Electrical oscillation with external excitation
Damped LC tank with external excitation We consider a resistor, an inductance and a cap in series excited by an external AC signal. This circuit is mathematically described by UGURULUC U0ηtRILI'1CQ Q''RLQ'1CLQU0Lηt cnRmZAAAAAADAAAABAAAAAIAAAAuLhcAAABfX0BVVEY4UHJlZmVycmVkTmFtZUBf XxMAAABfX0BQcmVmZXJyZWROYW1lQF9fAQAAAC77KgAADwAAAA8AAAAmAAAAAQAA AAAAAIBYGwAAkw8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA 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Solving the homogeneous part of the differential equation We solve the homogeneous part of the differential equation Eq. 1 first Q''RLQ'1CLQ0 using the approach QKeiσt Q'Kiσeiσt Q''Kσ2eiσt Substitution into the differential equation gives Kσ2eiσtRLKiσeiσt1CLKeiσt0 σ2RLiσ1CLKeiσt0 This equation is true for all t only if σ2RLiσ1CL0 is true. We solve this as follows: pRLi q1CL σ1p2p22q σ2p2p22q σ1RLi2RLi221CL σ1R2LiR2L21CL σ2R2LiR2L21CL We define ωR2L21CL and thus get the following two solutions: QKeiσ1t QKeiR2Liωt QKeR2Lteiωt QKeR2Ltωtiωt QKeiσ2t QKeiR2Liωt QKeR2Lteiωt QKeR2Ltωtiωt The general solution for the homogeneous differential equation can therefore be written as QAeR2LtωtiωtBeR2Ltωtiωt QeR2LtAωtAiωteR2LtBωtBiωt QeR2LtAωtAiωtBωtBiωt QeR2LtABωtiABωt or after replacing the factors QeR2LtK1ωtiK2ωt Since the differential equation is linear we can rewrite this as follows QeR2LtK1ωtK2ωt or even shorter like so QeR2LtKωtφ ωω02ρ2 ω021CL ρR2L with the two freely choosable constants K und φ. Please note the dependence of the resonance frequency from the resistance. fR12πR2L21CL { ordinateAuto = 0; xScale = 1; xmarkDensity = 7; yScale = 1; ymarkDensity = 4; } 0;70;0;100;0;;Fine 2013-05-29 16:36:11 C=50E-6; L=50E-3 JVBERi0xLjMKJcTl8uXrp/Og0MTGCjQgMCBvYmoKPDwgL0xlbmd0aCA1IDAgUiAv RmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeAFtnVuObLuOXf+jFdmCdLwf 39WCQjXhwIYN3Pqwq/+Ax5iUKK3IujbqaM/UkiiKpEjqEf/3599//u/P//i3/7r8 /PNfP5ef//rn58z/e1wvP9fH/ef//c+f//Vz/n3+/J+q82//wR//499+bh/+/POf P/fP08K/KFxO13MKQfzT5ecmcjlff9+PFw1e+WL9683fPo/f9+VCC5fH7/X8vt2o MbHT5/f5eTxvh3qf3+vr/Xl+Uose3r+fz/3zePx8nr/nx/kh8r7fHu/Xzz9i18v5 fDm9ael6vfDV8/dxfUH9+/fxeT/p7/P6PZ+vz6fI8/qGFL5b2C3/k9Lb7+d+udxo 6w+2vqXe/fd+/rzfYu/H83IRed1e5xvI8/X43Onz/gs5jzdtMeQ3vKbPx+/l/L5c qfX5XJ6fjPB6PvPfX2bi4RDlUYDn9XqD8/kq0OX2+7i83+9TWDqhz/NN94/f8/VB w5f77+VzZR7o/vm60WAjNHSngft5YScQWD0QOCYDnpLBV+fb5fG2+xvMft+eP/T/ +dBLMWpgNAADQhM9vpkmv31cbreM5Xb7XO8i79v5BVP+iMI/P/874nJ/PN8/l8vv 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Finding a particular solution for the inhomogeneous differential equation We find a particular solution for the inhomogeneous differential equation by trying a suitable approach. Q''RLQ'1CLQU0Lηt A suitable approach for the above equation is QtDeiηtδ Q'tDiηeiηtδ Q''tDη2eiηtδ Substituted into the differential equation we get Q''tRLQ't1CLQtU0Leiηt Dη2eiηtδRLDiηeiηtδ1CLDeiηtδU0Leiηt DeiηtδRLiη1CLη2U0Leiηt DeiδRLiη1CLη2U0L DeiδU0L1CLη2RLiη DeiδU01CLη2RLiηL1CLη2RLiη1CLη2RLiη DeiδU0L1CLη2RLiη1CLη22RLη2 DeiδU0L1CLη22RLη21CLη2RLηi The value D is the length of the complex number on the right. DU0L1CLη22RLη21CLη22RLη2 DU0L1CLη22RLη2 The angle δ is determined as follows: δDIDR δDID δDRD DRU01CLη2L1CLη22RLη2 DIU0RLηL1CLη22RLη2 DU0L1CLη22RLη2 δRηL1CLη2 δRLη δ1CLη2 QtDηtδ
