CN108116260A - 一种等功率的多单元双向无线电能传输系统参数配置方法 - Google Patents

一种等功率的多单元双向无线电能传输系统参数配置方法 Download PDF

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CN108116260A
CN108116260A CN201711401631.4A CN201711401631A CN108116260A CN 108116260 A CN108116260 A CN 108116260A CN 201711401631 A CN201711401631 A CN 201711401631A CN 108116260 A CN108116260 A CN 108116260A
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CN108116260B (zh
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王维
羊树文
杨靖宇
王�琦
胡敏强
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Nanjing Normal University
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    • BPERFORMING OPERATIONS; TRANSPORTING
    • B60VEHICLES IN GENERAL
    • B60LPROPULSION OF ELECTRICALLY-PROPELLED VEHICLES; SUPPLYING ELECTRIC POWER FOR AUXILIARY EQUIPMENT OF ELECTRICALLY-PROPELLED VEHICLES; ELECTRODYNAMIC BRAKE SYSTEMS FOR VEHICLES IN GENERAL; MAGNETIC SUSPENSION OR LEVITATION FOR VEHICLES; MONITORING OPERATING VARIABLES OF ELECTRICALLY-PROPELLED VEHICLES; ELECTRIC SAFETY DEVICES FOR ELECTRICALLY-PROPELLED VEHICLES
    • B60L55/00Arrangements for supplying energy stored within a vehicle to a power network, i.e. vehicle-to-grid [V2G] arrangements
    • BPERFORMING OPERATIONS; TRANSPORTING
    • B60VEHICLES IN GENERAL
    • B60LPROPULSION OF ELECTRICALLY-PROPELLED VEHICLES; SUPPLYING ELECTRIC POWER FOR AUXILIARY EQUIPMENT OF ELECTRICALLY-PROPELLED VEHICLES; ELECTRODYNAMIC BRAKE SYSTEMS FOR VEHICLES IN GENERAL; MAGNETIC SUSPENSION OR LEVITATION FOR VEHICLES; MONITORING OPERATING VARIABLES OF ELECTRICALLY-PROPELLED VEHICLES; ELECTRIC SAFETY DEVICES FOR ELECTRICALLY-PROPELLED VEHICLES
    • B60L53/00Methods of charging batteries, specially adapted for electric vehicles; Charging stations or on-board charging equipment therefor; Exchange of energy storage elements in electric vehicles
    • B60L53/10Methods of charging batteries, specially adapted for electric vehicles; Charging stations or on-board charging equipment therefor; Exchange of energy storage elements in electric vehicles characterised by the energy transfer between the charging station and the vehicle
    • B60L53/12Inductive energy transfer
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02TCLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00Road transport of goods or passengers
    • Y02T10/60Other road transportation technologies with climate change mitigation effect
    • Y02T10/70Energy storage systems for electromobility, e.g. batteries
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02TCLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00Road transport of goods or passengers
    • Y02T10/60Other road transportation technologies with climate change mitigation effect
    • Y02T10/7072Electromobility specific charging systems or methods for batteries, ultracapacitors, supercapacitors or double-layer capacitors
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02TCLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00Road transport of goods or passengers
    • Y02T10/60Other road transportation technologies with climate change mitigation effect
    • Y02T10/72Electric energy management in electromobility
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02TCLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation
    • Y02T90/10Technologies relating to charging of electric vehicles
    • Y02T90/14Plug-in electric vehicles
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02TCLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation
    • Y02T90/10Technologies relating to charging of electric vehicles
    • Y02T90/16Information or communication technologies improving the operation of electric vehicles

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  • Charge And Discharge Circuits For Batteries Or The Like (AREA)
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Abstract

本发明公开一种等功率的多单元双向无线电能传输系统参数配置方法,包括多单元双向无线电能传输系统参数设置和通过配置系统相关的电气参数,实现多单元之间的等功率传输。采用该方法可实现多单元具有相同的有功出力或拾取相同的有功功率,充分实现电能的双向利用,在无线V2G、无线V2H等应用场合,该参数配置方法简单多样且更易于实施,具有很好的经济性和实用性。

