NE 2 Chap. 1 Fundamental Concepts Important Laws in Electromagnetics Coulomb s Law (1785) Gauss s Law (1839) Ampere s Law (1827) Ohm s Law (1827) Kirchhoff s Law (1845) Biot-Savart Law (1820) Faradays Law (1831) Maxwell Equations(1873) The governing equations of macroscopic electromagnetic phenomena. Predict the existence of electromagnetic waves. Predict light to be electromagnetic waves. Verification of electromagnetic waves: Hertz (1887-1891) Radio communication: Marconi (1901) : Electric field intensity (V/m) : Electric flux density (A/m) : Magnetic field intensity (C/m 2 ) : Magnetic flux density (W/m 2 ) : Electric current density (A/m 2 ) : Volume electric charge density (C/m 3 )
NE 3 Continuity Equation Conservation of charges Integral Forms of Maxwell Equations : total magnetic flux. : total current. : total charge. Note: differential form only works when derivatives exist. Constituent relationship In general, In free space, where vacuum. (F/m): Permittivity or capacitivity of (H/m): Permeability or inductivity of vacuum.
NE 4 Note: In simple media, Not true when or are very large, or the time derivatives of or are large. In general, for linear material, Category of material: conductor: large,. Good or perfect conductor:. dielectric: small,. Good dielectric or insulator:. diamagnetic:, (of the order of 0.01 percent). paramagnetic:, (of the order of 0.01 percent). ferromagnetic:. Generalized Current Concept, where t, c and i are total, conduction and impressed currents. is magnetic current.
Vector Analysis NE 5 What is a coordinate system? Examples: Rectangular coordinate system Polar coordinate system y (a,b) b a x r (r, ) Suppose a coordinate system described by three ordered variable. When we say a point s coordinate is, we mean the point is located at the interception of the three surfaces defined by. When the three surfaces are always orthogonal, the coordinate system is called orthogonal coordinate systems. What is a vector? A vector is a quantity defined by a direction and a magnitude. What is a vector field? A vector field is a distribution of vectors whose directions and magnitudes are a function of location. Base vectors
NE 6 In order to define the direction everywhere, the surface normals of the constant surfaces at each point are used as base vectors to defined the direction of the vector. The directions of the surface normals are chosen to point to the direction at which the coordinate increases. In a general right-handed, orthogonal, curvilinear coordinate system, the base vectors are arranged in such a way that the following relations are satisfied:, Note that in general the base vectors are functions of coordinate. Then a vector field can be represented as! Vector Algebra " Addition and subtraction. " Dot product: " Cross product : " Identities - -! Orthogonal coordinate systems " Cartesian " Cylindrical
NE 7 " Spherical! Vector calculus " Integration " Gradient " Divergence " Curl Metric coefficient Some coordinate variables may not correspond to the actual length, therefore a conversion factor is needed to convert a differential change, say to a change in length by a factor, i.e., Line Integral Surface Integral Volume Integral! Cartesian coordinates
NE 8 " " " " " "! Cylindrical coordinates " " " " " Coordinate transform
NE 9! Spherical coordinates " " " " " " Coordinate transform
NE 10 Vector Calculus Integration! Volume integration! Line integration " Scalar: " Vector:! Surface integration:
NE 11 Gradient! " Cartesian: " Cylindrical: " Spherical:! " is normal to the surface " represent both the magnitude and the direction of the maximum space rate of increase of a scalar function. " Proof: From the above, the maximum of occurs when. Let be a tangential vector on the surface,, therefore.
EMT 12 Divergence! " Cartesian: " Cylindrical: " Spherical:! Definition:! Divergence Theorem
EMT 13 Curl! " Cartesian: " Cylindrical: " Spherical:! Definition:! Stoke s theorem:
EMT 14 Two Null Identities Ex: Ex: Helmhotz s Theorm Let In other words, if the curl and divergence of a vector field are known, the vector field can be uniquely determined within a constant.
