This document discusses different types of electromagnetic waves that can propagate in waveguides, including transverse electric (TE), transverse magnetic (TM), and transverse electromagnetic (TEM) modes. It focuses on analyzing TE and TM modes in rectangular waveguides. Only certain discrete modes are allowed to propagate depending on the waveguide dimensions and frequency. Both TE and TM modes have cutoff frequencies below which propagation does not occur. In an X-band rectangular waveguide, only the TE10 mode can propagate from 6.56 to 13.12 GHz, which is called the dominant mode.
The document discusses electromagnetic wave propagation in rectangular waveguides. It introduces transverse electric (TE), transverse magnetic (TM), and transverse electromagnetic (TEM) wave modes. For a rectangular waveguide, only discrete solutions exist for the transverse wavenumber due to the boundaries. The cutoff frequency is the minimum frequency at which a mode can propagate. For an X-band rectangular waveguide, only the TE10 mode can propagate between 8.2-12.5 GHz, while the first three TE modes could exist above 15.5 GHz.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.
The document discusses Legendre polynomials, which are special functions that arise in solutions to Laplace's equation in spherical coordinates. Some key points:
1) Legendre polynomials Pn(cosθ) are a set of orthogonal polynomials that satisfy Legendre's differential equation.
2) Pn(cosθ) can be defined using a generating function or by taking partial derivatives of 1/r.
3) Important properties of Legendre polynomials include P0(t)=1, Pn(1)=1, Pn(-1)=(-1)n, and a recurrence relation involving Pn+1, Pn, and their derivatives.
1) The document discusses dynamical Shiba states induced by a single precessing magnetic impurity in a superconductor.
2) A rotating wave transformation is used to transform the time-dependent Hamiltonian into a time-independent effective Hamiltonian.
3) This allows the problem to be solved similarly to the static case. Special cases are considered and show that the Shiba energy depends non-linearly on the driving frequency and precession angle.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
This document introduces the concept of γ-sαg*-semi Ti spaces where i = 0, 1/2, 1, 2. It defines γ-sαg*-semi open and closed sets. Properties of γ-sαg*-semi closure and γ-sαg*-semi generalized closed sets are discussed. It is shown that every γ-sαg*-semi generalized closed set is γ-semi generalized closed. The paper investigates when a space is a γ-sαg*-semi Ti space by looking at when γ-sαg*-semi generalized closed sets are γ-semi closed. It concludes that for each point x in a space, the singleton {x} is either γ-
11. gamma sag semi ti spaces in topological spacesAlexander Decker
This document introduces the concept of γ-sαg*-semi Ti spaces where i = 0, 1/2, 1, 2. It defines γ-sαg*-semi open and closed sets. Properties of γ-sαg*-semi closure and γ-sαg*-semi generalized closed sets are discussed. It is shown that every γ-sαg*-semi generalized closed set is γ-semi generalized closed. A subset A is γ-sαg*-semi generalized closed if and only if the intersection of A with the γ-sαg*-semi closure of each point in the γ-closure of A is non-empty. The γ-sαg*-semi closure of a set
The document discusses electromagnetic wave propagation in rectangular waveguides. It introduces transverse electric (TE), transverse magnetic (TM), and transverse electromagnetic (TEM) wave modes. For a rectangular waveguide, only discrete solutions exist for the transverse wavenumber due to the boundaries. The cutoff frequency is the minimum frequency at which a mode can propagate. For an X-band rectangular waveguide, only the TE10 mode can propagate between 8.2-12.5 GHz, while the first three TE modes could exist above 15.5 GHz.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
The document discusses partial differential equations (PDEs). It defines PDEs and gives their general form involving independent variables, dependent variables, and partial derivatives. It describes methods for obtaining the complete integral, particular solution, singular solution, and general solution of a PDE. It provides examples of types of PDEs and how to solve them by assuming certain forms for the dependent and independent variables and their partial derivatives.
The document discusses Legendre polynomials, which are special functions that arise in solutions to Laplace's equation in spherical coordinates. Some key points:
1) Legendre polynomials Pn(cosθ) are a set of orthogonal polynomials that satisfy Legendre's differential equation.
2) Pn(cosθ) can be defined using a generating function or by taking partial derivatives of 1/r.
3) Important properties of Legendre polynomials include P0(t)=1, Pn(1)=1, Pn(-1)=(-1)n, and a recurrence relation involving Pn+1, Pn, and their derivatives.
1) The document discusses dynamical Shiba states induced by a single precessing magnetic impurity in a superconductor.
2) A rotating wave transformation is used to transform the time-dependent Hamiltonian into a time-independent effective Hamiltonian.
