JART-TUD VCM memristor model

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Memristor Current Equation

Simplified memristor current ($I_{\rm M}$) expression

$$\begin{aligned}[b] I_{\rm M}\left(N_{\rm d},V_{\rm M}, d_{\rm r}, d_{\rm l}\right) = &\textcolor{green}{{p}_{\rm1}(V_{\rm M}, d_{\rm r}, d_{\rm l})}\left(\textcolor{green}{{p}_{\rm2}(V_{\rm M}, d_{\rm r}, d_{\rm l})}\left( e^{\frac{\left(\ln\frac{N_{\rm d}}{N_{\rm d,L}}-\textcolor{green}{{p}_{\rm3}(V_{\rm M}, d_{\rm r}, d_{\rm l})}\right)}{\textcolor{green}{{p}_{\rm4}(V_{\rm M}, d_{\rm r}, d_{\rm l})}}}-1\right)+\left(\ln\frac{N_{\rm d}}{N_{\rm d,L}}-\textcolor{green}{{p}_{\rm3}(V_{\rm M}, d_{\rm r}, d_{\rm l})}\right)\right) \\ &+ \frac{\textcolor{green}{{p}_{\rm5}(V_{\rm M}, d_{\rm r}, d_{\rm l})}}{\left(\textcolor{green}{{p}_{\rm6}(V_{\rm M},d_r,d_l)}+\textcolor{green}{{p}_{\rm7}(V_{\rm M},d_r,d_l)}\cdot\left(\textcolor{green}{{p}_{\rm8}(V_{\rm M},d_r,d_l)}e^{\ln\left(\frac{N_{\rm d}}{N_{\rm d,L}}\right)-\textcolor{green}{{p}_{\rm9}(V_{\rm M},d_r,d_l)}}\right)^{-\textcolor{green}{{p}_{\rm10}(V_{\rm M},d_r,d_l)}}\right)^{1/\textcolor{green}{{p}_{\rm11}(V_{\rm M},d_r,d_l)}}} \end{aligned}$$

$p_i$ functions

Applied Voltage V_M>0

$p_1(V_{\rm M}, d_{\rm r}, d_{\rm l}) = 0$

$p_2(V_{\rm M}, d_{\rm r}, d_{\rm l}) = 0$

$p_3(V_{\rm M}, d_{\rm r}, d_{\rm l}) = 0$

$p_4(V_{\rm M}, d_{\rm r}, d_{\rm l}) = 1$

$p_5(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{5,0|f}(d_{\rm r}, d_{\rm l})\color{black} - \color{blue}p_{5,1|f}(d_{\rm r}, d_{\rm l})\color{black} * e^{-\color{blue}p_{5,2|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}}$

$p_6(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{6,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \color{blue}p_{6,1|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}$

$p_7(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{7,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \color{blue}p_{7,1|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M} + \color{blue}p_{7,2|f}(d_{\rm r}, d_{\rm l})\color{black} * e^{-\color{blue}p_{7,3|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}}$

$p_8(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{8,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \color{blue}p_{8,1|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}$

$p_9(V_{\rm M}, d_{\rm r}, d_{\rm l}) = 0$

$p_{10}(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{10,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \color{blue}p_{10,1|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M} + \color{blue}p_{10,2|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}^2$

$p_{11}(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{11,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \color{blue}p_{11,1|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M} + \color{blue}p_{11,2|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}^2$

Applied Voltage V_M<0

$p_1(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{1,0|f}(d_{\rm r}, d_{\rm l})\color{black}\frac{\color{blue}p_{1,1|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M} + \color{blue}p_{1,2|f}(d_{\rm r}, d_{\rm l})\color{black} * V_{\rm M}^2}{1 + \color{blue}p_{1,3|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M} + \color{blue}p_{1,4|f}(d_{\rm r}, d_{\rm l})\color{black} * V_{\rm M}^2}$

$p_2(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{2,0|f}(d_{\rm r}, d_{\rm l})\color{black}$

$p_3(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{3,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \color{blue}p_{3,1|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}$

$p_4(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{4,0|f}(d_{\rm r}, d_{\rm l})\color{black} - \color{blue}p_{4,1|f}(d_{\rm r}, d_{\rm l})\color{black} * e^{-\color{blue}p_{4,2|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}}$

