Haus, Hermann A., and James R. Melcher. Electromagnetic Fields and Energy. Englewood Cliffs, NJ: Prentice-Hall, ISBN:

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1 MIT OpenCourseWre Hus, Hermnn A., nd Jmes R. Melcher. Electromgnetic Fields nd Energy. Englewood Cliffs, NJ: Prentice-Hll, ISBN: Plese use the following cittion formt: Hus, Hermnn A., nd Jmes R. Melcher, Electromgnetic Fields nd Energy. (Msschusetts Institute of Technology: MIT OpenCourseWre). (ccessed [Dte]). License: Cretive Commons Attribution-NonCommercil-Shre Alike. Also vilble from Prentice-Hll: Englewood Cliffs, NJ, ISBN: Note: Plese use the ctul dte you ccessed this mteril in your cittion. For more informtion bout citing these mterils or our Terms of Use, visit:

2 13 ELECTRODYNAMIC FIELDS: THE BOUNDARY VALUE POINT OF VIEW 13.0 INTRODUCTION In the tretment of EQS nd MQS systems, we strted in Chps. 4 nd 8, respectively, by nlyzing the fields produced by specified (known) sources. Then we recognized tht in the presence of mterils, t lest some of these sources were induced by the fields themselves. Induced surfce chrge nd surfce current densities were determined by mking the fields stisfy boundry conditions. In the volume of given region, fields were composed of prticulr solutions to the governing qusisttic equtions (the sclr nd vector Poisson equtions for EQS nd MQS systems, respectively) nd those solutions to the homogeneous equtions (the sclr nd vector Lplce eqution, respectively) tht mde the totl fields stisfy pproprite boundry conditions. We now embrk on similr pproch in the nlysis of electrodynmic fields. Chpter 12 presented study of the fields produced by specified sources (dipoles, line sources, nd surfce sources) nd obeying the inhomogeneous wve eqution. Just s in the cse of EQS nd MQS systems in Chp. 5 nd the lst hlf of Chp. 8, we shll now concentrte on solutions to the homogeneous source free equtions. These solutions then serve to obtin the fields produced by sources lying outside (mybe on the boundry) of the region within which the fields re to be found. In the region of interest, the fields generlly stisfy the inhomogeneous wve eqution. However in this chpter, where there re no sources in the volume of interest, they stisfy the homogeneous wve eqution. It should come s no surprise tht, following this systemtic pproch, we shll reencounter some of the previously obtined solutions. In this chpter, fields will be determined in some limited region such s the volume V of Fig The boundries might be in prt perfectly conducting in the sense tht on their surfces, E is perpendiculr nd the time vrying H is tngentil. The surfce current nd chrge densities implied by these conditions 1

3 2 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig Fields in limited region re in prt due to sources induced on boundries by the fields themselves. re not known until fter the fields hve been found. If there is mteril within the region of interest, it is perfectly insulting nd of piece wise uniform permittivity nd permebility µ.1 Sources J nd ρ re specified throughout the volume nd pper s driving terms in the inhomogeneous wve equtions, (12.6.8) nd ( ). Thus, the H nd E fields obey the inhomogeneous wve equtions. 2 H 2 H µ t 2 = J (1) 2 E µ 2 E = ρ J + µ t 2 t As in erlier chpters, we might think of the solution to these equtions s the sum of prt stisfying the inhomogeneous equtions throughout V (prticulr solution), nd prt stisfying the homogeneous wve eqution throughout tht region. In principle, the prticulr solution could be obtined using the superposition integrl pproch tken in Chp. 12. For exmple, if n electric dipole were introduced into region contining uniform medium, the prticulr solution would be tht given in Sec for n electric dipole. The boundry conditions re generlly not met by these fields. They re then stisfied by dding n pproprite solution of the homogeneous wve eqution. 2 In this chpter, the source terms on the right in (1) nd (2) will be set equl to zero, nd so we shll be concentrting on solutions to the homogeneous wve eqution. By combining the solutions of the homogeneous wve eqution tht stisfy boundry conditions with the source driven fields of the preceding chpter, one cn describe situtions with given sources nd given boundries. In this chpter, we shll consider the propgtion of wves in some xil direction long structure tht is uniform in tht direction. Such wves re used to trnsport energy long pirs of conductors (trnsmission lines), nd through 1 If the region is one of free spce, o nd µ µ o. 2 As pointed out in Sec. 12.7, this is essentilly wht is being done in stisfying boundry conditions by the method of imges. (2)

4 Sec TEM Wves 3 wveguides (metl tubes t microwve frequencies nd dielectric fibers t opticl frequencies). We confine ourselves to the sinusoidl stedy stte. Sections study two dimensionl modes between plne prllel conductors. This exmple introduces the mode expnsion of electrodynmic fields tht is nlogous to the expnsion of the EQS field of the cpcitive ttenutor (in Sec. 5.5) in terms of the solutions to Lplce s eqution. The principl nd higher order modes form complete set for the representtion of rbitrry boundry conditions. The exmple is model for strip trnsmission line nd hence serves s n introduction to the subject of Chp. 14. The higher order modes mnifest properties much like those found in Sec for hollow pipe guides. The dielectric wveguides considered in Sec explin the guiding properties of opticl fibers tht re of gret prcticl interest. Wves re guided by dielectric core hving permittivity lrger thn tht of the surrounding medium but possess fields extending outside this core. Such electromgnetic wves re guided becuse the dielectric core slows the effective velocity of the wve in the guide to the point where it cn mtch the velocity of wve in the surrounding region tht propgtes long the guide but decys in direction perpendiculr to the guide. The fields considered in Secs offer the opportunity to reinforce the notions of qusisttics. Connections between the EQS nd MQS fields studied in Chps. 5 nd 8, respectively, nd their corresponding electrodynmic fields re mde throughout Secs INTRODUCTION TO TEM WAVES The E nd H fields of trnsverse electromgnetic wves re directed trnsverse to the direction of propgtion. It will be shown in Sec tht such TEM wves propgte long structures composed of pirs of perfect conductors of rbitrry cross section. The prllel pltes shown in Fig re specil cse of such pir of conductors. The direction of propgtion is long the y xis. With source driving the conductors t the left, the conductors cn be used to deliver electricl energy to lod connected between the right edges of the pltes. They then function s prllel plte trnsmission line. We ssume tht the pltes re wide in the z direction compred to the spcing,, nd tht conditions imposed in the plnes y = 0 nd y = b re independent of z, so tht the fields re lso z independent. In this section, discussion is limited to either open electrodes t y = 0 or shorted electrodes. Techniques for deling with rbitrrily terminted trnsmission lines will be introduced in Chp. 14. The open or shorted terminls result in stnding wves tht serve to illustrte the reltionship between simple electrodynmic fields nd the EQS nd MQS limits. These fields will be generlized in the next two sections, where we find tht the TEM wve is but one of n infinite number of modes of propgtion long the y xis between the pltes. If the pltes re open circuited t the right, s shown in Fig , voltge is pplied t the left t y = b, nd the fields re EQS, the E tht results is x directed. (The pltes form prllel plte cpcitor.) If they re shorted t the right nd the fields re MQS, the H tht results from pplying current source t the left is z directed. (The pltes form one turn inductor.) We re now looking