3) This allows the problem to be solved similarly to the static case. Special cases are considered and show that the Shiba energy depends non-linearly on the driving frequency and precession angle.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
This document introduces the concept of γ-sαg*-semi Ti spaces where i = 0, 1/2, 1, 2. It defines γ-sαg*-semi open and closed sets. Properties of γ-sαg*-semi closure and γ-sαg*-semi generalized closed sets are discussed. It is shown that every γ-sαg*-semi generalized closed set is γ-semi generalized closed. The paper investigates when a space is a γ-sαg*-semi Ti space by looking at when γ-sαg*-semi generalized closed sets are γ-semi closed. It concludes that for each point x in a space, the singleton {x} is either γ-
11. gamma sag semi ti spaces in topological spacesAlexander Decker
This document introduces the concept of γ-sαg*-semi Ti spaces where i = 0, 1/2, 1, 2. It defines γ-sαg*-semi open and closed sets. Properties of γ-sαg*-semi closure and γ-sαg*-semi generalized closed sets are discussed. It is shown that every γ-sαg*-semi generalized closed set is γ-semi generalized closed. A subset A is γ-sαg*-semi generalized closed if and only if the intersection of A with the γ-sαg*-semi closure of each point in the γ-closure of A is non-empty. The γ-sαg*-semi closure of a set
This document discusses theta functions with spherical coefficients and their behavior under transformations of the modular group. It begins by defining spherical polynomials and proving that a polynomial is spherical of degree r if and only if it is a linear combination of terms of the form (ξ·x)r, where ξ has zero norm if r is greater than or equal to 2. It then defines theta functions associated with a lattice Γ, a point z, and a spherical polynomial P. The behavior of these theta functions under substitutions of the modular group is studied by applying the Poisson summation formula. A table of relevant Fourier transforms is also provided.
This document contains the solution to a problem involving a sequence of continuously differentiable functions defined by a recurrence relation. The solution shows that:
1) The sequence is monotonically increasing and bounded, so it converges pointwise to a limit function g(x).
2) The limit function g(x) is the unique fixed point of the operator defining the recurrence, and is equal to 1/(1-x).
3) Uniform convergence on compact subsets is proved using Dini's theorem and properties of the operator.
A partial differential equation contains one dependent variable and more than one independent variable. The partial derivatives of a function f(x,y) with respect to x and y at a point (x,y) are represented as ∂f/∂x and ∂f/∂y. Higher order partial derivatives can be found by taking partial derivatives multiple times with respect to the independent variables. The chain rule can be used to find partial derivatives when the dependent variable is a function of other variables that are themselves functions of the independent variables.
The document discusses generalizing the tropical semiring to higher dimensions. Specifically, it considers closed convex sets in Rn as elements, with the union operation defined as the convex hull of the union, and the sum operation defined as the Minkowski sum.
It first provides background on polyhedra, recession cones, and asymptotic cones. It then shows that the set of convex polytopes in Rn forms a semiring under these operations. The proof establishes that the union operation yields a closed convex set, and the sum and other axioms hold.
Subsequent sections will consider semirings of other closed convex sets in Rn, such as compact sets and those with a fixed recession cone
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
There are infinitely many maximal primitive positive clones in a diagonalizable algebra M* that serves as an algebraic model for provability logic GL. The paper constructs M* as the subalgebra of infinite binary sequences generated by the zero element (0,0,...). It defines primitive positive clones as sets of operations closed under existential definitions, and shows there are infinitely many maximal clones K1, K2, etc. that preserve the relations x = ¬Δi, x = ¬Δ2, etc. in M*. This provides a simple example of infinitely many maximal primitive positive clones.
The document summarizes Fourier series and related concepts:
1. It defines periodic functions and their periods, gives examples like sin x and cos x having period 2π.
2. It states Dirichlet's conditions for a function to have a Fourier series representation and defines the Fourier coefficients using Euler's formulas.
3. It explains that the Fourier series of an even function contains only cosine terms and of an odd function contains only sine terms.
4. It provides examples of even and odd functions and solves problems finding Fourier series representations of specific functions.
This document contains mathematical formula tables including:
1. Greek alphabet, indices and logarithms, trigonometric identities, complex numbers, hyperbolic identities, and series.
2. Derivatives of common functions, product rule, quotient rule, chain rule, and Leibnitz's theorem.
3. Integrals of common functions, double integrals, and the substitution rule for integrals.
The document discusses solving the two-dimensional Laplace equation to model steady heat flow problems. It presents:
1) The general boundary value problem (BVP) for the Laplace equation in a semi-infinite and finite lamina.