$p_5(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{5,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \color{blue}p_{5,1|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M} + \color{blue}p_{5,2|f}(d_{\rm r}, d_{\rm l})\color{black}*V_{\rm M}^2$

$p_6(V_{\rm M}, d_{\rm r}, d_{\rm l}) = 1$

$p_7(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{7,0|f}(d_{\rm r}, d_{\rm l})\color{black}$

$p_8(V_{\rm M}, d_{\rm r}, d_{\rm l}) = 1$

$p_9(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{9,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \frac{\color{blue}p_{9,1|f}(d_{\rm r}, d_{\rm l})\color{black} - \color{blue}p_{9,0|f}(d_{\rm r}, d_{\rm l})\color{black}}{1 + e^{\frac{V_{\rm M}-\color{blue}p_{9,2|f}(d_{\rm r}, d_{\rm l})\color{black}}{\color{blue}p_{9,3|f}(d_{\rm r}, d_{\rm l})\color{black}}}}$

$p_{10}(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{10,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \frac{\color{blue}p_{10,1|f}(d_{\rm r}, d_{\rm l})\color{black} - \color{blue}p_{10,0|f}(d_{\rm r}, d_{\rm l})\color{black}}{1 + e^{\frac{V_{\rm M}-\color{blue}p_{10,2|f}(d_{\rm r}, d_{\rm l})\color{black}}{\color{blue}p_{10,3|f}(d_{\rm r}, d_{\rm l})\color{black}}}}$

$p_{11}(V_{\rm M}, d_{\rm r}, d_{\rm l}) = \color{blue}p_{11,0|f}(d_{\rm r}, d_{\rm l})\color{black} + \frac{\color{blue}p_{11,1|f}(d_{\rm r}, d_{\rm l})\color{black} - \color{blue}p_{11,0|f}(d_{\rm r}, d_{\rm l})\color{black}}{1 + e^{\frac{V_{\rm M}-\color{blue}p_{11,2|f}(d_{\rm r}, d_{\rm l})\color{black}}{\color{blue}p_{11,3|f}(d_{\rm r}, d_{\rm l})\color{black}}}}$

$p_{i,j|f}$ functions

Applied Voltage V_M>0

$p_{5,1|f}(d_{\rm r}, d_{\rm l}) = p_{5,0|f}(d_{\rm r}, d_{\rm l})$

$p_{5,1|f}(d_{\rm r}, d_{\rm l}) = p_{5,1} + D_{p_{5,1,r}}*d_{\rm r} + D_{p_{5,1,l}}*d_{\rm l}$

$p_{5,2|f}(d_{\rm r}, d_{\rm l}) = p_{5,2}$

$p_{6,0|f}(d_{\rm r}, d_{\rm l}) = p_{6,0} + D_{p_{6,0,r}}*d_{\rm r}$

$p_{6,1|f}(d_{\rm r}, d_{\rm l}) = p_{6,1}$

$p_{7,0|f}(d_{\rm r}, d_{\rm l}) = p_{7,0} + D_{p_{7,0,r}}*d_{\rm r} + D_{p_{7,0,r2}}*d_{\rm r}^2 + D_{p_{7,0,l}}*d_{\rm l}$

$p_{7,1|f}(d_{\rm r}, d_{\rm l}) = p_{7,1}$

$p_{7,2|f}(d_{\rm r}, d_{\rm l}) = p_{7,2} + D_{p_{7,2,r}}*d_{\rm r} + D_{p_{7,2,r2}}*d_{\rm r}^2 + (D_{p_{7,2,l}} + D_{p_{7,2,l,r}}*d_{\rm r} + D_{p_{7,2,l,r2}}*d_{\rm r}^2) * d_{\rm l}$

