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Transmission Lines an d W aveguides Given a particular condu ctor geometry for a transmission line or waveg uide, only certain patterns of electric and mag netic fields (modes) can exist for propagating waves. The se modes must be solutions to the gove rning d ifferential equation ( wave equation) while satisfying t he appropriate boundary con ditions for the fi elds. Transmission line CTwo or more condu ctors (two- wire, co axial, etc.) . CCan define a unique current and voltage an d ch aract eristic impedan ce along the line (u se circu it equations).Waveguide CTypically one enclosed condu ctor (rectangular, circu lar, etc.) . CCannot define a unique voltage and current along the waveguide (must use field equations.) The propagating modes along the transmission line or waveguide may be classified according to whi ch field compone nts are present or not present in the wave. The field components in the direction of wave propagation are defined as longi tudinal compone nts while those perpendicular to the direction of propag ation are defi ned as transverse comp onents. Assuming the transmission line or waveguide is oriented with its axis along the z-axis (direction o f wave propagation), the modes may be classified as (1) Transverse electrom agnetic (TEM) modes - the electric and magnetic fields are transverse to the direction o f wave z zpropag ation with no longitudinal comp onents [E = H = 0]. TEM modes cannot exist on s ingle conduct or guiding structures. TEM modes are sometimes called transmission line modes since they are the dominant mod es on transmi ssion lines. Plane waves can also be classified as TEM modes. Quasi-TEM modes - modes which approximate true TEM modes when the frequency is sufficiently small. (2)Transvers e electric (TE) modes - the electric field is transver se to the direction of propagation (no longitudinal electric field compone nt) while the magnetic field has both transverse and z zlongitudinal comp onents [E = 0, H … 0]. (3) Transvers e magnetic (TM) modes - th e magnetic field is transver se to the d irection of propag ation (no longitudinal magnetic field component) while the electric field has both z ztransver se and longitudinal comp onents [H = 0, E … 0]. TE and TM mod es are commo nly referred t o as waveguide modes since they ar e the o nly mod es which can exist in an enclosed gu iding structure. TE an d TM modes ar e char acterized by a cutoff frequen cy below which they d o not propagate. TE and TM mod es can exist on transmi ssion lines but are generally undesirable. Transmi ssion lines ar e typically op erated at frequencies below the cutoff frequencies of TE and TM modes so t hat only the T EM mod e exi sts. (4) Hybrid modes (EH or HE m odes) - bot h the e lectric and z zmagn etic fields have l ongitudinal comp onents [H … 0, E … 0]. The longi tudinal electric field is dom inant in the EH mode while the longitudinal magnetic field is dominant in the HE mode. Hybrid modes are commonly found in waveguides with inhomogeneo us dielectrics and optical fibers. General Guided W ave Solutions We may write general solutions to the fields associated wi th the waves t hat propagate on a gu iding structure using Maxwell’s equations. We assume t he fol lowing about the guiding structure: (1) the guiding structure is infinitely long, oriented al ong the z- axis, and u niform along its length. (2)the guiding structure is constructed from ideal materials (conductors are PEC and insulators are lossless). (3) fields ar e time-harmonic. The field s of the guiding structure m ust satisfy the source free Maxwell’s equations given by For a wave propag ating along the guiding structure in the z-direction, the associated el ectric and mag netic fields may be w ritten as The vectors e(x,y) and h(x,y) represent the transverse field compone nts of zz zzthe w ave w hile vect ors e(x,y)a and h(x,y)a are the l ongitudinal comp onents of t he w ave. By exp anding the curl operator in r ectangular coordinates, and noting that the derivatives of the transverse components with respect to z can be eval uated as we can equate the vector components on each side of the equation to write the si x components of t he el ectric and mag netic field as Equa tions (1) and ( 2) are valid for any wave ( guided or ung uided) propagating in the z-direction in a source-free region wi th a propagation constant of j â. W e may use Equa tions (1) and ( 2) to solve for the longitudinal field components in terms of the transver se field components. cwhere k is the cutoff wavenum ber defined by The cutoff wavenum ber for the wave guiding structure is determined by the waven umber of the insulating medi um throug h which the w ave prop agates (k = ù%ì&å& ) and the propagation constant for the structure (j â). The equations for the transverse compone nts of the fields are valid for all of the modes defi ned previously. The se transver se field component equations can be speci alized for each one of these gu ided structure mod es. TEM Mode Using the general equations for t he transver se fields of guided waves [Equation (3)], we see that the tran sverse fields of a TEM mod e (defined zz cby E = H = 0) ar e no n-zero on ly when k = 0 . W hen the cutoff wavenum ber of the T EM mode is zero, an indeterminant form of (0/0) results for each of the transver se field equations. A zero-valued cutoff wavenum ber yields the following: The first equation above shows that the phase constant â of the TEM mode on a guiding structure is equivalent to the phase constant of a plane wave propagating