5 4 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig Plne prllel plte trnsmission line. for solutions to Mxwell s equtions (12.0.7) ( ) tht re similrly trnsverse to the y xis. E = E x i x ; H = H z i z (1) Fields of this form utomticlly stisfy the boundry conditions of zero tngentil E nd norml H (norml B) on the surfces of the perfect conductors. These fields hve no divergence, so the divergence lws for E nd H [(12.0.7) nd ( )] re utomticlly stisfied. Thus, the remining lws, Ampère s lw (12.0.8) nd Frdy s lw (12.0.9) fully describe these TEM fields. We pick out the only components of these lws tht re not utomticlly stisfied by observing tht E x / t drives the x component of Ampère s lw nd H z / t is the source term of the z component of Frdy s lw. H z E x = y t (2) E x H z = µ y t (3) The other components of these lws re utomticlly stisfied if it is ssumed tht the fields re independent of the trnsverse coordintes nd thus depend only on y. The effect of the pltes is to terminte the field lines so tht there re no fields in the regions outside. With Guss continuity condition pplied to the respective pltes, E x termintes on surfce chrge densities of opposite sign on the respective electrodes. σ s (x = 0) = E x ; σ s (x = ) = E x (4) These reltionships re illustrted in Fig The mgnetic field is terminted on the pltes by surfce current densities. With Ampère s continuity condition pplied to ech of the pltes, K y (x = 0) = H z ; K y (x = ) = H z (5)

6 Sec TEM Wves 5 Fig () Surfce chrge densities terminting E of TEM field between electrodes of Fig (b) Surfce current densities terminting H. these reltionships re represented in Fig b. We shll be interested primrily in the sinusoidl stedy stte. Between the pltes, the fields re governed by differentil equtions hving constnt coefficients. We therefore ssume tht the field response tkes the form H z = Re Ĥ z(y)e jωt ; E x = Re Ê x(y)e jωt (6) where ω cn be regrded s determined by the source tht drives the system t one of the boundries. Substitution of these solutions into (2) nd (3) results in pir of ordinry constnt coefficient differentil equtions describing the y dependence of E x nd H z. Without bothering to write these equtions out, we know tht they too will be stisfied by exponentil functions of y. Thus, we proceed to look for solutions where the functions of y in (6) tke the form exp( jk y y). H z = Re ĥ ze j(ωt kyy) ; E x = Re eˆ x e j(ωt kyy) (7) Once gin, we hve ssumed solution tking product form. Substitution into (2) then shows tht k y ê x = ĥ z (8) ω nd substitution of this expression into (3) gives the dispersion eqution k y = ±β; β ω ω µ = c For given frequency, there re two vlues of k y. A liner combintion of the solutions in the form of (7) is therefore (9) H z = Re [A + e jβy + A e jβy ]e jωt (10) The ssocited electric field follows from (8) evluted for the ± wves, respectively, using k y = ±β. E e jβy jβy ]e jωt x = Re µ/[a + A e (11) The mplitudes of the wves, A + nd A, re determined by the boundry conditions imposed in plnes perpendiculr to the y xis. The following exmple

7 6 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig () Shorted trnsmission line driven by distributed current source. (b) Stnding wve fields with E nd H shown t times differing by 90 degrees. (c) MQS fields in limit where wvelength is long compred to length of system. illustrtes how the imposition of these longitudinl boundry conditions determines the fields. It lso is the first of severl opportunities we now use to plce the EQS nd MQS pproximtions in perspective. Exmple Stnding Wves on Shorted Prllel Plte Trnsmission Line In Fig , the prllel pltes re terminted t y = 0 by perfectly conducting plte. They re driven t y = b by current source I d distributed over the width w. Thus, there is surfce current density K y = I d /w K o imposed on the lower plte t y = b. Further, in this exmple we will ssume tht distribution of sources is used in the plne y = b to mke this driving surfce current density uniform over tht plne. In summry, the longitudinl boundry conditions re E x(0, t) = 0 (12) H z( b, t) = Re Kˆ oe jωt (13) To mke E x s given by (11) stisfy the first of these boundry conditions, we must hve the mplitudes of the two trveling wves equl. A + = A (14) With this reltion used to eliminte A + in (10), it follows from (13) tht A + Kˆ o = (15) 2 cos βb We hve found tht the fields between the pltes tke the form of stnding wves. Re ˆ cos βy jωt H z = K o e (16) cos βb

