${q}_{1t}={\displaystyle {\int}_{0}^{{S}_{1t}}\text{d}{q}_{1}}={\displaystyle {\int}_{0}^{{S}_{1t}}{\sigma}_{1}\left({S}_{1}\right)\cdot \text{d}{S}_{1}}=q$ (3)

For plate 2, there are similarly

$\text{d}{q}_{2}={\sigma}_{2}\left({S}_{2}\right)\cdot \text{d}{S}_{2}$,${S}_{2}\in \left[0,{S}_{2t}\right]$

${E}_{ds2}=\frac{{\sigma}_{2}\left({S}_{2}\right)}{\epsilon}=\frac{\text{d}{q}_{2}}{\epsilon \cdot \text{d}{S}_{2}}$ (4)

${q}_{20}=0$

${q}_{2t}={\displaystyle {\int}_{0}^{{S}_{2t}}\text{d}{q}_{2}}={\displaystyle {\int}_{0}^{{S}_{2t}}{\sigma}_{2}\left({S}_{2}\right)\cdot \text{d}{S}_{2}}=q$ (5)

So, the electric field strength nearby plate 2 is ${E}_{dS2}=\frac{\text{d}{q}_{2}}{\epsilon \cdot \text{d}{S}_{2}}$. From the first

and the third assumption conditions, we known that: the total electric field lines ${E}_{dS2}$ emitted from plate 2, not only distribute to the area differential element $\text{d}{s}_{1}$ on plate 1 (which is charged with electric quantity $\text{d}{q}_{1}$ ), but also distribute to the rest of area (the total area corresponding to $\text{d}{s}_{2}$ ) on plate 1 (which is charged with total electric quantity q). Hence the area differential element $\text{d}{s}_{1}$

actually achieves a electric field strength ${E}_{dS1h}={E}_{dS2}\frac{\text{d}{q}_{1}}{q}$ which is derived from $\text{d}{s}_{2}$.

The assumption of equal electric density causes the remaining charge produced on one differential element of plate 1 which electric density is actually higher than plate 2. The remaining charge $\Delta q=\text{d}{q}_{1}-\text{d}{q}_{2}$ on one differential element of plate 1 $\text{d}{s}_{1}$ which satisfies $\text{d}{S}_{1}=\text{d}{S}_{2}=\text{d}S$.

So we can get the force produced by remaining charge $\Delta q$ in the assumption electric field ${E}_{dS1h}$

$\text{d}f=\Delta q\cdot {E}_{dS1h}=\Delta q\cdot {E}_{dS2}\frac{\text{d}{q}_{1}}{q}=\left(\text{d}{q}_{1}-\text{d}{q}_{2}\right)\cdot \frac{\text{d}{q}_{2}}{\epsilon \cdot \text{d}{S}_{2}}\cdot \frac{\text{d}{q}_{1}}{q}$ (6)

Integrating Equation (6), we have

$\begin{array}{c}f={\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}\text{d}f}}={\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}\left(\text{d}{q}_{1}-\text{d}{q}_{2}\right)\cdot \frac{\text{d}{q}_{2}}{\epsilon \cdot \text{d}{S}_{2}}}}\cdot \frac{\text{d}{q}_{1}}{q}\\ ={\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}\left[{\sigma}_{1}\left({S}_{1}\right)\cdot \text{d}{S}_{1}-{\sigma}_{2}\left({S}_{2}\right)\cdot \text{d}{S}_{2}\right]\cdot \frac{{\sigma}_{2}\left({S}_{2}\right)\cdot \text{d}{S}_{2}}{\epsilon \cdot \text{d}{S}_{2}}}}\cdot \frac{{\sigma}_{1}\left({S}_{1}\right)\cdot \text{d}{S}_{1}}{q}\end{array}$ (7)

Combining the initial condition $\text{d}{S}_{1}=\text{d}{S}_{2}=\text{d}S$, we can further simplify above equation

