Waveguide systems. S. Kazakov 19/10/2017, JAS 2017

Transcription

1 Waveguide systems S. Kazakov 19/10/017, JAS 017

2 What is waveguide systems? Let s define a waveguide system as everything between a source of electromagnetic power and power consumer. For example: the Sun is source of EM power (light), Earth - consumer, space - waveguide system. Fortunately the efficiency of this waveguide system (efficiency of transmission) is not too high, < 0.1%. We need a slightly more efficient waveguide systems in case of application in particles accelerators. Mission of waveguide system is to transmit and distribute the RF power from RF source effectively, reliably, without distortion (amplitude and phase) and safely (for RF source and other components). 19/10/017 "Waveguide systems", S. Kazakov, JAS 017

3 Effectively - losses of power/energy should be small enough during transmission. Reliably - without interruption. Without distortion - keeping the shape and phase of signals. Safely - abnormal situations(rf breakdown, for example) should not damage RF source or other components. 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 3

4 Exemplars of waveguide lines: Rectangular waveguides Coaxial lines Two-wire line Circular waveguides Coaxial cable Strip line 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 4

5 Elements/parts of waveguide systems Directional couplers Power splitters Windows Circulators Phase shifters Waveguide coaxial adapters 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 5

6 Same elements/parts of waveguide systems Bends 3-stub tuner Couplers for superconductive cavities. Matched loads TE10-TE01 SLED pulse Mode converters compression systems 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 6

7 Main phenomena, which can turn the mission into mission "impossible : Losses in walls and media of transmission lines. It reduces the efficiency and may cause thermal stresses, which can destroy parts of transmission lines. It applies especially to RF windows and circulators. Mismatching transmission line and consuming device. Mismatching some parts of transmission lines. It causes a reflection of RF power back to source. It reduces the efficiency and can cause a malfunction of RF source. RF breakdown. RF breakdown causes the deposition of high RF power/energy in small volume. It can destroy the elements of waveguide system. Multipactor (if transmission line medium is vacuum). Miltipactor absorbs RF power. It reduces the efficiency and can overheat the parts of waveguide system. Home task 1: to know (wiki) what is multipactor and try to estimate the RF power in rectangular waveguide when multipactor starts between wide sides. Length of sides wall of waveguide: a and b. a = b. Wavelength of RF power is λ = 3a. Neglect the influence of the magnetic field. Operating mode is TE 10. Try to find the absolute value of power (in W). 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 7

8 Basic theory of waveguides. Let s start from Maxwell s equations for empty media ( no external charges and currents) rot E = B t B = μ 0 μh μ 0 and ε 0 vacuum permeability and permittivity rot H = D t D = ε 0 εe μ and ε relative permeability and permittivity of media Transform the equations: rot rot E = rot B t rot rot H = rot D t E E = 1 c m H H = 1 c m E t E t c m = 1 μ 0 ε 0 με - velocity of light in the medium E = H = 0 no charges, no currents c m E = E t c m H = H t - wave equations 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 8

9 For time dependency e iωt E + k E = 0, H + k H = 0, k = ω c m Type equation here. Let s consider the structure homogenous in direction Z and will try to find solution in the form: A(x, y) e iγz e iωt E + χ E = 0, H + χ H = 0, E and H are functions of x, y only now, A x, y type x + y two dimentional Laplas operator, χ = k γ It easy to show (Home task ): E x = iγ E z χ x iωμ 0μ H z χ y E y = iγ E z χ y + iωμ 0μ H z χ x H x = iωε 0ε χ H y = iωε 0ε χ E z y iγ H z χ x E z x iγ H z χ y E x, E y, H x, H y can be determined through E z and H z E z + χ E z = 0, H z + χ H z = 0, E z and H z satisfy the boundary conditions. 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 9

10 Solutions can be divided into three classes: E z = 0, H z 0 H z = 0, E z 0 E z = 0, H z = 0 Transvers E, TE-waves (modes), (or H-waves), with boundary conditions on ideal conductor surfaces H z n, n - perpendicular to the surface. Transvers H, TM-waves (modes), (or E-waves), with boundary conditions on ideal conductor surfaces E z = 0. Transvers E and H, TEM waves (modes), with boundary conditions on ideal conductor surfaces E τ = 0, E τ tangential = 0 at the surface. From previous slide, χ = 0 in case of TEM nonzero solutions, or γ = k ω, γ = ±k. = v γ p - phase velocity in z-direction. For TEM waves v p = ±c m, phase velocity is equal to speed of light (in medium). In finite area, the solutions of two-dimensional equations are discrete sets of eigenfunctions and eigenvalues: E z + χ E z = 0, H z + χ H z = 0 E z = E i ( x, y), χ i = A i, H z = H k (x, y), χ k = B k 0 < A 1 < A i 0 < B 1 < B k 19/10/017 "Waveguide systems", S. Kazakov, JAS

