# 直流内部电感解析式DC Internal Inductance Analysis Formula

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In the super-large-scale integration, the influence of interconnecting wire inductance effect on the circuit performance is increasingly obvious as the clock frequency increases, the signal rise time decreases and the copper technology is adopted. Both the moment method and high-order basis function method are the numerical methods with the advantage of accurate computing result and disadvantage of low computing efficiency. Therefore, the paper proposes a analysis formula of the dc internal and the formula can improve the computing efficiency.

1. 引言

${L}_{in}=\frac{{\Psi }_{in}}{I}$${L}_{ext}=\frac{{\Psi }_{ext}}{I}$ (1)

$L={L}_{in}+{L}_{ext}$ (2)

${L}_{dc}={L}_{in}+{L}_{ext}$ (3)

Figure 1. Equivalent circuit for skin-effect

${L}_{hf}={L}_{ext}$${L}_{in}=0$ (4)

${L}_{hf}$ 为高频时电感值。该式表明外部电感等于高频率时的电感值。

${L}_{in}={L}_{dc}-{L}_{hf}$ (5)

${L}_{in}=\frac{{\mu }_{0}}{8\text{π}}=5×{10}^{-8}$ (6)

${L}_{in}={10}^{-7}\left[0.3+0.28{\text{e}}^{-0.14\frac{w}{t}}\right]$ (7)

2. 直流内部电感的数值计算结果

Figure 2. The direct inductance of different line

(a) (b) (c) (d)

Figure 3. The inductance of different line

${f}_{high}=1028075\cdot {f}_{b}=1028075\cdot \frac{{\omega }_{b}}{2\text{π}}$ (8)

$\text{error}=100×\left({L}_{in}^{{t}_{x}}-{L}_{in}^{{t}_{0}}\right)/{L}_{in}^{{t}_{0}}$ (9)

3. 直流内部电感计算公式

Figure 4. The internal inductance of different line

Figure 5. The relative error of internal inductance

${L}_{in}^{r}=\frac{1}{a+b{x}^{-1}-c{x}^{-2}+d{x}^{-3}}$ (10)

$b=8.0288×{10}^{5}$$c=-1.6278×{10}^{4}$$d=134.8696$

Figure 6. The error between Formula (10) and MOM

Figure 7. The error of the Formula (13)

$\left({x}_{0},{y}_{0}\right),\cdots ,\left({x}_{k},{y}_{k}\right)$

$L\left(x\right)=\underset{j=0}{\overset{k}{\sum }}{y}_{j}{l}_{j}\left(x\right)$ (11)

${l}_{j}\left(x\right)=\underset{i=0,i\ne j}{\overset{k}{\prod }}\frac{x-{x}_{i}}{{x}_{j}-{x}_{i}}=\frac{x-{x}_{0}}{{x}_{j}-{x}_{0}}\cdots \frac{x-{x}_{j-1}}{{x}_{j}-{x}_{j-1}}\frac{x-{x}_{j+1}}{{x}_{j}-{x}_{j+1}}\cdots \frac{x-{x}_{k}}{{x}_{j}-{x}_{k}}$ (12)

${L}_{in}^{l}={h}_{0}-{h}_{1}x+{h}_{2}{x}^{2}-{h}_{3}{x}^{3}+{h}_{4}{x}^{4}-{h}_{5}{x}^{5}+{h}_{6}{x}^{6}$ (13)

4. 结论

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