; \(K_{s}\) is the stiffness of the zero spring, N/mm; and \(F_{xev}\) is the steady-state flow force, N.  
According to the analysis above, ignoring the internal and external leakage and taking the voltage signal \(u\) of the solenoid proportional valves as the input and the linear effective displacement \(x_{ev}\) of the valve core as the output, the transfer function of the solenoid proportional valves can be expressed as:  
\[G_{ux}=\frac{X_{ev}(s)}{U(s)}=\frac{K_{i}}{(Ls+R_{1}+R_{2})\big{(}ms^{2}+cs+K_ {s}+K_{y}\big{)}+K_{si}K_{i}s} \tag{5}\]  
After the proportional amplifier is introduced, the overall transfer function of the system is:  
\[G_{\rm s}=\frac{X_{ev}(s)}{U(s)}=\frac{K_{a}K_{i}}{(Ls+R_{1}+R_{2})\big{(}ms^ {2}+cs+K_{s}+K_{y}\big{)}+K_{si}K_{i}s} \tag{6}\]