CN203827075U - Fractional-order serial-connected resonance wireless power transmission system - Google Patents

Abstract

The utility model provides a fractional-order serial-connected resonance wireless power transmission system which comprises a high-frequency power source, an emission portion, a reception portion and a load, the emission portion comprises a primary fractional order capacitor and a primary fractional order inductor with primary resistor, the primary fractional order capacitor and the primary fractional order inductor being serially connected; and the reception portion comprises a secondary fractional order capacitor and a secondary fractional order inductor with secondary resistor, the primary fractional order capacitor and the primary fractional order inductor being serially connected. The wireless power transmission system employs fractional order elements to achieve wireless power transmission, is simple in structure and increases dimensions for parameter design. The wireless power transmission system is completely different from a conventional power transmission system achieved through integer order elements.

Description

A kind of fractional order series resonance radio energy transmission system

Technical field

The utility model belongs to the field of wireless power transmission or wireless technology of transmission of electricity, particularly a kind of fractional order series resonance radio energy transmission system.

Background technology

Wireless power transmission or wireless technology of transmission of electricity were just attempted experimentally by novel tesla of U.S. utility (Nicola Tesla) before more than 100 years.2006, the researcher of the Massachusetts Institute of Technology (MIT) utilizes the resonance technique of physics successfully to light the bulb of a 60W with 40% efficiency apart from left and right at 2m, this experiment is not only the reproduction of tesla's experiment, another new breakthrough of wireless power transmission technology especially, and started the upsurge of wireless power transmission research.

Wireless power transmission technology is a kind of delivery of electrical energy mode of wide application prospect, there is the advantages such as safe, reliable, flexible, convenient, day by day be subject to the attention of countries in the world, and be more and more widely used in the various places that are not suitable for or are inconvenient to use wire contact electric energy transmitting, as occasions such as implantable medical device, mobile electronic product, robot, rail electric car power supplies, and be expected to can aspect small-power electronic product wireless charging, replace traditional plug charging in the near future.

Current radio energy transmission system is all realized based on integer rank inductance, electric capacity, and its resonance frequency only determines by inductance value and capacitance, and therefore, its system only need be considered parameter value, and without the exponent number of considering element, the degree of freedom of design is fewer.Meanwhile, the element of real system is fractional order in essence, but the most exponent number of using in current reality is close to 1, ignores completely for the situation of fractional order.Traditional modeling of passing through integer rank designs radio energy transmission system, and under certain conditions, theoretical and actual error may be very large.

The generation that derives from fractional calculus of fractional order device (as fractional order electric capacity and fractional order inductance) concept, and the concept of fractional calculus has had the history of more than 300 year, is almost born with integer rank calculus simultaneously.But due to fractional order more complicated, and never have good numerical analysis tools, therefore it is always in the theory analysis stage.In recent decades, due to the development of biotechnology, macromolecular material etc., it is found that integer rank calculus can not well explain nature exist phenomenon, therefore fractional calculus starts to be paid attention to, and starting to be applied to engineering field, its research at control field and application are day by day perfect.Meanwhile, the fractional order device at two ends is out manufactured in laboratory.But some special character of fractional order circuit and system are studied, and are not mentioned especially in the application in wireless power transmission field.

In view of current fractional order element or fractional order circuit huge advantage in some aspects, and it is not also applied to wireless power transmission field, is therefore necessary to propose a kind of fractional order series resonance radio energy transmission system.

Utility model content

The purpose of this utility model is to overcome above-mentioned the deficiencies in the prior art, and a kind of fractional order series resonance radio energy transmission system is provided.

The utility model is achieved through the following technical solutions:

A kind of fractional order series resonance radio energy transmission system, comprises high frequency power source V _s, radiating portion, receiving unit and load R _l, radiating portion comprises the former limit fractional order electric capacity being connected in series former limit fractional order inductance former limit fractional order inductance there is former limit resistance R _p; Receiving unit comprises the secondary fractional order electric capacity being connected in series with secondary fractional order inductance secondary fractional order inductance there is secondary resistance R _s.

Described a kind of fractional order series resonance radio energy transmission system, former limit fractional order electric capacity secondary fractional order electric capacity voltage, current differential relation all meet: phase relation meets: wherein, i _cfor fractional order capacitance current, v _cfor fractional order capacitance voltage, α is the exponent number of fractional order electric capacity, and 0 < α≤2, C ^αfor the value of fractional order electric capacity.

