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325_Sp2011_12_Inductance

325_Sp2011_12_Inductance - 12 Inductance and magnetic field...

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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 1 12. Inductance and magnetic field energy EE325 Mikhail Belkin
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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 2 Inductance We will learn in a few days that when the magnetic field flux through a conducting circuit changes, an ‘electromotive force’ or emf is generated in the circuit (this is Faraday Law: emf=-d Φ /dt) In the absence of permanent magnets and other ‘nonlinear’ magnetic materials, the magnetic field is linearly proportional to currents. It is important to know how a flux through a circuit is related to current in the circuit. We want to link current in the circuit and flux through the circuit “inductance” is a flux linkage concept Φ is the total flux that “links” a circuit I is the current flowing in the circuit units: Φ is from the magnetic field, measured in webers L (units): weber / amp = henry (H) L only depends on geometery and materials constants ( μ r ) Remember that for electric fields we had capacitance, that depends only on geometry and materials constants ( ε r ) capacitance was a way to link charge and voltage L I Φ =
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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 3 Flux through a circuit One loop is easy: How about a coil? We can calculate the magnetic field it produces from Biot-Saver law But what surface does it enclose? This is the surface (you may use a soap bubble concept and imagine a soap solution forming a continuous film having the wire as one continuous edge) When the coil loops are tight, we may write for the flux through the coil: where N is the number of loops B surface enclosed by the loop B dS Φ= r r 1 coil loop or N λ Φ = Φ ( ) 1 loop coil N L I Φ =
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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 4 Easier way to calculate inductance: use energy in circuits, we already know the magnetic energy should be but the energy using field concepts is so the inductance is given by volume must include ALL the fields (integration over the whole 3D space) 2 1 2 H W L I = 1 2 H volume W B Hdv = r r g ( 29 2 1 volume L A H dv I = ∇× r r g ( 29 { ( 29 ( 29 vector identity A H H A A H × = ∇× - ∇× r r r r r r g g g ( 29 ( 29 { J H A A H A H ∇× = ∇ × + ∇× ÷ r r r r r r r g g g ( 29 2 1 volume L A H A J dv I = × + r r r r g g 2 2 2 volume H B H dv W L I I = = r r g 2 2 H W L I = We can also express inductance using magnetic vector potential: next slide
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© Copyright Dean P. Neikirk 2004-2009 Mikhail Belkin, EE 325, ECE Dept., UT Austin 5 Another way to calculate inductance: using vector potential now apply the divergence theorem ( 29 2 1 volume L A H A J dv I = × + r r r r g g ( 29 closed volume surface S enclosed by S F dS F dv = r r r r g ( 29 2 1 volume S 0 since volume had to include EVERYTHING L A H dS A Jdv I = × +
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