Description

一种等功率的多单元双向无线电能传输系统参数配置方法
技术领域
本发明涉及一种等功率的多单元双向无线电能传输系统参数配置方法,适用于需要实现多单元相同有功出力或拾取相同有功功率的能量双向互动的无线电能传输领域,如无线V2G,无线V2H等应用场合。
背景技术
多单元双向无线传输系统在应用于电动汽车时,虽然能实现能量的双向流动,使得无线供电变得更高效有利,但是安装在电动汽车上的电池储能有限,当电动汽车采用双向无线电能传输系统时,电动汽车既可以作为电能的提供方又可以作为电能的接收方;无论是其对其他车辆进行放电工作,还是其他电动汽车对其进行充电供电,功率的调节都尤为关键。当其作为电源向其他电动汽车进行充电时,必须要在满足自身的功率需求情况下进行,同时要能够对放电的功率实时控制,避免过放现象;当其他电动汽车向其进行充电时,为了方便自身调节接收功率或主动控制接收功率以抑制过充现象。在多单元双向无线电能传输领域,如何通过配置系统参数定量表征单元间传输功率比需要展开研究在多单元双向无线电能传输领域,如何通过配置系统参数定量表征单元间传输功率比需要展开研究。
发明内容
发明目的:本发明公开了一种多单元双向无线电能传输系统的参数配置方法,通过该参数配置方法,系统能够实现任意两个功率发射单元相同出力或者任意两个单元接收相同的功率。
技术方案:本发明所述的一种等功率的多单元双向无线电能传输系统参数配置方法,包括以下步骤:
(1)多单元双向无线电能传输系统参数设置;
(2)通过配置系统相关的电气参数,实现多单元之间的等功率传输。
步骤(1)所述的多单元双向无线电能传输系统由两辆具有无线充放电功能的电动汽车以及无线充放电换能站组成;每辆电动汽车能量部分均包含有车载谐振器线圈、单相全桥式变换器、车载电池;无线充放电换能站包含有换能站侧功率线圈、单相全桥式变换器以及直流母线。
步骤(1)所述的参数设置包括:电动汽车谐振器参数一致,自感L2=L3,内阻R2=R3;电动汽车的车载谐振器线圈与换能站侧功率线圈隔空放置并存在耦合,但两辆电动汽车自身车载谐振器线圈不存在耦合;电动汽车的谐振线圈与换能站侧功率线圈之间的互感分别为M12、M13,M12与M13之间的比值为γ,γ=M12/M13,M12和M13的上限值和下限值分为为M12max、M13max以及M12min、M13min,γmax=M12max/M13min,γmin=M12min/M13max,并有γmin<1<γmax;单相全桥式变换器工作频率、车载谐振器线圈的谐振频率与换能站侧功率线圈谐振频率一致,记为f;电动汽车车载单相全桥变换器的控制器和换能站单相全桥变换器的控制器之间采用WIFI通讯;换能站直流母线电压、两电动汽车电池电压分别设置为V1、V2和V3;对应的单相全桥变换器出口电压相量分别为U1、U2和U3,相角分别为φ1、φ2和φ3;U1、U2和U3的上限值和下限值分别为U1max、U2max和U3max以及U1min、U2min和U3min。
所述步骤(2)包括以下步骤:
(21)将φ1、φ2和φ3分别配置成0,-π/2和0,配置互感比γ=1,即M12=M13,配置电压U1=0.5*(U1min+U1max);
(22)配置电压幅值U2=U3,且max(U2min,U3min)<U2=U3<min(U2max,U3max);
(23)电压幅值互感M12与电压U2之间满足电压幅值U3的上下限值满足:
(25)电压幅值互感M12与电压U3之间满足电压幅值U2的上下限值满足:
有益效果:与现有技术相比,本发明的有益效果:1、通过换能站向两电动汽车无线充电并且两电动汽车接收功率相等,充分实现电能的双向利用;2、通过装有该系统参数电动汽车之间的双向传输,能够有效解决电能在传输中只能单向传输、效率低的问题;3、提出把车载谐振器线圈与换能站功率谐振器线圈隔空放置并存在耦合但两辆电动汽车自身车载谐振器线圈不存在耦合,考虑实际中车主对电动车在无线充电的灵活需求,设置能够实现表征单元间传输功率比的最恰当参数;4、在无线V2G、无线V2H等应用场合,该参数配置方法简单多样且更易于实施,具有很好的经济性和实用性,在实际中实施起来非常方便。