Described a kind of fractional order series resonance radio energy transmission system, former limit fractional order inductance secondary fractional order inductance voltage, current differential relation all meet: phase relation meets: wherein, v _lfor the voltage of fractional order inductance, i _lfor the electric current of fractional order inductance, β is the exponent number of fractional order inductance, and 0 < β≤2, L ^βfor the value of fractional order inductance.

In described a kind of fractional order series resonance radio energy transmission system, it between radiating portion and receiving unit, is the wireless power transmission that the mode that is coupled by fractional order circuit series resonance realizes.

Operation principle of the present utility model is: radiating portion and receiving unit are respectively by former limit fractional order electric capacity former limit fractional order inductance former limit resistance R _p, secondary fractional order electric capacity secondary fractional order inductance secondary resistance R _sform fractional order RLC series resonant circuit, the mode that radiating portion and receiving unit are coupled by resonance realizes the wireless transmission of electric energy.

Compared with prior art, the utlity model has following advantage:

1, wireless power transmission simple in structure, to adopt fractional order element to realize, is different from radio energy transmission system in the past completely, has increased the degree of freedom that parameter is selected.

2, by the exponent number of selection element, can greatly reduce the resonance frequency of radio energy transmission system, thereby reduce the requirement to power electronic device, be very beneficial for the design of real system.

3, by selecting suitable fractional order exponent number, can make through-put power larger.

Brief description of the drawings

Fig. 1 is the schematic diagram of fractional order series resonance radio energy transmission system of the present utility model.

Fig. 2 is the equivalent circuit diagram of former secondary parameter Fig. 1 when consistent.

Fig. 3 is α=1.2, the power output of β=0.9 o'clock and the relation curve of frequency f.

Fig. 4 is α=1.2, the efficiency of transmission of β=0.9 o'clock and the relation curve of frequency f.

Fig. 5 is α=0.8, the power output of β=0.9 o'clock and the relation curve of frequency f.

Fig. 6 is α=1.2, the power output of β=1.5 o'clock and the relation curve of frequency f.

Fig. 7 is α=0.8, the power output of β=1.5 o'clock and the relation curve of frequency f.

Fig. 8 is the equivalent circuit diagram of Fig. 1 generally.

Specific embodiments

Below in conjunction with accompanying drawing, the concrete enforcement of utility model is further described, but enforcement of the present utility model and protection are not limited to this.

Embodiment

As shown in Figure 1, for the schematic diagram of fractional order series resonance radio energy transmission system of the present utility model, below in conjunction with this figure, operation principle of the present utility model and method for designing are described.In Fig. 1, high frequency power source V _s, former limit fractional order electric capacity former limit fractional order inductance with former limit resistance R _pconnect and compose successively series resonance; Secondary fractional order electric capacity secondary fractional order inductance secondary resistance R _swith load R _lconnect and compose successively series resonance; Radiating portion and receiving unit are realized wireless power transmission by mutual inductance M.For the convenience of analyzing, make former limit fractional order electric capacity with secondary fractional order electric capacity parameter is consistent, and omits upper and lower mark, is designated as C; Make former limit fractional order inductance with secondary fractional order inductance parameter is consistent, and omits upper and lower mark, is designated as L; Make former limit resistance R _pwith secondary resistance R _sfor R.Can obtain the Fractional Differential Equation of system:

$v_S=v_{C1}+L\frac{d^{\mathrm{\&beta;}}{i}_1}{{\mathrm{dt}}^{\mathrm{\&beta;}}}+M\frac{d^{\mathrm{\&beta;}}{i}_2}{{\mathrm{dt}}^{\mathrm{\&beta;}}}+i_{1}R$

$0=v_{C2}+M\frac{d^{\mathrm{\&beta;}}{i}_1}{{\mathrm{dt}}^{\mathrm{\&beta;}}}+L\frac{d^{\mathrm{\&beta;}}{i}_2}{{\mathrm{dt}}^{\mathrm{\&beta;}}}+i_{2}R+i_2{R}_L$