2) The separation of variables method to obtain solutions as a sum of products of ordinary differential equations.
3) Applying boundary conditions to determine constants and obtain the general solution for temperature distribution.
4) Examples of applying the method to specific BVPs for steady heat flow, including plates with various boundary temperature profiles and geometries.
The document discusses partial differentiation and its applications. It covers functions of two variables, first and second partial derivatives, and applications including the Cobb-Douglas production function and finding marginal productivity from a production function. Examples are provided to demonstrate calculating partial derivatives of various functions and applying partial derivatives in contexts like production analysis.
This document defines continuity and uniform continuity of functions. A function f is continuous on a set S if small changes in the input x result in small changes in the output f(x). A function is uniformly continuous if the same relationship holds for all inputs and outputs simultaneously, not just for a fixed input. Several examples are provided to illustrate the difference. The key difference is that a continuous function may depend on the specific input point, while a uniformly continuous function does not. Functions that satisfy a Lipschitz inequality are proven to be uniformly continuous.
The document defines and discusses universal algebras. A universal algebra is a set with operations and relations defined on it. A subalgebra is a subset closed under the operations. The subalgebra generated by a set A, denoted [[A]], is the smallest subalgebra containing A. Morphisms between universal algebras are functions that preserve the operations and relations. Important concepts include products, quotients, and compatibility of operations. Examples of simple universal algebras with constants and 1-ary or 2-ary operations are given.
On the Application of a Classical Fixed Point Method in the Optimization of a...BRNSS Publication Hub
This work on classical optimization reveals the Newton’s fixed point iterative method as involved in
the computation of extrema of convex functions. Such functions must be differentiable in the Banach
space such that their solution exists in the space on application of the Newton’s optimization algorithm
and convergence to the unique point is realized. These results analytically were carried as application
into the optimization of a multieffect evaporator which reveals the feasibility of theoretical and practical
optimization of the multieffect evaporator.
Fourier series can be used to represent periodic and discontinuous functions. The document discusses:
1. The Fourier series expansion of a sawtooth wave, showing how additional terms improve the accuracy of the representation.
2. How Fourier series are well-suited to represent periodic functions over intervals like [0,2π] since the basis functions are also periodic.
3. An example of using Fourier series to analyze a square wave, finding the coefficients for its expansion in terms of sines and cosines.
The document discusses Euler's generalization of Fermat's Little Theorem to composite moduli called the Theorem of Euler-Fermat. It explains that for any integer a coprime to a composite number m, a raised to the totient function of m (φ(m)) is congruent to 1 modulo m. It also provides formulas for calculating the totient function for prime powers and products of coprime integers. The Chinese Remainder Theorem, which states that a system of congruences with coprime moduli always has a solution, is introduced as well.
The document discusses the chain rule and Euler's theorem.
It explains the chain rule for functions of single, multiple, and general variables. The chain rule gives rules for finding the derivative of a composite function.
It also explains that if a function is homogeneous of degree k, its partial derivatives will be homogeneous of degree k-1. Euler's theorem relates the values of a homogeneous function to the values of its partial derivatives. The theorem is extended to functions of multiple variables.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
Partial derivatives are used to calculate the rate of change of a function of two or more variables with respect to one variable, while holding the other variables constant. The partial derivative of z with respect to x, denoted ∂z/∂x, is defined as the limit of the difference quotient as Δx approaches 0, while holding y constant. Similarly, the partial derivative of z with respect to y, denoted ∂z/∂y, is defined as the limit of the difference quotient as Δy approaches 0, while holding x constant. Notations for higher order partial derivatives are also introduced. An example problem finds the first and second order partial derivatives of the function z=x^2y^3+6
Waveguiding Structures Part 1 (General Theory).pptxPawanKumar391848
1. The document discusses general theory and notation for waveguiding structures, including rectangular and circular waveguides. It presents the vector Helmholtz and scalar Helmholtz equations, which describe electromagnetic wave propagation in homogeneous, isotropic, source-free media.
2. Guided waves can be divided into transverse electric (TE), transverse magnetic (TM), and hybrid modes. The transverse field components are related to the longitudinal components through Maxwell's equations.
3. For rectangular and circular waveguides, the Helmholtz equations need only be solved for the longitudinal electric and magnetic fields to determine the guided wave solutions. Boundary conditions require the fields to be zero at the conducting walls of the wave
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
This document discusses electromagnetic plane waves in various materials. It contains:
1) Derivations of plane wave solutions to Maxwell's equations and expressions for the electric and magnetic fields.