EMT 15 Power Relationship From vector identity or for simply medium or in integral form That is, where : supplied energy ( ) : flow out energy ( ), Poynting vector. : dissipated energy ( ) : stored electric and magnetic energy ( ) : total stored electric and magnetic energy in a closed surface.! Boundary Conditions
EMT 16! Time-Harmonic Fields Time-harmonic: : a real function in both space and time. : a real function in space. : a complex function in space. A phaser. Thus, all derivative of time becomes. For a partial deferential equation, all derivative of time can be replace with, and all time dependence of can be removed and becomes a partial deferential equation of space only. Representing all field quantities as
EMT 17, then the original Maxwell s equation becomes! Power Relationship! Poynting vector: Complex Constitutive Parameters Similarly,
EMT 18 DC approximation Let, Note, : loss due to free charge. : loss due to bound charge. Conductor: Dielectric: Correction to power relationship: Homogeneous: is independent of position. Isotropic: is independent of direction. Linear: the relationship between and is linear. Simple medium: Homogeneous, isotropic, and linear. In general,
EMT 19 Anisotropic: Nonlinear: Inhomogeneous:
EMT 20
Chap. 2 Introduction to Waves EMT 21 The Wave Equation Assume simple media and source free ( ). Taking curl to the first two equations, we have. Let (wave number), then (Complex vector wave equation) Applying vector identity, we have, In Cartesian coordinates, assume only exist, Assume is independent of x and y, then The solutions are ( real)..
Consider the minus solution. In time domain, EMT 22 Constant phase, intrinsic impedance In general, where The solution is a uniform plane wave.
EMT 23 On Wave in General A wave function can be specified in complex domain as below: where : the magnitude, real, : the phase, real. The corresponding time domain wave function is Equal phase surface are defined as Definitions: 1. Plane, cylindrical, or spherical waves: equal phase surfaces are planes, cylinders, or spheres. 2. Uniform waves: amplitude is constant over
EMT 24 the equal phase surface. 3. Wave normal: surface normals of the equiphase surfaces. is the direction and is the curve along which the phase changes most rapidly. 4. Phase constant: the rate at which the phase decreases in some direction is called the phase constant in that direction (note: phase constant is not necessary a constant). Phase constant can be written in vector form as. The maximum phase constant is therefore. 5. Phase velocity: the speed the constant phase surface moves at in a given direction. The instantaneous equiphase surface of a wave is For ant increment, the change in is To keep the phase constant for an incremental increase in time, corresponding incremental change in is necessary. We have. In cartesian coordinates The phase velocity along a wave normal is
EMT 25 and is the smallest. Alternatively, the wave function can be expressed as where is a complex function. A complex propagation constant can be defined as where is the vector phase constant and is the vector attenuation constant. 6. Wave impedance: the ratios of components of to. Follow right-hand cross-product rule of component rotated into. For example,,, Example:
EMT 26 Wave in Perfect Dielectrics Energy relations Define velocity of propagation of energy as Then, In general, or Standing Waves Let and in time domain,,
EMT 27 Properties: 1. The field oscillates in amplitude in stead of traveling; Hence the name standing wave. 2. E reaches maximum when H reaches minimum. In other words, E and H are out of phase. 3. Planes of zero E and H are fixed in space. Zeros of E and H are separated by quarter- wavelength.
EMT 28 Polarization In general, If and, then 1. Elliptically polarized: in general. 2. Linearly polarized:. 3. Circularly polarized: Ex: (RHC)
EMT 29 No change in energy and power densities with time or space, steady power flow. Ex: (circular polarized standing-wave) 1. and are always parallel to each other. 2. and rotate about the axis as time progresses. 3. Amplitude is independent of time. 4. Energy and power densities are independent of time.
EMT 30
EMT 31 Oblique incidents Perpendicular Polarization (TE),,,, The above fields must satisfy Boundary conditions,,. which lead to..