11 k = χ + γ ( χ A i B k ) k = ω c m - wave number, χ - transverse wave number, γ - longitudinal wave number γ = ω c m χ For some ω c = χ c m, γ = 0 γ = ω ω c c m For ω > ω c, γεr and wave can propagate along axis Z. For ω < ω c, γεim and wave cannot propagate along axis Z. Wave exponentially decay ( or rise) along axis Z. ω c - cutoff frequency For each mode i i χ i 0 there is a cutoff frequency ω ci. Mode i cannot propagate in waveguide if ω < ω ci There is two smallest member in set 0 < χ 1 < χ A i B k which determine cutoff frequencies ω c1 and ω c If operating frequency ω c1 < ω < ω c, only one mode can propagate in waveguide. This mode is called dominant mode. Single mode regime is typical way to use waveguides. 19/10/017 "Waveguide systems", S. Kazakov, JAS

12 E x = iωμ 0μ χ E y = + iωμ 0μ χ TE- modes Hz y Hz x H x = iγ Hz χ x H y = iγ Hz χ y Transverse fields E x = iγ Ez χ x E y = iγ Ez χ y TH - modes H x = iωε 0ε χ H y = iωε 0ε χ Ez y Ez x It is easy to notice that E τ = Z w H τ n z, n z unit vector in z direction. Z w = ωμ 0μ γ in case of TE-mode Z w = γ ωε 0 ε in case of TM-mode. Z w Wave impedance of mode. In waveguide without losses: γ R, Z w R in case of propagating modes. γ > 0, Z w > 0 - propagating in positive direction. γ < 0, Z w < 0 propagating in negative direction. γ Im, Z w Im in case of non-propagating modes Im γ < 0 propagating in positive direction. Im γ > 0 propagating in negative direction. 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 1

13 Z TE = ωμ 0μ = γ μ 0 μ ε 0 ε 1 ω c ω = Z m 1 ω c ω Z TM = γ ωε 0 ε = μ 0μ ε 0 ε 1 ω c ω = Z m 1 ω c ω Z m wave impedance of the media. Wave impedance of vacuum Z 0 = Ohm Z TEM = Z m ( χ = 0 ω c = 0) Do not confuse with line (caxial) impedance! Transmission line impedance = U I ( not E H ) U~ න E dl l1 I~ ර Hdl l Z TEM = Z m Z TE > Z m Z TM < Z m 19/10/017 "Waveguide systems", S. Kazakov, JAS

14 For TE and TM mode the phase velocity is: v p = ω γ = c m 1 ω c ω v p > c m c m - speed of light in media. Who told that the speed of light is the greatest in the universe?! We undermine the foundations of modern physics. Really, information (energy) propagates in waveguide line with group velocity. Group velocity in Z direction: v g = dω dγ Home task 3 : save the physics. Show that v g c m and v p v g = c m In case of TEM modes, ω c = 0, v p = v g = c m 19/10/017 "Waveguide systems", S. Kazakov, JAS

15 TEM modes. χ = 0 turns equations E + χ E = 0, H + χ H = 0, in to: E = 0, H = 0, with boundary conditions E t = 0, H n = 0 at surface of ideal conductors. These equations coincide with equations for static electric and magnetic fields. It means, TEM modes can exist in configurations where static fields can exist only. TEM mode can exist in multiply-connected configuration only. Z No TEM modes Z Z Yes, TEM modes! Z Multiply-connecting of configuration is not enough for TEM mode existing. Medium between conductors must be homogenous in all directions (not only in Z-direction as for TE and TM modes). 19/10/017 "Waveguide systems", S. Kazakov, JAS

16 Rectangular waveguide, TE modes (E z = 0) H z = H 0 cos mπx a H x = ih 0 mπ a γ mπ a nπ + b cos nπy b e iγz sin mπx a cos nπy b e iγz b H y = ih 0 mπ a γ nπ b nπ + b cos mπx a sin nπy b e iγz a Z E x = ih 0 H x = ih 0 mπ a ωμ 0 μ nπ b + nπ b ωμ 0 μ mπ a mπ nπ + a b cos mπx a sin mπx a sin nπy b cos nπy b e iγz e iγz m number of field variations along wide side n - number of field variations along narrow side m, n 0,1,, m + n 1 Mode notation: TE mn 19/10/017 "Waveguide systems", S. Kazakov, JAS

17 Rectangular waveguide.te modes (E z = 0) γ = ± k mπ a nπ b, k = ω c m Cut off frequency of TE mn (or H mn ) mode: γ = ± ω c c m mπ a nπ b = 0 ω c = c m mπ a + nπ b Lowest frequency when m=1, n=0. ω c10 = c mπ a λ c10 = πc m ω c10 = a Waves with wavelength λ > a cannot propagate in rectangular waveguide. 19/10/017 "Waveguide systems", S. Kazakov, JAS

