Lecture 6: Electromagnetic Power Outline 1. Power and energy in a circuit 2. Power and energy density in a distributed system 3. Surface Impedance September 27, 2005 Massachusetts Institute of Technology 6.763 2005 Lecture 6 Power in a Circuit We WmEnergy Power: Image removed for copyright reasons. 18, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 6 Please see: Figure 2.1, page 1Average Power for a Sinusoidal Drive which gives Massachusetts Institute of Technology 6.763 2005 Lecture 6 The time average power is Power is a bilinear term, not a linear one, so must use real variables, The time average power is then Average Power for a Sinusoidal Drive R jωL1 jωC Massachusetts Institute of Technology 6.763 2005 Lecture 6 2Power in Distributed Systems E. H. where we have used Use the full Maxwell’s Equations, Massachusetts Institute of Technology 6.763 2005 Lecture 6 Therefore, we have found ( σ0, µ, ε) we wm energy density Joule heating For a sinusoidal drive: and Massachusetts Institute of Technology 6.763 2005 Lecture 6 Poynting’s Theorem Poynting’s theorem, with For a linear, isotropic, homogenous ohmic medium3Maxwell’s equations still give But for a superconductor Therefore, wK Massachusetts Institute of Technology 6.763 2005 Lecture 6 Poynting’s Theorem for a Superconductor Kinetic Energy Density Kinetic energy of a superelectron and Massachusetts Institute of Technology 6.763 2005 Lecture 6 Superelectron density With Energy is also stored in the kinetic energy of the superelectrons 4Averaged Poynting Vector For a sinusoidal drive: and Energy in a superconductor is dissipated through the normal channel Massachusetts Institute of Technology 6.763 2005 Lecture 6 Power Loss in a Slab Normal Metal Superconductor Image removed for copyright reasons. Please see: Figure 2.13, page 43, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 6 5For a unit δ << a, or λ << a Image removed for copyright reasons. Please see: Figure 2.13, page 43, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 6 area, the time averaged power is In the bulk approximation, where For a normal metal: and surface resistance The current density is given by The power dissipated per unit area is Image removed for copyright reasons. Please see: Figure 3.19, page 110, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 6 Surface Resistance: normal metal For an area on the surface of The current is 6Surface Impedance: Normal Metal In the bulk approximation, where δ << a, or λ << a, a surface impedance can be defined from as For the normal metal, jωLs Rs Ls Rs Massachusetts Institute of Technology 6.763 2005 Lecture 6 Surface Impedance: Superconductor For the superconductor, with λ << δ, to lowest order Rs~ ω2 Ls Λ(T) 1/σ0(T) For Rs >> ω Ls For Pb at 2K and 100 MHz, Rs = 10-10 Ohm/ , and Q of cavity = 1010 Massachusetts Institute of Technology 6.763 2005 Lecture 6