in a region c haracterized by the same medium between the condu ctors of the guiding structure. The second equation shows that the cutoff fr equency of a TE M mode is 0 Hz. This mea ns that TEM mod es can be propagated at an y non-zero frequency assuming the guiding stru cture can sup port a TE M mod e. Relationships between the transverse fields of the TEM mode can be determined by returning to the source-free Maxwell’s equation results for zzguided wave s [Equa tions (1) and (2)] and setting E = H = 0 and â = k. Note that the ratios of the TEM electric and magnetic field components define wave impedances which are equal to those of equivalent plane waves. The previous results can combined to yield The fields of the TEM mode must also satisfy the respective wave equation: where In rectangular coordinates, the vector Laplacian operator is By separating the rectangul ar coordinate compone nts in the wave equation, x y xywe find that each of the fi eld components F 0 (E, E, H, H) must then satisfy the same equation [Helmholtz equation]. so that the T EM field components must satisfy This result can be w ritten in compact form as twhere L defi nes t he transver se Laplacian operator which in rectangular2 coordinates is According to the previous result, the transverse fields of the TEM mode must satisfy Laplace’s equation wi th bo undary condi tions defined by the conductor geometry of t he guiding structure, just like the static fields w hich would exist on the guiding stru cture fo r f = 0. Thus, the T EM transver se field vectors e(x,y) and h(x,y) are identical to the static fields for the transmi ssion line. This allows us t o solve for the st atic fields of a given guiding structure geometry (Laplace’s equation) to determine the fields of the T EM mod e. TE Modes The transverse fields of TE m odes are found by simplifying the zgeneral guided wave equations in (3) with E = 0. The resulting transver se field s for TE modes are cThe cutoff wavenumber k must be non-zero to yield bounde d solutions for the transve rse field components of TE mod es. This mea ns that we must operate the guiding structure above the correspond ing cutoff frequency for the particular TE mode to propagate. Note that all of the transverse field components of the TE modes can be determined once the single zlongitudinal component (H) is found . The longi tudinal field compone nt zH must satisfy the w ave equ ation so that Given the basic form of the guided wave magnetic field we may write The equation above represents a reduced Helmholtz equation whi ch can be zsolved for h(x,y) based on the boundary conditions of the guiding structure zgeometry. Once h(x,y) is found , the longi tudinal magnetic field is known, and a ll of the transverse field compone nts are found by evaluating the derivatives in Equation (4). The wave impedance for TE modes is fo und from Equation (4): Note that the T E wave i mpedan ce is a function of fr equency. TM Modes The transverse fields of TM modes are found by simplifying the zgeneral guided wave equations in (3) with H = 0. The resulting transver se field s for TM modes are cThe cutoff wavenumber k must also be non-zero to yi eld bounded solutions for the transver se field comp onents of TM mod es so that we must operate the guiding structure above the correspond ing cutoff frequency for the particular TMmode to propagate. Note that all of the transverse field components of the TMmodes can be determined once the single zlongitudinal component (E) is found . The longi tudinal field compone nt zE must satisfy the w ave equ ation so that Given the basic form of the guided wave electric field we may write The equation above represents a reduced Helmholtz equation whi ch can be zsolved for e(x,y) based on the boundary conditions of the guiding structure zgeometry. Once e(x,y) is found , the longi tudinal magnetic field is known, and a ll of the transverse field compone nts are found by evaluating the derivatives in Equation (5). The wave impedance for TM modes is fo und from Equation (5): Note that the T M wave i mpedan ce is also a funct ion of fr equency. Parallel Plate Waveguide The parallel plate waveguide is formed by two condu cting p lates of width w separated by a di stance d as shown bel ow. This “waveg uide” can support TEM, TE and TM mod es. The following assumptions are made in the determination o f the various modes on the parallel plate waveg uide: (1)The waveguide is in finite in length (no reflec tions). (2)The waveguide conductors are PEC’s and the dielectric is lossless. (3) The plate width is much larger than the plate separation (w >> d) so that the v ariation o f the fields with respect to x may be neglected. Parallel Plate Waveguide TEM Mode We have previously shown that the transverse fields of the TEM mode on a general wave guiding structure are equal to the correspond ing static fields of the structure. The electrostatic field of the parallel plate waveg uide (w >> d) is equivalent to that found in the ideal parallel plate capacitor. The vector electrostatic field in the ideal parallel plate capacitor is Thus, the transver se electric field function for the T EM mod e e(x,y) in the parallel plate waveguide is Given â=k for TEM waves, the overall electric field vector for the TEM mode on the parallel plate waveguide is A similar proced ure could be followed for the magn etic field of the parallel plate waveguide TEM mode (determine the magnetostatic field within the parallel plate waveguide given oppositely directed DC currents in the two plates). However, w e have already determined a simple