18 Rectangular waveguide, TM modes (H z = 0). E z = E 0 sin mπx a sin nπy b e iγz E x = ie 0 mπ a γ mπ a nπ + b cos mπx a sin nπy b e iγz b E y = ie 0 H x = ie 0 H x = ie 0 mπ a mπ a γ nπ b nπ + b ωε 0 ε nπ b + nπ b ωε 0 ε mπ a mπ nπ + a b sin mπx a sin mπx a cos mπx a cos nπy b cos nπy b sin nπy b e iγz e iγz e iγz Z a m number of field variations along wide side n - number of field variations along narrow side m, n 1,, (m 0)&(n 0) Mode notation: TM mn 19/10/017 "Waveguide systems", S. Kazakov, JAS

19 Rectangular waveguide, TM modes (H z = 0). γ = ± k mπ a nπ b, k = ω c m Cut off frequency of TM mn (or E mn ) mode: γ = ± ω c c m mπ a nπ b = 0 ω c = c m mπ a + nπ b Lowest frequency when m=1, n=1. ω c11 = c m π a + π b λ c11 = πc m ω c11 = ab a + b 19/10/017 "Waveguide systems", S. Kazakov, JAS

20 Lowers modes in a rectangular waveguide: Solid lines lines of electric field, dashed lines lines of magnetic field. 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 0

21 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 1

22 Standard dimensions of rectangular waveguides. b a You cannot transmit RF power through rectangle waveguide if frequency less then 30 MHz no standard waveguide exists. Biggest a = 3 = 584 mm Smallest a = = 1.3 mm 19/10/017 "Waveguide systems", S. Kazakov, JAS 017

23 Circle waveguide, radius R, TE modes (E z = 0). J nm Bessel function, J m derivative of Bessel function, B mn n root of J m, J m B mn = 0 χ mn = B mn R Θ mα = cos mα sin mα, sin mα Θ mα = cos mα, m = 0,1.. n = 1,,. H z = H 0 J m χ mn r Θ mα e iγz H r = ih 0 γ χ J m χ mn r Θ mα e iγz H α = ih 0 mγ χ r J m χ mn r Θ mα e iγz γ = ± k χ mn = ± ω c B mn m R Cut off frequency: Cut off wavelength: ω c = c mb mn R λ c = πc m ω c = πr B mn E r = ih 0 mωμ 0 μ χ r J m χ mn r Θ mα e iγz E α = ih 0 ωμ 0 μ χ J m χ mn r Θ mα e iγz B mn : m\n /10/017 "Waveguide systems", S. Kazakov, JAS 017 3

24 Circle waveguide, radius R, TM modes (H z = 0). J m Bessel function, J m derivative of Bessel function, A mn n root of J m, J m A mn = 0 χ mn = A mn R Θ mα = cos mα sin mα sin mα, Θ mα = m = 0,1.. n = 1,,. cos mα, E z = E 0 J m χ mn r Θ mα e iγz γ = ± k χ mn = ± ω c A mn m R E r = ie 0 γ χ J m χ mn r Θ mα e iγz E α = ie 0 mγ χ r J m χ mn r Θ mα e iγz Cut off frequency: Cut off wavelength: ω c = c ma mn R λ c = πc m ω c = πr A mn H r = ie 0 mωε 0 ε χ r J m χ mn r Θ mα e iγz H α = ie 0 ωε 0 ε χ J m χ mn r Θ mα e iγz A mn : m\n /10/017 "Waveguide systems", S. Kazakov, JAS 017 4

25 Mode notation: TE mn (H mn ), TM mn (E mn ) First index, m number of angle variations, clear from Θ mα = cos mα sin mα Second index, n number of variations along radius (radius variations) Mode TE 11 TM 01 TE 1 TE 01 TM 11 TE 31 TM 1 TE 41 TE 1 TM 0 λ c R Waves with λ > R cannot propagate in circular waveguide. 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 5

26 Lower modes in a circular waveguide: Solid lines lines of electric field, dashed lines lines of magnetic field. 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 6

27 TEM modes. TEM modes can propagate in multiply-connected configuration bounded by perfect conductor. But it is not enough: the medium between the conductors must be homogeneous in all directions. Most often used TEM transmission line is coaxial line and, for lower frequencies, is two-wire line. Because the TEM fields are similar to static field we can introduce voltage and current of line and its impedance as their ratio: C E U = න dl integral between conductors, I = ර H dl C C 1 1 integral around (internal) conductor Z l = U I It easy to find that Z l of coaxial line is: Z l = μ 0μ ε 0 ε 1 π ln R R 1 = μ ε 60ln R R 1 Ohm For vacuum/air filled coaxial line Z l = 60ln R μ = 1, ε = 1 R 1 Ohm Impedance of line depends on line geometry. Do not confuse with wave impedance E H which always equals to 19/10/017 "Waveguide systems", S. Kazakov, JAS μ 0 μ ε 0 ε

28 Some modes of coaxial lines. Not only TEM modes can propagate in coaxial lines. Home task 4: Find Z l (or ratio R ΤR 1 ) when: 1) losses in coaxial line are minimal ) Max. strength of E-field is minimal Power and outer radius are fixed. 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 8

