Fields and Waves I

1. Fields and Waves I Lecture 16 Faraday’s Law K. A. Connor Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute, Troy, NY Y. Maréchal Power Engineering Department Institut National Polytechnique de Grenoble, France

2. These Slides Were Prepared by Prof. Kenneth A. Connor Using Original Materials Written Mostly by the Following: • Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY • J. Darryl Michael – GE Global Research Center, Niskayuna, NY • Thomas P. Crowley – National Institute of Standards and Technology, Boulder, CO • Sheppard J. Salon – ECSE Department, Rensselaer Polytechnic Institute, Troy, NY • Lale Ergene – ITU Informatics Institute, Istanbul, Turkey • Jeffrey Braunstein – Chung-Ang University, Seoul, Korea Materials from other sources are referenced where they are used. Those listed as Ulaby are figures from Ulaby’s textbook. Fields and Waves I

3. Overview • Review • Ampere’s Law • Magnetic Flux • Magnetic Vector Potential • Faraday’s Law • EMF • Induced Voltage/Current • Moving Magnet or Loop • Inductance • Self Inductance • Mutual Inductance • Quiz review Fields and Waves I

4. Ampere’s Law Maxwell’s Equations: Ampere’s Law Integral form Magnetostatics Electrostatics Fields and Waves I

5. Use right-hand rule • thumb along • fingers are in & B-Fields & wraps around Direction of http://encarta.msn.com/media_701504656_761566543_-1_1/Right-Hand_Rule.html multiple wires or segments - use superposition Fields and Waves I

6. Magnetic Flux & Magnetic Vector Potential Magnetic Vector potential definition: Flux definition: Alternative way to find FLUX Magnetic FLUX Fields and Waves I

7. Example – Field Due To Several Wires What is the direction of B and A at the 4 indicated points? z-direction is into the page Fields and Waves I

8. Field Due To Long Straight Wire First, determine the magnetic field due to a long straight wire carrying a current I. (See Example 5-5 of Ulaby) Line for Ampere’s Law I http://www.ee.surrey.ac.uk/Workshop/advice/coils/terms.html Fields and Waves I

9. Long Straight Wire The magnetic vector potential can be determined from first principles or from the magnetic field. We will do the latter. Specifying the zero reference will determine this constant From the curl expression Note that the vector potential is always in the direction of the current Fields and Waves I

10. Magnetic Vector Potential Direction All currents are in the z-direction and, thus, the vector potential will also be in the z-direction. Its sign is arbitrary, since we are free to select the reference potential point anywhere. That is, we could chose all potentials to be positive or all to be negative. Fields and Waves I

11. 1 2 Magnetic Field Direction At point 1 (point 3 has to opposite sign): 4 3 Fields and Waves I

12. 1 2 Magnetic Field Direction At point 2: 4 3 Fields and Waves I

13. Faraday’s Law and dynamic fields Faraday’s Law

14. Faraday’s Law In electrostatics, we used: Faraday’s Law comes from Maxwell’s equation: • inductors • transformers • motors • generators • noise Applications: Fields and Waves I

15. Use right hand rule for and Faraday’s Law - concept of EMF Time varying flux through a coil is the electromotive force The emf is similar to a VOLTAGE Orientation issues : Fields and Waves I

16. Faraday’s Law – various types of EMF What does the flux derivative means ? The emf may come from: • A dynamic field and a stationary loop • A moving loop in a static field • Both moving loop and dynamic field Fields and Waves I

17. d l 50 W B 10 cos( w t) V I I I I I 1 MHz ~ 1 m H solenoid coil 1 function generator 2 Faraday’s Law : dynamic field experiment Assume that we hook up the experiment as shown where the 1 micro Henry inductor is connected across the output of the function generator and monitor the output of the generator using one scope channel. Then, place a coil facing the inductor and connect it to the other scope channel. An induced voltage is observed at the ends of the pickup coil (in phase with the generator) Fields and Waves I

18. Transformers : Faraday’s law with dynamic fields • A huge range in sizes http://www.transformerfactory.com/e1-model-small-power-transformer-1va-70a.html http://www.meppi.com/Products/Transformers/Power/Pages/Core-formTransformers.aspx http://en.ferilex.eu/transformers.html Fields and Waves I

19. Faraday’s Law: moving loop experiment A loop falls through the magnetic field between two pole faces at a constant velocity, u0. A current is flowing in the loop as it pass through the magnets Fields and Waves I