2) Examples calculating wave properties like wavelength, propagation velocity, and Poynting vector for different materials and field configurations.
3) Exercises finding field values at specific points given propagating plane wave descriptions.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
This document discusses theta functions with spherical coefficients and their behavior under transformations of the modular group. It begins by defining spherical polynomials and proving that a polynomial is spherical of degree r if and only if it is a linear combination of terms of the form (ξ·x)r, where ξ has zero norm if r is greater than or equal to 2. It then defines theta functions associated with a lattice Γ, a point z, and a spherical polynomial P. The behavior of these theta functions under substitutions of the modular group is studied by applying the Poisson summation formula. A table of relevant Fourier transforms is also provided.
This document contains the solution to a problem involving a sequence of continuously differentiable functions defined by a recurrence relation. The solution shows that:
1) The sequence is monotonically increasing and bounded, so it converges pointwise to a limit function g(x).
2) The limit function g(x) is the unique fixed point of the operator defining the recurrence, and is equal to 1/(1-x).
3) Uniform convergence on compact subsets is proved using Dini's theorem and properties of the operator.
A partial differential equation contains one dependent variable and more than one independent variable. The partial derivatives of a function f(x,y) with respect to x and y at a point (x,y) are represented as ∂f/∂x and ∂f/∂y. Higher order partial derivatives can be found by taking partial derivatives multiple times with respect to the independent variables. The chain rule can be used to find partial derivatives when the dependent variable is a function of other variables that are themselves functions of the independent variables.
The document discusses generalizing the tropical semiring to higher dimensions. Specifically, it considers closed convex sets in Rn as elements, with the union operation defined as the convex hull of the union, and the sum operation defined as the Minkowski sum.
It first provides background on polyhedra, recession cones, and asymptotic cones. It then shows that the set of convex polytopes in Rn forms a semiring under these operations. The proof establishes that the union operation yields a closed convex set, and the sum and other axioms hold.
Subsequent sections will consider semirings of other closed convex sets in Rn, such as compact sets and those with a fixed recession cone
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
There are infinitely many maximal primitive positive clones in a diagonalizable algebra M* that serves as an algebraic model for provability logic GL. The paper constructs M* as the subalgebra of infinite binary sequences generated by the zero element (0,0,...). It defines primitive positive clones as sets of operations closed under existential definitions, and shows there are infinitely many maximal clones K1, K2, etc. that preserve the relations x = ¬Δi, x = ¬Δ2, etc. in M*. This provides a simple example of infinitely many maximal primitive positive clones.
The document summarizes Fourier series and related concepts:
1. It defines periodic functions and their periods, gives examples like sin x and cos x having period 2π.
2. It states Dirichlet's conditions for a function to have a Fourier series representation and defines the Fourier coefficients using Euler's formulas.
3. It explains that the Fourier series of an even function contains only cosine terms and of an odd function contains only sine terms.
4. It provides examples of even and odd functions and solves problems finding Fourier series representations of specific functions.
This document contains mathematical formula tables including:
1. Greek alphabet, indices and logarithms, trigonometric identities, complex numbers, hyperbolic identities, and series.
2. Derivatives of common functions, product rule, quotient rule, chain rule, and Leibnitz's theorem.
3. Integrals of common functions, double integrals, and the substitution rule for integrals.
The document discusses solving the two-dimensional Laplace equation to model steady heat flow problems. It presents:
1) The general boundary value problem (BVP) for the Laplace equation in a semi-infinite and finite lamina.
2) The separation of variables method to obtain solutions as a sum of products of ordinary differential equations.
3) Applying boundary conditions to determine constants and obtain the general solution for temperature distribution.
4) Examples of applying the method to specific BVPs for steady heat flow, including plates with various boundary temperature profiles and geometries.
The document discusses partial differentiation and its applications. It covers functions of two variables, first and second partial derivatives, and applications including the Cobb-Douglas production function and finding marginal productivity from a production function. Examples are provided to demonstrate calculating partial derivatives of various functions and applying partial derivatives in contexts like production analysis.
This document defines continuity and uniform continuity of functions. A function f is continuous on a set S if small changes in the input x result in small changes in the output f(x). A function is uniformly continuous if the same relationship holds for all inputs and outputs simultaneously, not just for a fixed input. Several examples are provided to illustrate the difference. The key difference is that a continuous function may depend on the specific input point, while a uniformly continuous function does not. Functions that satisfy a Lipschitz inequality are proven to be uniformly continuous.