relationship between the transverse electric an d magnetic fields of a general TEM mode: where The transver se magn etic field function of the T EM mod e becomes and the overall magnetic field vector is Note that both the electric field and magnetic field of the parallel plate waveg uide TE M mode are indepen dent of the transver se directions (x and y). Thus , these fields are uniform over the cross-section of the waveguide as shown be low. The current directions in the two plates correspond to the direction of propag ation assumed for the w aves. If we con sider the parallel plate waveguide as a transmission line o(carrying the TEM mode), the characteristic impedance Z of this line is defined as the ratio of voltage to current (for the respective forward and reverse traveling waves) at any po int on the line: For the infinite length line, we have only forward traveling waves . The voltage and current of these waves are defined according to where the path L goes from the lower plate to the upp er plate and the contour C encloses the upper conductor. The evaluation of the integrals yields The phase velocity of the TEM mode on the parallel plate waveguide is given by Parallel Plate Waveg uide T E Modes The go verning partial differential equation for the lo ngitudinal magnetic field function of the TE mode on a general wave guiding structure is cwhere k = k ! â. For the parallel plate waveguide with (w >> d), we22 2 assume that the variation of the fields with respect to x is negligible which yields The general solution to the equation for the longitudinal magnetic field function is such that the longi tudinal magnetic field is given by The con stants A and B are found be applying the appropriate b oundary conditions for the longitudinal and transver se fields w ithin the w aveg uide. For the parallel plate waveguide, with PEC’s at y = 0 a nd y = d, the TE boundary conditions are zThe x compone nt of the TE m ode electric field is related to H by Application of the T E boundary con ditions gives This yields nnwhere B is an amplitude constant associated with the discrete TE mod e. zThe remaining transverse fields are related to H by nThe propagation constant of the TE mode is nThe cutoff frequency for the TE mode is found according to the value of the propagation constant. Note that These attenuated m odes are called evanescent mod es. Th e cut off nfrequencies for the TE propagating m odes are defined by The phase constant â may be exp ressed in terms of t he cutoff frequency as From the equation for â in the parallel plate waveguide, we see that â < k. nThe wave impedance of the TE mode is Parallel Plate Waveg uide T M Modes The go verning partial differential equation for the lo ngitudinal electric field function of the TM mode on a general wave guiding structure is cwhere k = k ! â. For the parallel plate waveguide with (w >> d), we22 2 assume that the variation of the fields with respect to x is negligible which yields The general solution to the equation for the longitudinal electric field function is such that the longi tudinal electric field is given by The con stants A and B are found be applying the appropriate b oundary conditions for the longitudinal and transver se fields w ithin the w aveg uide. For the parallel plate waveguide, with PEC’s at y = 0 a nd y = d, the TM boundary conditions are zThe x compone nt of the TM mode electric field is related to E by zApplication o f the TM bou ndary condi tion o n E gives This yields nnwhere A is an am plitude constant associated with the discrete TM mod e. zThe remaining transverse fields are related to E by c nGiven that k = nð/d is the same cutoff wavenum ber found for the TE nmodes, the propagation constant and the cutoff frequency of the TM mode nare the same a s that for the T E mod e. nnSince the TE and TM modes hav e the same c utoff fr equency ( degenerate modes), we cannot propagate one mode without the other. Not e that the nfields of the parallel plate waveguide TM mode with n = 0 are identical to those of the T EM mod e. nThe wave impedance of the TM mode is Waveguide Phase Velocity and Wavelength nnThe phase velocity for the TE and TM modes on the parallel plate waveguide is cThe phase vel ocity for propag ating mod es (f >f) is actually larger than the phase velocity of a plane wave propag ating in a medium defi ned by ( ì,å). In the case of the plane wave, the phase velocity (the speed at which the points of constant phase on t he wave move) is equal to the velocity of the wave. For the waveg uide, the phase vel ocity is not equal to the speed at which the overall wave propagates along the guide (this velocity is kno w n as t he group velocity and will be defined later). To physically interpret the phase velocity, we consider the fields of the parallel plate waveguide which vary as The overall waves within the parallel plate waveguide are mathematically equivalent to two plane w aves pr opagating along the w aveg uide in at angles defined by t he !y and + z comp onents of t he fi rst terms and t he + y and + z comp onents of t he secon d terms. Thus, the w ave on the parallel plate waveguide looks like the superposition of two plane waves bei ng reflected between t he two plates as they p ropagate. For propagating modes within a waveguide, the phase constant is grelated to the w avelength within the w aveg uide (ë) by The guide wavelength is the distance between equiphase planes along the direction o f propagation (z-axis). For the parallel plate waveguide, the guide w avelength is where ë is the wavelength of a plane wave propagating in a medium cdefined