Combining the general and the particular solution We combine the general solution for the homogeneous differential equation QeR2LtKωtφ ωω02ρ2 ω01CL ρR2L and the particular solution of the inhomogeneous differential equation QtDηtδ DU0L1CLη22RLη2 δRηL1CLη2 into a total solution by summation: QteR2LtKωtφDηtδ Since the e-Function approaches 0 for large values of t only the particular solution remains after an initiation period. QtDηtδ We devide by C to get the voltage in the cap, take first derivatives to get the current and take the second derivative to get the voltage over the coil. QtDηtδ Q'tDηηtδ Q''tDη2ηtδ UCtQtC ULtLQ''t ItQ't δRηL1CLη2 We determine the peak voltage in the cap and the peak current with respect to f (resonance frequency): QtDηtδ UsfDC UsfU0LC1CLη22RLη2 UsfU0LC1CL2πf222πfRL2 ItDηηtδ IsfDη IsfU0L1CLη22RLη2η Isf2πfU0L1CL2πf222πfRL2 If we excite the series tank with its resonance frequency we get high voltages over the components and a very high current only limited by the resistance of the wire. { ordinateAuto = 0; xScale = 1; xmarkDensity = 4; yScale = 1; ymarkDensity = 5; } 0;150;0;50;0;;Fine 2013-05-29 23:08:49 C=50E-6; L=100E-3; R=1; U_0=1; Factor I_s(f): 10.00 JVBERi0xLjMKJcTl8uXrp/Og0MTGCjQgMCBvYmoKPDwgL0xlbmd0aCA1IDAgUiAv RmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeAFsvU2uZD2Obdn3UfgI/F37 N2vHCAo5hEAVsoCIxqucP1BrbZKSzF9ENvK723V0JIqktkge2f/+/X/9/t+//9c/ /ufy+5//8/vy+3/++fuH/7t/Pr+vj/vv/+///v3//P758/z9/1abf/wX//hf//h9 e/253N+3F41+//v3/XXnP3796/f9yWM/v/mPIP4TyE3kcr/9eT8vt9/XK/D+6+2/ 3Z5/fl7P1+/L9c/9/rl8ftFiYZc/r8vrPu2ut8/t9wXs9nqn0f1yvQS4vz70wGOP 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Energy Consideration The power going into the circuit is given by P1Tt0TUGtIt P1Tt0TU0ηtDηηtδ PU0DηTt0Tηtηtδ PU0DηTt0T2ηtδ2ηtδ4η PU0DηT2ηTδ2ηTδ4ηδ4η PU0D4Tδ2ηTδ2ηTδ η2πT T2πη PU0D42πηδ2η2πηδ2η2πηδ This gets us the following expression for the input power. PinU0Dη8πδ4πδ4πδ δRηL1CLη2 DU0L1CLη22RLη2 The circulating power in the circuit is defined as PcircttWt with Wt12CUCt212LIt2 UCtDCηtδ ItDηηtδ PcircttWt Pcirctt12CUCt212LIt2 Pcirctt12CDCηtδ212LDηηtδ2 Pcirct12CtDCηtδ212LtDηηtδ2 PcirctCDCηtt0ηtδDCηtδLDηηtt0ηtδDηηtδ PcirctCDCηηtδDCηtδLDη2ηtδDηηtδ PcirctDηηtδDCηtδLDη2ηtδDηηtδ PcirctDηDCηtδηtδLDη2Dηηtδηtδ PcirctDηDCLDη2Dηηtδηtδ PcirctD2η1CLη2ηηtδηtδ PcirctD2η1CLη2ηtδηtδ DU0L1CLη22RLη2 δRηL1CLη2 We have scaled Pin with a factor of 100 in the figure above. The Input power is very small compared to the circulating power for R0.25Ohm. { ordinateAuto = 0; xScale = 1; xmarkDensity = 5; yScale = 1; ymarkDensity = 4; } 0;25;-2;2;-3;;Fine 2013-05-30 00:13:26 C=50E-6; L=50E-3; R=0.25; U₀=12; η=314; Factor P_in: 100.00 JVBERi0xLjMKJcTl8uXrp/Og0MTGCjQgMCBvYmoKPDwgL0xlbmd0aCA1IDAgUiAv RmlsdGVyIC9GbGF0ZURlY29kZSA+PgpzdHJlYW0KeAF1nVmuNLmRpd9jFXcFt2Ie nnMFBS0hUQ01kHqo1v6B/r5jRjoZWSUBkv/n0jkYbeIhnfHfP//5898///HHvy8/ f/775/Lz7z9/zvz3cb38XB/3n//3Xz//5+f8+/z5v1Xmj3/wx3/88XP9/Fyu159/ /dw/zzz9dbp/Lj+Xy+vnr4n5V7Cr2OV6/r3z9w/v9PP1IX65/77OlzcPl9/P5/b4 +ddJ7P7gNbHb+51i19/H9U2rC8QLTyq5XM6/59e9arv+vujQ+fP7ud2f1HSzgluQ 5/1hlbff9/1VyPX8ev/8CXb/vZ2fYu/fl2Xuv4+bhfn39U6fB3K6nF+/z+vt3m91 qScdyeit58Iwz8/f+znv0drtao8ev58r0rE1+vS8fajr8Xu/X/jj5fr7Pjs4kOv5 4pCuv5dnRtKI711+79d31dWlLr/nx+daNV2cCcqIpKb341Gjm9jz9/w+OwuX38vj TTP08/IZgs6ACziN5lCBrdDj9mRGfe3+sJuX3+fbkYOcHxd0KNibP4Ax5Nft5Bxf f8+fu/J9MNBzje/2auR1udNhyjwf9VaAEhSV2oPHr1hN50f5WNHj48Tcfp+vIPTp 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