29 Losses in waveguides. Two reasons of losses losses in the medium (in dielectric) and ohmic losses in conductors. Losses in medium: Example: conducive medium causes losses because of ohmic losses in medium. Suppose conductivity of media is σ and density of current j = σe. Maxwell equation for fields with time dependence e iωt will look like: rot H = iωε 0 εe + j = iωε 0 εe + σe = iω ε 0 ε i σ ω E Thus, introducing complex ε ሶ = ε 0 ε i σ = ω ε iε we can describe a lossy media. ε ε 1 dielectric, Another example: ε and μ describes of polarization of medium. In real world the polarization follows the field with some delay. For tome dependence e iωt it means phase of D shifted by angle α relatively E: D = ε 0 εe iα E Again, we come to complex ε ሶ = ε iε = ε 0 ε (cos α i sin α) D = εሶ Ee iα σ ωε 0 ε 1 ε tg α = ε ε tg α loss tangents of medium ε 1 conductor, σ ωε 0 ε 1 α angle of electrical losses (magnetic in case of μ) 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 9

30 TEM waves in lossy medium. Wave number in lossy medium is complex as well: γ ሶ = γ iγ γ ሶ = ሶ k = ω μሶ ε ሶ = ω μ 1 i μ μ ε 1 i ε ε = ω μ 1 i tg μ ε 1 i tg ε σ For case of dielectric 1, tg ωε 0 ε μ 1, tg ε 1 γ ω μ ε 1 i tg μ + tg ε k 1 i tg μ + tg ε E z = Ee γz ሶ = e k tg μ+tg ε z e ikz Wave decay along z as e k tg μ+tg ε z, wave impedance Z μ Case of conductor, tg ε = γ ሶ = γ iγ ω 1 i μ ε tg ε σ ωε 0 ε 1 (tg μ = 0 for simplicity) ε γ γ k tg ε = k σ ωε 0 ε = ωμ 0μσ, dimension (1/L). δ = Weave impedance of TEM wave in conductor Z c = μ 0μ i σ ω ωμ 0 μσ = 1 + i - skin depth ωμ 0 μ σ 19/10/017 "Waveguide systems", S. Kazakov, JAS

31 Copper, F = 1GHz σ = s δ = m Perpendicular incidence of TEM on the conductive surface: Inside conductor ratio E H = Z c = 1 + i ωμ 0 μ σ The ratio at the surface is the same because of continues of E τ and H τ Non-perpendicular incidence on the conductive surface: Because of σ ω ε 0ε ε 0 ε media adjacent to conductor,wave in conductor is perpendicular to the surface. With high accuracy E H = 1 + i ωμ 0 μ σ E τ ωμ 0 μ = 1 + i at the surface H τ σ inside of conductor and 19/10/017 "Waveguide systems", S. Kazakov, JAS

32 Losses in conductor Density of Power flow in to conductor (Poynting vector): തP = 1 Re(E τh τ ) = 1 H τ ωμ 0μ σ = H τ δσ തP period average power density, δ skin depth For σ 1 തP = H δσ H τ H, H field on the surface, H τ tangential field on the surface Home task 5: calculate the efficiency of copper mirror in vacuum for 100 GHz perpendicular wave. Now we can calculate losses in waveguides. 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 3

33 Losses in waveguides. In case of losses the propagation value γ becomes complex γ = γ iγ Power flow through cross section at z, P(z) = Re න(E H )ds = Re[e iγz e iγ z න(E τ H τ )ds] = P 0 e γ z Decreasing power at length z: തP d തP dz z = γ തP(z) z In the same time തP = തP L, തP L power of losses at z. തP L d തP L z = pҧ dz L z, γ = p L ҧ തP p L ҧ power losses per unit of length 19/10/017 "Waveguide systems", S. Kazakov, JAS

34 Losses in dielectric media: ω p L ҧ = lim z 0 න V ε E E + μ H H dv = ω න S ε E E + μ H H ds ω γ S ε E E + μ H H ds d = Re S (E τ H τ ) ds Losses on the metal surface: p L ҧ = 1 σδ ර l H dl σ conductivity, δ skin depth l perpendicular to Z direction contour on the conductor surface γ l H dl c = σδ Re S (E τ H τ ) ds 19/10/017 "Waveguide systems", S. Kazakov, JAS

35 Losses in rectangular waveguide Notations: k = ω c, χ = mπ m a + nπ b, Z = μ 0μ ε 0 ε, R s = 1 σδ = ωμ 0μ σ surface resistance ε 0n = 1, for n = 0 and ε 0n = for n > 0 TE modes: γ = bz R s 1 χ k 1 + b a χ k + b a ε 0n χ m ab + n a k m b + n a TM modes: γ = bz R s 1 χ k m b 3 + n a 3 m b a + n a 3 19/10/017 "Waveguide systems", S. Kazakov, JAS

