UPB ETTI O DROSU Electrical Engineering 2 Lecture

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

Flow of Electromagnetic Power n n n 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. Lecture 11 3

Derivation of Poynting’s Theorem n Poynting’s theorem concerns the conservation of energy for a given volume in space. n Poynting’s theorem is a consequence of Maxwell’s equations. 4 Lecture 11

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) n Time-Domain Maxwell’s curl equations in differential form 5 Lecture 11

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) n Recall a vector identity n Furthermore, 6 Lecture 11

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) 7 Lecture 11

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) n Integrating over a volume V bounded by a closed surface S, we have 8 Lecture 11

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) n Using the divergence theorem, we obtain the general form of Poynting’s theorem 9 Lecture 11

Derivation of Poynting’s Theorem in the Time Domain (Cont’d) n For simple, lossless media, we have n Note that 10 Lecture 11

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

Derivation of Poynting’s Theorem in the Frequency Domain (Cont’d) n Time-Harmonic Maxwell’s curl equations in differential form for a simple medium 12 Lecture 11

Derivation of Poynting’s Theorem in the Frequency Domain (Cont’d) n Poynting’s theorem for a simple medium 13 Lecture 11

Physical Interpretation of the Terms in Poynting’s Theorem n The terms represent the instantaneous power dissipated in the electric and magnetic conductivity losses, respectively, in volume V. 14 Lecture 11

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) n The terms represent the instantaneous power dissipated in the polarization and magnetization losses, respectively, in volume V. 15 Lecture 11

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) n Recall that the electric energy density is given by n Recall that the magnetic energy density is given by 16 Lecture 11

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) n Hence, the terms represent the total electromagnetic energy stored in the volume V. 17 Lecture 11

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) n The term represents the flow of instantaneous power out of the volume V through the surface S. 18 Lecture 11

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) n The term represents the total electromagnetic energy generated by the sources in the volume V. 19 Lecture 11

Physical Interpretation of the Terms in Poynting’s Theorem (Cont’d) n 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 Lecture 11

Poynting Vector in the Time Domain n We define a new vector called the (instantaneous) Poynting vector as • The Poynting vector has units of W/m 2. n n 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. 21 Lecture 11

Time-Average Poynting Vector n The time-average Poynting vector can be computed from the instantaneous Poynting vector as period of the wave 22 Lecture 11

Time-Average Poynting Vector (Cont’d) n The time-average Poynting vector can also be computed as phasors 23 Lecture 11

Time-Average Poynting Vector for a Uniform Plane Wave n Consider a uniform plane wave traveling in the +z-direction in a lossy medium: 24 Lecture 11

Time-Average Poynting Vector for a Uniform Plane Wave (Cont’d) n The time-average Poynting vector is 25 Lecture 11

Time-Average Poynting Vector for a Uniform Plane Wave (Cont’d) n For a lossless medium, we have 26 Lecture 11

Reflection and Transmission of Waves at Planar Interfaces medium 1 medium 2 incident wave transmitted wave reflected wave 27 Lecture 11

Normal Incidence on a Lossless Dielectric Consider both medium 1 and medium 2 to be lossless dielectrics. n 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. n No loss of generality is suffered if we assume that the electric field of the incident wave is in the x-direction. n 28 Lecture 11

Normal Incidence on a Lossless Dielectric (Cont’d) x medium 1 medium 2 z z=0 29 Lecture 11

Normal Incidence on a Lossless Dielectric (Cont’d) n Incident wave known 30 Lecture 11

Normal Incidence on a Lossless Dielectric (Cont’d) n Reflected wave unknown 31 Lecture 11

Normal Incidence on a Lossless Dielectric (Cont’d) n Transmitted wave unknown 32 Lecture 11

Normal Incidence on a Lossless Dielectric (Cont’d) n The total electric and magnetic fields in medium 1 are 33 Lecture 11

Normal Incidence on a Lossless Dielectric (Cont’d) n The total electric and magnetic fields in medium 2 are 34 Lecture 11

Normal Incidence on a Lossless Dielectric (Cont’d) n To determine the unknowns Er 0 and Et 0, we must enforce the BCs at z = 0: 35 Lecture 11

Normal Incidence on a Lossless Dielectric (Cont’d) n From the BCs we have or 36 Lecture 11

Reflection and Transmission Coefficients n Define the reflection coefficient as n Define the transmission coefficient as 37 Lecture 11

Reflection and Transmission Coefficients n Note also that(Cont’d) The definitions of the reflection and transmission coefficients do generalize to the case of lossy media. n For lossless media, G and t are real. n n For lossy media, G and t are complex. 38 Lecture 11

Traveling Waves and Standing Waves n The total field in medium 1 is partially a traveling wave and partially a standing wave. n The total field in medium 2 is a pure traveling wave. 39 Lecture 11

Traveling Waves and Standing Waves (Cont’d) n The total electric field in medium 1 is given by standing wave traveling wave 40 Lecture 11

Traveling Waves and Standing Waves: Example x medium 1 medium 2 z z=0 41 Lecture 11

Traveling Waves and Standing Waves: Example (Cont’d) 1. 4 Normalized E field 1. 3 1. 2 1. 1 1 0. 9 0. 8 0. 7 0. 6 -2 -1. 5 -1 -0. 5 z/l 0 42 0 0. 5 1 Lecture 11

Standing Wave Ratio n The standing wave ratio is defined as n In this example, we have 43 Lecture 11

Time-Average Poynting Vectors 44 Lecture 11

Time-Average Poynting Vectors (Cont’d) We note that 45 Lecture 11

Time-Average Poynting Vectors (Cont’d) n Hence, Power is conserved at the interface. 46 Lecture 11

Oblique Incidence at a Dielectric Interface 47 Lecture 11

Oblique Incidence at a Dielectric Interface: Parallel Polarization (TM to z) 48 Lecture 11

Oblique Incidence at a Dielectric Interface: Parallel Polarization (TM to z) 49 Lecture 11

Oblique Incidence at a Dielectric Interface: Perpendicular Polarization (TE to z) 50 Lecture 11

Oblique Incidence at a Dielectric Interface: Perpenidcular Polarization (TM to z) 51 Lecture 11

Brewster Angle n The Brewster angle is a special angle of incidence for which G=0. n For dielectric media, a Brewster angle can occur only for parallel polarization. 52 Lecture 11

Critical Angle n The critical angle is the largest angle of incidence for which k 2 is real (i. e. , a propagating wave exists in the second medium). n For dielectric media, a critical angle can exist only if e 1>e 2. 53 Lecture 11