$i_1=C\frac{d^{\mathrm{\&alpha;}}{v}_{c1}}{{\mathrm{dt}}^{\mathrm{\&alpha;}}}$

$i_2=C\frac{d^{\mathrm{\&alpha;}}{v}_{c2}}{{\mathrm{dt}}^{\mathrm{\&alpha;}}}$

In formula, v _sfor the transient expression form of high frequency power source, i ₁for former limit loop current, i ₂for secondary loop current, v _c1for former limit fractional order capacitance voltage, v _c2for secondary fractional order capacitance voltage.The differential equation of said system can be obtained by Laplace transform:

V _S(s)＝V _C1(s)+s ^βLI ₁(s)+s ^βMI ₂(s)+I ₁(s)R

0＝V _C2(s)+s ^βMI ₁(s)+s ^βLI ₂(s)+I ₂(s)R+I ₂(s)R _L

I ₁(s)＝s ^αCV _C1(s)

I ₂(s)＝s ^αCV _C2(s)

Symbol in above equation group is Laplace transform form, has one-to-one relationship, i.e. I with the differential equation of system ₁for former limit loop current, I ₂for secondary loop current, V _c1for former limit fractional order capacitance voltage, V _c2for secondary fractional order capacitance voltage.In frequency domain, there is s=j ω.Defined loop impedance:

$Z_{11}=R+{\left(\mathrm{j\&omega;}\right)}^{\mathrm{\&beta;}}L+\frac{1}{{\left(\mathrm{j\&omega;}\right)}^{\mathrm{\&alpha;}}C}=1/Y_{11}$

$Z_{22}=R+R_L{\left(\mathrm{j\&omega;}\right)}^{\mathrm{\&beta;}}L+\frac{1}{{\left(\mathrm{j\&omega;}\right)}^{\mathrm{\&alpha;}}C}=1/Y_{22}$

In formula, Z ₁₁for former limit impedance loop, Z ₂₂for secondary impedance loop.

Solve:

$I_1=\frac{V_S{Z}_{22}}{Z_{11}{Z}_{22}-{\left(\mathrm{j\&omega;}\right)}^{2\mathrm{\&beta;}}{M}^2}=\frac{V_S}{Z_{11}-{\left(\mathrm{j\&omega;}\right)}^{2\mathrm{\&beta;}}{M}^2{Y}_{22}}$

$I_2=-\frac{V_S{\left(\mathrm{j\&omega;}\right)}^{\mathrm{\&beta;}}M/Z_{11}}{Z_{22}-{\left(\mathrm{j\&omega;}\right)}^{2\mathrm{\&beta;}}{M}^2{Y}_{11}}$

Can make the equivalent circuit diagram of Fig. 1 according to the expression formula of above electric current, as shown in Figure 2.Equiva lent impedance Z in Fig. 2 _eqfor:

$Z_{\mathrm{eq}}=-{\left(\mathrm{j\&omega;}\right)}^{2\mathrm{\&beta;}}{M}^2{Y}_{11}+R_S+{\left(\mathrm{j\&omega;}\right)}^{\mathrm{\&beta;}}L+\frac{1}{{\left(\mathrm{j\&omega;}\right)}^{\mathrm{\&alpha;}}C}$

Can obtain power output P _oexpression formula be:

$\begin{array}{c}{P}_o=I_2^2{R}_L{\left|\frac{V_S{\left(\mathrm{j\&omega;}\right)}^{\mathrm{\&beta;}}M}{{-Z}_{11}{Z}_{22}+{\left(\mathrm{j\&omega;}\right)}^{2\mathrm{\&beta;}}{M}^2}\right|}^2{R}_L\\ ={\left|\frac{V_S{\mathrm{\&omega;}}^{\mathrm{\&beta;}}M(\cos \frac{\mathrm{\&beta;\&pi;}}{2}+j\sin \frac{\mathrm{\&alpha;\&pi;}}{2})}{\begin{array}{c}-[R+{\mathrm{\&omega;}}^{\mathrm{\&beta;}}L(\cos \frac{\mathrm{\&beta;\&pi;}}{2}+j\sin \frac{\mathrm{\&beta;\&pi;}}{2})+\frac{1}{{\mathrm{\&omega;}}^{\mathrm{\&alpha;}}C}(\cos \frac{\mathrm{\&alpha;\&pi;}}{2}-j\sin \frac{\mathrm{\&alpha;\&pi;}}{2})]g[R+R_L+\\ {\mathrm{\&omega;}}^{\mathrm{\&beta;}}L(\cos \frac{\mathrm{\&beta;\&pi;}}{2}+j\sin \frac{\mathrm{\&beta;\&pi;}}{2})+\frac{1}{{\mathrm{\&omega;}}^{\mathrm{\&alpha;}}C}(\cos \frac{\mathrm{\&alpha;\&pi;}}{2}-j\sin \frac{\mathrm{\&alpha;\&pi;}}{2})]+{\mathrm{\&omega;}}^{2\mathrm{\&beta;}}{M}^2(\cos \mathrm{\&beta;\&pi;}+j\sin \mathrm{\&beta;\&pi;})\end{array}}\right|}^2{R}_L\end{array}$

