Analysis of Periodically Time-Varying Systems

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Dedication......Page 4
Preface......Page 6
Contents......Page 8
List of Symbols......Page 12
Part I Theory and Techniques......Page 16
1.1 The Nature of Systems with Periodically Time-Varying Parameters......Page 18
1.2 1831-1887 Faraday to Rayleigh-Early Experimentalists and  Theorists......Page 22
1.3 1918-1940 The First Applications......Page 24
1.4 Second Generation Applications......Page 25
1.6 Commonplace Illustrations of Parametric Behaviour......Page 27
References for Chapter 1......Page 29
Problems......Page 31
2.1 Hill Equations......Page 32
2.2 Matrix Formulation of Hill Equations......Page 33
2.3 The State Transition Matrix......Page 34
2.4 Floquet Theory......Page 35
2.5 Second Order Systems......Page 37
2.6 Natural Modes of Solution......Page 38
2.7 Concluding Comments......Page 39
Problems......Page 40
3.1 Solutions Over One Period of the Coefficient......Page 42
3.2 The Meissner Equation......Page 43
3.3 Solution at Any Time for a Second Order Periodic Equation......Page 44
3.4 Evaluation of ?n, 0)', m Integral......Page 45
3.6 The Hill Equation with a Sawtooth Waveform Coefficient......Page 47
3.6.2 The Wronskian Matrix with z Zero......Page 50
3.7 The Hill Equation with a Positive Slope, Sawtooth Waveform  Coefficient......Page 51
3.8 The Hill Equation with a Triangular Coefficient......Page 52
3.10 Bessel Function Generation......Page 53
3.11 The Hill Equation with a Repetitive Exponential Coefficient......Page 54
3.12 The Hill Equation with a Coefficient in the Form of a Repetitive  Sequence of Impulses......Page 55
3.14 Response to a Sinusoidal Forcing Function......Page 56
3.15 Phase Space Analysis......Page 59
3.16 Concluding Comments......Page 61
References for Chapter 3......Page 63
Problems......Page 64
4.1 Types of Stability......Page 65
4.2 Stability Theorems for Periodic Systems......Page 66
4.3.1 Stability and the Characteristic Exponent......Page 67
4.3.2 The Meissner Equation......Page 68
4.3.3 The Hill Equation with an Impulsive Coefficient......Page 71
4.3.6 Hill Determinant Analysis......Page 72
4.3.7 Parametric Frequencies for Second Order Systems......Page 77
4.4.1 Hill Determinant Analysis for General Order Systems......Page 78
4.4.2 Residues of the Hill Determinant for q -+ 0......Page 81
4.4.4 Stability Diagrams for General Order Systems......Page 82
4.5.1 Nature of the Basis Solutions......Page 83
4.5.2 P Type Solutions......Page 84
4.5.4 N Type Solutions......Page 85
4.5.6 The Modes of a Second Order Periodic System......Page 86
4.5.7 Boundary Modes......Page 87
4.5.9 Modes for Systems of General Order......Page 88
4.5.10 Coexistence......Page 89
4.6 Short Time Stability......Page 90
Problems......Page 94
5.1.1 Second Order Systems......Page 96
5.2 A Modelling Philosophy for Intractable Hill Equations......Page 99
5.3 The Frequency Spectrum of a Periodic Staircase Coefficient......Page 100
5.4.2 Trapezoidal Models......Page 102
5.7 Models for Nonlinear Hill Equations......Page 103
5.8 A Note on Discrete Spectral Analysis......Page 104
5.9 Concluding Remarks......Page 105
Problems......Page 106
6.1.1 Periodic Solutions......Page 108
6.1.2 Mathieu Functions of Fractional Order......Page 110
6.1.4 Limitations of the Classical Method of Treatment......Page 111
6.2 Numerical Solution of the Mathieu Equation......Page 113
6.3.1 Rectangular Waveform Models......Page 114
6.3.2 Trapezoidal Waveform Models......Page 115
6.3.3 Staircase Waveform Models......Page 116
6.3.4 Performance Comparison of the Models......Page 117
6.4.1 The Lossless Mathieu Equation......Page 118
6.4.2 The Damped (Lossy) Mathieu Equation......Page 120
References for Chapter 6......Page 121
Problems......Page 122
Part II Applications......Page 124
7.1 The Quadrupole Mass Spectrometer......Page 126
7.1.1 Spatially Linear Electric Fields......Page 127
7.1.2 The Quadrupole Mass Filter......Page 128
7.1.3 The Monopole Mass Spectrometer......Page 132
7.1.5 Simulation of Quadrupole Devices......Page 135
7.2 Dynamic Buckling of Structures......Page 138
7.3 Elliptical Waveguides......Page 142
7.3.1 The Helmholtz Equation......Page 143
7.3.2 Rectangular Waveguides......Page 144
7.3.3 Circular Waveguides......Page 146
7.3.4 Elliptical Waveguides......Page 148
7.3.5 Computation of the Cut-off Frequencies for an Elliptical Waveguide......Page 151
7.4 Wave Propagation in Periodic Media......Page 152
7.4.1 Pass and Stop Bands......Page 153
7.4.2 The w - f3, (Brillouin) Diagram......Page 155
7.4.3 Electromagnetic Wave Propagation in Periodic Media......Page 158
7.4.4 Guided Electromagnetic Wave Propagation in Periodic Media......Page 159
7.4.5 Electrons in Crystal Lattices......Page 160
7.4.6 Other Examples of Waves in Periodic Media......Page 164
7.5 Electric Circuit Applications......Page 165
7.5.1 Degenerate Parametric Amplification......Page 166
7.5.3 Nondegenerate Parametric Amplification......Page 169
7.5.4 Parametric Up Converters......Page 170
7.5.5 N-path Networks......Page 173
References for Chapter 7......Page 177
Problems......Page 180
Appendix Bessel Function Generation by Chebyshev Polynomial Methods......Page 183
References for Appendix......Page 184
Subject Index......Page 186