Prof. David R. Jackson ECE Dept. Fall 2014 Notes 31 ECE 2317 Applied Electricity and Magnetism 1

Inductance The unit normal is chosen from a “right hand rule for the inductor” : I S The fingers are in the direction of the current reference direction, and the thumb is in the direction of the unit normal. Definition of inductance: Single turn coil 2 Note: Please see the Appendix for a summary of right-hand rules. The current I produces a flux though the loop. Magnetic flux through loop: Note: L is always positive

N-Turn Solenoid The definition of inductance is now 3 N I We assume here that the same flux cuts though each of the N turns.

Example Find L N turns z I a LsLs so 4 Note: We neglect “fringing” here and assume the same magnetic field as if the solenoid were infinite.

Example N turns z I a LsLs Final result: 5 Note: L is increased by using a high-permeability core!

Example  Find H inside toroid  Find L 6 Note: This is a practical structure, in which we do not have to neglect fringing and assume that the length is infinite. Toroidal Inductor I N turns

Example (cont.) 7

Assume A is the cross-sectional area: a A 8 The radius R is the average radius (measured to the center of the toroid). I N turns Example (cont.)

Hence so 9 We then have N turns I C r 

Example (cont.) Hence 10 S a I N turns R

Example Coaxial cable Ignore flux inside wire and shield Find L l (inductance per unit length) A short is added at the end to form a closed loop. S h + - I I z Center conductor Note: S is the surface shaded in green. 11 I I h a b z

Example (cont.) 12 S h + - I I z Center conductor

Example (cont.) 13 S h + - I I z Center conductor

Example (cont.) Hence Per-unit-length: 14 I I h a b z

Example (cont.) Hence: 15 Recall: From previous notes: (intrinsic impedance of free space) Note:  r =1 for most practical coaxial cables.

Voltage Law for Inductor C + - v (t)v (t) B 16 Apply Faraday’s law: Note: There is no electric field inside the wire. PEC wire: Hence Note: The unit normal (and hence the direction of the flux) is chosen from the right-hand rule according to the direction of the path C.

Note: We are using the “active” sign convention. Voltage Law for Inductor (cont.) i C + - v B so or 17 Note: The direction of the current is chosen to be consistent with the direction of the defined flux, according to the RH rule for the inductor.

Note: This assumes a passive sign convention. To agree with the usual circuit law (passive sign convention), we change the reference direction of the voltage drop. (This introduces a minus sign.) i + - v B Voltage Law for Inductor (cont.) Passive sign convention (for loop) 18

Energy Stored in Inductor i t = 0 L + v(t) Hence we have 19 + v -

Energy Formula for Inductor We can write the inductance in terms of stored energy as: Next, we use We then have 20 This gives us an alternative way to calculate inductance.

Example Find L Solenoid From a previous example, we have Energy formula: Hence, we have 21 N LsLs z a

Example Coaxial cable Find L l 22 We ignore magnetic stored energy from fields inside the conductors. We would have these at DC, but not at high frequency due to the skin effect. I I h b z a

Example (cont.) Per-unit-length: 23 Note: We could include the inner wire region if we had a solid wire at DC. We could even include the energy stored inside the shield at DC. These contributions would be called the “internal inductance” of the coax. I I h b z a

Appendix 24 In this appendix we summarize the various right-hand rules that we have seen so far in electromagnetics.

Summary of Right-Hand Rules Faraday’s law: Stokes’ theorem: Fingers are in the direction of the path C, the thumb gives the direction of the unit normal. Fingers are in the direction of the path C, the thumb gives the direction of the unit normal (the reference direction for the flux). (fixed path) Ampere’s law: Fingers are in the direction of the path C, the thumb gives the direction of the unit normal (the reference direction for the current enclosed). 25 The following two RH rules follow from Stokes’s theorem.

Summary of Right-Hand Rules (cont.) Inductor definition: Fingers are in the direction of the current I and the thumb gives the direction of the unit normal (the reference direction for the flux). Magnetic field law: For a wire or a current sheet or a solenoid, the thumb is in the direction of the current and the fingers give the direction of the magnetic field. (For a current sheet or solenoid the fingers are simply giving the overall sense of the direction.) 26