UPB / ETTI O.DROSU Electrical Engineering 2 Lecture 11: Electromagnetic Power Flow; Reflection And Transmission Of Normally and Obliquely Incident Plane Waves; Useful Theorems 1

Lecture 11 Objectives To study electromagnetic power flow; reflection and transmission of normally and obliquely incident plane waves; and some useful theorems. 2

Flow of Electromagnetic Power Electromagnetic waves transport throughout space the energy and momentum arising from a set of charges and currents (the sources). If the electromagnetic waves interact with another set of charges and currents in a receiver, information (energy) can be delivered from the sources to another location in space. The energy and momentum exchange between waves and charges and currents is described by the Lorentz force equation. 3

Derivation of Poynting’s Theorem Poynting’s theorem concerns the conservation of energy for a given volume in space. Poynting’s theorem is a consequence of Maxwell’s equations. 4

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Time-Domain Maxwell’s curl equations in differential form 5

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Recall a vector identity Furthermore, 6

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) 7

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Integrating over a volume V bounded by a closed surface S, we have 8

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Using the divergence theorem, we obtain the general form of Poynting’s theorem 9

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) For simple, lossless media, we have Note that 10

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) Hence, we have the form of Poynting’s theorem valid in simple, lossless media: 11

Derivation of Poynting’s Theorem in the Frequency Domain (Cont’d) Time-Harmonic Maxwell’s curl equations in differential form for a simple medium 12

Derivation of Poynting’s Theorem in the Frequency Domain (Cont’d) Poynting’s theorem for a simple medium 13

Physical Interpretation of the Terms in Poynting’s Theorem represent the instantaneous power dissipated in the electric and magnetic conductivity losses, respectively, in volume V. 14

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) represent the instantaneous power dissipated in the polarization and magnetization losses, respectively, in volume V. 15

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) Recall that the electric energy density is given by Recall that the magnetic energy density is given by 16

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) Hence, the terms represent the total electromagnetic energy stored in the volume V. 17

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) represents the flow of instantaneous power out of the volume V through the surface S. 18

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) represents the total electromagnetic energy generated by the sources in the volume V. 19

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) In words the Poynting vector can be stated as “The sum of the power generated by the sources, the imaginary power (representing the time-rate of increase) of the stored electric and magnetic energies, the power leaving, and the power dissipated in the enclosed volume is equal to zero.” 20

Poynting Vector in the Time Domain We define a new vector called the (instantaneous) Poynting vector as The Poynting vector has the same direction as the direction of propagation. The Poynting vector at a point is equivalent to the power density of the wave at that point. The Poynting vector has units of W/m2. 21

Time-Average Poynting Vector The time-average Poynting vector can be computed from the instantaneous Poynting vector as period of the wave 22

Time-Average Poynting Vector (Cont’d) The time-average Poynting vector can also be computed as phasors 23

Time-Average Poynting Vector for a Uniform Plane Wave Consider a uniform plane wave traveling in the +z-direction in a lossy medium: 24

Time-Average Poynting Vector for a Uniform Plane Wave (Cont’d) The time-average Poynting vector is 25

Time-Average Poynting Vector for a Uniform Plane Wave (Cont’d) For a lossless medium, we have 26

Reflection and Transmission of Waves at Planar Interfaces medium 2 medium 1 incident wave reflected wave transmitted wave 27

Normal Incidence on a Lossless Dielectric Consider both medium 1 and medium 2 to be lossless dielectrics. Let us place the boundary between the two media in the z = 0 plane, and consider an incident plane wave which is traveling in the +z-direction. No loss of generality is suffered if we assume that the electric field of the incident wave is in the x-direction. 28

Normal Incidence on a Lossless Dielectric (Cont’d) x medium 1 medium 2 z z = 0 29

Normal Incidence on a Lossless Dielectric (Cont’d) Incident wave known 30

Normal Incidence on a Lossless Dielectric (Cont’d) Reflected wave unknown 31

Normal Incidence on a Lossless Dielectric (Cont’d) Transmitted wave unknown 32

Normal Incidence on a Lossless Dielectric (Cont’d) The total electric and magnetic fields in medium 1 are 33

Normal Incidence on a Lossless Dielectric (Cont’d) The total electric and magnetic fields in medium 2 are 34

Normal Incidence on a Lossless Dielectric (Cont’d) To determine the unknowns Er0 and Et0, we must enforce the BCs at z = 0: 35

Normal Incidence on a Lossless Dielectric (Cont’d) From the BCs we have or 36

Reflection and Transmission Coefficients Define the reflection coefficient as Define the transmission coefficient as 37

Reflection and Transmission Coefficients (Cont’d) Note also that The definitions of the reflection and transmission coefficients do generalize to the case of lossy media. For lossless media, G and t are real. For lossy media, G and t are complex. 38

Traveling Waves and Standing Waves The total field in medium 1 is partially a traveling wave and partially a standing wave. The total field in medium 2 is a pure traveling wave. 39

Traveling Waves and Standing Waves (Cont’d) The total electric field in medium 1 is given by traveling wave standing wave 40

Traveling Waves and Standing Waves: Example medium 1 medium 2 z z = 0 41

Traveling Waves and Standing Waves: Example (Cont’d) -2 -1.5 -1 -0.5 0.5 1 0.6 0.7 0.8 0.9 1.1 1.2 1.3 1.4 z/ l Normalized E field 42

Standing Wave Ratio The standing wave ratio is defined as In this example, we have 43

Time-Average Poynting Vectors 44

Time-Average Poynting Vectors (Cont’d) We note that 45

Time-Average Poynting Vectors (Cont’d) Hence, Power is conserved at the interface. 46

Oblique Incidence at a Dielectric Interface 47

Oblique Incidence at a Dielectric Interface: Parallel Polarization (TM to z) 48

Oblique Incidence at a Dielectric Interface: Parallel Polarization (TM to z) 49

Oblique Incidence at a Dielectric Interface: Perpendicular Polarization (TE to z) 50

Oblique Incidence at a Dielectric Interface: Perpenidcular Polarization (TM to z) 51

Brewster Angle The Brewster angle is a special angle of incidence for which G=0. For dielectric media, a Brewster angle can occur only for parallel polarization. 52

Critical Angle The critical angle is the largest angle of incidence for which k2 is real (i.e., a propagating wave exists in the second medium). For dielectric media, a critical angle can exist only if e1>e2. 53

Some Useful Theorems The reciprocity theorem Image theory The uniqueness theorem 54

Image Theory for Current Elements above a Infinite, Flat, Perfect Electric Conductor magnetic actual sources images 55

Image Theory for Current Elements above a Infinite, Flat, Perfect Magnetic Conductor electric magnetic actual sources h h images 56