36 γ, relative units γ of dominant TE 10 mode H 10 mode : γ = bz R s 1 χ k 1 + b a χ k + 1 χ k = γ = bz R s 1 χ k 1 + b χ a k Typical attenuation in rectangular waveguide 19/10/017 f fc "Waveguide systems", S. Kazakov, JAS

37 Losses in circular waveguides: γ, relative units Notations: The same as for rectangular waveguides, except χ χ mn = B mn R χ mn = A mn R TE modes: γ = for TE mn modes for TM mn modes RZ R s 1 χ mn k (J m B mn = 0) (J m A mn = 0) χ mn k + m m B mn R radius of waveguide Attenuations in circular waveguide TM modes: γ = RZ R s 1 χ mn k TE 0n - low attenuated modes TE(M) mn modes, m 0 TE 0n modes TE 01 mode is used often to transmute power for long distance f fc 19/10/017 "Waveguide systems", S. Kazakov, JAS

38 Orthogonality of waveguide modes. s some mode in waveguide, E s, H s - fields of this mode, γ s longitudinal wave number Mode propagates in positive Z direction if Im γ s < 0, ( there is losses in waveguide or f < f c ) And mode s propagates in negative Z direction if Im γ s > 0, For each mode s there is mode s, which propagates in opposite Z direction and γ s = γ s Let s take two modes s and p, p s. It is easy to show that (using Lorentz reciprocity theorem ): J s,p = න E s H p E p H s d ԦS = 0 - orthogonality of waveguide modes. S S transvers cross-section of waveguide. If p = s, J s, s = N s 0, N s norm of mode s. Dimension of N s is power. N s = න S E s H s E s H s d ԦS 19/10/017 "Waveguide systems", S. Kazakov, JAS

39 Some other useful relations: For p s and p s true: J s,p = න E s H p + E p H s d ԦS = 0 S Combining with J s,p = S E s H p E p H s d ԦS = 0 we have: J s,p = න Remaining, in waveguide: S E s H p d ԦS = 0 E s = e iγ sz E τs x, y + n z E z x, y H p = e iγ pz H τp x, y + n z H z x, y, from this: J s,p = න E s H p d ԦS = න E s H p n z ds = න E τs H τp d ԦS = 0 S S S We know, in waveguide E τ = Z w H τ n z, and: න E τs E τp ds = 0 න H τs H τp ds = 0 S S In waveguide without losses: න E τs H τp d ԦS = න E τs E τp ds = න H τs H τρ ds = 0 S S S 19/10/017 "Waveguide systems", S. Kazakov, JAS

40 Mode orthogonality helps us to find fields in waveguide excited by electrical and magnetic currents. Suppose we have waveguide with electrical and magnetic currents j e, j m distributed in waveguide volume between z 1 and z, z 1 < z. Let s find amplitudes modes exited by these currents. Let s write exited field as series of waveguide modes propagating in negative and positive directions -S j e j m S E = C s E s + C s E s H = C s H s + C s H s High order modes Z 1 Z High order modes For z z 1 only modes s propagating in negative direction exist: E = C s E s H = C s H s For z z only modes s propagating in positive direction exist: E = C s E s H = C s H s Writing Lorentz reciprocity theorem for E, H and E p, H p, then for E, H and E p, H p, we will have (modes p and p): ර E H p E p H d ԦS = න Ԧj e E p Ԧj m H p dv ර E H p E p H d ԦS = න Ԧj e E p Ԧj m H p dv S V S V 19/10/017 "Waveguide systems", S. Kazakov, JAS

41 Taking into account the orthogonality of modes, we will have: C p = 1 N p න V Ԧj e E p Ԧj m H p dv C p = 1 N p න V Ԧj e E p Ԧj m H p dv Home task 6: Find amplitude of TE10 mode in rectangular waveguide excited by current j, frequency ω, see the drawing: 19/10/017 "Waveguide systems", S. Kazakov, JAS

42 General remarks: Circuit analyses of waveguide lines. Typical scheme of waveguide using: usually only one, dominant mode can propagate in waveguide. EM fields close to non-regular parts of waveguides (place of connection to RF source or connection to load) are complicated and combination of many modes. But all modes except dominant mode decay very fast and in regular part only dominant modes exist, propagating from RF source (direct mode or wave) and propagating from load to RF source(reflected mode or wave). Ԧe τ,s, h τ,s, Ԧe τ,s = Z w h τ,s n z -transverse fields of direct mode s. Transvers fields of reflected s mode can be written as: Ԧe τ, s = Ԧe τ,s, h τ, s = h τ,s, Ԧe τ, s = Z w h τ, s n z Transvers field in regular part of waveguide can be presented as sum of direct and reflected waves: E τ = C s Ԧe s e iγ sz + C s Ԧe s e iγ sz E τ = C s Ԧe s e iγ sz 1 + ρ 0 e iγ sz H τ = C s h s e iγ sz C s h s e iγ sz H τ = C s h s e iγ sz 1 ρ 0 e iγ sz 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 4

