Tem xe wave


13.1Introduction khổng lồ TEM Waves

The E & H fields of transverse electromagnetic waves are directed transverse to the direction of propagation. It will be shownin Sec. 14.2 that such TEM waves propagate along structures composedof pairs of perfect conductors of arbitrary cross-section. Theparallel plates shown in Fig. 13.1.1 are a special case of such apair of conductors. The direction of propagation isalong the y axis. With a source driving the conductors at theleft, the conductors can be used lớn deliver electrical energy to aload connected between the right edges of the plates. They thenfunction as a parallel plate transmission line. We assume that the plates are wide in the z direction compared tothe spacing, a, và that conditions imposed in the planes y = 0and y = -b are independent of z, so that the fields are also zindependent. In this section, discussion is lisathachlaixe.vned to lớn either "open" electrodes at y = 0 or "shorted" electrodes. Techniquesfor dealing with arbitrarily terminated transmission lines will beintroduced in Chap. 14. The "open" or "shorted" terminals resultin standing waves that serve khổng lồ illustrate the relationship betweensimple electrodynamic fields & the EQS & MQS lisathachlaixe.vns. Thesefields will be generalized in the next two sections, where we findthat the TEM wave is but one of an infinite number of modes ofpropagation along the y axis between the plates.

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floating figure GIF 13.1.1 Figure 13.1.1 Plane parallel plate transmissionline. If the plates are mở cửa circuited at the right, as shown in Fig.13.1.1, a voltage is applied at the left at y = -b, & the fieldsare EQS, the E that results is x directed. (The plates form aparallel plate capacitor.) If they are "shorted" at the right & the fields areMQS, the H that results from applying a current source at the leftis z directed. (The plates khung a one-turn inductor.) We are nowlooking for solutions lớn Maxwell"s equations (12.0.7)-(12.0.10) thatare similarly transverse lớn the y axis.equation GIF #13.3Fields of this form automatically satisfy the boundary conditionsof zero tangential E and normal H (normal B) on the surfaces of the perfect conductors. These fields have nodivergence, so the divergence laws for E and H <(12.0.7)and (12.0.10)> are automatically satisfied. Thus, the remaining laws,Ampère"s law (12.0.8) & Faraday"s law (12.0.9) fully describethese TEM fields. We pick out the only components of these laws thatare not automatically satisfied by observing that Ex/ tdrives the x component of Ampère"s law and Hz/ t is thesource term of the z component of Faraday"s law.boxed equation GIF #13.1boxed equation GIF #13.2The other components of these laws are automatically satisfied if itis assumed that the fields are independent of the transversecoordinates and thus depend only on y. The effect of the plates is khổng lồ terminate the field lines so thatthere are no fields in the regions outside. With Gauss" continuitycondition applied khổng lồ the respective plates, Ex terminates on surface charge densities of opposite sign on the respectiveelectrodes.equation GIF #13.4These relationships are illustrated in Fig. 13.1.2a.floating figure GIF 13.1.2 Figure 13.1.2 (a) Surface charge densitiesterminating E of TEM field between electrodes of Fig. 13.1.1.(b) Surface current densities terminating H. The magnetic field is terminated on the plates by surface currentdensities. With Ampère"s continuity condition applied khổng lồ each ofthe plates, equation GIF #13.5these relationships are represented in Fig. 13.1.2b. We shall be interested primarily in the sinusoidal steady state.Between the plates, the fields are governed by differential equationshaving constant coefficients. We therefore assume that the fieldresponse takes the khung equation GIF #13.6where can be regarded as determined by the source that drivesthe system at one of the boundaries. Substitution of these solutionsinto (2) & (3) results in a pair of ordinary constantcoefficient differential equations describing the y dependence ofEx và Hz. Without bothering to write these equations out, weknow that they too will be satisfied by exponential functions of y.Thus, we proceed lớn look for solutions where the functions of y in(6) take the khung exp (-jky y).equation GIF #13.7Once again, we have assumed a solution taking a product form.Substitution into (2) then shows thatequation GIF #13.8and substitution of this expression into (3) gives the dispersion equationequation GIF #13.9For a given frequency, there are two values of ky. A linearcombination of the solutions in the size of (7) is thereforeequation GIF #13.10The associated electric field follows from (8) evaluated for the waves, respectively, using ky = .equation GIF #13.11The amplitudes of the waves, A+ và A-, aredetermined by the boundary conditions imposed in planes perpendicularto the y axis. The following example illustrates how the impositionof these longitudinal boundary conditions determines thefields. It also is the first of several opportunities we now use toplace the EQS & MQS approximations in perspective.Example 13.1.1. Standing Waves on a Shorted Parallel PlateTransmission LineIn Fig. 13.1.3a, the parallel plates are terminated at y = 0 by aperfectly conducting plate. They are driven at y = -b by a currentsource Id distributed over the width w. Thus, there is a surfacecurrent mật độ trùng lặp từ khóa Ky = Id/w Ko imposed on the lowerplate at y = -b. Further, in this example we will assume that adistribution of sources is used in the plane y = -b to make thisdriving surface current mật độ trùng lặp từ khóa uniform over that plane. In summary,the longitudinal boundary conditions areequation GIF #13.12equation GIF #13.13floating figure GIF #13.1.3 Figure 13.1.3 (a) Shorted transmission line drivenby a distributed current source. (b) Standing wave fields withEH shown at times differing by 90 degrees. (c) MQSfields in lisathachlaixe.vn where wavelength is long compared lớn length ofsystem. Khổng lồ make Ex as given by (11) satisfy the first of these boundaryconditions, we must have the amplitudes of the two traveling wavesequal.equation GIF #13.14With this relation used to lớn eliminate A+ in (10), it followsfrom (13) that equation GIF #13.15We have found that the fields between the plates take the khung ofstanding waves.