8 Sec TEM Wves sin βy E x = Re jkˆ o µ/ e jωt (17) cos βb 7 Note tht E nd H re 90 out of temporl phse. 3 When one is t its pek, the other is zero. The distributions of E nd H shown in Fig b re therefore t different instnts in time. Every hlf wvelength π/β from the short, E is gin zero, s sketched in Fig b. Beginning t distnce of qurter wvelength from the short, the mgnetic field lso exhibits nulls t hlf wvelength intervls. Adjcent peks in given field re 180 degrees out of temporl phse. The MQS Limit. If the driving frequency is so low tht wvelength is much longer thn the length b, we hve 2πb = βb 1 (18) λ In this limit, the fields re those of one turn inductor. Tht is, with sin(βy) βy nd cos(βy) 1, (16) nd (17) become H z Re Kˆ oe jωt (19) E x Re Kˆ ojωµye jωt (20) The mgnetic field intensity is uniform throughout nd the surfce current density circultes uniformly round the one turn loop. The electric field increses in liner fshion from zero t the short to mximum t the source, where the source voltge is jωt dλ v(t) = E x ( b, t)dx = Re Kˆ ojωµbe = (21) dt 0 To mke it cler tht these re the fields of one turn solenoid (Exmple 8.4.4), the flux linkge λ hs been identified s where L is the inductnce. di λ = L ; i = Re ˆKo we jωt ; L = dt bµ w (22) The MQS Approximtion. In Chp. 8, we would hve been led to these sme limiting fields by ssuming t the outset tht the displcement current, the term on the right in (2), is negligible. Then, this one dimensionl form of Ampère s lw nd (1) requires tht H z H 0 y 0 H z = H z (t) = Re Kˆ oe jωt (23) If we now use this finding in Frdy s lw, (3), integrtion on y nd use of the boundry condition of (12) gives the sme result for E s found tking the lowfrequency limit, (20). 3 In mking this nd the following deductions, it is helpful to tke Kˆ o s being rel.

9 8 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig () Open circuit trnsmission line driven by voltge source. (b) E nd H t times tht differ by 90 degrees. (c) EQS fields in limit where wvelength is long compred to b. In the previous exmple, the longitudinl boundry conditions (conditions imposed t plnes of constnt y) could be stisfied exctly using the TEM mode lone. The short t the right nd the distributed current source t the left ech imposed condition tht ws, like the TEM fields, independent of the trnsverse coordintes. In lmost ll prcticl situtions, longitudinl boundry conditions which re independent of the trnsverse coordintes (used to describe trnsmission lines) re pproximte. The open circuit termintion t y = 0, shown in Fig , is cse in point, s is the source which in this cse is not distributed in the x direction. If longitudinl boundry condition is independent of z, the fields re, in principle, still two dimensionl. Between the pltes, we cn therefore think of stisfying the longitudinl boundry conditions using superposition of the modes to be developed in the next section. These consist of not only the TEM mode considered here, but of modes hving n x dependence. A detiled evlution of the coefficients specifying the mplitudes of the higher order modes brought in by the trnsverse dependence of longitudinl boundry condition is illustrted in Sec There we shll find tht t low frequencies, where these higher order modes re governed by Lplce s eqution, they contribute to the fields only in the vicinity of the longitudinl boundries. As the frequency is rised beyond their respective cutoff frequencies, the higher order modes begin to propgte long the y xis nd so hve n influence fr from the longitudinl boundries. Here, where we wish to restrict ourselves to situtions tht re well described by the TEM modes, we restrict the frequency rnge of interest to well below the lowest cutoff frequency of the lowest of the higher order modes. Given this condition, end effects re restricted to the neighborhood of longitudinl boundry. Approximte boundry conditions then determine the distribution of the TEM fields, which dominte over most of the length. In the open

10 Sec TEM Wves 9 Fig The surfce current density, nd hence, H z go to zero in the vicinity of the open end. circuit exmple of Fig , ppliction of the integrl chrge conservtion lw to volume enclosing the end of one of the pltes, s illustrted in Fig , shows tht K y must be essentilly zero t y = 0. For the TEM fields, this implies the boundry condition 4 H z (0, t) = 0 (24) At the left end, the verticl segments of perfect conductor joining the voltge source to the prllel pltes require tht E x be zero over these segments. We shll show lter tht the higher order modes do not contribute to the line integrl of E between the pltes. Thus, in so fr s the TEM fields re concerned, the requirement is tht V d V d (t) = E x ( b, t)dx E x ( b, t) = (25) 0 Exmple Stnding Wves on n Open Circuit Prllel Plte Trnsmission Line Consider the prllel pltes open t y = 0 nd driven by voltge source t y = b. Boundry conditions re then H z (0, t) = 0; E x ( b, t) = Re Vˆd e jωt / (26) Evlution of the coefficients in (10) nd (11) so tht the boundry conditions in (26) re stisfied gives Vˆd A + = A = 2 cos βb /µ (27) It follows tht the TEM fields between the pltes, (10) nd (11), re Vˆd sin βy jωt H z = Re j /µ e (28) cos βb Vˆd cos βy jωt E x = Re e (29) cos βb These distributions of H nd E re shown in Fig t times tht differ by 90 degrees. The stnding wve is similr to tht described in the previous exmple, except tht it is now E rther thn H tht peks t the open end. 4 In the region outside, the fields re not confined by the pltes. As result, there is ctully some rdition from the open end of the line, nd this too is not represented by (24). This effect is smll if the plte spcing is smll compred to wvelength.

11 10 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 The EQS Limit. In the low frequency limit, where the wvelength is much longer thn the length of the pltes so tht βb 1, the fields given by (28) nd (29) become Vˆd jωt H z Re j ωye (30) Vˆd jωt E x Re e (31) At low frequencies, the fields re those of cpcitor. The electric field is uniform nd simply equl to the pplied voltge divided by the spcing. The mgnetic field vries in liner fshion from zero t the open end to its pek vlue t the voltge source. Evlution of H z t z = b gives the surfce current density, nd hence the current i, provided by the voltge source. Note tht this expression implies tht i = Re jω bw Vˆde jωt (32) dq bw i = ; q = CV d ; C = (33) dt so tht the limiting behvior is indeed tht of plne prllel cpcitor. EQS Approximtion. How would the qusisttic fields be predicted in terms of the TEM fields? If qusisttic, we expect the system to be EQS. Thus, the mgnetic induction is negligible, so tht the right hnd side of (3) is pproximted s being equl to zero. E x E 0 y 0 (34) It follows from integrtion of this expression nd using the boundry condition of (26b) tht the qusisttic E is V d E x = (35) In turn, this result provides the displcement current density in Ampère s lw, the right hnd side of (2). H z d Vd (36) y dt The right hnd side of this expression is independent of y. Thus, integrtion with respect to y, with the constnt of integrtion evluted using the boundry condition of (26), gives d y H z V d (37) dt For the sinusoidl voltge drive ssumed t the outset in the description of the TEM wves, this expression is consistent with tht found in tking the qusisttic limit, (30).