The document defines and discusses universal algebras. A universal algebra is a set with operations and relations defined on it. A subalgebra is a subset closed under the operations. The subalgebra generated by a set A, denoted [[A]], is the smallest subalgebra containing A. Morphisms between universal algebras are functions that preserve the operations and relations. Important concepts include products, quotients, and compatibility of operations. Examples of simple universal algebras with constants and 1-ary or 2-ary operations are given.
On the Application of a Classical Fixed Point Method in the Optimization of a...BRNSS Publication Hub
This work on classical optimization reveals the Newton’s fixed point iterative method as involved in
the computation of extrema of convex functions. Such functions must be differentiable in the Banach
space such that their solution exists in the space on application of the Newton’s optimization algorithm
and convergence to the unique point is realized. These results analytically were carried as application
into the optimization of a multieffect evaporator which reveals the feasibility of theoretical and practical
optimization of the multieffect evaporator.
Fourier series can be used to represent periodic and discontinuous functions. The document discusses:
1. The Fourier series expansion of a sawtooth wave, showing how additional terms improve the accuracy of the representation.
2. How Fourier series are well-suited to represent periodic functions over intervals like [0,2π] since the basis functions are also periodic.
3. An example of using Fourier series to analyze a square wave, finding the coefficients for its expansion in terms of sines and cosines.
The document discusses Euler's generalization of Fermat's Little Theorem to composite moduli called the Theorem of Euler-Fermat. It explains that for any integer a coprime to a composite number m, a raised to the totient function of m (φ(m)) is congruent to 1 modulo m. It also provides formulas for calculating the totient function for prime powers and products of coprime integers. The Chinese Remainder Theorem, which states that a system of congruences with coprime moduli always has a solution, is introduced as well.
The document discusses the chain rule and Euler's theorem.
It explains the chain rule for functions of single, multiple, and general variables. The chain rule gives rules for finding the derivative of a composite function.
It also explains that if a function is homogeneous of degree k, its partial derivatives will be homogeneous of degree k-1. Euler's theorem relates the values of a homogeneous function to the values of its partial derivatives. The theorem is extended to functions of multiple variables.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
Partial derivatives are used to calculate the rate of change of a function of two or more variables with respect to one variable, while holding the other variables constant. The partial derivative of z with respect to x, denoted ∂z/∂x, is defined as the limit of the difference quotient as Δx approaches 0, while holding y constant. Similarly, the partial derivative of z with respect to y, denoted ∂z/∂y, is defined as the limit of the difference quotient as Δy approaches 0, while holding x constant. Notations for higher order partial derivatives are also introduced. An example problem finds the first and second order partial derivatives of the function z=x^2y^3+6
Waveguiding Structures Part 1 (General Theory).pptxPawanKumar391848
1. The document discusses general theory and notation for waveguiding structures, including rectangular and circular waveguides. It presents the vector Helmholtz and scalar Helmholtz equations, which describe electromagnetic wave propagation in homogeneous, isotropic, source-free media.
2. Guided waves can be divided into transverse electric (TE), transverse magnetic (TM), and hybrid modes. The transverse field components are related to the longitudinal components through Maxwell's equations.
3. For rectangular and circular waveguides, the Helmholtz equations need only be solved for the longitudinal electric and magnetic fields to determine the guided wave solutions. Boundary conditions require the fields to be zero at the conducting walls of the wave
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
This document discusses electromagnetic plane waves in various materials. It contains:
1) Derivations of plane wave solutions to Maxwell's equations and expressions for the electric and magnetic fields.
2) Examples calculating wave properties like wavelength, propagation velocity, and Poynting vector for different materials and field configurations.
3) Exercises finding field values at specific points given propagating plane wave descriptions.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document examines the advantages and disadvantages of unconditionally stable FDTD approaches.
This document discusses unconditionally stable finite-difference time-domain (FDTD) methods for solving Maxwell's equations numerically. It outlines FDTD algorithms such as Yee's method from 1966 which discretize the equations on a staggered grid. It also discusses the von Neumann stability analysis and compares implicit Crank-Nicolson and alternating-direction implicit methods to conventional explicit FDTD methods. The document notes the advantages of unconditionally stable methods but also mentions potential disadvantages.
This document describes uniform plane waves propagating in lossless media. It discusses:
1) Uniform plane waves have no dependence on the transverse coordinates x and y, and depend only on the longitudinal coordinate z and time t. Maxwell's equations simplify for such waves.
2) The electric and magnetic fields can be separated into forward and backward propagating waves. The forward wave E+ depends only on z-ct, while the backward wave E- depends only on z+ct.
3) General solutions are given as a superposition of forward and backward waves. The Poynting vector and energy density contain contributions from both waves.