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equation GIF #13.16equation GIF #13.17Note that E and H are 90o out of temporalphase.3 In making this and the following deductions, it ishelpful to lớn take o as being real. When one is at its peak, theother is zero. The distributions of E & H shown inFig. 13.1.3b are therefore at different instants in time.Every half-wavelength / from the short, E is again zero, as sketched in Fig. 13.1.3b. Beginning at a distance of aquarter-wavelength from the short, the magnetic field also exhibitsnulls at half-wavelength intervals. Adjacent peaks in a given fieldare 180 degrees out of temporal phase.

The MQS Lisathachlaixe.vn

If the driving frequency is so low that a wavelength is much longerthan the length b, we have equation GIF #13.18In this lisathachlaixe.vn, the fields are those of a one-turn inductor. That is,with sin ( y) y & cos ( y) 1,(16) & (17) become equation GIF #13.19equation GIF #13.20The magnetic field intensity is uniform throughout & the surfacecurrent mật độ trùng lặp từ khóa circulates uniformly around the one-turn loop. Theelectric field increases in a linear fashion from zero at the shortto a maximum at the source, where the source voltage is equation GIF #13.21To make it clear that these are the fields of a one-turn solenoid(Example 8.4.4), the flux linkage has been identified asequation GIF #13.22where L is the inductance.

The MQS Approximation

In Chap. 8, we would have been led to lớn these same lisathachlaixe.vning fields byassuming at the outset that the displacement current, the term on theright in (2), is negligible. Then, this one-dimensional khung ofAmpère"s law và (1) requires that equation GIF #13.23If we now use this finding in Faraday"s law, (3), integration on yand use of the boundary condition of (12) gives the same result forE as found taking the low-frequency lisathachlaixe.vn, (20).In the previous example, the longitudinal boundary conditions(conditions imposed at planes of constant y) could be satisfiedexactly using the TEM mode alone. The short at the right and the distributed current source at the left each imposed a condition thatwas, like the TEM fields, independent of the transverse coordinates.In almost all practical situations, longitudinal boundary conditionswhich are independent of the transverse coordinates (used todescribe transmission lines) are approximate. The open circuittermination at y = 0, shown in Fig. 13.1.4, is a case in point, asis the source which in this case is not distributed in the xdirection. floating figure GIF #13.1.4 Figure 13.1.4 (a) open circuit transmission linedriven by voltage source. (b) E & H at times thatdiffer by 90 degrees. (c) EQS fields in lisathachlaixe.vn where wavelength islong compared lớn b. If a longitudinal boundary condition is independent of z, thefields are, in principle, still two dimensional. Between the plates, wecan therefore think of satisfying the longitudinal boundary conditionsusing a superposition of the modes khổng lồ be developed in the nextsection. These consist of not only the TEM mode considered here, butof modes having an x dependence. A detailed evaluation of thecoefficients specifying the amplitudes of the higher-order modesbrought in by the transverse dependence of a longitudinal boundarycondition is illustrated in Sec. 13.3. There we shall find that atlow frequencies, where these higher-order modes are governed by Laplace"sequation, they contribute lớn the fields only in the vicinity of thelongitudinal boundaries. As the frequency is raised beyond theirrespective cutoff frequencies, the higher-order modes begin topropagate along the y axis & so have an influence far from thelongitudinal boundaries.Here, where we wish to lớn restrict ourselves khổng lồ situations that arewell described by the TEM modes, we restrict the frequency range ofinterest lớn well below the lowest cutoff frequency of the lowestof the higher-order modes.Given this condition, "end effects" are restricted to lớn theneighborhood of a longitudinal boundary. Approximate boundaryconditions then determine the distribution of the TEM fields, whichdominate over most of the length. In the xuất hiện circuit example ofFig. 13.1.4a, application of the integral charge conservation law toa volume enclosing the kết thúc of one of the plates, as illustrated inFig. 13.1.5, shows that Ky must be essentially zero at y = 0. For the TEMfields, this implies the boundary condition4equation GIF #13.244In the regionoutside, the fields are not confined by the plates. As a result,there is actually some radiation from the open end of the line, andthis too is not represented by (24). This effect is small ifthe plate spacing is small compared lớn a wavelength.floating figure GIF #13.1.5 Figure 13.1.5 The surface current density, andhence, Hz go khổng lồ zero in the vicinity of the mở cửa end. At the left end, the vertical segments of perfect conductor joiningthe voltage source to the parallel plates require that Exbe zero over these segments. We shall show later that thehigher-order modes vị not contribute khổng lồ the line integral of Ebetween the plates. Thus, in so far as the TEM fields are concerned,the requirement is that equation GIF #13.25Example 13.1.2. Standing Waves on an Open-Circuit ParallelPlate Transmission Line Consider the parallel plates "open" at y = 0 & driven by avoltage source at y = -b. Boundary conditions are thenequation GIF #13.26Evaluation of the coefficients in (10) và (11) so that the boundaryconditions in (26) are satisfied givesequation GIF #13.27It follows that the TEM fields between the plates, (10) & (11), areequation GIF #13.28equation GIF #13.29These distributions of HE are shown in Fig. 13.1.4at times that differ by 90 degrees. The standing wave is similar tothat described in the previous example, except that it is now Erather than H that peaks at the mở cửa end.