12 Sec Prllel Plte Modes 11 Demonstrtion Visuliztion of Stnding Wves A demonstrtion of the fields described by the two previous exmples is shown in Fig A pir of sheet metl electrodes re driven t the left by n oscilltor. A fluorescent lmp plced between the electrodes is used to show the distribution of the rms electric field intensity. The gs in the tube is ionized by the oscillting electric field. Through the field induced ccelertion of electrons in this gs, sufficient velocity is reched so tht collisions result in ioniztion nd n ssocited opticl rdition. Wht is seen is time verge response to n electric field tht is oscillting fr more rpidly thn cn be followed by the eye. Becuse the light is proportionl to the mgnitude of the electric field, the observed 0.75 m distnce between nulls is hlf wvelength. It cn be inferred tht the genertor frequency is f = c/λ = /1.5 = 200 MHz. Thus, the frequency is typicl of the lower VHF television chnnels. With the right end of the line shorted, the section of the lmp ner tht end gives evidence tht the electric field there is indeed s would be expected from Fig b, where it is zero t the short. Similrly, with the right end open, there is pek in the light indicting tht the electric field ner tht end is mximum. This is consistent with the picture given in Fig b. In going from n open to shorted condition, the positions of pek light intensity, nd hence of pek electric field intensity, re shifted by λ/4.

13 12 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig () Plne prllel perfectly conducting pltes. (b) Coxil geometry in which z independent fields of () might be pproximtely obtined without edge effects TWO DIMENSIONAL MODES BETWEEN PARALLEL PLATES This section trets the boundry vlue pproch to finding the fields between the perfectly conducting prllel pltes shown in Fig Most of the mthemticl ides nd physicl insights tht come from study of modes on perfectly conducting structures tht re uniform in one direction (for exmple, prllel wire nd coxil trnsmission lines nd wveguides in the form of hollow perfectly conducting tubes) re illustrted by this exmple. In the previous section, we hve lredy seen tht the pltes cn be used s trnsmission line supporting TEM wves. In this nd the next section, we shll see tht they re cpble of supporting other electromgnetic wves. Becuse the structure is uniform in the z direction, it cn be excited in such wy tht fields re independent of z. One wy to mke the structure pproximtely uniform in the z direction is illustrted in Fig b, where the region between the pltes becomes the nnulus of coxil conductors hving very nerly the sme rdii. Thus, the difference of these rdii becomes essentilly the spcing nd the z coordinte mps into the φ coordinte. Another wy is to mke the pltes very wide (in the z direction) compred to their spcing,. Then, the fringing fields from the edges of the pltes re negligible. In either cse, the understnding is tht the field excittion is uniformly distributed in the z direction. The fields re now ssumed to be independent of z. Becuse the fields re two dimensionl, the clssifictions nd reltions given in Sec nd summrized in Tble serve s our strting point. Crtesin coordintes re pproprite becuse the pltes lie in coordinte plnes. Fields either hve H trnsverse to the x y plne nd E in the x y plne (TM) or hve E trnsverse nd H in the x y plne (TE). In these cses, H z nd E z re tken s the functions from which ll other field components cn be derived. We consider sinusoidl stedy stte solutions, so these fields tke the form jωt H z = Re Ĥ z (x, y)e (1) jωt E z = Re Ê z (x, y)e (2)

14 Sec Prllel Plte Modes 13 These field components, respectively, stisfy the Helmholtz eqution, (12.6.9) nd ( ) in Tble , nd the ssocited fields re given in terms of these components by the remining reltions in tht tble. Once gin, we find product solutions to the Helmholtz eqution, where H z nd E z re ssumed to tke the form X(x)Y (y). This formlism for reducing prtil differentil eqution to ordinry differentil equtions ws illustrted for Helmholtz s eqution in Sec This time, we tke more mture pproch, bsed on the observtion tht the coefficients of the governing eqution re independent of y (re constnts). As result, Y (y) will turn out to be governed by constnt coefficient differentil eqution. This eqution will hve exponentil solutions. Thus, with the understnding tht k y is yet to be determined constnt (tht will turn out to hve two vlues), we ssume tht the solutions tke the specific product forms Ĥ z = ĥ z(x)e jk yy (3) Ê z = ê z (x)e jk yy Then, the field reltions of Tble become (4) TM Fields: where p 2 ω 2 µ k 2 y d 2 ĥ z dx 2 2ˆ + p h z = 0 (5) k y ˆ ê x = ω h z (6) 1 dĥ z ê y = jω dx (7) TE Fields: where q 2 ω 2 µ k 2 y d 2 ê z 2 + q ê z = 0 (8) dx 2 ĥ x = k y ê z (9) ωµ 1 dê z ĥ y = jωµ dx The boundry vlue problem now tkes clssic form fmilir from Sec Wht vlues of p nd q will mke the electric field tngentil to the pltes zero? For the TM fields, ê y = 0 on the pltes, nd it follows from (7) tht it is the derivtive of H z tht must be zero on the pltes. For the TE fields, E z must itself be zero t the pltes. Thus, the boundry conditions re (10) TM Fields: dĥ z (0) = 0; dx dĥ z () = 0 (11) dx