Note that the condition holds. If at (Brewster (polarization) angle), then EMT 32 Therefore,, or. No solution for. Snell s Law Definition of refraction index Total reflection occurs at critical angle. When, is real. At angle larger than critical angle, surface wave exists in dielectric. The wave decays inside the dielectric. For, the propagation constant in media 2 becomes, Note that the minus sign is chosen such that the resulting field quantity in media 2 won t grow to
infinity. In view of the phase term of the propagation wave in media 2, the wave decay in the media in direction. EMT 33
EMT 34 Parallel Polarization (TM),,,..,., Note that the condition holds. If at Brewster angle, then.
EMT 35 Therefore, or. For nonmagnetic media,, Applications: Polarization separation, anti-flare glasses
EMT 36
TEM Waves Assume and dependence of the form. Substitute to Maxwell s equations, we have EMT 37 These lead to 1. The propagation constant of any TEM wave is the intrinsic propagation constant of the media. Also, 2.. The z-directed wave impedance of any TEM wave is the intrinsic wave impedance of the medium. Let, then from wave equation we have. Similarly, The boundary conditions at perfect conductors are Also,
EMT 38 Therefore, there exists unique such that. 3. The boundary-value problem for and is the same as the 2-dimensional electrostatic and magnetostatic problem. Thus, static capacitances and inductances can be used for transmission lines even though the field is time-harmonic. 4. The conductor must be perfect, otherwise will exist.
EMT 39 Multiplying both, we have. Equating both, we have
EMT 40 Radiation r potential) (vector potential) (scala Let, then (Helmholtz equation, or complex wave equation) Assume the source is an z-directed infinite small current element or electric dipole of moment Il located at the origin in free space. Then, outside the source,. Due to spherical symmetry,. We have. The solutions are. Choose the minus solution due to out-going wave assumption. Then,. Since (Spherical wave)
EMT 41 1. Near field approximation. Quasi-static.( ) No power flow. 2. Far field approximation.( ) TEM wave. (HW#1)
Chapter 3 Some Theorems and Concepts EMT 42 Duality Due to symmetry of Maxwell s Equations, systematic exchange of variables leads to the same form of equations. Thus the solutions will be of the same form. Suppose we have as our source producing and in medium and a source producing and in medium. If is made to be equal to in terms of functional form, and,, then Example:
EMT 43 Example: The magnetic field of an electric dipole is The electric field due to a magnetic dipole is While for a electric current loop,. Therefore, an electric current loop can be considered equivalent to a magnetic dipole according to the following equation. (Homework #2) Uniqueness Theorem Let excitation be two sets of solutions of the same, then Similar to the way Poynting Theorem is proved,
EMT 44 The surface integral vanishes if 1., tangential electric field specified, or 2., tangential magnetic field specified. then. Assuming that real medium is always lossy, then for the second term to be zero, it is necessary that everywhere in. Lossless case can be consider as the limit of the lossy case. Therefore, in a enclosed volume, if the source in the volume and the tangential fields on the boundary are the same, the fields are the same everywhere inside the volume. If the given sources is in an unbounded lossy region, the surface integral term vanish too. Thus, the uniqueness theorem still hold. Image Theory An application of Uniqueness Theorem.
