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(Hình 1.19): một phần bao gồm các yếu tố tuyến tính và một phần chỉ bao gồm những phi tuyến, điện áp tại các cổng kết nối được thể hiện bằng cách mở rộng Fourier series: Chuyển đổi đạt được là tương tự như đối với nhân đôi duy nhất kết thúc, và công suất đầu ra cao hơn 3 dB, cung cấp năng lượng đầu vào tương ứng cao hơn được cung cấp, không có cải tiến phù hợp với thu được. . | 28 NONLINEAR ANALYSIS METHODS Figure 1.18 Currents and voltages in the example circuit at the sampling times only as computed from Fourier transform Figure 1.19 A general nonlinear network as partitioned for the harmonic balance analysis harmonics and a real phasor at DC n 0 . The continuous curves in the previous figures have been plotted by means of eq. 1.69 and eq. 1.73 once the values of the phasors are known. In general terms a nonlinear circuit is divided into two parts connected by M ports Figure 1.19 a part including only linear elements and a part including only nonlinear ones the voltages at the connecting ports are expressed by Fourier series expansions r N Vm t Re EV . ejnM0t m n e _n 0 Vm 0 Re N 5 V m n _n 1 ejn M0t Vm 0 N Y Vm .n cos n 01 - Vimn sin n 01 m 1 . M 1.81 n 1 SOLUTION THROUGH SERIES EXPANSION 29 The voltages at the connecting ports are the unknowns of Kirchhoff s node equations. In our formulation the unknowns are actually the phasors that appear in their Fourier series expansion since the series is truncated they are M 2N 1 . In vector form r i r i r V - LV1 0 VM 0 V1 1 V1 1 VM 1 VM 1 V1 N V1 N VM N VM n 1.82 The linear part of the circuit is replaced by its Norton equivalent the currents flowing into it are computed by simple multiplication of the still unknown vector of the voltage phasors by the Norton equivalent admittance matrix plus the known Norton equivalent current sources due to the input signal Figure 1.20 . 1L - Y V Il 0 1.83 where ĩ r r i r i r 1 LI1 0 L M 0 L I1 1 L 0 1 L IM 1 L 7M 1 L n.N.L Ii N L IM N L ỊM.N.l 1.84 When the voltages and currents are ordered as in eqs. 1.82 and 1.84 the admit- tance matrix relative to the linear subcircuit is block-diagonal - Yl 0 0 0 0 0 0 Y l mo 0 0 0 0 1 85 _ 0 0 0 Yl Nmo where Yl m y11 M _ yM1 m y1M M yMM to _ 1 86 is the m X m standard linear admittance matrix of the linear subnetwork at frequency u . Figure 1.20 Currents and voltages for the harmonic balance analysis 30 NONLINEAR .