15 14 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 TE Fields: Fig Dependence of fundmentl fields on x. ê z (0) = 0; ê z () = 0 (12) To check tht ll of the conditions re indeed met t the boundries, note tht if (11) is stisfied, there is neither tngentil E nor norml H t the boundries for the TM fields. (There is no norml H whether the boundry condition is stisfied or not.) For the TE field, E z is the only electric field, nd mking E z =0 on the boundries indeed gurntees tht H x = 0 there, s cn be seen from (9). Representing the TM modes, the solution to (5) is liner combintion of sin(px) nd cos(px). To stisfy the boundry condition, (11), t x = 0, we must select cos(px). Then, to stisfy the condition t x =, it follows tht p = p n = nπ/, n = 0, 1, 2,... hˆ z cos p n x (13) nπ p n =, n = 0, 1, 2,... (14) These functions nd the ssocited vlues of p re clled eigenfunctions nd eigenvlues, respectively. The solutions tht hve been found hve the x dependence shown in Fig From the definition of p given in (5), it follows tht for given frequency ω (presumbly imposed by n excittion), the wve number k y ssocited with the n th mode is ω2 µ (nπ/) 2 ; ω 2 µ > (nπ/) 2 k y ±β n ; β n j (nπ/) 2 ω 2 µ; ω 2 µ < (nπ/) 2 (15) Similr resoning identifies the modes for the TE fields. Of the two solutions to (8), the one tht stisfies the boundry condition t x = 0 is sin(qx). The second boundry condition then requires tht q tke on certin eigenvlues, q n. ê z sin q n x (16) nπ q n = (17)

16 Sec Prllel Plte Modes 15 The x dependence of E z is then s shown in Fig b. Note tht the cse n = 0 is excluded becuse it implies solution of zero mplitude. For the TE fields, it follows from (17) nd the definition of q given with (8) tht 5 ω2 µ (nπ/) 2 ; ω 2 µ > (nπ/) 2 k y ±β n ; β n j (nπ/) 2 ω 2 µ; ω 2 µ < (nπ/) 2 (18) In generl, the fields between the pltes re liner combintion of ll of the modes. In superimposing these modes, we recognize tht k y = ±β n. Thus, with coefficients tht will be determined by boundry conditions in plnes of constnt y, we hve the solutions TM Modes: TE Modes: H z =Re A + e jβ oy o o e e jβ ny + A jβ oy + A nπ (19) jωt + A + n n e jβ ny cos x e n=1 + e jβ ny + C jβ nπ jωt E z = Re Cn n e n y sin x e (20) n=1 We shll refer to the n th mode represented by these fields s the TM n or TE n mode, respectively. We now mke n observtion bout the TM 0 mode tht is of fr reching significnce. Its distribution of H z hs no dependence on x [(13) with p n = 0]. As result, E y = 0 ccording to (7). Thus, for the TM 0 mode, both E nd H re trnsverse to the xil direction y. This specil mode, represented by the n = 0 terms in (19), is therefore the trnsverse electromgnetic (TEM) mode fetured in the previous section. One of its most significnt fetures is tht the reltion between frequency ω nd wve number in the y direction, k y, [(15) with n = 0] is k y = ±ω µ = ±ω/c, the sme s for uniform electromgnetic plne wve. Indeed, s we sw in Sec. 13.1, it is uniform plne wve. The frequency dependence of k y for the TEM mode nd for the higher order TM n modes given by (15) re represented grphiclly by the ω k y plot of Fig For given frequency, ω, there re two vlues of k y which we hve clled ±β n. The dshed curves represent imginry vlues of k y. Imginry vlues correspond to exponentilly decying nd growing solutions. An exponentilly growing solution is in fct solution tht decys in the y direction. Note tht the switch from exponentilly decying to propgting fields for the higher order modes occurs t the cutoff frequency 1 nπ ω cn = µ 5 For the prticulr geometry considered here, it hs turned out tht the eigenvlues p n nd q n re the sme (with the exception of n = 0). This coincidence does not occur with boundries hving other geometries. (21)

17 16 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig Dispersion reltion for TM modes. Fig Dispersion reltion for TE modes. The velocity of propgtion of points of constnt phse (for exmple, point t which field component is zero) is ω/k y. Figure emphsizes tht for ll but the TEM mode, the phse velocity is function of frequency. The eqution relting ω to k y represented by this figure, (15), is often clled the dispersion eqution. The dispersion eqution for the TE modes is shown in Fig Although the field distributions implied by ech brnch re very different, in the cse of the plne prllel electrodes considered here, the curves re the sme s those for the TM n=0 modes. The next section will provide greter insight into the higher order TM nd TE modes.

18 Sec TE nd TM Stnding Wves TE AND TM STANDING WAVES BETWEEN PARALLEL PLATES In this section, we delve into the reltionship between the two dimensionl higherorder modes derived in Sec nd their sources. The exmples re chosen to relte directly to cse studies treted in qusisttic terms in Chps. 5 nd 8. The mtching of longitudinl boundry condition by superposition of modes my t first seem to be purely mthemticl process. However, even qulittively it is helpful to think of the influence of n excittion in terms of the resulting modes. For qusisttic systems, this hs lredy been our experience. For the purpose of estimting the dependence of the output signl on the spcing b between excittion nd detection electrodes, the EQS response of the cpcitive ttenutor of Sec. 5.5 could be pictured in terms of the lowest order mode. In the electrodynmic situtions of interest here, it is even more common tht one mode domintes. Above its cutoff frequency, given mode cn propgte through wveguide to regions fr removed from the excittion. Modes obey orthogonlity reltions tht re mthemticlly useful for the evlution of the mode mplitudes. Formlly, the mode orthogonlity is implied by the differentil equtions governing the trnsverse dependence of the fundmentl field components nd the ssocited boundry conditions. For the TM modes, these re (13.2.5) nd ( ). TM Modes: where d 2 ĥ zn + p 2 nĥ zn = 0 (1) dx 2 dĥ zn () = 0; dx dĥzn dx (0) = 0 nd for the TE modes, these re (13.2.8) nd ( ). TE Modes: where d 2 ê zn + q n 2 ê zn = 0 (2) dx 2 ê zn () = 0; ê zn (0) = 0 The word orthogonl is used here to men tht hˆ zn hˆ zm dx = 0; n = m (3) 0 0 ê zn ê zm dx = 0; n = m (4) These properties of the modes cn be seen simply by crrying out the integrls, using the modes s given by ( ) nd ( ). More fundmentlly, they cn be deduced from the differentil equtions nd boundry conditions themselves, (1) nd (2). This ws illustrted in Sec. 5.5 using rguments tht re directly pplicble here [(5.5.20) (5.5.26)].

