Generating Electromagnetic Waves

Transcription

1 Generating Electromagnetic Waves Outline Parabolic antenna for communicating with spacecraft, Canberra, Australia Image is in the public domain. Uniform Plane Waves Energy Transported by EM Waves (Poynting Vector) Current Sheet & Quasi-static Approximation Transient Analysis for Current Sheet Antenna 1

2 Uniform Plane Wave 100,000 km per div 1 V/m per div Estimate the wavevector k? Estimate the angular frequency? Estimate the magnitude of the electric field. Estimate the magnitude of the magnetic field. 2

3 Coupling of Electric and Magnetic Fields Maxwell s Equations couple H and E fields E dl = d ( C dt S B da) H dl C = J S d da + dt ɛe da Source free The Wave Equation 3

4 Recall Electromagnetic Energy Storage Magnetic Electric Magnetic machine Electric machine 4

5 Power Flow of a Uniform Plane Wave Poynting Vector 5

6 Euler s Formula This gives us the famous identity known as Euler s formula: e iy = cos(y)+i sin(y) From this, we get two more formulas: iy iy e + e iy e e iy cos(y) = sin(y) = 2 2i Exponential functions are often easier to work with than sinusoids, so these formulas can be useful. The following property of exponentials is still valid for complex z: e z 1+z 2 = e z 1 e z 2 Using the formulas on this page, we can prove many common trigonometric identities. Proofs are presented in the text. 6

7 Modern Version of Steinmetz Analysis 1. Begin with a time-dependent analysis problem posed in terms of real variables. 2. Replace the real variables with variables written in terms of complex exponentials; e jωt is an eigenfunction of linear timeinvariant systems. 3. Solve the analysis problem in terms of complex exponentials. 4. Recover the real solution from the results of the complex analysis. 7

8 How Are Uniform EM Plane Waves Launched? Generally speaking, electromagnetic waves are launched by time-varying charge distributions and currents. Man-made systems that launch waves are often called antennas. Uniform plane waves are launched by current sheets. E H K n 8

9 Static Magnetic Field from Current Sheet uniform DC surface current K = Kẑ ŷ ẑ ˆx C H dl = J S d da + dt ɛe da d ˆx ˆx H = E What is the -field? 9

10 Magnetic Field Above/Below a Sheet of Current flowing in the ẑ direction with current density K d uniform DC surface current K = Kẑ ŷ ˆx ẑ ˆx As seen end on, the current sheet may be thought of as a combination of parallel wires, each of which produces its own H field. These fields combine, so that the total H field above and below the current sheet is directed in and direction, respectively. H dl H ˆx In between the wires the fields cancel H = 10

11 Transient Magnetic Field from Current Sheet Turn on current (red) at t =0 t =0 For now lets note that H can t propagate further than distance ct o away from the current sheet in time t o Don t worry about the shape of H yet. Which way does E point? 11

12 S Time-Varying Magnetic Field from Current Sheet K = K ŷ = K cos(ωt y s K o ŷ )ˆ Close to the sheet (quasi-static) H (z = x o ) = + 2 K 0 cos( ω t) S Ey ( Δ z ) = Δzω ( ) 0 sin( t μ o K 2 ω ) ˆx ẑ 2 Δzω H (Δ ( = 1 x z) H x 0) K 2 0 cos(ωt) c Δzω 2 H ( Δ z) = + 1 K co s(ωt) 1 = + 1 K x cos(ωt kz) c 1 H x (z) = + 2 K 0 cos( ωt kz) 12 μ E 0 y = 2 c K 0 cos( ωt kz)

13 Time-Varying Magnetic Field from Current Sheet S ˆx K ŷ S K = K s ŷ = K o cos(ωt)ˆ y η 0K 2 S = E H = 0 cos 2 (ωt 4 Average intensity ẑ I = S ˆ = η 0K0 n 2 8 kz)ˆ z Why do we care about plane waves? 13

14 Examples of the Wireless Energy Transfer An electric toothbrush uses traditional magnetic induction to recharge its batteries, avoiding the need for exposed electrical contacts. A microwave oven utilizes microwave radiation to cook food. Nikola Tesla s Wardenclyffe tower built on Long Island, NY in This tower was intended to implement Tesla s vision of transmitting power and information around the world. The tower was destroyed in MRI machines use magnetic resonance imaging to produce diagnostic images of soft tissue. On June 5, 1975 NASA JPL Goldstone demonstrated directed radiative microwave power transmission successfully transferring 34kW of electrical power over a distance of 1.5km at 82% efficiency. All images are in the public domain. 14

15 Wireless Energy Transfer 30kHz 100kHz -100MHz 300MHz Radio Waves Microwaves 300GHz 800MHz -2GHz 2.45GHz GHz Radio Wi-Fi Microwave oven Tokamak Tokamak image by Sheliak (Princeton Plasma Physics Laboratory) < on Wikimedia, radio, TV and cellphone images are in the public domain 15

16 Cell Phones 30kHz 300MHz Radio Waves Microwaves 300GHz 800MHz -2GHz Current / Planned Technologies Band Frequency (MHz) SMR iden and AMPS, GSM, IS-95 (CDMA), IS-136 (D-AMPS), 3G Cellular and GSM, IS-95 (CDMA), IS-136 (D-AMPS), 3G PCS and G, 4G, MediaFlo, DVB-H 700 MHz Unknown 1.4 GHz and G, 4G AWS and G BRS/EBS Table from 16

17 Inductive Charging Primary coil Secondary coil 17

18 Magnetic Field in a Uniform Electromagnetic Plane Wave B x In free space where 18

19 Uniform Electromagnetic Plane Waves In Materials B x Inside a material where is known as the phase velocity 19

20 Index of Refraction νλ = v p = c/n frequency wavelength When propagating in a material, c c/n λ 0 λ 0 /n k 0 n k 0 Material n Vacuum 1 Air Water liquid Water ice 1.31 Diamond Silicon 3.96 at 5 x Hz λ (m) Radio waves Hard X-rays IR UV Microwaves E(t, z) =Re{E 0 e j(ωt k 0nz) } Visible Soft X-rays Gamma rays E(t, z) =Re{E 0 e j(ωt k z) } 20

21 check Absorption Photograph by Hey Paul on Flickr. Why are these stained glasses different colors? E(t, z) =Re{ E e αz/2 e j(ωt 0 k 0 nz) } Tomorrow: lump refractive index and absorption into a Absorption Refractive complex refractive index ñ coefficient index 21

22 Key Takeaways Propagation velocity Direction of propagation given by E H Energy stored in electric field per unit volume at any instant at any point is equal to energy stored in magnetic field Instantaneous value of the Poynting vector given by E 2 /η = ηh 2 = E H When propagating in a material, c c/n λ0 λ 0 /n k 0 n k 0 E(t, z) =Re{E e αz/2 e j(ωt k 0nz) } Antennas use time-varying current to send Absorption Refractive EM waves. Antennas receive EM waves coefficient index because oscillating fields produce timevarying current. 0 22

23 MIT OpenCourseWare Electromagnetic Energy: From Motors to Lasers Spring 2011 For information about citing these materials or our Terms of Use, visit: