Internal inductance - Revision history

Admin at 10:30, 26 January 2023

2023-01-26T10:30:39Z

Admin: /* Normalized internal inductance */

2012-08-12T18:20:12Z

Normalized internal inductance

← Older revision		Revision as of 20:20, 12 August 2012
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	which is equal to <math>l_i</math> assuming the plasma has a perfect toroidal shape, <math>V = \pi a^2 \cdot 2 \pi R_0</math>.<ref>[[Effective plasma radius]]</ref>		which is equal to <math>l_i</math> assuming the plasma has a perfect toroidal shape, <math>V = \pi a^2 \cdot 2 \pi R_0</math>.<ref>[[Effective plasma radius]]</ref>

	The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile).		== Relation to current profile ==

			The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile): a small value of <math>l_i</math> corresponds to a broad current profile.

	== References ==		== References ==
	<references />		<references />

Admin: /* Normalized internal inductance */

2012-08-10T20:15:33Z

Normalized internal inductance

← Older revision		Revision as of 22:15, 10 August 2012
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	(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.		(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.

	~~Alternatively~~, ~~sometimes the internal inductance per unit length is used, defined as~~<ref name="Freidberg"/>		Using Ampère's Law (<math>2 \pi a B_\theta(a) = \mu_0 I</math>), one obtains <ref name="Freidberg"/>
	:<math>l_i' = \frac{L_i}{2\pi R_0}\frac{4\pi}{\mu_0} = \frac{2L_i}{\mu_0R_0}</math>		:<math>l_i = \frac{L_i}{2\pi R_0}\frac{4\pi}{\mu_0} = \frac{2L_i}{\mu_0R_0}</math>
	where ''R<sub>0</sub>'' is the [[Toroidal coordinates\|major radius]], and similar for the external inductance.		where ''R<sub>0</sub>'' is the [[Toroidal coordinates\|major radius]], and similar for the external inductance.
	~~Using Ampère's Law (<math>2 \pi a B_\theta(a) = \mu_0 I</math>), one finds <math>l_i = 2 \pi l_i'</math>.~~

	The [[ITER]] design uses the following approximate definition:<ref>[[doi:10.1088/0029-5515/48/12/125002\|G.L. Jackson, T.A. Casper, T.C. Luce, et al., ''ITER startup studies in the DIII-D tokamak'', Nucl. Fusion '''48''', 12 (2008) 125002]]</ref>		The [[ITER]] design uses the following approximate definition:<ref>[[doi:10.1088/0029-5515/48/12/125002\|G.L. Jackson, T.A. Casper, T.C. Luce, et al., ''ITER startup studies in the DIII-D tokamak'', Nucl. Fusion '''48''', 12 (2008) 125002]]</ref>

Admin: /* Normalized internal inductance */

2012-08-10T10:49:54Z

Normalized internal inductance

← Older revision		Revision as of 12:49, 10 August 2012
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	In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.		In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.
	In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>
	:<math>l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \~~int_P~~{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}</math>		:<math>l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \int_0^a{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}</math>
	(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.		(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.

Admin: /* Normalized internal inductance */

2012-08-10T09:52:20Z

Normalized internal inductance

← Older revision		Revision as of 11:52, 10 August 2012
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	The [[ITER]] design uses the following approximate definition:<ref>[[doi:10.1088/0029-5515/48/12/125002\|G.L. Jackson, T.A. Casper, T.C. Luce, et al., ''ITER startup studies in the DIII-D tokamak'', Nucl. Fusion '''48''', 12 (2008) 125002]]</ref>		The [[ITER]] design uses the following approximate definition:<ref>[[doi:10.1088/0029-5515/48/12/125002\|G.L. Jackson, T.A. Casper, T.C. Luce, et al., ''ITER startup studies in the DIII-D tokamak'', Nucl. Fusion '''48''', 12 (2008) 125002]]</ref>
	:<math>l_i(3) = \frac{2 V \left \langle B_\theta^2 \right \rangle}{\mu_0^2I^2 R_0}</math>		:<math>l_i(3) = \frac{2 V \left \langle B_\theta^2 \right \rangle}{\mu_0^2I^2 R_0}</math>
	which is equal to <math>l_i</math> ~~asuming~~ the plasma has a perfect toroidal shape, <math>V = \pi a^2 \cdot 2 \pi R_0</math>.<ref>[[Effective plasma radius]]</ref>		which is equal to <math>l_i</math> assuming the plasma has a perfect toroidal shape, <math>V = \pi a^2 \cdot 2 \pi R_0</math>.<ref>[[Effective plasma radius]]</ref>

	The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile).		The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile).