EMT 45
EMT 46 Equivalence Principle Another application of Uniqueness Theorem In a volume, two sets of fields produced by two sets of sources are equivalent(the same) if the tangential fields on the boundary of the volume and the sources in the volume are the same. Case 1. Zero field replacement
EMT 47 Case 2. Exchanging two fields Case 3. Perfect electric or magnetic conductor replacement
EMT 48 Field in Half-space Approximate: The Induction Theorem
EMT 49 Define: Total field : the field generated by the source with the obstacle present. Incident field : the field generated by the sources with the obstacle absent. Scattered field : the difference of the total and incident field, that is, Let exist inside the obstacle and exist outside the obstacle. By equivalence principle, must exist on the surface of the obstacle and satisfy Usually, are known. However, the field generated by with the obstacle present is not known. Approximation is needed. Example: Scattering by a conducting plane
EMT 50 Approximate the finite plate to infinite plate, then image theory can be applied to get at far field where A is the area of the plate. Effective area or radar cross section Reciprocity Theorem Two sets of solutions with two sets of excitation in the same space: satisfying Maxwell s Equations
EMT 51 Then by vector identity, we have Note that for the above equation to be true, it is required that, that is, and. 1. No sources 2. Bound by a perfect conductor 3. Unbounded Example: proof of circuit reciprocity. For instance, in a two port network,
EMT 52 Since. Therefore, Example: the receiving pattern and transmission pattern of an antenna are the same. Let be the antenna in interest. When is excited by current causing voltage at antenna, is proportional to the transmission pattern of antenna. When is excited by current causing voltage at antenna, is proportional to the receiving pattern of antenna. By reciprocity,. Thus, the receiving pattern and transmission pattern are the same. Example: an electric current impressed along the
EMT 53 surface of a perfect conductor radiated no field. Proof: Let be on the conductor and anywhere outside the conductor. Then Green s Functions Reciprocity relationships are formulas symmetrical in two field-source pairs. Mathematical statements of reciprocity are called Green s theorems. From identity apply divergence theory, we have Green s first identity. Interchange and, then subtract each other, we have Green s second identity Similarly, based on vector identity we have Green s first vector identity and second vector identity
EMT 54 Ex: Let. If, then is the vector potential generated by an infinite small current at. Then, any in a source free region enclosed by can be express as (Use Green s second vector identity) Homework #3: prove Eq. (3-53) Here the field of a point source is called the Green s function.
EMT 55 Ex: Suppose in a volume bounded by surface where the tangential electric field or magnetic field vanishes, the vector potential and the source satisfy the following equation and the required boundary condition. If is a unit delta source located at pointing in direction, that is. Let the solution be. Then, From linear system point of view, for any arbitrary, since. That is, is superposition of, thus, the solution is also a superposition of. Therefore, the solution is. Tensor Green s Functions In general, caused by point source at can be expressed in tensor form as
EMT 56 where is called tensor Green s functions. In rectangular coordinate, the equation can be expressed in matrix form as or The electric field generated by arbitrary source expressed as can be Ex: derive free space Green s function from following equations. Alternatively, consider be a linear operator. Define inner product. Let and are the solutions of the wave equations in satisfying the required boundary conditions. From Green s identity. Then,
EMT 57 The surface integral vanishes due to the boundary condition since the stronger condition is or. Or, if the volume is unbounded,, the surface integral will vanish too. Too sum up,. That is is a self-adjoint operator. Then,
EMT 58 That is, Therefore, we can write or in dyadic form.
EMT 59 Integral Equations An integral equation is one for which the unknown quantity appears in an integrand. Ex: the induction theorem of Fig. 3-16. Let be the tenser Green s function of generated by. Then, the total scattered field for the problem of is. Since and on the conductor surface, then for on S. Ex: inhomogeneous material. Assume,. In the obstacle Let, then the obstacle can be removed and the whole space can be treated as free space. Thus, free space Green s function can be used. Inside the obstacle, we have
EMT 60
EMT 61 Solutions in a Homogeneous Source-free Region Because sourceless and homogeneous, we have and Also, and satisfy How to divide the field between and? Let us choose an arbitrary direction, say, and construct the vector potential as. Then, we have Since, it is called transverse magnetic to z (TM).
EMT 62 Like wise, if choose, we have the dual case, Since, it is called transverse electric to z (TE). Note that and satisfy Now suppose we have a field neither TE nor TM, we can determine a and according to and. Thus, an arbitrary field in a homogeneous sourcefree region can always be expressed as the sum of a TM field and a TE field. In general, let be a constant vector, then If not sourceless in the region, then
EMT 63
EMT 64 The Radiated Field Consider far field: At far field, HW#3 Prove the above two equations. HW#4 Problem 3-7, Babinet s principle.