by ( ì,å). For propagating mod es (f >f) , the guide wavelength is is longe r than the correspond ing p lane wave wavelength. We may also de fine a so-called cutoff waveleng th using the plane wave equation The lowest frequency TE or TM mode (their cutoff frequencies are the same ) are the n = 1 modes where which gives a cutoff wavelength of 11for the T E and TM mod es in a p arallel plate waveg uide. Condu ctor and Dielectric Los ses in a W aveguide All of the previous waveguide equations were derived assuming the waveguide consists o f perfect conductors an d dielec trics. W hen losses are incorporated, the form of t he z-dependent wave propagation terms must be modified accor dingly: where the complex propagation constant ã may be w ritten as cdwhere á and á account for the con ductor and dielectric losses respect ively. To de termine the condu ctor losses, we may employ the perturbation method which requires that we kno w the field distribution wi thin the waveg uide. Since different modes hav e different field distributions, each mode w ill have a di fferent attenuation constant due to conductor losses. If the w aveg uide is characterized by a homogeneous dielectric, then the attenuation constant due to dielectric losses can be determined without having to know the exact distribution of fi elds w ithin the w aveg uide. Dielectric Lo sses If we account for dielectric losses only within the waveguide, the propag ation constant becom es If the mag netic loss i n the dielectric is assumed to be negli gible, then The propagation constant can t hen be w ritten as orowhere k = ùìåå is the square of the real wavenumber for the dielectric22 without loss. The equat ion for the propagation constant can be r ewritten as If the dielectric losses are small (the loss t angent is small), then we may apply the following series approximation to the square root term in the propagation constant expression: This yields Thus , the attenuation constant due to dielectric losses for a TE or TM mode in any waveguide is found to be For the TEM mode, â = k so that Conductor Losses According to the perturbation m ethod, the attenuation constant due to conductor losses is owhere P is the power flow along the waveguide given by 1l(S defines t he cr oss-sectional surface of t he w aveg uide) and where P is the power dissipated per unit length of the waveguide given by 2(S defines the surface area of the waveguide for a unit length l). Examp le (conductor loss i n a p arallel plate waveg uide - TM mod es) The power flow along the parallel plate waveguide for a TM mode is given by The power dissipated per unit length of the parallel plate waveguide TM mode is According to the perturbation method, the attenuation constant due to conductor losses for the parallel plate waveguide TM mode is Rectangu lar Waveguide The rectangular waveg uide can s upport only TE and TM mod es given only a single condu ctor. The rectangul ar cross-section (a > b) allows for single-mode operation. Rectangular Waveg uide T E mod es The longi tudinal magnetic field function for the TE m odes within the rectangular waveg uide mu st satisfy cwhere k = k ! â. The magnetic field function m ay be determined using22 2 the separation of variab les tec hnique by assuming a solution of the form Inserting the assumed sol ution into the g overning par tial differential equation yields Dividing by X(x)Y(y) gives The first two terms in (1) are each dep endent on only one var iable. In order for (1) to be satisfied for every x and y within the waveguide, each of the first two terms must be equal to a constant. The original second order partial differential equation dependent on two variables has been separated into two secon d order pu re differential equations each dependent on only one variable. The general solutions to the two separate differential eq uations are The resulting longitudinal magn etic field fu nction for the rectangular waveguide TE modes is The longitudinal magnetic field is(1) The TE boundary conditions for the rectangular w aveguide are where The application o f the bou ndary condi tions yields The resulting produc t of the constants B and D into combined into one mnconstant (A). The tran sverse components of the magnetic field are mnThe index de signation for the discrete TE m odes is TE . Note that the case of n = m = 0 is not allowed si nce t his would mak e all of the transver se field compone nts zero. mnThe phase constant and the cutoff wavenumber of the TE mode are mncThe phase con stant is real if k > k (unattenuated propagating m odes) and mncimagi nary if k < k (evanescent modes). The cutoff frequency for the TE mn mn c mod e is defi ned by k = k which gives Note that we may again wr ite the pha se constant in terms of the correspon ding cutoff fr equency mnThe wave impedance of the TE mode is and the guide wavelength is The cutoff wavelength is given by Rectangular Waveg uide T M mod es The longi tudinal electric field function for the TM modes within the rectangular waveg uide mu st satisfy cwhere k = k ! â. Note that this is the same partial differential equation22 2 that the longi tudinal magnetic field satisfies in the TE c ase. Thus , the TM magn etic field function may be det ermined using the same t echn ique used in the TE case (separation of variab les) by assuming a solution of the form The resulting differential eq uations for the component functions are with solutions of the form The r esulting longitudinal electric field fu nction fo r the r ectangular waveguide TM modes is The longitudinal electric field is The TM boundary conditions for the rectangular w aveguide are We will find t hat satisfaction o f the bou ndary condi tions on t he longi tudinal electric field automatically satisfies the bou ndary condi tions on the transver se fields. xy Note that the values of k and k are identical to those of the correspond ing TE mod es. The resulting produ ct of the constants A and C into combined mninto one constant (B). The resulting transver se fields ar e The pha se constant, cutoff waven umber, cutoff frequency, guide mnwavelength, and cutoff guide wavelength for the TM mode are all mnidentical to those of the corr espon ding TE mod e. mnThe wave impedance of the TM mode is Rectangular W aveguide M odes The fol lowing TE and TM mod es can prop agate in a r ectangular waveg uide: mn TE n = 0, 1, 2...m = 0, 1, 2...