$p_{7,3|f}(d_{\rm r}, d_{\rm l}) = p_{7,3}$

$p_{8,0|f}(d_{\rm r}, d_{\rm l}) = p_{8,0} + D_{p_{8,0,l}}*d_{\rm l}$

$p_{8,1|f}(d_{\rm r}, d_{\rm l}) = p_{8,1}$

$p_{10,0|f}(d_{\rm r}, d_{\rm l}) = p_{10,0} + D_{p_{10,0,l}}*d_{\rm l}$

$p_{10,1|f}(d_{\rm r}, d_{\rm l}) = p_{10,1}$

$p_{10,2|f}(d_{\rm r}, d_{\rm l}) = p_{10,2}$

$p_{11,0|f}(d_{\rm r}, d_{\rm l}) = p_{11,0} + D_{p_{11,0,l}}*d_{\rm l}$

$p_{11,1|f}(d_{\rm r}, d_{\rm l}) = p_{11,1}$

$p_{11,2|f}(d_{\rm r}, d_{\rm l}) = p_{11,2}$

Applied Voltage V_M<0

$p_{1,0|f}(d_{\rm r}, d_{\rm l}) = p_{1,0} + D_{p_{1,0,r}}*d_{\rm r} + D_{p_{1,0,l}}*d_{\rm l}$

$p_{1,1|f}(d_{\rm r}, d_{\rm l}) = p_{1,1} + D_{p_{1,1,r}}*d_{\rm r} + D_{p_{1,1,l}}*d_{\rm l}$

$p_{1,2|f}(d_{\rm r}, d_{\rm l}) = p_{1,2} + D_{p_{1,2,r}}*d_{\rm r} + D_{p_{1,2,l}}*d_{\rm l}$

$p_{1,3|f}(d_{\rm r}, d_{\rm l}) = p_{1,3} + D_{p_{1,3,r}}*d_{\rm r} + D_{p_{1,3,l}}*d_{\rm l}$

$p_{1,4|f}(d_{\rm r}, d_{\rm l}) = p_{1,4} + D_{p_{1,4,r}}*d_{\rm r} + D_{p_{1,4,l}}*d_{\rm l}$

$p_{2,0|f}(d_{\rm r}, d_{\rm l}) = p_{2,0}$

$p_{3,0|f}(d_{\rm r}, d_{\rm l}) = p_{3,0} + D_{p_{3,0,r}}*d_{\rm r} + D_{p_{3,0,l}}*d_{\rm l}$

$p_{3,1|f}(d_{\rm r}, d_{\rm l}) = p_{3,1} + D_{p_{3,1,r}}*d_{\rm r} + D_{p_{3,1,l}}*d_{\rm l}$

$p_{4,0|f}(d_{\rm r}, d_{\rm l}) = p_{4,0}$

$p_{4,1|f}(d_{\rm r}, d_{\rm l}) = p_{4,1} + D_{p_{4,1,r}}*d_{\rm r} + D_{p_{4,1,l}}*d_{\rm l}$

$p_{4,2|f}(d_{\rm r}, d_{\rm l}) = p_{4,2}$

$p_{5,0|f}(d_{\rm r}, d_{\rm l}) = 0$

$p_{5,1|f}(d_{\rm r}, d_{\rm l}) = p_{5,1} + D_{p_{5,1,r}}*d_{\rm r} + D_{p_{5,1,l}}*d_{\rm l}$

$p_{5,2|f}(d_{\rm r}, d_{\rm l}) = p_{5,2} + D_{p_{5,2,r}}*d_{\rm r} + D_{p_{5,2,r2}}*d_{\rm r}^2$

$p_{7,0|f}(d_{\rm r}, d_{\rm l}) = p_{7,0} + D_{p_{7,0,r}}*d_{\rm r} + D_{p_{7,0,l}}*d_{\rm l}$

$p_{9,0|f}(d_{\rm r}, d_{\rm l}) = p_{9,0} + D_{p_{9,0,r}}*d_{\rm r} + D_{p_{9,0,l}}*d_{\rm l}$