Admin at 09:48, 10 August 2012

2012-08-10T09:48:37Z

← Older revision		Revision as of 11:48, 10 August 2012
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	The [[ITER]] design uses the following approximate definition:<ref>[[doi:10.1088/0029-5515/48/12/125002\|G.L. Jackson, T.A. Casper, T.C. Luce, et al., ''ITER startup studies in the DIII-D tokamak'', Nucl. Fusion '''48''', 12 (2008) 125002]]</ref>		The [[ITER]] design uses the following approximate definition:<ref>[[doi:10.1088/0029-5515/48/12/125002\|G.L. Jackson, T.A. Casper, T.C. Luce, et al., ''ITER startup studies in the DIII-D tokamak'', Nucl. Fusion '''48''', 12 (2008) 125002]]</ref>
	:<math>l_i(3) = \frac{2 V \left \langle B_\theta^2 \right \rangle}{\mu_0^2I^2 R_0}</math>		:<math>l_i(3) = \frac{2 V \left \langle B_\theta^2 \right \rangle}{\mu_0^2I^2 R_0}</math>
	which is equal to <math>l_i</math> asuming the plasma has a perfect toroidal shape, <math>V = \pi a^2 \cdot 2 \pi R_0</math>.<ref>[[Effective plasma radius]]		which is equal to <math>l_i</math> asuming the plasma has a perfect toroidal shape, <math>V = \pi a^2 \cdot 2 \pi R_0</math>.<ref>[[Effective plasma radius]]</ref>

	The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile).		The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile).

Admin: /* Normalized internal inductance */

2012-08-10T09:48:02Z

Normalized internal inductance

← Older revision		Revision as of 11:48, 10 August 2012
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	In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.		In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.
	In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>
	:<math>l_i = \frac{2 \pi \~~int_P{~~B_\theta^2(\~~rho)~~ \~~rho d\rho}~~}{~~\pi a^2~~ B_\theta^2(a)} = \frac{\~~left~~ \~~langle~~ B_\theta^2 \~~right~~ \~~rangle_P~~}{B_\theta^2(a)}</math>		:<math>l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \int_P{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}</math>
	(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.		(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.

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	where ''R<sub>0</sub>'' is the [[Toroidal coordinates\|major radius]], and similar for the external inductance.		where ''R<sub>0</sub>'' is the [[Toroidal coordinates\|major radius]], and similar for the external inductance.
	Using Ampère's Law (<math>2 \pi a B_\theta(a) = \mu_0 I</math>), one finds <math>l_i = 2 \pi l_i'</math>.		Using Ampère's Law (<math>2 \pi a B_\theta(a) = \mu_0 I</math>), one finds <math>l_i = 2 \pi l_i'</math>.

			The [[ITER]] design uses the following approximate definition:<ref>[[doi:10.1088/0029-5515/48/12/125002\|G.L. Jackson, T.A. Casper, T.C. Luce, et al., ''ITER startup studies in the DIII-D tokamak'', Nucl. Fusion '''48''', 12 (2008) 125002]]</ref>
			:<math>l_i(3) = \frac{2 V \left \langle B_\theta^2 \right \rangle}{\mu_0^2I^2 R_0}</math>
			which is equal to <math>l_i</math> asuming the plasma has a perfect toroidal shape, <math>V = \pi a^2 \cdot 2 \pi R_0</math>.<ref>[[Effective plasma radius]]

	The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile).		The value of the normalized internal inductance depends on the current density profile in the toroidal plasma (as it produces the <math>B_\theta(\rho)</math> profile).