(m = n … 0) mn TM n = 1, 2, 3...m = 1, 2, 3... The cutoff frequencies for these modes are defined by According to the cutoff frequency equation, the cutoff frequencies of both 10 01the TE and TE modes are less than that of the lowest order TM mode 11 10(TM). Given a > b for the rectangular waveg uide, the TE has t he lowest cutoff frequency of any of the rectangul ar waveguide modes and is thus the 10 01dominant mod e. Note that the TE and TE modes are degen erate mod es for a square waveguide. The rectangular waveguide allows one to operate 10at a frequency abo ve the cutoff of the do minant TE mod e but below that of the next highest mode to achieve single mode operation. A waveguide operating at a frequency where more than one mode propagates is said to be overm oded. General Equations for W ave Guiding Structures in Cylindrical Coordinates The general equations for the TEM, TE and TM mod es of cylindrical wave gui ding s tructures (circular waveguide, coaxial transmission line, etc.) shoul d be defined in cylindrical coordinates. Assuming the axis of the guiding structure lies along the z-axis, the general expressions for the cylindrical coordinate fields may be w ritten as where the v ectors e(ñ,ö) and h(ñ,ö) represen t the transverse fie ld zz zzcomp onents of t he w ave while the vect ors e(ñ,ö)a and h(ñ,ö)a are the longitudinal components of the wave. Inserting the field expressions into the source-free Maxwell’s equations, expanding the curl operators in cylindrical coordinates, and equating compone nts yields Equations (1) and (2) may be used to solve for the longitudinal field comp onents in terms of t he transver se field components. where TEM Mode zzThe TEM mode fi eld relationships are found by inserting E = H = 0 and â = k into Equations (1) and (2). zTE Modes (E = 0 in the general transver se field expressions) zTM Modes (H = 0 in the general transver se field expressions) Cylindrical Waveguide The cylindrical waveg uide can support only TE and TM mod es given only a single conductor. Cylindrical Waveg uide T E mod es The longi tudinal magnetic field of the TE m odes within the cylindrical waveg uide mu st satisfy where zInserting the expression for H into the differential equation yields cwhere k = k ! â. The magnetic field function m ay be determined using22 2 the separation of variab les tec hnique by assuming a solution of the form Inserting the assumed sol ution into the g overning par tial differential equation yields Dividing by R(ñ)P(ö) gives We multiply the exp ression abo ve b y ñ in order to make the third term2 dependent on ö only. The result is According to the separ ation of var iables techn ique, we may set the ö- ödependent term in (1) equal to a constant (!k). The resulting differential2 equation d efining P(ö) is which has the general solution o f öThe function P(ö) must be periodic in ö so that k must be an integer (n).(1) öReplacing the third term in (1) with !k = !n gives22 Equation (3) is known as Bessel’s equat ion which has sol utions known as Bessel functions. We may write the general solution to Bes sel’s equation as where nc cJ(kñ) - nth order Bessel function o f the first kind (argum ent = kñ) nc cY(kñ) - nth order Bessel function of the second kind (argum ent = kñ) The B essel function of t he secon d kind app roach es 4 as its argument approach es zero. Since the circular waveg uide fi elds mu st be b ounded at the origin (ñ = 0), th en the constant D must be zero. The resulting longi tudinal magnetic field function f or the cylindrical waveguide TE modes is The longitudinal magnetic field is Since there is no l ongi tudinal electric field for the TE m odes, the only TE boundary condition for the cylindrical waveguide is where(2) Using the chain rule, the partial derivative of the Bessel function m ay be written as ncwhere JN(kñ) denotes the derivative with respect to the Bessel function argument . The TE boundary con dition becom es Thus , the TE m odes of the cylindrical waveguide are defined by If we defi ne the mth zero of derivative of the nth order Bessel function as nm nmpN, then the TE mode cutoff wavenum ber is found by nmThe resulting transverse fields of the TE mod es are nmThe cutoff frequency of the TE mode is given by nmThe wavenum ber for the TE mod e is nmThe wave impedance of the TE mode is Cylindrical Waveg uide T M mod es The longitudinal electric field of the TM modes within the cyl indrical waveg uide mu st satisfy where The l ongi tudinal electric field function of the TM modes satisfies the same differential equation as the magn etic field of the TE mod es. Thus, we may zwrite the solution for e(ñ,ö) as The longitudinal electric field is The TM boundary conditions for the cylindrical waveguide are Just as in the cas e of t he rectangular waveg uide T M mod es, we find that zenforcement