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Electromagnetics ENGR 367

Electromagnetics ENGR 367. Inductance. Introduction. Question: What physical parameters determine how much inductance a conductor or component will have in a circuit? Answer: It all depends on current and flux linkages!. Flux Linkage. Definition:

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Electromagnetics ENGR 367

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  1. Electromagnetics ENGR 367 Inductance

  2. Introduction • Question: What physical parameters determine how much inductance a conductor or component will have in a circuit? • Answer: It all depends on current and flux linkages!

  3. Flux Linkage • Definition: the magnetic flux generated by a current that passes through one or more conducting loops of its own or another separate circuit • Mathematical Expression:

  4. Types of Inductance • Self-Inductance (L): whenever the flux linkage of a conductor or circuit couples with itself • Mutual Inductance (M): if the flux linkage of a conductor or circuit couples with another separate one

  5. Self-Inductance • Formula by Definition • Applies to linear magnetic materials only • Units:

  6. Inductance of Coaxial Cable • Magnetic Flux • Inductance (as commonly used in transmission line theory)

  7. Inductance of Toroid • Magnetic Flux Density • Magnetic Flux • If core small vs. toroid

  8. Inductance of Toroid • Inductance • Result assumes that no flux escapes through gaps in the windings (actual L may be less) • In practice, empirical formulas are often used to adjust the basic formula for factors such as winding (density) and pitch (angle) of the wiring around the core

  9. Alternative Approaches • Self-inductance in terms of • Energy • Vector magnetic potential (A) • Estimate by Curvilinear Square Field Map method

  10. Inductance of a Long Straight Solenoid • Energy Approach • Inductance

  11. Internal Inductance of a Long Straight Wire • Significance: an especially important issue for HF circuits since • Energy approach (for wire of radius a)

  12. Internal Inductance of a Long Straight Wire • Expressing Inductance in terms of energy • Note: this result for a straight piece of wire implies an important rule of thumb for HF discrete component circuit design: “keep all lead lengths as short as possible”

  13. Example of Calculating Self-Inductance • Exercise 1 (D9.12, Hayt & Buck, 7th edition, p. 298) Find: the self-inductance of a) a 3.5 m length of coax cable with a = 0.8 mm and b = 4 mm, filled with a material for which r = 50.

  14. Example of Calculating Self-Inductance • Exercise 1 (continued) Find: the self-inductance of b) a toroidal coil of 500 turns, wound on a fiberglass form having a 2.5 x 2.5 cm square cross section and an inner radius of 2.0 cm

  15. Example of Calculating Self-Inductance • Exercise 1 (continued) Find: the self-inductance of c) a solenoid having a length of 50 cm and 500 turns about a cylindrical core of 2.0 cm radius in which r = 50 for 0 <  < 0.5 cm and r = 1 for 0.5 <  < 2.0 cm

  16. Example of Estimating Inductance: Structure with Irregular Geometry • Exercise 2: Approximate the inductance per unit length of the irregular coax by the curvilinear square method

  17. Mutual Inductance • Significant when current in one conductor produces a flux that links through the path of a 2nd separate one and vice versa • Defined in terms of magnetic flux (m)

  18. Mutual Inductance • Expressed in terms of energy • Thus, mutual inductances between conductors are reciprocal

  19. Example of Calculating Mutual Inductance • Exercise 3 (D9.12, Hayt & Buck, 7/e, p. 298) • Given: 2 coaxial solenoids, each l = 50 cm long 1st: dia. D1= 2 cm, N1=1500 turns, core r=75 2nd: dia. D2=3 cm, N2=1200 turns, outside 1st • Find: a) L1=? for the inner solenoid

  20. Example of Calculating Mutual Inductance • Exercise 3 (continued) • Find: b) L2 = ? for the outer solenoid • Note: this solenoid has inner core and outer air filled regions as in Exercise 1 part c), so it may be treated the same way!

  21. Example of Calculating Mutual Inductance • Exercise 3 (continued) • Find: M = ? between the two solenoids

  22. Summary • Inductance results from magnetic flux (m) generated by electric current in a conductor • Self-inductance (L) occurs if it links with itself • Mutual inductance (M) occurs if it links with another separate conductor • The amount of inductance depends on • How much magnetic flux links • How many loops the flux passes through • The amount of current that generated the flux

  23. Summary • Inductance formulas may be derived from • Direct application of the definition • Energy approach • Vector Potential Method • The self-inductance of some common structures with sufficient symmetry have an analytical result • Coaxial cable • Long straight solenoid • Toroid • Internal Inductance of a long straight wire

  24. Summary • Numerical inductance may be evaluated by • Calculation by an analytical formula if sufficient information is known about electric current, dimensions and permeability of material • Approximation based on a curvilinear square method if axial symmetry exists (uniform cross section) and a magnetic field map is drawn

  25. References • Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: Bangkok, 2006. • Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: Bangkok, 1999. • Wentworth, Fundamentals of Electromagnetics with Engineering Applications, John Wiley & Sons, 2005.

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