20. Generators : Faraday’s Law Hoover Dam http://isu.indstate.edu/jspeer/conservation/ Fields and Waves I http://www.wenzelontheweb.de/Hoover%20Dam.htm

21. Faraday’s Law and dynamic fields Dynamic fields

22. 50 W l d B 10 cos( w t) V I I I I I 1 MHz ~ 1 m H solenoid function generator coil 1 2 Example 1 • For the solenoid, inside and 0 outside • n is the number of turns per unit length • a is the radius of the solenoid and the coil Fields and Waves I

23. Example 1 • Circuit analysis. What are the current I and voltage V through the inductor? • What is the flux,  = B ds, through the loop? Do this analytically and then obtain a numerical value for n = 1560 and solenoid radius a = 2.5 mm. Pay attention to the signs/direction of dl and ds. • c. What is the emf induced around the loop? Again do an analytical calculation, but then plug in the numbers from above. • 1) At t=0+, does a scope read V1 - V2 > or < 0? • 2) If the clip leads were connected through a low impedance, which way would current flow at t=0+? • d. Sketch emf and  vs time. What is the flux when the emf is largest? Fields and Waves I

24. Example 1 Fields and Waves I

25. Example 1 Fields and Waves I

26. B I I I I I Faraday’s Law and Lenz’s law Previous result : t=0+, flux decreasing, I as shown Lenz’s law : “The current in the loop is always in such a direction as to oppose the change of magnetic flux that produced it.” l 1 2 Low impedance Output Ulaby High impedance Output Fields and Waves I

27. Faraday’s Law and dynamic fields Moving conductors

28. Previous example had: In moving loop example: ,but changes with time is a function of t Faraday’s Law for moving loop 0 Fields and Waves I

29. sliding bar B-field into page At time t1 At time t2 = t1 + Dt sliding bar Faraday’s Law for moving loop Fields and Waves I

30. Faraday’s Law for moving loop Approximate flux derivative as: Dt A general form: Alternate expression Fields and Waves I

31. Example 2 A loop falls through the magnetic field between two pole faces at a constant velocity, u0. Assume that the magnetic field is B0 between the pole faces and that the fringe fields are 0. Plot the flux through the loop,  =B  ds, as a function of time. Calculate the emf around the loop for all times by derivation of the flux. Calculate the emf around the loop for all times indicated using the uxB. If the loop is connected across a low impedance output, will the current be in the clockwise direction, 0, or in the counter-clockwise direction? Fields and Waves I

32. Example 2 Fields and Waves I

33. Example 2 Fields and Waves I

34. Faraday’s Law and dynamic fields Inductances

35. , creates • coil of wire with • wire loop intersects • this creates Inductances • self inductance - e.g. inductors Two types of Inductances: • mutual inductance - e.g. transformers Self Inductance: http://www.gaussbusters.com/ppm93.html Fields and Waves I

36. Inductor Geometric parameters for a solenoidal inductor http://www3.telus.net/chemelec/Calculators/Helical-Coil-Calc.htm Fields and Waves I

37. Example 3 : Solenoid Inductance • Consider a solenoid with N turns, length l , and radius a . Assume the current is sinusoidal with a frequency f and ignore fringing effects. • What emf, E  dl is induced around the solenoid (include all turns)? • The "voltage" across an inductor is the emf (with care taken about signs). Find the solenoid inductance by substituting the absolute value of the emf in part b. for the voltage in V = L dI/dt. • What is the flux linkage through all N turns? • Calculate L = Flux/I and compare with your answer to part c. Fields and Waves I

38. Example 3 Fields and Waves I

39. not true for finite solenoid and will need: Self inductance Two ways to calculate the inductance: • Calculate the emf then use = L dI/dt. • Calculate the total flux linkage and use L = Total Flux / I or Things to remember : The flux linkage, • only if all loops intersect same flux Fields and Waves I

40. , because Note: To calculate L, don’t need Faraday’s Law just need: is independent of I Self inductance • x N, because • x N, because thus L depends on materials (through m) and geometry (like C) Fields and Waves I

41. Coil 1 Coil 2 Mutual inductance Current through Coil 1 induces e.m.f. in Coil 2 Mutual Inductance: Fields and Waves I

42. Mutual Inductance Coil 1 Coil 2 Mutual inductance where, Also, And, Fields and Waves I