The document provides information about decomposing electromagnetic fields in waveguides into longitudinal and transverse components. It introduces the key concepts of cutoff frequency, cutoff wavelength, propagation constant, transverse impedances, and relates them through important equations. Several types of waveguide modes (TEM, TE, TM, hybrid) are also defined based on which field components are nonzero.
Microwave is a descriptive term used to identify EM waves in the frequency spectrum ranging approximately from 1 GHz to 30 GHz.
This corresponds to wavelength from 30 cm to 1 cm (λ = c/f).Sometimes higher frequency ranging upto 600 GHz are also called Microwaves
EWS
ECM
ESM
ECCM
Microwave Heating
Industrial Heating
Microwave oven
Industrial,Scientific,Medical
Linear Acclerators
Plasma Containment
Radio Astronomy
Whole body cancer theraphy
The document introduces Maxwell's equations in differential geometric formulation. It defines the electromagnetic tensor F, which combines the electric and magnetic fields E and B into a single matrix. F can be represented as a 2-form comprising forms E and B. The Hodge star operator establishes duality between forms. Applying the exterior derivative d to F yields Maxwell's equations dF=0 and applying it to the Hodge dual d*F yields the other set of Maxwell's equations d*F=0, showing the equivalence between the differential geometric and standard formulations.
11.solution of linear and nonlinear partial differential equations using mixt...Alexander Decker
This document discusses solving linear and nonlinear partial differential equations using a combination of the Elzaki transform and projected differential transform methods.
[1] It introduces the Elzaki transform and defines its properties for handling partial derivatives.
[2] It then introduces the projected differential transform method and its fundamental theorems for decomposing nonlinear terms.
[3] As an example application, it shows how to use the two methods together to solve a general nonlinear, non-homogeneous partial differential equation with initial conditions. The Elzaki transform is used to obtain an expression in terms of the nonlinear terms, which are then decomposed using the projected differential transform method.
Solution of linear and nonlinear partial differential equations using mixture...Alexander Decker
This document discusses solving linear and nonlinear partial differential equations using a combination of the Elzaki transform and projected differential transform methods.
[1] It introduces the Elzaki transform and defines its properties for solving partial differential equations. Properties include formulas for the Elzaki transform of partial derivatives.
[2] It then introduces the projected differential transform method and defines its basic properties and theorems for decomposing nonlinear terms.
[3] As an example application, it shows how to use the two methods together to solve a general nonlinear, non-homogeneous partial differential equation with initial conditions. The Elzaki transform is used to obtain an expression for the solution, then the projected differential transform decomposes the
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Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Alexander Decker
The document presents a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. The homotopy perturbation method is used to handle the nonlinear terms, while the Elzaki transform is applied to reformulate the equations in terms of transformed variables, obtaining a series solution via inverse transformation. The method is demonstrated to be effective for both homogeneous and non-homogeneous nonlinear partial differential equations. Key steps include using integration by parts to obtain Elzaki transforms of partial derivatives and defining a convex homotopy to reformulate the equations for the homotopy perturbation method.
11.homotopy perturbation and elzaki transform for solving nonlinear partial d...Alexander Decker
The document discusses using a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. It begins by introducing the homotopy perturbation method and Elzaki transform individually. It then presents the homotopy perturbation Elzaki transform method, which applies Elzaki transform to reformulate the problem before using homotopy perturbation method to obtain approximations of the solution as a series. Finally, it applies the new combined method to solve an example nonlinear partial differential equation.
1. The document provides solutions to homework problems involving partial differential equations.
2. Problem 1 solves the wave equation utt = c2uxx using d'Alembert's formula to find the solution u(x,t).
3. Problem 2 proves that if the initial conditions φ and ψ are odd functions, then the solution u(x,t) is also an odd function.
The document defines and provides theory on triple integrals. It discusses that triple integrals are used to calculate volumes or integrate over a fourth dimension based on three other dimensions. Examples are provided on setting up and evaluating triple integrals in rectangular, cylindrical, and polar coordinates. Specific steps are outlined for solving triple integrals over different shapes and order of integration.
1. The document discusses various electromagnetic boundary conditions including:
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2. Whites, EE 481 Lecture 10 Page 2 of 10
• Transverse Electric (TE), or
• Transverse Magnetic (TM).