of the bou ndary condition o n E automatically sa tisfies transverse field bounda ry condi tion. Application of the boundary condi tion zon E yields Thus , the TM modes of the cylindrical waveguide are defined by nmIf we defi ne the mth zer o of the nth order Bessel function as p, then the nmTM mode cutoff wavenum ber is found by nmThe resulting transverse fields of the TM mod es are nmThe cutoff frequency of the TM mode is given by nmThe wavenum ber for the TM mod e is nmThe wave impedance of the TM mode is Coaxi al Tran smission Li ne The coaxial transmi ssion line can support TEM, TE and TM mod es. However, it is normally operated at frequencies where only the TEM mode (transmi ssion line mo de) propagates. Coaxial Line TE M Modes As previously illustrated in the parallel plate waveguide example, the transver se fields of the T EM mod e may be defined as where e(ñ,ö) and h(ñ,ö) are equivalent to the electrostatic and m agnetostatic fields for the coaxial geometry. The electrostatic field is that of a coaxial capacitor with a voltage of oV between t he cyl indrical conductors. The magnetostatic field is that of a coaxial transmission line carrying DC currents in opposite directions. For the magnetostatic field to be the static limit of a + z directed w ave on the transmi ssion line, the curr ent in the inner condu ctor must be outward. The TEM field s within the coaxial transmissio n line are The TEM mode represents the dom inant mode in a coaxial line with a cutoff fr equency of ze ro. However, higher order TE and TM mod es can propag ate in the coax ial line at suffi ciently high frequencies. Coaxial Line TE Modes The longitudinal magn etic field of the TE mod es within the coax ial transmi ssion line mu st satisfy where which leads to the same general separation of variables solution found for the ci rcular waveg uide. Note that the ñ-dependent term includes both Bessel functions of t he fi rst kind a nd second kind. We cannot eliminate the Bessel function o f the second kind from the sol ution since ñ = 0 is not in the domain of interest [a < ñ < b] for the coaxial transmission line. The general solution for the longitudinal magn etic field function becom es while the longitudinal magnetic field is Since there is no l ongi tudinal electric field for the TE modes, the only TE boundary conditions for the coaxial transmissio n line are where The TE boundary conditions yield This linear system of equations has a non trivial solution o nly when the determinant is zero: cn mThe roots of this characteristic equation ( k=qN) defi ne cutoff nmwavenum bers of the TE mod es for the coax ial line w here m is the index on the roots and n is the order of the Bessel functions in the characteristic equation. Equivalently, we may write which gives The resulting longitudinal magnetic field is where the constant CN has been i ncorporated into the con stants A and B. nmThe resulting transverse fields of the TE mod es are nmThe cutoff frequency of the TE mode may be defined in terms of the roots to the characteristic equation. nmThe wavenum ber for the TE mod e is nmThe wave impedance of the TE mode is Coaxial Line TM mod es The longitudinal electric field of the TM mod es within the coax ial transmi ssion line mu st satisfy where The longitudinal electric field fu nction of t he coax ial line T M mod es satisfies the same differential equation a s the magnetic field of the TE zzmodes. Thus, we may write the solutions for e(ñ,ö) and E(ñ,ö,z) as The TM boundary conditions for the coaxial line are zAgain, we find t hat enforcement of the bou ndary condi tion o n E automatically satisfies transverse field bounda ry condi tion. Application of zthe bou ndary condi tion o n E yields This linear system of equations has a non trivial solution o nly when the determinant is zero: cn mThe roots of this characteristic equation ( k=q) define cutoff nmwavenum bers of the TM modes for the coaxial line. Equivalently, we may write which gives The resulting longitudinal electric field is and the tran sverse field s are nmThe cutoff frequency of the TM mode is given by nmThe wavenum ber for the TM mod e is nmThe wave impedance of the TM mode is Ground ed Di electric Slab W aveguide The groun ded dielectric slab waveg uide is the basis for many p lanar transmission lines (microstrip, stripline, etc.). This waveguide suppo rts the propag ation of surface w aves which are guided along the dielectric interface. The ground ed dielectric slab can suppor t TE and TM modes but not the TEM mode since there is only one con ductor. Actually, the conductor is not necessary for the TE and TM modes to propagate. A simple dielectric slab without a ground plane can also suppo rt TE a nd TM modes. Assumptions: i. The grounded dielectric slab is of infinite ext ent in the y and z directions. ii.The no nmagnetic dielectric and the surrounding air are lossless. iii.Waves pr opagate in the z direction (e ).!jâz iv. Fields decay expon entially away from the slab (the fields of the propagating w ave a re concentrated wi thin the dielectric. The anal ysis of t he grounded dielectric slab is som ewhat different than that of the rectangul ar and circular waveguides and coaxial transmission line given the inhomogeneo us dielectric throug h which the w ave prop agates. Grounded D ielectric Slab T M mod es The TM longitudinal elec tric field associate d with any uniform guiding structure carrying a wave i n the z direction may be w ritten as where the el ectric field function must satisfy cwith k = k ! â. Given the symmetry of the dielectric slab, there should22 2 be no v