43 ρ 0 = C s Ԧe s e iγ sz C s Ԧe s e iγ sz z=0 = C sh s e iγsz C s h s e iγ sz z=0 = C s C s coefficient of reflection at cross-section z = 0 ρ z = C s Ԧe s e iγ sz C s Ԧe s e iγ sz z=z = C sh s e iγsz C s h s e iγ sz z=z = ρ 0 e iγ sz coefficient of reflection at cross-section z 1 + ρ 1 ρ Standing wave ratio (SWR), ρ = 0, SWR = 1 pure traveling wave ρ = 1, SWR = pure standing wave Wave impedance in cross-section z: Z z = E τ z H τ z = Z 1 + ρ z w 1 ρ z We know Z w R in waveguide without losses. Check that Z z Im if ρ = 1. Z z Im - means there is no power flow through cross-section z (pure standing wave). 19/10/017 "Waveguide systems", S. Kazakov, JAS

44 Suppose there is waveguide system with include several waveguide connected through some volume, which can include conductors or dielectrics, isotropic or anisotropic. Let s choose in each waveguide crosssection plane S 1, S,, S n For each plane i introduce set of transvers electrical and magnetic fields Ԧe k,i, h k,i. s mode in wavegide i, k = 1,,, Fields are octagonal and let s chose normalization condition: න S i Ԧe k,i h k,i ds = δ k,m Z k Z k න S i e k,i e m,i ds = δ k,m Z k න S i h k,i h m,i ds = δ k,m Z k Any transvers fields in cross-section S i of waveguide i can be extended in series: E i = a k,i e k,i k=1 H i = b k,i e k,i k=1 Series are infinite, but if cross-section Si is in regular waveguide, far from irregularities the number if terms can be limited by propagating and weakly decay modes only, assuming that the remaining modes are stormily damped. Simples case with single propagating mode and E i a i e i, H i b i h i. To simplify the notation, we assume that the series are finite. 19/10/017 "Waveguide systems", S. Kazakov, JAS

45 Let s make vectors: Ԧa = a 1,1, a,1,, a 1,n,, a k,n and Ԧb = b 1,1, b,1,, b 1,n,, b k,n. If all medias in waveguide system are isotropic and passive (no sources), Ԧa and Ԧb related as: Ԧa = Z Ԧb, Ԧb = Y Ԧa Z impedance matrix, Y admittance matrix. Fields in cross-section can be written as superposition of direct and reflected waves: E i = k=1 (c + k,i e k,i +c k,i e k,i ) H i = k=1 (c + k,i h k,i c k,i h k,i ) From this: Ԧa = Ԧc + + Ԧc Ԧ b = Ԧc + Ԧc Ԧc + and Ԧc related as: Ԧc = S Ԧc +, S Scattering Matrix Ԧc + - describes the waves coming into device from. Ԧc - describes the waves coming from device from, scattered waves. 19/10/017 "Waveguide systems", S. Kazakov, JAS

46 Relations between matrixes : From Ԧc + + Ԧc = Z Ԧc + Ԧc and Ԧc = S Ԧc + it folows I + S = Z I S and: Z = I + S I S 1, S = Z + I 1 Z I, Y = I S I + S 1, S = Y + I 1 I Y, Y = Z 1 Properties of matrixes In case of isotropic and passive medias all matrixes Z, Y, S are symmetric: m i.j = m j,m If there is no losses ( and passive ), all elements of matrix Z and Y are imaginary, Re m i,k = 0 In case no losses, and passive, the scattering matric is unitary: S T = S 1 Unitarily of matrix S is consequence of energy conservation low. Matrix is unitary for isotropic and anisotropic cases. 19/10/017 "Waveguide systems", S. Kazakov, JAS

47 S-matrixes are useful when you combine a complex system of simpler elements, whose S-matrixes are known or can be easy calculated. In this case you do not need to solve electrodynamic (fields) problem for whole system. You can divide system on simple elements, find S-matrixes and works with S-matrix only solving systems of liner equations. This approach provides you a full information about transmitting properties, like a bandwidth, losses, etc. Some times it is enough to solve the problem. But it provides pore information about electromagnetic field inside the system. Full electrodynamic task can be solved later, when you synthesize system with desired transiting properties. Example of matrixes: Waveguide with length L, without losses and with longitudinal wave number γ z S = 0 e iγ zl e iγ zl 0 Z = i cos γ zl sin γ z L i sin γ z L i sin γ z L i cos γ zl sin γ z L Y = i cos γ zl sin γ z L i sin γ z L i sin γ z L i cos γ zl sin γ z L For L = Τ πn γ z : S = Z and Y matrixes are indefinite. 19/10/017 "Waveguide systems", S. Kazakov, JAS