19 18 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig Configurtion for excittion of TM wves. The following two exmples illustrte how TE nd TM modes cn be excited in wveguides. In the qusisttic limit, the configurtions respectively become identicl to EQS nd MQS situtions treted in Chps. 5 nd 8. Exmple Excittion of TM Modes nd the EQS Limit In the configurtion shown in Fig , the prllel pltes lying in the plnes x = 0 nd x = re shorted t y = 0 by perfectly conducting plte. The excittion is provided by distributed voltge sources driving perfectly conducting plte in the plne y = b. These sources constrin the integrl of E cross nrrow insulting gps of length Δ between the respective edges of the upper plte nd the djcent pltes. All the conductors re modeled s perfect. The distributed voltge sources mintin the two dimensionl chrcter of the fields even s the width in the z direction becomes long compred to wvelength. Note tht the configurtion is identicl to tht treted in Sec Therefore, we lredy know the field behvior in the qusisttic (low frequency) limit. In generl, the two dimensionl fields re the sum of the TM nd TE fields. However, here the boundry conditions cn be met by the TM fields lone. Thus, we begin with H z, ( ), expressed s single sum. (A + e jβ ny jβ n y nπ jωt H z = Re n + A n e ) cos x e (5) n=0 This field nd the ssocited E stisfy the boundry conditions on the prllel pltes t x = 0 nd x =. Boundry conditions re imposed on the tngentil E t the longitudinl boundries, where y = 0 E x (x, 0, t) = 0 (6)

20 Sec TE nd TM Stnding Wves 19 nd t the driving electrode, where y = b. We ssume here tht the gp lengths Δ re smll compred to other dimensions of interest. Then, the electric field within ech gp is conservtive nd the line integrl of E x cross the gps is equl to the gp voltges ±v. Over the region between x = Δ nd x = Δ, the perfectly conducting electrode mkes E x = 0. Δ E x (x, b, t)dx = v; E x (x, b, t)dx = v (7) Δ 0 Becuse the longitudinl boundry conditions re on E x, we substitute H z s given by (5) into the x component of Frdy s lw [(12.6.6) of Tble ] to obtin E x = Re n=0 β n (A + e jβ ny A jβ nπ jωt n n e n y cos x e (8) ω To stisfy the condition t the short, (6), A + = A nd (8) becomes n n 2jβn A + nπ jωt E x = Re n sin β n y cos x e (9) ω n=0 This set of solutions stisfies the boundry conditions on three of the four boundries. Wht we now do to stisfy the lst boundry condition differs little from wht ws done in Sec The A + n s re djusted so tht the summtion of product solutions in (9) mtches the boundry condition t y = b summrized by (7). Thus, we write (9) with y = b on the right nd with the function representing (7) on the left. This expression is multiplied by the m th eigenfunction, cos(mπx/), nd integrted from x = 0 to x =. mπx 2jβnA n + Ê x(x, b) cos dx = sin β n b ω 0 0 n=0 (10) nπ mπ cos x cos xdx Becuse the intervls where Ê x(x, b) is finite re so smll, the cosine function cn be pproximted by constnt, nmely ±1 s pproprite. On the right hnd side of (10), we exploit the orthogonlity condition so s to pick out only one term in the infinite series. vˆ[ 1 + cos mπ] = 2jβ m sin β mb A + m ω 2 (11) Of the infinite number of terms in the integrl on the right in (10), only the term where n = m hs contributed. The coefficients follow from solving (11) nd replcing m n. 0; n even A + n = 2ωvˆ ; n odd jβ n sin β n b (12)

21 20 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 With the coefficients A + n = An now determined, we cn evlute ll of the fields. Substitution into (5), nd (8) nd into the result using (12.6.7) from Tble gives 4jωvˆ cos β n y nπ jωt H z = Re cos x e (13) E x = Re n=1 odd β n sin β nb 4ˆ v sin β n y nπ jωt cos x e (14) sin β n b n=1 odd 4nπ vˆ cos β ny nπ jωt E y = Re sin x e (15) (β n ) sin β n b n=1 odd Note the following spects of these fields (which we cn expect to see in Demonstrtion ). First, the mgnetic field is directed perpendiculr to the x y plne. Second, by mking the excittion symmetric, we hve eliminted the TEM mode. As result, the only modes re of order n = 1 nd higher. Third, t frequencies below the cutoff for the TM 1 mode, β y is imginry nd the fields decy in the y direction. 6 Indeed, in the qusisttic limit where ω 2 µ (π/) 2, the electric field is the sme s tht given by tking the grdient of (5.5.9). In this sme qusisttic limit, the mgnetic field would be obtined by using this qusisttic E to evlute the displcement current nd then solving for the resulting mgnetic field subject to the boundry condition tht there be no norml flux density on the surfces of the perfect conductors. Fourth, bove the cutoff frequency for the n = 1 mode but below the cutoff for the n = 2 mode, we should find stnding wves hving wvelength 2π/β 1. Finlly, note tht ech of the expressions for the field components hs sin(β nb) in its denomintor. With the frequency djusted such tht β n = nπ/b, this function goes to zero nd the fields become infinite. This resonnce condition results in n infinite response, becuse we hve pictured ll of the conductors s perfect. It occurs when the frequency is djusted so tht wve reflected from one boundry rrives t the other with just the right phse to reinforce, upon second reflection, the wve currently being initited by the drive. The following experiment gives the opportunity to probe the fields tht hve been found in the previous exmple. In prcticl terms, the structure considered might be prllel plte wveguide. Demonstrtion Evnescent nd Stnding TM Wves The experiment shown in Fig is designed so tht the field distributions cn be probed s the excittion is vried from below to bove the cutoff frequency of the TM 1 mode. The excittion structures re designed to give fields pproximting those found in Exmple For convenience, = 4.8 cm so tht the excittion frequency rnges bove nd below cut off frequency of 3.1 GHz. The genertor is modulted t n udible frequency so tht the mplitude of the detected signl is converted to loudness of the tone from the loudspeker. In this TM cse, the driving electrode is broken into segments, ech insulted from the prllel pltes forming the wveguide nd ech ttched t its center to 6 sin(ju) = j sinh(u) nd cos(ju) = cosh(u)