Admin at 09:02, 10 August 2012

2012-08-10T09:02:47Z

← Older revision		Revision as of 11:02, 10 August 2012
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	It can be shown that<ref>P.M. Bellan, ''Fundamentals of Plasma Physics'', Cambridge University Press (2006) ISBN 0521821169</ref><ref>[[:Wikipedia:Inductance]]</ref>		It can be shown that<ref>P.M. Bellan, ''Fundamentals of Plasma Physics'', Cambridge University Press (2006) ISBN 0521821169</ref><ref>[[:Wikipedia:Inductance]]</ref>
	:<math>W = \frac12 L I^2</math>		:<math>W = \frac12 L I^2</math>

			== Internal inductance of a plasma ==

	The ''internal'' inductance is defined as the part of the inductance obtained by integrating over the plasma volume ''P'' <ref name="Freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>:		The ''internal'' inductance is defined as the part of the inductance obtained by integrating over the plasma volume ''P'' <ref name="Freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>:
	:<math>\frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r}</math>		:<math>\frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r}</math>

Admin: /* Normalized internal inductance */

2012-08-10T08:40:40Z

Normalized internal inductance

← Older revision		Revision as of 10:40, 10 August 2012
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	In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.		In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.
	In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>
	:<math>l_i = \frac{2 \pi \int_P{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)} = \frac{\left \langle B_\theta^2 \right \~~rangle~~}{B_\theta^2(a)}</math>		:<math>l_i = \frac{2 \pi \int_P{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)} = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)}</math>
	(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.		(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.

Admin: /* Normalized internal inductance */

2012-08-10T08:39:42Z

Normalized internal inductance

← Older revision		Revision as of 10:39, 10 August 2012
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	In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.		In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.
	In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>
	:<math>l_i = \frac{2 \pi \int_P{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)} </math>		:<math>l_i = \frac{2 \pi \int_P{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)} = \frac{\left \langle B_\theta^2 \right \rangle}{B_\theta^2(a)}</math>
	(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a'').		(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.

	Alternatively, sometimes the internal inductance per unit length is used, defined as<ref name="Freidberg"/>		Alternatively, sometimes the internal inductance per unit length is used, defined as<ref name="Freidberg"/>

← Older revision		Revision as of 12:30, 26 January 2023
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	On the other hand, the energy contained in the magnetic field produced by the loop is		On the other hand, the energy contained in the magnetic field produced by the loop is
	:<math>W = \int{\frac{B^2}{2\mu_0} d\vec r}</math>		:<math>W = \int{\frac{B^2}{2\mu_0} d\vec r}</math>
	It can be shown that<ref>P.M. Bellan, ''Fundamentals of Plasma Physics'', Cambridge University Press (2006) ISBN 0521821169</ref><ref>[[:Wikipedia:Inductance]]</ref>		It can be shown that<ref>P.M. Bellan, ''Fundamentals of Plasma Physics'', Cambridge University Press (2006) {{ISBN\|0521821169}}</ref><ref>[[:Wikipedia:Inductance]]</ref>
	:<math>W = \frac12 L I^2</math>		:<math>W = \frac12 L I^2</math>

	== Internal inductance of a plasma ==		== Internal inductance of a plasma ==

	The ''internal'' inductance is defined as the part of the inductance obtained by integrating over the plasma volume ''P'' <ref name="Freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) ISBN 0521851076</ref>:		The ''internal'' inductance is defined as the part of the inductance obtained by integrating over the plasma volume ''P'' <ref name="Freidberg">J.P. Freidberg, ''Plasma physics and fusion energy'', Cambridge University Press (2007) {{ISBN\|0521851076}}</ref>:
	:<math>\frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r}</math>		:<math>\frac12 L_i I^2 = \int_P{\frac{B^2}{2\mu_0} d\vec r}</math>
	Its complement is the external inductance (''L = L<sub>i</sub> + L<sub>e</sub>'').		Its complement is the external inductance (''L = L<sub>i</sub> + L<sub>e</sub>'').
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	In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.		In a [[tokamak]], the field produced by the plasma current is the ''poloidal'' magnetic field ''B<sub>θ<sub>'', so only this field component enters the definition.
	In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) ISBN 3540242171</ref>		In this context, it is common to use the ''normalized'' internal inductance<ref>K. Miyamoto, ''Plasma Physics and Controlled Nuclear Fusion'', Springer-Verlag (2005) {{ISBN\|3540242171}}</ref>
	:<math>l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \int_0^a{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}</math>		:<math>l_i = \frac{\left \langle B_\theta^2 \right \rangle_P}{B_\theta^2(a)} = \frac{2 \pi \int_0^a{B_\theta^2(\rho) \rho d\rho}}{\pi a^2 B_\theta^2(a)}</math>
	(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.		(for circular cross section plasmas with [[Toroidal coordinates\|minor radius]] ''a''), where angular brackets signify taking a mean value.