$\begin{array}{c}f={\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}\left[{\sigma}_{1}\left({S}_{1}\right)\cdot \text{d}{S}_{2}-{\sigma}_{2}\left({S}_{2}\right)\cdot \text{d}{S}_{2}\right]\cdot \frac{{\sigma}_{2}\left({S}_{2}\right)\cdot \text{d}{S}_{2}}{\epsilon \cdot \text{d}{S}_{2}}\cdot \frac{{\sigma}_{1}\left({S}_{1}\right)\cdot \text{d}{S}_{1}}{q}}}\\ ={\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}\frac{{\sigma}_{1}\left({S}_{1}\right)\cdot {\sigma}_{2}\left({S}_{2}\right)}{\epsilon q}\left[{\sigma}_{1}\left({S}_{1}\right)-{\sigma}_{2}\left({S}_{2}\right)\right]\cdot \text{d}{S}_{2}\text{d}{S}_{1}}}\\ =\frac{1}{\epsilon q}{\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}{\sigma}_{1}\left({S}_{1}\right){\sigma}_{2}\left({S}_{2}\right)\cdot \left[{\sigma}_{1}\left({S}_{1}\right)-{\sigma}_{2}\left({S}_{2}\right)\right]\cdot \text{d}{S}_{2}\text{d}{S}_{1}}}\end{array}$ (8)

So we get the lift force formula in general form.

3. Similarity Principle

Equation (8) of calculating lift force is based on knowing electric density distribution function $\sigma \left(S\right)$. But in most situations, the electric density distribution is normally unknown. In this case, we should use theoretical combining with experimental method based on similarity principle of dimensional analysis to figure out the final force produced between two plates. The detail steps should be as follows.

1) Establishing a smaller model in scale $1:a$ $\left(a>1\right)$ that is similar to the original asymmetric capacitor.

2) Applying a voltage to the miniature model of which the value is $1/a$ of the original.

3) Electrifying the asymmetric capacitor, and actually measuring the lift force produced by the miniature model which is ${f}_{2}$.

4) Achieving the lift force of original model which is equal to square value of the small similar one, using magnifying method ${f}_{1}={a}^{2}{f}_{2}$.

Before calculating the final lift force of the right side lifter (refer Figure 2), we may measure lift force of a small scale model similar to it firstly, and then achieve the lift force of original model by scale magnifying method. If we measured the lift force is 0.2 N on the left side lifter, the lift force of original right side lifter (10 times larger than the left side one) is ${f}_{1}={a}^{2}{f}_{2}={10}^{2}\times 0.2=20\text{\hspace{0.17em}}\text{N}$.

4. Affiliated Verification

A brief verification is performed by simplifying Equation (8) from general to special case. If the both plates of an ideal asymmetric capacitor are distributed with uniform charge, there are following relations.

Firstly, we can take a part area from plate 1 ${S}_{1p}$ whose electric quantity is

Figure 2. Scale magnifying method to calculate lift force of arbitrarily shaped asymmetric capacitor (or called lifter).

${q}_{1p}={\displaystyle {\int}_{0}^{{S}_{1p}}{\sigma}_{1}\left({S}_{1}\right)\cdot \text{d}{S}_{1}}=\frac{{S}_{1p}}{{S}_{1t}}\cdot q$ (9)

So we get the distribution function of electric charge density that is

${\sigma}_{1}\left({S}_{1}\right)=\frac{{q}_{1p}}{{S}_{1p}}=\frac{q}{{S}_{1t}}$ (10)

Secondly, we can take a part area from plate 2 ${S}_{2p}$ whose electric quantity is

${q}_{2p}={\displaystyle {\int}_{0}^{{S}_{2p}}{\sigma}_{2}\left({S}_{2}\right)\cdot \text{d}{S}_{2}}=\frac{{S}_{2p}}{{S}_{2t}}\cdot q$ (11)

So we get the distribution function of electric charge density that is

${\sigma}_{2}\left({S}_{2}\right)=\frac{{q}_{2p}}{{S}_{2p}}=\frac{q}{{S}_{2t}}$ (12)

Lastly, plugging the Equations (10) and (12) of special case of uniformly distribution of electric charge into Equation (8) of general case of nonuniform distribution of electric charge, we have