48 Usual task, when you design a waveguide system, is to make it matched or, in other words, without reflection. Reflected wave (power) reduce the efficiency, increase losses and fields strength, and affects to RF source. System is matched at port i if scattering matrix element S ii = 0. Let s consider three port power divider. Power coming to port 1 and divides between ports,3. Suppose we would like design divider without losses and matched at all ports. First case media inside device is isotropic. S-matrix looks like this: 1 S33+S31+S3 S = 0 s 1 s 13 s 1 0 s 3 s 31 s 3 0 s 11 = s = s 33 = 0 device is matched at all port s 1 = s 1, s 13 = s 31, s 3 = s 3 - isotropic conditions 1 S11+S1+S13 Port 1? Port 3 S11=S=S33=0? Port Unitarity: 1 S+S1+S3 s 1 s 1 + s 31 s 31 = s 1 s 1 + s 3 s 3 = s 13 s 13 + s 3 s 3 = 1 s 31 s 3 = 0, s 1 s 13 = 0, s 3 s 1 = 0 1 Port 3 S33+S31+S3 There is only one solution matrix S = 0 1 S11+S1+S13 Port 1? Port S11=S=S33=0? 19/10/017 "Waveguide systems", S. Kazakov, JAS S+S1+S3

49 Conclusion: Three port divider, matched, isotropic and without losses, is impossible. Some of these conditions must not be satisfied: no losses, isotropic and not-matched, isotropic, matched and lossy or nolosses, matched and anisotropic. Let s consider anisotropic case. We have the same equations except symmetry: S = 0 s 1 s 13 s 1 0 s 3 s 31 s 3 0 s 11 = s = s 33 = 0 device is matched at all port. Unitarity: s 31 s 3 = 0, s 1 s 13 = 0, s 3 s 1 = 0 s 1 s 1 + s 31 s 31 = s 1 s 1 + s 3 s 3 = s 13 s 13 + s 3 s 3 = 1 Suppose s 1 = 1. Therefor s 3 = 0 and s 13 = 1. Also s 31 = s 1 = 0 and s 3 = 1. Сcorrectly choosing the reference planes in the ports, we can assume s 1 = s 3 = s 13 = 1. And matrix will look like: S = We get matrix of three port matched circulator without losses. 19/10/017 "Waveguide systems", S. Kazakov, JAS

50 Tree port circulator: Typical scheme of circulator connection with RF source: Reflected power does not affect the source. It improves stability of source. 19/10/017 "Waveguide systems", S. Kazakov, JAS

51 Locking frequency and phase of magnetron trough circulator. Magnetron is generator / oscillator (not amplifier) with single output (no input). Own frequency and phase of magnetron are not defined and stable enough for particle accelerators application. But magnetron is attractive device because it is, probably, the cheapest powerful RF source (minimum $/kw). Using circulator, it is possible to lock frequency and phase of magnetron. Injected signal defines frequency and phase. 19/10/017 "Waveguide systems", S. Kazakov, JAS

52 We found: 3- port divider, matched, isotropic and without losses, is impossible. But solution exists for 4-port device. It is, for example, two hole directional coupler: S matrix of ideal directional coupler looks like this (based on symmetry and unitality) : S = 0 ae iφ ae iφ 0 0 be iθ be iθ 0 0 be iθ be iθ 0 0 ae iφ ae iφ 0 a Re, b Re b = 1 a φ θ = ± n 1 π, n = 1,, Phases in port and 4, 1 and 3 shifted ±90 O For some reference planes in ports S-matrix can looks like: S = 0 a a 0 0 ib ib 0 0 ib ib 0 0 a a 0 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 5

53 For a = b = 1 we have 3-dB directional coupler, a frequently used device in waveguide systems (some times it is called hybrid ). Instantaneous electric field pattern. Typical view of 3-D directional coupler: Note, fields in port and 4 shifted 90 O. Patten of average electric field: 19/10/017 "Waveguide systems", S. Kazakov, JAS

54 There is interesting example of 3-D directional coupler, which we cannot miss. This is so-called Magic-T Why it is magic? My guess, because the port and 4 are located directly opposite each other and they are decoupled. RF power propagates from port (4) in to ports 1 and 3, but not in port 4 (). It is real magic! 19/10/017 "Waveguide systems", S. Kazakov, JAS

55 It easy to understand how it works qualitatively using symmetry property of configuration: Step 1. Let s consider the excitation from port 1. Because of symmetry of excitation and configuration, we can divide geometry by Magnetic wall. It is clear from symmetry, TE10 mode can not propagate in port 3 (magnetic field, no H field along waveguide). Waveguide is cut-off, for other modes. No fields in port 3. E-field. Fields in port and 3 are symmetric, have the same phase. No field in por3. No reflection from port Average E-field. Sign of reflection 19/10/017 "Waveguide systems", S. Kazakov, JAS

56 E-field. Fields in port and 3 are antisymmetric, have the opposite phases. No field in por3. Step. The excitation from port 3. Because of symmetry of excitation and configuration, we can divide geometry by Electric wall (electric wall is equivalent to perfect conductor the same boundary conditions). It is clear from symmetry, TE10 mode can not propagate in port 1. Waveguide(s), divided by E-wall is cut-off, all modes. No fields in port 1. No reflection from port Sign of reflection Average E-field. 19/10/017 "Waveguide systems", S. Kazakov, JAS

