Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

More documents

Recommendations

Info

Magnetic fields and tokamak plasmas Alan Wootton vector potential height z (m) 0.15 0.2 0.3 0.4 exact approximate major radius R (m) major radius R (m) Figure 1.8. The exact (solid line), the approximate Figure 1.9. Contours of vector potential zero order (dashed line) and the approximate first from the exact solution (thin dark line), order (dotted line) solutions in the z = 0 plane for the zero order approximation (thick gray the vector potential from a circular current filament line) and including first order corrections with current I = 1/µ. (intermediate thickness dark line) for a circular current filament with current I = 1/µ. Contours of potential equal to 0.15, 0.2, 0.3 and 0.4 are shown. Mutual inductance We shall use mutual and self inductances, often between circular filaments. They can be used to derive the vertical field necessary to maintain a plasma equilibrium, and to analyze axisymmetric instabilities. We are interested in the relationships between mutual and self inductances and fluxes, and how to write energy in terms of mutual and self inductances. The mutual inductance M 12 between two circuits is defined as the flux N 12 through circuit 1 produced by unit current in circuit 2. Then A 2 l 1 I 2 M 12 = ∫ dl 1 1.24 with A 2 the total vector potential due to current 2 in circuit 2. That is, M 12 = µ dl 1 dl 2 4π ∫ ∫ = M l 1 l2 21 1.25 r The electromotance through circuit 1 due to a current I 2 in circuit 2 is 16
Magnetic fields and tokamak plasmas Alan Wootton ε 1 = M 12 dI 2 dt 1.26 The total energy (in a volume V) associated with two circuits is W t = 1 B 1 + B 2 2µ ∫ ( ) •( B 1 + B 2 )dV V = 1 2 B 1 2µ ∫ dV + 1 2 B V 2 2µ ∫ dV + 1 B V 1 • B 2 µ ∫ dV V 1.27 The first two terms represent the energy required to establish the currents I 1 and I 2 producing the fields B 1 and B 2 in circuits 1 and 2. The third term is the energy used in bringing the two circuits together. This mutual energy between the two circuits W 12 is W 12 = M 12 I 1 I 2 = 1 B 1 B 2 dV µ ∫ 1.28 V Self inductance The magnetic energy density W of a single circuit carrying current I 1 is used to define the self inductance L 11 of the circuit: W = 1 ∫ dV = 1 2 L I 2 11 1 2µ V B2 1.29 To maintain the current I 1 a power source must, in each second, do an amount of work ε 1 I 1 = I 1 dN 1 dt 1.30 (because ε = dN/dt) in addition to working against resistance Ω. The stored energy per second in the magnetic field equals dW/dt, so that I 1 dN 1 dt = L 11 I 1 dI 1 dt 1.31 i.e. N 11 = L 11 I 1 1.32 Therefore we can also define the self inductance of a circuit through the change in flux linking that circuit when the current changes by one unit: 17
Page 1 and 2: Magnetic fields and tokamak plasmas
Page 15: Magnetic fields and tokamak plasmas
Page 67 and 68:
Magnetic fields and tokamak plasmas
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
show all

Magnetic Fields and Magnetic Diagnostics for Tokamak Plasmas

Create successful ePaper yourself

Delete template?

Save as template?