Circuit Model Analysis of General Charge-controlled Memristor Based on Hyperbolic Functions
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摘要: 目前，忆阻器模拟器的研究主要集中在磁控忆阻器，对荷控忆阻器模拟器的研究不多，双曲函数型的荷控忆阻器模拟器也很少涉及。因此，该文提出一种基于双曲函数的通用型荷控忆阻器模拟器。模拟器通过电压-电流的相互转换电路，实现电路中电压和电流信号之间的相互转换，再通过电路中对应的电路模块对产生的信号进行计算，最终得到通用型双曲荷控忆阻器模型。模拟器能够实现双曲正弦、双曲余弦以及双曲正切函数对应的荷控忆阻器模型。通用型双曲函数荷控忆阻器模拟器对应的等效电路，主要由运算放大器、电阻、电容、三极管等基本元件组成。分析模拟器在不同幅值以及不同频率的输入信号下的伏安特性曲线，得出荷控忆阻器模拟器符合记忆元件的基本特性。该文提出的通用型双曲函数荷控忆阻器模型，对忆阻器模型的发展具有一定的参考意义。Abstract: At present, most of the researches on the memristor simulators are flux-controlled, there are few researches on the charge-controlled memristor simulator, and the hyperbolic function simulator is seldom mentioned. Therefore, a general-purpose simulator of charge-controlled memristor based on hyperbolic function is proposed. The simulator realizes the conversion between voltage and current signals in the circuit through the voltage-current mutual conversion circuit, and calculates the generated signals through the corresponding module in the circuit, and finally obtains the universal hyperbolic charge-controlled memristor model. The simulator can realize the charge-controlled memristor corresponding to hyperbolic sine, hyperbolic cosine and hyperbolic tangent function. The equivalent circuit of the general-purpose hyperbolic function charge-controlled memristor simulator is mainly composed of operational amplifier, resistor, capacitor, triode and other basic components. By analyzing the volt-ampere characteristic curves of the simulator at different amplitudes and different frequencies, it is concluded that the simulator conforms to the basic characteristics of memory devices. The model of hyperbolic charge-controlled memristor presented in this paper has a certain reference significance for the development of memristor model.
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表 1 双曲正弦荷控忆阻器数学模型和电路模型
开关
状态u1状态 忆阻数学模型 忆阻器电路模型 S1断开
S2闭合u1$ \ne $0 $M(q) = a \cdot \sinh (b + cq) + {{p} }$ $\begin{gathered} M(q) = \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_8} } }\frac{ {l{R_{39} } } }{ { {R_{38} } } }{R_5} \cdot \sinh \left[\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{16} } } }{u_1} \right.\\ \left. + \frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{15} } } }\rho q\right] + \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_9} } }{R_7} \\ \end{gathered}$ S1闭合
S2闭合u1$ = $0 $M(q) = a \cdot \sinh (cq + d{q^2}) + {{p} }$ $\begin{gathered} M(q) = \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_8} } }\frac{ {l{R_{39} } } }{ { {R_{38} } } }{R_5} \cdot \sinh \left[\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{15} } } }\rho q \right. \\ \left. + \frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{14} } } }{\rho ^2}{q^2}\right] + \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_9} } }{R_7} \\ \end{gathered}$ S1闭合
S2断开u1$ \ne $0 $M(q) = a \cdot \sinh (b + d{q^2}) + {{p} }$ $\begin{gathered} M(q) = \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_8} } }\frac{ {l{R_{39} } } }{ { {R_{38} } } }{R_5} \cdot \sinh \left[\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{16} } } }{u_1} \right. \\ \left. + \frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{14} } } }{\rho ^2}{q^2}\right] + \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_9} } }{R_7} \\ \end{gathered}$ S1闭合
S2闭合u1$ \ne $0 $\begin{gathered} M(q) = a \cdot \sinh (b + cq + d{q^2}) \\ {\text{ } } + {{p} } \\ \end{gathered}$ $\begin{gathered} M(q) = \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_8} } }\frac{ {l{R_{39} } } }{ { {R_{38} } } }{R_5} \cdot \sinh \left[\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{16} } } }{u_1} \right. \\ \left.\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{15} } } }\rho q{\text{ + } }\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{14} } } }{\rho ^2}{q^2}\right] + \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_9} } }{R_7} \\ \end{gathered}$ 
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表 2 双曲余弦荷控忆阻器数学模型和电路模型
开关
状态u1状态 忆阻数学模型 忆阻器电路模型 S1断开