Input power expression formula is:

$P_{\mathrm{in}}=\mathrm{Re}\left(V_S{I}_1^{*}\right)$

Or

$\begin{array}{c}{P}_{\mathrm{in}}=I_1^2\left|\mathrm{Re}\left(Z_{11}\right)\right|+I_2^2\left|\mathrm{Re}(Z_{22}-R_L)\right|+P_o\\ =I_1^2\left|(R+{\mathrm{\&omega;}}^{\mathrm{\&beta;}}L\cos \frac{\mathrm{\&beta;\&pi;}}{2}+\frac{1}{{\mathrm{\&omega;}}^{\mathrm{\&alpha;}}C}\cos \frac{\mathrm{\&alpha;\&pi;}}{2})\right|+I_2^2\left|(R+{\mathrm{\&omega;}}^{\mathrm{\&beta;}}L\cos \frac{\mathrm{\&beta;\&pi;}}{2}+\frac{1}{{\mathrm{\&omega;}}^{\mathrm{\&alpha;}}C}\cos \frac{\mathrm{\&alpha;\&pi;}}{2})\right|+P_o\end{array}$

System efficiency of transmission η is:

$\begin{array}{c}\mathrm{\&eta;}=\frac{P_o}{P_{\mathrm{in}}}=\frac{I_2^2{R}_L}{I_1^2\mathrm{Re}\left(Z_{11}\right)+I_2^2\mathrm{Re}(Z_{22}-R_L)+P_o}\\ =\frac{I_2^2{R}_L}{(I_1^2+I_2^2)\left|(R+{\mathrm{\&omega;}}^{\mathrm{\&beta;}}L\cos \frac{\mathrm{\&beta;\&pi;}}{2}+\frac{1}{{\mathrm{\&omega;}}^{\mathrm{\&alpha;}}C}\cos \frac{\mathrm{\&alpha;\&pi;}}{2})\right|+P_o}\end{array}$

From the expression formula of power output, the size of power output is mainly relevant with β with mutual inductance M, operating angle frequencies omega, fractional order exponent number α.Research below, the impact of operating angle frequency on power output, other parameters remain unchanged.Power output P _odiagonal frequencies ω carries out differentiate, and to make its derivative be zero, can be in the hope of the angular frequency extreme point of power output, and this angular frequency is:

${\mathrm{\&omega;}}_r={\left(\frac{\sin \left(\frac{\mathrm{\&alpha;\&pi;}}{2}\right)}{\mathrm{LC}\sin \left(\frac{\mathrm{\&beta;\&pi;}}{2}\right)}\right)}^{\frac{1}{\mathrm{\&alpha;}+\mathrm{\&beta;}}},(\mathrm{\&alpha;},\mathrm{\&beta;}\&NotEqual;2)$

Be the resonance angular frequency of fractional order series resonance radio energy transmission system.Work as α, β=2, the input impedance of fractional order series resonance radio energy transmission system is pure real number, and irrelevant with operating frequency.From above formula, the resonance angular frequency of series resonance is not only relevant with inductance value and capacitance, and relevant with the exponent number of fractional order electric capacity and fractional order inductance.And the situation on traditional integer rank is only relevant with inductance value and electric capacity.The situation of below dividing is discussed the impact of fractional order exponent number on systematic function:

1), as α >1, when β <1, as an example, the design parameter of fractional order series resonance radio energy transmission system is: V _s=10V, L=100 μ H, C=0.2533nF, R _l=12 Ω, coupling coefficient k=0.1(and mutual inductance M=k × L), α=1.2, β=0.9, R=0.5 Ω.The relation curve of power output and frequency f (dotted portion) as shown in Figure 3.In order to show advantage of the present utility model, for the situation on integer rank, i.e. α=1, β=1, other parameters are consistent.Equally, the power output of the radio energy transmission system on integer rank and the curve of frequency f (solid line part) also as shown in Figure 3.The power output on comparison score rank and integer rank and the relation curve of frequency f can find out, the power output of fractional order is greater than the situation on integer rank, and the resonance frequency of fractional order is less than the situation on integer rank.The resonance frequency that calculates new fractional-order system according to theory is 0.47MHz, conform to, and the resonance frequency on traditional integer rank is 1MHz with the simulation result of Fig. 3.Can find out in addition, fractional order the in the situation that of coupling coefficient k=0.1, there is not frequency splitting phenomenon, and there is frequency splitting phenomenon in integer rank system.In the time that coupling coefficient further increases, as k=0.5, also frequency of occurrences division of new fractional-order system, but its transmission power ratio integer rank system is larger.From relatively, the utility model fractional order series resonance radio energy transmission system has huge advantage with respect to the system on integer rank.

Now the efficiency of transmission of system and the curve of frequency f are as shown in Figure 4.As shown in Figure 4, be a bit larger tham 50% in the efficiency of transmission at peak power output place, and maximum transmitted efficiency obtaining being greater than resonance frequency place, is also that through-put power and efficiency of transmission can not obtain maximum simultaneously.

2), as α <1, when β <1, as an example, the design parameter of fractional order series resonance radio energy transmission system is: V _s=10V, L=100 μ H, C=0.2533nF, R _l=1000 Ω, coupling coefficient k=0.5(and mutual inductance M=k*L), α=0.8, β=0.9, R=0.5 Ω.The relation curve of power output and frequency f as shown in Figure 5.As shown in Figure 5, efficiency of transmission is in this case very low, should avoid the appearance of this situation when design.

3), as α >1, when β >1, as an example, the design parameter of fractional order series resonance radio energy transmission system is: V _s=10V, L=100 μ H, C=0.2533nF, R _l=5000 Ω, coupling coefficient k=0.5(and mutual inductance M=k*L), α=1.2, β=1.5, R=0.5 Ω.The relation curve of power output and frequency f as shown in Figure 6.As shown in Figure 6, efficiency of transmission is in this case very low, should avoid the appearance of this situation when design.

4), as α <1, when β >1, as an example, the design parameter of fractional order series resonance radio energy transmission system is: V _s=10V, L=100 μ H, C=0.2533nF, R _l=5000 Ω, coupling coefficient k=0.5(and mutual inductance M=k*L), α=0.8, β=1.5, R=0.5 Ω.The relation curve of power output and frequency f as shown in Figure 7.As shown in Figure 7, substantially can not realize in this case the transmission of power, when design, should avoid the appearance of this situation.

Situation described above is applicable equally for the situation of α=β.

Further, for situation more generally, the i.e. different situation of the parameter of system, the system row that Fig. 1 is described are write the differential equation, can obtain:

$v_S=v_{C1}+L_P^{\mathrm{\&beta;}}\frac{d^{{\mathrm{\&beta;}}_1}{i}_1}{{\mathrm{dt}}^{{\mathrm{\&beta;}}_1}}+M\frac{d^{{\mathrm{\&beta;}}_2}{i}_2}{{\mathrm{dt}}^{{\mathrm{\&beta;}}_2}}+i_1{R}_P$

$0=v_{C2}+M\frac{d^{{\mathrm{\&beta;}}_1}{i}_1}{{\mathrm{dt}}^{{\mathrm{\&beta;}}_1}}+L_S^{\mathrm{\&beta;}}\frac{d^{{\mathrm{\&beta;}}_2}{i}_2}{{\mathrm{dt}}^{{\mathrm{\&beta;}}_2}}+i_2{R}_S+i_2{R}_L$

$i_1=C_P^{\mathrm{\&alpha;}}\frac{d^{{\mathrm{\&alpha;}}_1}{v}_{c1}}{{\mathrm{dt}}^{{\mathrm{\&alpha;}}_1}}$

$i_2=C_S^{\mathrm{\&alpha;}}\frac{d^{{\mathrm{\&alpha;}}_2}{v}_{c2}}{{\mathrm{dt}}^{{\mathrm{\&alpha;}}_2}}$

In formula, β ₁for the exponent number of former limit fractional order inductance, β ₂for the exponent number of secondary fractional order inductance, α ₁for the exponent number of former limit fractional order electric capacity, α ₁for the exponent number of secondary fractional order electric capacity.