$p_{9,1|f}(d_{\rm r}, d_{\rm l}) = p_{9,1}$

$p_{9,2|f}(d_{\rm r}, d_{\rm l}) = p_{9,2}$

$p_{9,3|f}(d_{\rm r}, d_{\rm l}) = p_{9,3} + D_{p_{9,3,r}}*d_{\rm r}$

$p_{10,0|f}(d_{\rm r}, d_{\rm l}) = p_{10,0}$

$p_{10,1|f}(d_{\rm r}, d_{\rm l}) = p_{10,1} + D_{p_{10,1,r}}*d_{\rm r}$

$p_{10,2|f}(d_{\rm r}, d_{\rm l}) = p_{10,2}$

$p_{10,3|f}(d_{\rm r}, d_{\rm l}) = p_{10,3}$

$p_{11,0|f}(d_{\rm r}, d_{\rm l}) = p_{11,0} + D_{p_{11,0,r}}*d_{\rm r}$

$p_{11,1|f}(d_{\rm r}, d_{\rm l}) = p_{11,1}$

$p_{11,2|f}(d_{\rm r}, d_{\rm l}) = p_{11,2}$

$p_{11,3|f}(d_{\rm r}, d_{\rm l}) = p_{11,3}$

Fitting parameter values

Applied Voltage V_M>0

Parameter	Vm	d_r	d_l
p1	0
p2	0
p3	0
p4	1
p5	p5,1=1.3769e-03 p5,2=8.1819e-02	Dp5,1,r=2.3087e-04	Dp5,1,l=1.3293e-07
p6	p6,0=1.9687e-01 p6,1=-2.1833e-02	Dp6,0,r=2.6129e-02
p7	p7,0=-9.7606e+01 p7,1=7.8250e+00 p7,2=9.9296e+01 p7,3=7.1092e-02	Dp7,0,r=-7.4338e-01 Dp7,0,r2=1.1713e-01 Dp7,2,r=6.9547e-01 Dp7,2,r2=-1.3724e-01 Dp7,2,l,r=-9.0456e-03 Dp7,2,l,r2=-1.2221e-03	Dp7,0,l=2.4377e+00 Dp7,2,l=-2.3728e+00
p8	p8,0=1.1713e-01 p8,1=8.1370e-02		Dp8,0,l=-3.8320e-03
p9	0
p10	p10,0=9.7733e-01 p10,1=3.5214e-02 p10,2=1.2856e-02		Dp10,0,l=5.9623e-05
p11	p11,0=9.4207e-01 p11,1=3.8953e-02 p11,2=2.3436e-02		Dp11,0,l=-6.7239e-04

Applied Voltage V_M<0

Parameter	Vm	d_r	d_l
p1	p1,0=1.1830e+00 p1,1=-2.7034e-03 p1,2=-4.5379e-06 p1,3=9.9115e-01 p1,4=4.4093e-01	Dp1,0,r=-6.2246e-02 Dp1,1,r=-1.8077e-04 Dp1,2,r=1.7313e-04 Dp1,3,r=5.7155e-03 Dp1,4,r=-9.7198e-04	Dp1,0,l=1.1419e-01 Dp1,1,l=-2.0831e-04 Dp1,2,l=-8.9677e-05 Dp1,3,l=-2.3237e-02 Dp1,4,l=-1.8507e-03
p2	-2.5955e+03
p3	p3,0=6.8845e+00 p3,1=-5.8995e-01	Dp3,0,r=1.2536e-01 Dp3,1,r=6.5498e-02	Dp3,0,l=2.5983e-01 Dp3,1,l=8.5666e-02
p4	p4,0=2.5890e+03 p4,1=-2.9537e+00 p4,2=-5.4031e-01	Dp4,1,r=8.2522e-02	Dp4,1,l=-7.2255e-02
p5	p5,0=0 p5,1=6.4705e-04 p5,2=5.1529e-05	Dp5,1,r=1.5169e-05 Dp5,2,r=6.7042e-07 Dp5,2,r2=1.0756e-06	Dp5,1,l=1.3260e-06
p6	1
p7	1.1708e-01	Dp7,0,r=4.8662e-04	Dp7,0,l=3.7351e-03
p8	1
p9	p9,0=3.9052e+00 p9,1=9.6130e+00 p9,2=-4.5637e-01 p9,3=1.4310e+00	Dp9,0,r=-5.4723e-01 Dp9,3,r=3.6000e-01	Dp9,0,l=3.6802e-02
p10	p10,0=4.6925e-01 p10,1=3.4731e+00 p10,2=-1.1871e+00 p10,3=5.6947e-01	Dp10,1,r=1.1444e-02
p11	p11,0=1.0667e+01 p11,1=1.2812e-01 p11,2=7.4414e-01 p11,3=4.2381e-01	Dp11,0,r=3.6290e-01