S2闭合u1$ \ne $0 $M(q) = a'' \cdot \cosh (b'' + c''q) + {{p} }''$ $ \begin{gathered} M(q) = \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_8}}}\frac{{g{R_{33}}}}{{{R_{32}}}}{R_5} \cdot \cosh \left[\frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{16}}}}{u_1} \right.\\ \left. + \frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{15}}}}\rho q \right.] + \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_9}}}{R_7} \\ \end{gathered} $ S1闭合
S2闭合u1$ = $0 $\begin{gathered} M(q) = a'' \cdot \cosh (c''q + d''{q^2}) + \\ {{ p} }'' \\ \end{gathered}$ $ \begin{gathered} M(q) = \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_8}}}\frac{{{\text{g}}{R_{33}}}}{{{R_{32}}}}{R_5} \cdot \cosh \left.[\frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{15}}}}\rho q \right. \\ \left. + \frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{14}}}}{\rho ^2}{q^2}\right] + \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_9}}}{R_7} \\ \end{gathered} $ S1闭合
S2断开u1$ \ne $0 $M(q) = a'' \cdot \cosh (b'' + d''{q^2}) + {{p} }''$ $\begin{gathered} M(q) = \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_8} } }\frac{ { {\text{g} }{R_{33} } } }{ { {R_{32} } } }{R_5} \cdot \cosh \left[\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{16} } } }{u_1} \right. \\ \left. + \frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{14} } } }{\rho ^2}{q^2}\right] + \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_9} } }{R_7} \\ \end{gathered}$ S1闭合
S2闭合u1$ \ne $0 $\begin{gathered} M(q) = a'' \cdot \cosh (b'' + c''q + \\ {\text{ } }d''{q^2}) + {{p} }'' \\ \end{gathered}$ $ \begin{gathered} M(q) = \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_8}}}\frac{{{\text{g}}{R_{33}}}}{{{R_{32}}}}{R_5} \cdot \cosh \left[\frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{16}}}}{u_1} \right.\\ \left.\frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{15}}}}\rho q{\text{ + }}\frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{14}}}}{\rho ^2}{q^2}\right] + \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_9}}}{R_7} \\ \end{gathered} $
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表 3 双曲正切荷控忆阻器数学模型和电路模型
开关
状态u1状态 忆阻数学模型 忆阻器电路模型 S1断开
S2闭合u1$ \ne $0 $ M(q) = a''' \cdot \tanh (b''' + c'''q) $ $ \begin{gathered} M(q) = \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_8}}}\frac{{l{R_{32}}{R_{39}}}}{{{\text{g}}{R_{33}}{R_{38}}}}z{R_5} \cdot \tanh \left[\frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{16}}}}{u_1} \right. \\ \left. + \frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{15}}}}\rho q \right] \\ \end{gathered} $ S1闭合
S2闭合u1$ = $0 $ M(q) = a''' \cdot \tanh (c'''q + d'''{q^2}) $ $ \begin{gathered} M(q) = \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_8}}}\frac{{l{R_{32}}{R_{39}}}}{{{\text{g}}{R_{33}}{R_{38}}}}z{R_5} \cdot \tanh \left[\frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{15}}}}\rho q \right.\\ \left. + \frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{14}}}}{\rho ^2}{q^2}\right] \\ \end{gathered} $ S1闭合
S2断开u1$ \ne $0 $ M(q) = a''' \cdot \tanh (b''' + d'''{q^2}) $ $ \begin{gathered} M(q) = \frac{{{R_{12}}}}{{{R_{11}}}}\frac{{{R_{10}}}}{{{R_8}}}\frac{{l{R_{32}}{R_{39}}}}{{{\text{g}}{R_{33}}{R_{38}}}}z{R_5} \cdot \tanh \left[\frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{16}}}}{u_1} \right. \\ \left. + \frac{{{R_{24}}}}{{\left( {{R_{23}} + {R_{24}}} \right){U_T}}}\frac{{{R_{19}}}}{{{R_{18}}}}\frac{{{R_{17}}}}{{{R_{14}}}}{\rho ^2}{q^2}\right] \\ \end{gathered} $ S1闭合
S2闭合u1$ \ne $0 $ \begin{gathered} M(q) = a''' \cdot \tanh (b''' + c'''q + \\ {\text{ }}d'''{q^2}) \\ \end{gathered} $ $\begin{gathered} M(q) = \frac{ { {R_{12} } } }{ { {R_{11} } } }\frac{ { {R_{10} } } }{ { {R_8} } }\frac{ {l{R_{32} }{R_{39} } } }{ { {\text{g} }{R_{33} }{R_{38} } } }z{R_5} \cdot \tanh \left[\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{16} } } }{u_1} \right. \\ \left.\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{15} } } }\rho q{\text{ + } }\frac{ { {R_{24} } } }{ {\left( { {R_{23} } + {R_{24} } } \right){U_T} } }\frac{ { {R_{19} } } }{ { {R_{18} } } }\frac{ { {R_{17} } } }{ { {R_{14} } } }{\rho ^2}{q^2}\right] \\ \end{gathered}$
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