Above-mentioned equation group is carried out to Laplace transform, can obtain:

$V_S\left(s\right)=V_{C1}\left(s\right)+s^{{\mathrm{\&beta;}}_1}{L}_P^{\mathrm{\&beta;}}{I}_1\left(s\right)+s^{{\mathrm{\&beta;}}_2}{\mathrm{MI}}_2\left(s\right)+I_1\left(s\right)R_P$

$0=V_{C2}\left(s\right)+s^{{\mathrm{\&beta;}}_1}{\mathrm{MI}}_1\left(s\right)+s^{{\mathrm{\&beta;}}_2}{L}_S^{\mathrm{\&beta;}}{I}_2\left(s\right)+I_2\left(s\right)R_S+I_2\left(s\right)R_L$

$I_1\left(s\right)=s^{{\mathrm{\&alpha;}}_1}{C}_P^{\mathrm{\&alpha;}}{V}_{C1}\left(s\right)$

$I_2\left(s\right)=s^{{\mathrm{\&alpha;}}_2}{C}_S^{\mathrm{\&alpha;}}{V}_{C2}\left(s\right)$

According to above analysis, same defined loop impedance (is ignored capacitance C ^αwith inductance value L ^βsubscript):

$z_{11}=R_P+{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1}{L}_P+\frac{1}{{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&alpha;}}_1}{C}_P}=1/Y_{11}$

$Z_{22}=R_S+{R_L+\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_2}{L}_S+\frac{1}{{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&alpha;}}_2}{C}_S}=1/Y_{22}$

Solve:

$I_1=\frac{V_S{Z}_{22}}{Z_{11}{Z}_{22}-{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1+{\mathrm{\&beta;}}_2}{M}^2}=\frac{V_S}{Z_{11}-{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1+{\mathrm{\&beta;}}_2}{M}^2{Y}_{22}}$

$I_2=-\frac{V_S{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1}M/Z_{11}}{Z_{22}-{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1+{\mathrm{\&beta;}}_2}{M}^2{Y}_{11}}$

Can make more generally equivalent circuit diagram of Fig. 1 according to the expression formula of above electric current, as shown in Figure 8.Equiva lent impedance Z in Fig. 8 _eqfor:

$Z_{\mathrm{eq}}=-{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1+{\mathrm{\&beta;}}_2}{M}^2{Y}_{11}+R_S+{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_2}{L}_S+\frac{1}{{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&alpha;}}_2}{C}_S}$

The expression formula that can obtain power output is:

$P_o=I_2^2{R}_L={\left|\frac{V_S{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1}M}{{-Z}_{11}{Z}_{22}+{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1+{\mathrm{\&beta;}}_2}{M}^2}\right|}^2{R}_L$

Input power expression formula is:

$P_{\mathrm{in}}=\mathrm{Re}\left(\frac{V_S^2}{|Z_{11}-{\left(\mathrm{j\&omega;}\right)}^{{\mathrm{\&beta;}}_1+{\mathrm{\&beta;}}_2}{M}^2{Y}_{22}|}\right)$

System efficiency of transmission is:

$\mathrm{\&eta;}=\frac{P_o}{P_{\mathrm{in}}}$

Important parameter of resonant circuit is exactly quality factor, below the quality factor of fractional order RLC series circuit is analyzed.Being commonly defined as of quality factor q:

$Q=\frac{{\mathrm{\&omega;}}_r}{{\mathrm{\&omega;}}_{3d{B}_H}-{\mathrm{\&omega;}}_{3d{B}_L}}$

In formula, for larger 3dB frequency, for less 3dB frequency.