22 Sec TE nd TM Stnding Wves 21 Fig Demonstrtion of TM evnescent nd stnding wves. coxil line from the genertor. The segments insure tht the fields pplied to ech prt of the electrode re essentilly in phse. (The cbles feeding ech segment re of the sme length so tht signls rrive t ech segment in phse.) The width of the structure in the z direction is of the order of wvelength or more to mke the fields two dimensionl. (Remember, in the vicinity of the lowest cutoff frequency, is bout one hlf wvelength.) Thus, if the feeder were ttched to contiguous electrode t one point, there would be tendency for stnding wves to pper on the excittion electrode, much s they did on the wire ntenne in Sec In the experiment, the segments re bout qurter wvelength in the z direction but, of course, bout hlf wvelength in the x direction. In the experiment, H is detected by mens of one turn coil. The voltge induced t the terminls of this loop is proportionl to the mgnetic flux perpendiculr to the loop. Thus, for the TM fields, the loop detects its gretest signl when it is plced in n x y plne. To void interference with E, the coxil line connected to the probe s well s the loop itself re kept djcent to the conducting wlls (where H z peks nywy). The sptil fetures of the field, implied by the normlized ω versus k y plot of Fig , cn be seen by moving the probe bout. With the frequency below cutoff, the field decys in the y direction. This exponentil decy or evnescence decreses to liner dependence t cutoff nd is replced bove cutoff by stnding wves. The vlue of k y t given frequency cn be deduced from the experiment by mesuring the qurter wve distnce from the short to the first null in the mgnetic field. Note tht if there re symmetries in the excittion tht result in excittion of the TEM mode, the stnding wves produced by this mode will tend to obscure

23 22 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 the TM 1 mode when it is evnescent. The TEM wves do not hve cutoff! As we hve seen once gin, the TM fields re the electrodynmic generliztion of two dimensionl EQS fields. Tht is, in the qusisttic limit, the previous exmple becomes the cpcitive ttenutor of Sec We hve more thn one reson to expect tht the two dimensionl TE fields re the generliztion of MQS systems. First, this ws seen to be the cse in Sec. 12.6, where the TE fields ssocited with given surfce current density were found to pproch the MQS limit s ω 2 µ ky 2. Second, from Sec. 8.6 we know tht for every two dimensionl EQS configurtion involving perfectly conducting boundries, there is n MQS one s well. 8 In prticulr, the MQS nlog of the cpcitor ttenutor is the configurtion shown in Fig The MQS H field ws found in Exmple In treting MQS fields in the presence of perfect conductors, we recognized tht the condition of zero tngentil E implied tht there be no time vrying norml B. This mde it possible to determine H without regrd for E. We could then dely tking detiled ccount of E until Sec Thus, in the MQS limit, system involving essentilly two dimensionl distribution of H cn (nd usully does) hve n E tht depends on the third dimension. For exmple, in the configurtion of Fig , voltge source might be used to drive the current in the z direction through the upper electrode. This current is returned in the perfectly conducting shped wlls. The electric fields in the vicinities of the gps must therefore increse in the z direction from zero t the shorts to vlues consistent with the voltge sources t the ner end. Over most of the length of the system, E is cross the gp nd therefore in plnes perpendiculr to the z xis. This MQS configurtion does not excite pure TE fields. In order to produce (pproximtely) two dimensionl TE fields, provision must be mde to mke E s well s H two dimensionl. The following exmple nd demonstrtion give the opportunity to further develop n pprecition for TE fields. Exmple Excittion of TE Modes nd the MQS Limit An idelized configurtion for exciting stnding TE modes is shown in Fig As in Exmple , the perfectly conducting pltes re shorted in the plne y = 0. In the plne y = b is perfectly conducting plte tht is segmented in the z direction. Ech segment is driven by voltge source tht is itself distributed in the x direction. In the limit where there re mny of these voltge sources nd perfectly conducting segments, the driving electrode becomes one tht both imposes z directed E nd hs no z component of B. Tht is, just below the surfce of this electrode, we z is equl to the sum of the source voltges. One wy of pproximtely relizing this ideliztion is used in the next demonstrtion. Let Λ be defined s the flux per unit length (length tken long the z direction) into nd out of the enclosed region through the gps of width Δ between the driving electrode nd the djcent edges of the plne prllel electrodes. The mgnetic field 7 The exmple which ws the theme of Sec. 5.5 might eqully well hve been clled the microwve ttenutor, for section of wveguide operted below cutoff is used in microwve circuits to ttenute signls. 8 The H stisfying the condition tht n B = 0 on the perfectly conducting boundries ws obtined by replcing Φ A z in the solution to the nlogous EQS problem.

24 Sec TE nd TM Stnding Wves 23 Fig fields. Two dimensionl MQS configurtion tht does not hve TE Fig Idelized configurtion for excittion of TE stnding wves. norml to the driving electrode between the gps is zero. Thus, t the upper surfce, H y hs the distribution shown in Fig Frdy s integrl lw pplied to the contour C of Fig nd to similr contour round the other gp shows tht E z(x, b, t) = d Λ Ê z = jωλˆ (16) dt