附图说明
图1为多单元双向无线电能传输系统拓扑结构图;
图2为多单元双向无线电能传输系统等效电路图。
具体实施方式
下面结合附图对本发明作更进一步的说明:
图1为多单元双向无线电能传输系统拓扑结构图,该多单元双向无线电能传输系统由具有无线充/放电功能的电动汽车2和电动汽车3以及无线充放电换能站1组成;每辆电动汽车的能量部分均包含有车载谐振器线圈4、单相全桥式变换器6、车载电池等关键设备,换能站侧功率线圈5自感为L1,等效串联电阻为R1;电动汽车2与3的谐振线圈自感及等效串联电阻分别为L2、R2及L3、R3;电动汽车2与3的谐振器参数一致,自感L2=L3,内阻R2=R3;换能站包含有换能站侧功率线圈5、单相全桥式变换器6以及直流母线7等关键设备。每辆电动汽车的车载谐振器线圈4均与换能站侧功率线圈5隔空放置并存在耦合但两辆电动汽车自身车载谐振器线圈4不存在耦合,电动汽车2谐振线圈与换能站功率线圈之间的互感为M12,电动汽车3谐振线圈与换能站功率线圈之间的互感为M13,M12与M13之间的比值为γ,γ=M12/M13,M12和M13的上限值和下限值分为为M12max、M13max以及M12min、M13min,γmax=M12max/M13min,γmin=M12min/M13max,并有γmin<1<γmax。单相全桥式变换器工作频率、车载谐振器线圈的谐振频率与换能站侧功率线圈谐振频率一致,记为f,且ω=2πf;电动汽车车载单相全桥变换器的控制器和换能站单相全桥变换器的控制器之间采用WIFI通讯;换能站直流母线电压、两电动汽车电池电压分别设置为V1、V2和V3;对应的单相全桥变换器出口电压相量分别为U1、U2和U3,相角分别为φ1、φ2和φ3;U1、U2和U3的上限值和下限值分别为U1max、U2max和U3max以及U1min、U2min和U3min。
根据双向无线电能传输系统的工作原理,可将该多单元双向无线电能传输系统等效成如附图2所示的等效电路图。
在附图2中,电压与电流为非关联参考方向,设置φ1=-π/2、φ2=0、φ3=0,配置互感比γ=1,即M12=M13。电动汽车2与3及换能站发出的有功功率分别为:
P1=[k0U1U21U1 2+k2U1U3]/Γ
P2=[λ0U2 2-k0U1U2-k1U2U3]/Γ
P3=[-k1U2U3-k2U1U32U3 2]/Γ
式中相关参数λ0=ω2M132+R1R3、λ1=R2R3、λ2=ω2M122+R1R2、k0=ωM12R3、k1=ω2M12M13、k2=ωM13R2、Γ=ω2M122R3+ω2M132R2+R1R2R3。
当U2=U3时,P1>0,P2<0,P3<0,P2/P3=γ2;当U2>U3时,P1>0,P2>0,P3<0,此时
当U2<U3时,P1>0,P2<0,P3>0,此时
选择下列三种参数配置方式之一实现等有功功率,具体包括1)调节电动汽车2与3的单相全桥变换器出口电压U2和U3使得max(U2min,U3min)<U2=U3<min(U2max,U3max)实现换能站向电动汽车2与3无线充电并且电动汽车2与3接收功率相等。
或2)配置互感M12与电压U2的参数,满足再配置U3的限值电压U3min和U3max使其满足
设置电压U3幅值,使其满足
此时U3<U2,实现电动汽车2与换能站有相同有功出力,并且能量流向电动汽车3。
或3)配置互感M12与电压U3的参数,满足
再配置U2的限制电压U2min和U2max使其满足
设置电压U2幅值,使其满足
此时U2<U3,实现电动汽车3与换能站有相同有功出力,并且能量流向电动汽车2。
以上所述仅是本发明的优选实施方式,应当指出:对于本技术领域的普通技术人员来说,在不脱离本发明原理的前提下,还可以做出若干改进和润饰,这些改进和润饰也应视为本发明的保护范围。