Proceeding from the Maxwell curl equations:
ˆ
x yˆ ˆ
z
∂ ∂ ∂
∇ × E = − jωμ H ⇒ = − jωμ H
∂x ∂y ∂z
Ex Ey Ez
∂Ez ∂E y
or ˆ
x: − = − jωμ H x
∂y ∂z
⎛ ∂E ∂E ⎞
y : − ⎜ z − x ⎟ = − jωμ H y
ˆ
⎝ ∂x ∂z ⎠
∂E y ∂Ex
ˆ
z: − = − jωμ H z
∂x ∂y
However, the spatial variation in z is known so that
∂ ( e− jβ z )
= − j β ( e− jβ z )
∂z
Consequently, these curl equations simplify to
∂Ez
+ j β E y = − jωμ H x (3.3a),(1)
∂y
∂E
− z − j β Ex = − jωμ H y (3.3b),(2)
∂x
∂E y ∂Ex
− = − jωμ H z (3.3c),(3)
∂x ∂y
3. Whites, EE 481 Lecture 10 Page 3 of 10
We can perform a similar expansion of Ampère’s equation
∇ × H = jωε E to obtain
∂H z
+ j β H y = jωε Ex (3.4a),(4)
∂y
∂H z
− jβ H x − = jωε E y (3.4b),(5)
∂x
∂H y ∂H x
− = jωε Ez (3.5c),(6)
∂x ∂y
Now, (1)-(6) can be manipulated to produce simple algebraic
equations for the transverse (x and y) components of E and H .
For example, from (1):
j ⎛ ∂Ez ⎞
Hx = + jβ Ey ⎟
ωμ ⎜ ∂y
⎝ ⎠
Substituting for Ey from (5) we find
j ⎡ ∂Ez 1 ⎛ ∂H z ⎞ ⎤
Hx = ⎢ + jβ ⎜ − jβ H x − ⎟⎥
ωμ ⎣ ∂y jωε ⎝ ∂x ⎠ ⎦
j ∂Ez β2 j β ∂H z
= + 2 Hx − 2
ωμ ∂y ω με ω με ∂x
j ⎛ ∂Ez ∂H z ⎞
or, Hx = ⎜ ωε −β ⎟ (3.5a),(7)
kc2 ⎝ ∂y ∂x ⎠
where kc2 ≡ k 2 − β 2 and k 2 = ω 2 με . (3.6)
Similarly, we can show that
4. Whites, EE 481 Lecture 10 Page 4 of 10
j ⎛ ∂Ez ∂H z ⎞
Hy = − ωε +β
kc2 ⎜ ∂y ⎟
(3.5b),(8)
⎝ ∂x ⎠
− j ⎛ ∂Ez ∂H z ⎞
Ex = ⎜ β + ωμ ⎟ (3.5c),(9)
kc2 ⎝ ∂x ∂y ⎠
j ⎛ ∂E ∂H z ⎞
Ey = − β z + ωμ
kc2 ⎜ ∂x ⎟
(3.5d),(10)
⎝ ∂y ⎠
Most important point: From (7)-(10), we can see that all
transverse components of E and H can be determined from
only the axial components Ez and H z . It is this fact that allows
the mode designations TEM, TE, and TM.
Furthermore, we can use superposition to reduce the complexity
of the solution by considering each of these mode types
separately, then adding the fields together at the end.
TE Modes and Rectangular Waveguides
A transverse electric (TE) wave has Ez = 0 and H z ≠ 0 .
Consequently, all E components are transverse to the direction
of propagation. Hence, in (7)-(10) with Ez = 0 , then all
transverse components of E and H are known once we find a
solution for only H z . Neat!
5. Whites, EE 481 Lecture 10 Page 5 of 10
For a rectangular waveguide, the solutions for Ex , E y , H x , H y ,
and H z are obtained in Section 3.3 of the text. The solution and
the solution process are interesting, but not needed in this
course.
What is found in that section is that
⎛ mπ ⎞ ⎛ nπ ⎞
2 2
m, n = 0,1,…
kc ,mn = ⎜ ⎟ +⎜ ⎟ (11)
⎝ a ⎠ ⎝ b ⎠ ( m = n ≠ 0)
Therefore, β = β mn = k 2 − kc2,mn (12)
These m and n indices indicate that only discrete solutions for
the transverse wavenumber (kc) are allowed. Physically, this
occurs because we’ve bounded the system in the x and y
directions. (A vaguely similar situation occurs in atoms, leading
to shell orbitals.)
Notice something important. From (11), we find that m = n = 0
means that kc ,00 = 0 . In (7)-(10), this implies infinite field
amplitudes, which is not a physical result. Consequently, the
m = n = 0 TE or TM modes are not allowed.