25 24 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig Equivlent boundry conditions on norml H nd tngentil E t y = b. Thus, either the norml B or the tngentil E on the surfce t y = b is specified. The two must be consistent with ech other, i.e., they must obey Frdy s lw. It is perhps esiest in this cse to del directly with E z in finding the coefficients ppering in ( ). Once they hve been determined (much s in Exmple ), H follows from Frdy s lw, ( ) nd ( ) of Tble jΛˆω sin β my mπx jωt E z = Re sin e (17) mπ sin βm b H x = Re H y = Re m=1 odd m=1 odd 4β m Λˆ cos β m y mπx jωt sin e (18) µmπ sin β m b 4ˆ m=1 odd Λ sin β m y mπx jωt cos e (19) µ sin β mb In the qusisttic limit, ω 2 µ (mπ/) 2, this mgnetic field reduces to tht found in Exmple A few observtions my help one to gin some insights from these expressions. First, if the mgnetic field is sensed, then the detection loop must hve its xis in the x y plne. For these TE modes, there should be no signl sensed with the xis of the detection loop in the z direction. This probe cn lso be used to verify tht H norml to the perfectly conducting surfces is indeed zero, while its tngentil vlue peks t the short. Second, the sme decy of the fields below cutoff nd ppernce of stnding wves bove cutoff is predicted here, s in the TM cse. Third, becuse E is perpendiculr to plnes of constnt z, the boundry conditions on E, nd hence H, re met, even if perfectly conducting pltes re plced over the open ends of the guide, sy in the plnes z = 0 nd z = w. In this cse, the guide becomes closed pipe of rectngulr cross section. Wht we hve found re then subset of the three dimensionl modes of propgtion in rectngulr wveguide. Demonstrtion Evnescent nd Stnding TE Wves The pprtus of Demonstrtion is ltered to give TE rther thn TM wves by using n rry of one turn inductors rther thn the rry of cpcitor pltes. These re shown in Fig

26 Sec Rectngulr Wveguide Modes 25 Fig Demonstrtion of evnescent nd stnding TE wves. Ech member of the rry consists of n electrode of width 2Δ, driven t one edge by common source nd shorted to the perfectly conducting bcking t its other edge. Thus, the mgnetic flux through the closed loop psses into nd out of the guide through the gps of width Δ between the ends of the one turn coil nd the prllel plte (verticl) wlls of the guide. Effectively, the integrl of E z creted by the voltge sources in the idelized model of Fig is produced by the integrl of E z between the left edge of one current loop nd the right edge of the next. The current loop cn be held in the x z plne to sense H y or in the y z plne to sense H x to verify the field distributions derived in the previous exmple. It cn lso be observed tht plcing conducting sheets ginst the open ends of the prllel plte guide, mking it rectngulr pipe guide, leves the chrcteristics of these two dimensionl TE modes unchnged RECTANGULAR WAVEGUIDE MODES Metl pipe wveguides re often used to guide electromgnetic wves. The most common wveguides hve rectngulr cross sections nd so re well suited for the explortion of electrodynmic fields tht depend on three dimensions. Although we confine ourselves to rectngulr cross section nd hence Crtesin coordintes, the clssifiction of wveguide modes nd the generl pproch used here re eqully pplicble to other geometries, for exmple to wveguides of circulr cross section. The prllel plte system considered in the previous three sections illustrtes

27 26 Electrodynmic Fields: The Boundry Vlue Point of View Chpter 13 Fig Rectngulr wveguide. much of wht cn be expected in pipe wveguides. However, unlike the prllel pltes, which cn support TEM modes s well s higher order TE modes nd TM modes, the pipe cnnot trnsmit TEM mode. From the prllel plte system, we expect tht wveguide will support propgting modes only if the frequency is high enough to mke the greter interior cross sectionl dimension of the pipe greter thn free spce hlf wvelength. Thus, we will find tht guide hving lrger dimension greter thn 5 cm would typiclly be used to guide energy hving frequency of 3 GHz. We found it convenient to clssify two dimensionl fields s trnsverse mgnetic (TM) or trnsverse electric (TE) ccording to whether E or H ws trnsverse to the direction of propgtion (or decy). Here, where we del with threedimensionl fields, it will be convenient to clssify fields ccording to whether they hve E or H trnsverse to the xil direction of the guide. This clssifiction is used regrdless of the cross sectionl geometry of the pipe. We choose gin the y coordinte s the xis of the guide, s shown in Fig If we focus on solutions to Mxwell s equtions tking the form H y = Re ĥ y (x, z)e j(ωt k yy) E y = Re ê y (x, z)e j(ωt kyy) (2) then ll of the other complex mplitude field components cn be written in terms of the complex mplitudes of these xil fields, H y nd E y. This cn be seen from substituting fields hving the form of (1) nd (2) into the trnsverse components of Ampère s lw, (12.0.8), (1) jk y ĥ z ĥ y z = jωê x (3)

28 Sec Rectngulr Wveguide Modes 27 ĥ y + jk y ĥ x = jωê z (4) x nd into the trnsverse components of Frdy s lw, (12.0.9), jk y ê z ê y z = jωµĥ x (5) ê y + jk y ê x = jωµĥ z (6) x If we tke ĥ y nd eˆ y s specified, (3) nd (6) constitute two lgebric equtions in the unknowns eˆ x nd ĥ z. Thus, they cn be solved for these components. Similrly, ĥ x nd ê z follow from (4) nd (5). ĥ y ê y ĥ x = jk y jω /(ω 2 µ ky 2 ) (7) x z ĥ y ê y ĥ z = jk y + jω /(ω 2 µ ky 2 ) z x (8) ĥ y ê y ê x = jωµ jk y /(ω 2 µ k 2 y ) z x (9) ĥ y ˆ e y ê z = jωµ jk y /(ω 2 µ k 2 x y ) z (10) We hve found tht the three dimensionl fields re superposition of those ssocited with E y (so tht the mgnetic field is trnsverse to the guide xis ), the TM fields, nd those due to H y, the TE modes. The xil field components now ply the role of potentils from which the other field components cn be derived. We cn use the y components of the lws of Ampère nd Frdy together with Guss lw nd the divergence lw for H to show tht the xil complex mplitudes eˆ y nd ĥ y stisfy the two dimensionl Helmholtz equtions. TM Modes (H y = 0): where nd 2 ê y 2 ê y x 2 + z 2 + p 2 ê y = 0 (11) 2 p = ω 2 µ ky 2 TE Modes (E y = 0): 2 ĥ y 2 ĥ y + + q 2 ĥ y = 0 (12) x 2 z 2