Claims (4)

1.一种等功率的多单元双向无线电能传输系统参数配置方法,其特征在于,包括以下步骤:
(1)多单元双向无线电能传输系统参数设置;
(2)通过配置系统相关的电气参数,实现多单元之间的等功率传输。
2.根据权利要求1所述的多单元双向无线电能传输系统参数配置方法,其特征在于,步骤(1)所述的多单元双向无线电能传输系统由电动汽车(2)与电动汽车(3)两辆具有无线充放电功能的电动汽车以及无线充放电换能站(1)组成;每辆电动汽车能量部分均包含有车载谐振器线圈(4)、单相全桥式变换器(6)、车载电池;无线充放电换能站(3)包含有换能站侧功率线圈(5)、单相全桥式变换器(6)以及直流母线(7)。
3.根据权利要求1所述的多单元双向无线电能传输系统参数配置方法,其特征在于,步骤(1)所述的参数设置包括:电动汽车(2)与(3)的谐振器参数一致,自感L2=L3,内阻R2=R3;电动汽车的车载谐振器线圈(4)均与换能站侧功率线圈(5)隔空放置并存在耦合,但两辆电动汽车自身车载谐振器线圈(4)不存在耦合;电动汽车(2)谐振线圈(4)与换能站侧功率线圈(5)之间的互感为M12,电动汽车(3)谐振线圈(4)与换能站侧功率线圈(5)之间的互感为M13,M12与M13之间的比值为γ,γ=M12/M13,M12和M13的上限值和下限值分为为M12max、M13max以及M12min、M13min,γmax=M12max/M13min,γmin=M12min/M13max,并有γmin<1<γmax;单相全桥式变换器(6)工作频率、车载谐振器线圈(4)的谐振频率与换能站侧功率线圈(5)谐振频率一致,记为f;电动汽车车载单相全桥变换器(6)的控制器和换能站单相全桥变换器(6)的控制器之间采用WIFI通讯;换能站直流母线(7)电压、电动汽车(2)电池电压以及电动汽车(3)电池电压分别设置为V1、V2和V3;对应的单相全桥变换器(6)出口电压相量分别为U1、U2和U3,相角分别为φ1、φ2和φ3;U1、U2和U3的上限值和下限值分别为U1max、U2max和U3max以及U1min、U2min和U3min。
4.根据权利要求1所述的多单元双向无线电能传输系统参数配置方法,其特征在于,所述步骤(2)包括以下步骤:
(21)将φ1、φ2和φ3分别配置成0,-π/2和0,配置互感比γ=1,即M12=M13,配置电压U1=0.5*(U1min+U1max);
(22)配置电压幅值U2=U3,且max(U2min,U3min)<U2=U3<min(U2max,U3max);
(23)电压幅值互感M12与电压U2之间满足电压幅值U3的上下限值满足:
<mrow> <msub> <mi>U</mi> <mrow> <mn>3</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>&amp;le;</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;M</mi> <mrow> <mn>12</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msubsup> <mi>U</mi> <mrow> <mn>2</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>U</mi> <mn>1</mn> </msub> <msub> <mi>U</mi> <mrow> <mn>2</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>/</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;M</mi> <mrow> <mn>12</mn> <mi>min</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>U</mi> <mrow> <mn>2</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>&amp;rsqb;</mo> </mrow>
<mrow> <msub> <mi>U</mi> <mrow> <mn>3</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>&amp;GreaterEqual;</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;M</mi> <mrow> <mn>12</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msubsup> <mi>U</mi> <mrow> <mn>2</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>U</mi> <mn>1</mn> </msub> <msub> <mi>U</mi> <mrow> <mn>2</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>/</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;M</mi> <mrow> <mn>12</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>U</mi> <mrow> <mn>2</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>&amp;rsqb;</mo> <mo>;</mo> </mrow>
(24)电压幅值互感M12与电压U3之间满足电压幅值U2的上下限值满足:
<mrow> <msub> <mi>U</mi> <mrow> <mn>2</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>&amp;le;</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;M</mi> <mrow> <mn>12</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msubsup> <mi>U</mi> <mrow> <mn>3</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>U</mi> <mn>1</mn> </msub> <msub> <mi>U</mi> <mrow> <mn>3</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>/</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;M</mi> <mrow> <mn>12</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>U</mi> <mrow> <mn>3</mn> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>&amp;rsqb;</mo> </mrow>
<mrow> <msub> <mi>U</mi> <mrow> <mn>2</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>&amp;GreaterEqual;</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;M</mi> <mrow> <mn>12</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msubsup> <mi>U</mi> <mrow> <mn>3</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>2</mn> <msub> <mi>U</mi> <mn>1</mn> </msub> <msub> <mi>U</mi> <mrow> <mn>3</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>&amp;rsqb;</mo> <mo>/</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;omega;M</mi> <mrow> <mn>12</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>U</mi> <mrow> <mn>3</mn> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>&amp;rsqb;</mo> <mo>.</mo> </mrow>
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