One exception might occur if Ez = H z = 0 since this leads to
indeterminate forms in (7)-(10). However, it can be shown that
inside hollow metallic waveguides when both m = n = 0 and
6. Whites, EE 481 Lecture 10 Page 6 of 10
Ez = H z = 0 , then E = H = 0 . This means there is no TEM
mode.
Consequently, EM waves will propagate only when the
frequency is “large enough” since there is no TEM mode.
Otherwise β will be imaginary ( β → − jα ), leading to pure
attenuation and no propagation of the wave e − j β z → e −α z .
This turns out to be a general result. That is, for a hollow
conductor, EM waves will propagate only when the frequency is
large enough and exceeds some lower threshold. This minimum
propagation frequency is called the cutoff frequency f c ,mn .
It can be shown that guided EM waves require at least two
distinct conductors in order to support wave propagation all the
way down to 0+ Hz.
The cutoff frequencies for TE modes in a rectangular waveguide
are determined from (12) with β = 0 to be
1 ⎛ mπ ⎞ ⎛ nπ ⎞
2 2
m, n = 0,1,…
f c ,mn = ⎜ ⎟ +⎜ ⎟ (3.84),(13)
2π με ⎝ a ⎠ ⎝ b ⎠ (m = n ≠ 0)
In other words, these are the frequencies where β mn = 0 and
wave propagation begins when the frequency slightly exceeds
f c ,mn .
7. Whites, EE 481 Lecture 10 Page 7 of 10
For an X-band rectangular waveguide, the cross-sectional
dimensions are a = 2.286 cm and b = 1.016 cm. Using (13):
TEm,n Mode Cutoff Frequencies
m n fc,mn (GHz)
1 0 6.562
2 0 13.123
0 1 14.764
1 1 16.156
In the X-band region ( ≈ 8.2-12.5 GHz), only the TE10 mode can
propagate in the waveguide regardless of how it is excited.
(We’ll also see shortly that no TM modes will propagate either.)
This is called single mode operation and is most often the
preferred application for hollow waveguides.
On the other hand, at 15.5 GHz any combination of the first
three of these modes could exist and propagate inside a metal,
rectangular waveguide. Which combination actually exists will
depend on how the waveguide is excited.
Note that the TE11 mode (and all higher-ordered TE modes)
could not propagate. (We’ll also see next that no TM modes will
propagate at 15.5 GHz either.)
8. Whites, EE 481 Lecture 10 Page 8 of 10
TM Modes and Rectangular Waveguides
Conversely to TE modes, transverse magnetic (TM) modes have
H z = 0 and Ez ≠ 0 .
The expression for the cutoff frequencies of TM modes in a
rectangular waveguide
⎛ mπ ⎞ ⎛ nπ ⎞
2 2
1
f c ,mn = ⎜ ⎟ +⎜ ⎟ m, n = 1, 2… (14)
2π με ⎝ a ⎠ ⎝ b ⎠
is very similar to that for TE modes given in (13).
It can be shown that if either m = 0 or n = 0 for TM modes,
then E = H = 0 . This means that no TM modes with m = 0 or
n = 0 are allowable in a rectangular waveguide.
For an X-band waveguide:
TMm,n Mode Cutoff Frequencies
m n fc,mn (GHz)
1 1 16.156
1 2 30.248
2 1 19.753
Therefore, no TM modes can propagate in an X-band
rectangular waveguide when f < 16.156 GHz.
9. Whites, EE 481 Lecture 10 Page 9 of 10
Dominant Mode
Note that from 6.56 GHz ≤ f ≤ 13.12 GHz in the X-band
rectangular waveguide, only the TE 10 mode can propagate. This
mode is called the dominant mode of the waveguide.
See Fig 3.9 in the text for plots of the electric and magnetic
fields associated with this mode.
TEM Mode
The transverse electric and magnetic (TEM) modes are
characterized by Ez = 0 and H z = 0 .
In order for this to occur, it can be shown from (3.4) and (3.5)
that it is necessary for f c = 0 . In other words, there is no cutoff
frequency for waveguides that support TEM waves.
Rectangular, circular, elliptical, and all hollow, metallic
waveguides cannot support TEM waves.
It can be shown that at least two separate conductors are
required for TEM waves. Examples of waveguides that will
10. Whites, EE 481 Lecture 10 Page 10 of 10
allow TEM modes include coaxial cable, parallel plate
waveguide, stripline, and microstrip.
Microstrip is the type of microwave circuit interconnection that
we will use in this course. This “waveguide” will support the
“quasi-TEM” mode, which like regular TEM modes has no non-
zero cutoff frequency.
However, if the frequency is large enough, other modes will
begin to propagate on a microstrip. This is usually not a
desirable situation.