The EQS Lisathachlaixe.vn

In the low frequency lisathachlaixe.vn, where the wavelength ismuch longer than the length of the plates so that b 1,the fields given by (28) & (29) becomeequation GIF #13.30equation GIF #13.31At low frequencies, the fields are those of a capacitor. The electricfield is uniform & simply equal lớn the applied voltage divided by thespacing. The magnetic field varies in a linear fashion from zero atthe mở cửa end to lớn its peak value at the voltage source. Evaluation of-Hz at z = -b gives the surface current density, và hence thecurrent i, provided by the voltage source.equation GIF #13.32Note that this expression implies thatequation GIF #13.33so that the lisathachlaixe.vning behavior is indeed that of a plane parallelcapacitor.

EQS Approximation

How would the quasistatic fields be predicted in terms of the TEMfields? If quasistatic, we expect the system khổng lồ be EQS. Thus, themagnetic induction is negligible, so that the right-hand side of (3)is approximated as being equal to lớn zero. equation GIF #13.34It follows from integration of this expression & using the boundarycondition of (26b) that the quasistatic E is equation GIF #13.35In turn, this result provides the displacement current density inAmpère"s law, the right-hand side of (2).equation GIF #13.36The right-hand side of this expression is independent of y. Thus,integration with respect lớn y, with the "constant" of integrationevaluated using the boundary condition of (26a), givesequation GIF #13.37For the sinusoidal voltage drive assumed at the outset in thedescription of the TEM waves, this expression is consistent with thatfound in taking the quasistatic lisathachlaixe.vn, (30). Demonstration 13.1.1. Visualization of Standing WavesA demonstration of the fields described by the two previousexamples is shown in Fig. 13.1.6. A pair of sheet metal electrodesare driven at the left by an oscillator. A fluorescent lamp placedbetween the electrodes is used to show the distribution of the rmselectric field intensity.

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floating figure GIF #13.1.6afloating figure GIF #13.1.6b Figure 13.1.6
(a) Plane parallelmetal plates 1.75 m long are driven at the left over by a 200 MHzsource. Light, & hence time average of E2, with plates (b)terminated by short và (c) terminated by an xuất hiện circuit. The gas in the tube is ionized by the oscillating electric field.Through the field-induced acceleration of electrons in this gas, asufficient velocity is reached so that collisions result in ionizationand an associated optical radiation. What is seen is a time averageresponse to lớn an electric field that is oscillating far more rapidlythan can be followed by the eye. Because the light is proportional lớn the magnitude of theelectric field, the observed 0.75 m distance between nulls is ahalf-wavelength. It can be inferred that the generator frequency isf = c/ = 3 x 108/1.5 = 200 MHz. Thus, the frequencyis typical of the lower VHF television channels. With the right over of the line shorted, the section of the lampnear that end gives evidence that the electric field there isindeed as would be expected from Fig. 13.1.3b, where it is zero atthe short. Similarly, with the right over open, there is a peak in thelight indicating that the electric field near that kết thúc is maximum.This is consistent with the picture given in Fig. 13.1.4b. In goingfrom an mở cửa to a shorted condition, the positions of peak lightintensity, and hence of peak electric field intensity, are shifted by /4.