ariation in the fields of the propagating wav e with respect to y. The gove rning differential equation for the longi tudinal electric field function of the w ave beco mes With no y-variation in the fields, the only non-zero transverse components xyin the T M waves ar e E and H. Given the inhomogeneo us di electric, the equation above must be applied to the in dividual h omogeneous dielectric and air r egions which are characterized by d ifferent waven umbers. cThis also means t hat the cutoff waven umbers k in the two regions are different. If we assume propagation in the dielectric and attenuation in the air, then Given the signs on the cutoff wavenum bers, the governing equations in the air an d dielec tric reg ions are The general solutions to these equations are The boundary conditions for the grounded dielec tric slab are Bound ary condi tion (1) ensures that the tangential electric field on the surface of t he ground plane i s zero. Boundary con dition (2) ensure s that the fields in the air region decay to zero as one gets very far away from the dielectric slab. Bound ary condi tions (3) and (4) enforce the continuity of the tangential electric and mag netic fields acr oss the ai r-dielectric interface. Enforcement of the first two bo unda ry condi tions yields Enforcement of the remaining two boun dary condi tions gives the following results. Dividing the equation resulting from the enforcement of B.C. (3) by the equation from B.C. (4) gives cAt this point we have t wo unknowns (k and h) but only one equation. We cmay w rite a s econ d equation in terms of k and h by sol ving for â in the cutoff wavenum ber equations. Thi s yields(1) If we mul tiply equ ation (1) by d and equation (2) by d, we obtain two2 cequations that can be sol ved graphically on a plot of hd verses kd. For valid solutions, h must be positive. The two intersections represent one nmode. As the electrical thickness of the slab grows, mo re TM modes are 0possible. The TM mode is the dom inant mode with a cutoff frequency of nzero. The TM mode begins to propagate when the radius of the circle nequals nð. The refore, the cutoff frequency for the TM mode may be written as nThe resulting fields for the TM mode of t he g rounded d ielectric slab 21waveguide may be found by writing the constant B in terms of A as given by boundary condition (3). Grounded D ielectric Slab T E mod es The TE longitudinal magnetic field associate d with any uniform guiding structure carrying a wave i n the z direction may be w ritten as where the mag netic field function must satisfy given no y-variation in the fields. With no y-variation in the fields, the only yxnon-zero transver se comp onents in the T E waves ar e E and H. The governing equations for the longi tudinal magnetic field in the air and dielec tric reg ions are where the wavenumber in the air region defines a propagating wave while the w aven umber in the the dielectric defi nes an at tenuated w ave. The general solutions to these equations are The TE boundary conditions for the grounded dielec tric slab are where 1Enforcement of boundary condition (1) yields A = 0 whi le the enforcement 2of boundary con dition (2) yields A = 0. Enforcement of the remaining two boundary con ditions gives t he fol lowing results. Dividing the equation resulting from the enforcement of B.C. (4) by the equation from B.C. (3) gives cJust as in the T M case, we may write a s econ d equation in terms of k and h by solving for â in the cutoff wavenum ber equations. Thi s yields If we mul tiply equ ation (1) by d and equation (2) by d, we obtain two2 cequations that can be sol ved graphically on a plot of hd verses kd.(1) 1As the electrical thickness of the slab grows, the first TE m ode (TE) mode propagates when the radius of the circle becomes greater than ð/2. Note 0that there is no TE mode (a mode with a cutoff frequency of zero) as in the ncase of the TM modes. The general TE mode propagates when the radius of the circle grows larger than (2n!1)ð/2. Therefore, the cutoff frequency nfor the T E mod e may be w ritten as nThe resulting fields for the TE mod e of t he g rounded d ielectric slab 21waveguide may be found by writing the constant B in terms of A as given by boundary condition (4). Stripline and Microstrip Stripline and m icrostrip are two commonly use d wave gui ding structures in low-power applications at microwave fr equencies. The planar structure of these devices allows them to be f abricated us ing the same techn iques in maki ng printed ci rcuits. Th e gener al structures of t hese devices are shown below. The st ripline con figuration can be vi ewed as a flattened coaxia l line. The stripline supports the same TEM mode as the coaxial line with electric fields lines t hat emanat e from the inner con ductor to the g round planes (outer con ductor) and magn etic field lines w hich enc ircle t he i nner conductor. Even though the st ripline may no t be to tally encl osed l ike a coaxial line, if the width of the ground plane is made large in comparison to the center condu ctor width W (about 5W), then the fields at the o uter edges of the st ripline ar e small. Thus, stripline has the same advan tage as coax in that there is little radiation from the line. The microstrip configuration is unlike the stripline given that the transverse fields of the propagating wav e are not confined to the homogeneous dielectric. A portion of the transverse fields lie in the air above the dielectric slab. This inhomogeneous dielec tric prevents a p ure TEM mode from propagating b ut a so-called quas i-TEM mode does propagate. This