$\begin{array}{c}f=\frac{1}{\epsilon q}{\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}{\sigma}_{1}\left({S}_{1}\right){\sigma}_{2}\left({S}_{2}\right)\cdot \left[{\sigma}_{1}\left({S}_{1}\right)-{\sigma}_{2}\left({S}_{2}\right)\right]\cdot \text{d}{S}_{2}\text{d}{S}_{1}}}\\ =\frac{1}{\epsilon q}{\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}\frac{q}{{S}_{1t}}\frac{q}{{S}_{2t}}\cdot \left(\frac{q}{{S}_{1t}}-\frac{q}{{S}_{2t}}\right)\cdot \text{d}{S}_{2}\text{d}{S}_{1}}}\\ =\frac{1}{\epsilon q}\cdot \frac{q}{{S}_{1t}}\frac{q}{{S}_{2t}}\cdot \left(\frac{q}{{S}_{1t}}-\frac{q}{{S}_{2t}}\right){\displaystyle {\int}_{0}^{1t}{\displaystyle {\int}_{0}^{2t}\text{d}{S}_{2}\text{d}{S}_{1}}}\\ =\frac{{q}^{2}}{\epsilon}\cdot \frac{1}{{S}_{1t}}\frac{1}{{S}_{2t}}\cdot \left(\frac{1}{{S}_{1t}}-\frac{1}{{S}_{2t}}\right)\cdot {S}_{2t}{S}_{1t}\\ =\frac{{q}^{2}}{\epsilon}\cdot \left(\frac{1}{{S}_{1t}}-\frac{1}{{S}_{2t}}\right)\end{array}$ (13)

Therefore, we can finally find that the lastly corollary is as the same as the former formula for calculating lift force of electric charge at ideal uniformly distribution in special case.

5. Conclusion

From above derivation, we get the general calculation formula of lift force of asymmetric capacitor. And by further simplification, we finally find the formula in ideal case is the same as previous deduction. The general calculation method can help us to get the lift force of asymmetric capacitor precisely in any shape. It is really an exciting method which can universally solve the hard thrust calculation problem completely.

Acknowledgements

The authors gratefully acknowledge the support of the Thirteenth Five-Year Plan of Hefei Institute of Physical Science of Chinese Academy of Science (Grant No. Y86CT21051 “Electric and Magnetic Propulsion System”), the Research Activity Funding of Postdoctoral Fellow of Anhui Province (Grant No. 2018B250 “High-energy Ions Accelerated Thruster”), the Natural Science Research Project of Anhui Education Department (Grant No. KJ2018A0725 “the Uniformity Optimization and Software Development for MRI Magnets”), and a portion of this work was supported by the High Magnetic Field Laboratory of Anhui Province.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Zhang, Y., Cheng, X.Y., Huang, P.C., Jiang, D.H., Jiang, S.L., Qian, X.X., Ding, H.W., Kuang, G.L. and Chen, W.G. (2020) Theoretical Calculation of Lift Force for General Electric Asymmetric Capacitor Loaded by High Voltage. Engineering, 12, 41-46. https://doi.org/10.4236/eng.2020.121004

References

- 1. Zhang, J.T., et al. (2011) A High-Performance Asymmetric Supercapacitor Fabricated with Graphene-Based Electrodes. Energy & Environmental Science, 4, 4009-4015. https://doi.org/10.1039/c1ee01354h
- 2. Lezana, P., Aguilera, R. and Quevedo, D.E. (2009) Model Predictive Control of an Asymmetric Flying Capacitor Converter. IEEE Transactions on Industrial Electronics, 56, 1839-1846. https://doi.org/10.1109/TIE.2008.2007545
- 3. Bahder, T. and Fazi, Ch. (2003) Force on an Asymmetrical Capacitor. Army Report ARL-TR-3005. Army Res. Lab., Adelphi.
- 4. Brown, T.T. (1928) A Method of and an Apparatus or Machine for Producing Force or Motion. UK Patent, No. 300311.
- 5. Primas, J., Malík, M., Jašíková, D. and Kopecky, V. (2010) Force on High Voltage Capacitor with Asymmetrical Electrodes. Processing of the WASET 2010 Conference, Amsterdam, 335-339.
- 6. Canning, F.X., Melcher, C. and Winet, E. (2004) Asymmetrical Capacitors for Propulsion. Institute for Scientific Research, Inc., Fairmont, West Virginia.
- 7. Long, K. (2012) The Role of Speculative Science in Driving Technology. In: Deep Space Propulsion, Springer, New York, NY, 287-304. https://doi.org/10.1007/978-1-4614-0607-5_16