For α=β > 1 & 2Q ₀>=| cos (0.5 α π) | situation, can obtain 3dB frequency according to following two formulas, that is:

$(x_{3\mathrm{dB}}+\frac{Q_0^2}{x_{3\mathrm{dB}}})=-\cos \left(\frac{\mathrm{\&alpha;\&pi;}}{2}\right)+\sqrt{{\mathrm{\&delta;}}_1}=\mathrm{\&lambda;}$

${\mathrm{\&delta;}}_1=({4Q}_0+1)(1+{\cos }^2\left(\frac{\mathrm{\&alpha;\&pi;}}{2}\right))+{8Q}_0\cos \left(\frac{\mathrm{\&alpha;\&pi;}}{2}\right)$

In formula, integer rank quality factor can solve thus:

$x_{3\mathrm{dB}}=0.5[\mathrm{\&lambda;}\&PlusMinus;\sqrt{{\mathrm{\&lambda;}}^2-{4Q}_0^2}]=\frac{1}{{\mathrm{\&omega;}}_{3\mathrm{dB}\mathrm{1,2}}^{\mathrm{\&alpha;}}\mathrm{RC}}$

Further obtaining 3dB frequency is:

${\mathrm{\&omega;}}_{{3\mathrm{dB}}_{L,H}}=\sqrt[\mathrm{\&alpha;}]{\frac{R}{2L}[\mathrm{\&lambda;}\&PlusMinus;\sqrt{{\mathrm{\&lambda;}}^2-{4Q}_0^2}]}$

Half-power frequency (being 3dB frequency) with the ratio of resonance frequency is:

$\frac{{\mathrm{\&omega;}}_{{3\mathrm{dB}}_{L,H}}}{{\mathrm{\&omega;}}_r}=\sqrt[\mathrm{\&alpha;}]{\frac{\mathrm{\&lambda;m}\sqrt{{\mathrm{\&lambda;}}^2-{4Q}_0^2}}{{2Q}_0}}$

Therefore obtaining quality factor is:

$Q=\frac{{\left({2Q}_0\right)}^{\frac{1}{\mathrm{\&alpha;}}}}{{(\mathrm{\&lambda;}+\sqrt{{\mathrm{\&lambda;}}^2-{4Q}_0^2})}^{\frac{1}{\mathrm{\&alpha;}}}-{(\mathrm{\&lambda;}-\sqrt{{\mathrm{\&lambda;}}^2-{4Q}_0^2})}^{\frac{1}{\mathrm{\&alpha;}}}}$

In the time of α=1, Q=Q ₀.

From quality factor defined above, there is very large difference in radio energy transmission system of the present utility model and traditional integer rank radio energy transmission system system, and the advantage of the utility model system is apparent.

Above-described embodiment is preferably execution mode of the utility model; but execution mode of the present utility model is not limited by the examples; other any do not deviate from change, the modification done under Spirit Essence of the present utility model and principle, substitutes, combination, simplify; all should be equivalent substitute mode, within being included in protection range of the present utility model.

Claims (3)

1. a fractional order series resonance radio energy transmission system, comprises high frequency power source (V _s), radiating portion, receiving unit and load (R _l), it is characterized in that radiating portion comprises the former limit fractional order electric capacity being connected in series former limit fractional order inductance former limit fractional order inductance there is former limit resistance (R _p); Receiving unit comprises the secondary fractional order electric capacity being connected in series with secondary fractional order inductance secondary fractional order inductance there is secondary resistance (R _s).

2. a kind of fractional order series resonance radio energy transmission system according to claim 1, is characterized in that former limit fractional order electric capacity secondary fractional order electric capacity voltage, current differential relation all meet: phase relation meets: wherein, i _cfor fractional order capacitance current, v _cfor fractional order capacitance voltage, α is the exponent number of fractional order electric capacity, and 0 < α≤2, C ^αfor the value of fractional order electric capacity.

3. a kind of fractional order series resonance radio energy transmission system according to claim 1, is characterized in that former limit fractional order inductance secondary fractional order inductance voltage, current differential relation all meet: phase relation meets: wherein, v _lfor the voltage of fractional order inductance, i _lfor the electric current of fractional order inductance, β is the exponent number of fractional order inductance, and 0 < β≤2, L ^βfor the value of fractional order inductance.