quasi-TEM mode is actually a hybrid mode which propagates at relati vely low frequencies and has transverse field s which are essentially the same as the TEM mode that would propagate if the dielectric surround ing the upp er condu ctor were uniform. In actuality, the stripline “TEM” mod e is also a qu asi-TEM mod e (hybrid) but typically has a lower cutoff frequency than that of the microstrip. Microstrip offers the advantage of an easy connection of devices to the line at any point given the expos ed upper conductor. Thi s is not the case for stripline si nce t he inner conductor is enclosed. Stripline offers a higher bandwi dth than an equivalent microstrip since the pure TEM mode has a cutoff frequency of zero. Both stripline and microstrip are relatively low power structures because of the field di stributions which result around the condu ctor configurations. T he tran sverse field s of the propagating modes are concentrated ar ound the stripline center conductor and the mi crostrip upper condu ctor where relatively large field magnitudes are encou ntered. The field levels that a given configuration c an ha ndle are limited by the breakdown voltage of the surrounding dielectric. Waveguides are capable of handling higher power levels because the fields are spread more evenly throug h the cr oss-section of the st ructure. Stripline C haracteristics In order to simplify the analysis of the stripline, we consider a shielded stripline configuration a s shown be low. Thi s configuration approximates that of the actual stripline configuration assuming that a is large in comparison to W. The analysis of this simplified structure is still quite involved. The shielded stripline configuration can support the pure TEM mode since it is totally enclosed. We can determine the characteristic impedance of this structure if a n analytical expression for the per-unit- length capacitance can be found. The characteristic impedance of the transmission line is given by The phase velocity within the shielded stripline is found according to Thus, we may write In the determination of the shi elded stripline capaci tance, we assume t hat the dielectric within the st ripline is lossless and the thickness of the cent er conductor is zero. The resulting characteristic impedance is where K is the elliptic integral of the first kind and By applying a simple curve fit to the analytical results, we find ewhere W is the effective width of the center condu ctor given by For stripline design, we normally know what characteristic impedance is oorequired and thus need to know W/b in terms of Z rather than Z in terms of W/b. Thus, so lving the equations above for W/b yields The at tenuation constant in a stripline (with a center conductor of thickness t) due to conductor losses only may be w ritten as where Microst rip Characteristics The inhomogeneous dielectric of the microstrip geometry along with the fact that the quasi-static mode is a hybrid mode complicates the analysis of the mi crostrip geomet ry. However, if the mo de is truly quasi-static, then we should be able to accurately app roximate the phase vel ocity and phase constant of the propagating wav es for a microstrip using the standard TEM eequations with an effective dielectric constant å. where the effective dielectric constant depends on the microstrip substrate thickness d and the condu ctor width W and satisfies the relation The fact that the effective dielectric constant lies between that of air and the dielectric means t hat the eff ective dielectric const ant depends on how much of the propagating m ode fields lie within the dielectric and air regions. The effective dielectric constant is approximated by Note that a ver y wide con ductor with a very thin su bstrate (d << W) approximates the parallel plate waveguide yielding an effective dielectric constant of The char acteristic impedan ce of t he mi crostrip may b e written as Solving this equation forW/d yields where The attenuation constant due to conductor loss in the microstrip geometry is swhere R is the surface resistiv ity of the conductor. Dispersion and Group Velocity In general, TE and TM waves of different frequencies propag ate at different velocities and are attenuated at different rates on a wave guiding structure (dispersion). For exam ple, the phase velocities of both TE a nd TM modes in a rec tangular w aveguide are On the other hand, the phase velocity of TEM waves (such as those on a lossless or low-loss coax ial transmission l ine) are independent of frequency: Thus , there is typically no s ignificant dispersion f or low-loss guiding structures of r easonable length which utilize TEM waves. Dispersion is a concern when mul ti-frequency or broadband signals are propagated using a TE or TM mod e. Th ese types of si gnals suffer distortion as they propagate along the structure since different components of the signals propag ate at different velocities. As an example, an amplitude-modul ation (AM) signal is the sum of scsignal (information) frequencies (ù) and a ca rrier frequency ( ù). The envelope of the carrier signal is th e signal information carried by the signal. The phase velocity of the carrier signal is given by The envelope (information) propag ates a what is called the group velocity g(v) which is defined by Example Determine the g roup velocity for a given mode in an air-filled rectangular waveg uide. Note that while the phase velocity within the air-filled wave guide can be greater than the speed of light, the information (envelope) travels at the group velocity which is le ss than the speed of light.