57 Step 3. Matching the ports 1 and 3. There are standing waves (reflections) in port 1 and port3. We have to match ports to eliminate the reflections. These ports are decoupled: when one port is excited there is no fields in another one. It means the ports can be matched independently matching elements in one port do not affect to another port. Matching irises No reflection signs Symmetric solution Antisymmetric solution 19/10/017 "Waveguide systems", S. Kazakov, JAS

58 Step 4. Combine symmetric and antisymmetric solutions and reverse the time. In combine solution the fields are summarized (the same phases) in port and subtracted (opposite phases) in port 4 or contrary. Thus, we have double power in port and zero power in port 4. S-matrix of Magic-T with length of ports: S = i 1 i i 1 i /10/017 "Waveguide systems", S. Kazakov, JAS

59 Configuration with 3-dB coupler which allows to control frequency, phase and power at the input of accelerating cavity. 19/10/017 "Waveguide systems", S. Kazakov, JAS

60 New element is phase shifter. Phase velocity in waveguide depends on waveguide geometry and properties of medium of waveguide. Changing the geometry or property of medium we can change the electric length of waveguide or, in other words, relative (to input) phase of RF signal at the output of waveguide. It principle of operation of waveguide. γ = ± k π Longitudinal wave number γ of dominant mode TE, k = ω 10 in rectangular waveguide: a Phase difference between input and output (electrical length) is γl, L length of waveguide. Changing the waveguide width a we can change the phase. Phase shifter, DESY design. When wall moves to center of waveguides the electrical length of device becomes shorter. Phase shifter, KEK design. When insert moves to center of waveguides the electrical length of device becomes longer. c m 19/10/017 "Waveguide systems", S. Kazakov, JAS

61 Matched load, what inside? Waveguide matched load. Coaxial water load. Matched load imitates infinite waveguide. There is no reflection from ideal load. Lossy dielectric inside rectangular waveguide. Dielectric has shape of long cone to avoid reflection (the waveguide parameters change adiabatically). 19/10/017 "Waveguide systems", S. Kazakov, JAS

62 Typical configurations of matched loads Lossy ferroelectric in rectangular waveguide: Compact waveguide water load. Dielectric constant of water ε w 80, dielectric constant of Al O 3 ε c 9 ε w. Thus quarter wavelength ceramics matches water very good. λ 4 Ceramics Al O 3 Water 19/10/017 "Waveguide systems", S. Kazakov, JAS 017 6

63 Mode converters. Mode converter is device which transform mode of one type to mode of another type and does it with high efficiency (~ 100%). Let s consider the example of design of mode converter which transform dominant mode TE 10 in rectangular waveguide to mode TE 01 in circular waveguide. TE 10 TE 01 - is interesting mode. It is low losses mode and can be used for power transmission for long distance. Second, this mode has no electrical field on the metal surface. It is convenient for high power RF windows, no electric field at the place of brazing ceramics to the metal. Third, there is only azimuthal electrical current on the surface. Narrow azimuthal slots do not affect the performance of waveguide. (For example, one part of waveguide can be easily rotate relatively another one). Some times ago (when I was young) there was no simple and practical TE 10 TE 01 mode convertors. We found several simple and compact solutions (thanks HFSS!). 19/10/017 "Waveguide systems", S. Kazakov, JAS

64 Deaign GHz TE 10 TE 01 mode converter, logic of design. Let s start from TE 01 mode and take circle wave guide with diameter where TE 01 can propagate, but small enough to avoid higher modes. Diameter 40mm is good number for 11.4 GHz. Let s squeeze it at both side symmetrically. No perturbation no mode conversion. D40mm metallic cylinder. Pure TE01 at input will be converted to combination of only two mode (because of symmetry) TE01 and TE1 + TE01 TE01 TE1 19/10/017 "Waveguide systems", S. Kazakov, JAS

65 But the same combination of mode we can get from transition of rectangular waveguide to cylindrical waveguide if we excite TE0 mode in rectangular one: + TE01 TE1 TE01 19/10/017 "Waveguide systems", S. Kazakov, JAS

66 Combining this two part, after optimization of petameters (dimensions), we have mode converter from TE0 to TE01 (not from TE10 to TE01 yet!): Pure TE01 Pure T0 To get TE10-TE0 converter we need to design TE10 to TE0 mode converter in rectangular waveguides. It is easy. 19/10/017 "Waveguide systems", S. Kazakov, JAS

67 This simpler configuration transforms TE01 mode in to TE0 mode: Pure TE10 mode Pure TE0 mode Combining all parts we have compact and simple TE10-TE01 mode converters: 19/10/017 "Waveguide systems", S. Kazakov, JAS

68 Recommended books: Foundations for microwave engineering, R. E. Collin Field theory of Guided Waves, R. E. Collin Microwave Circuits, J. L. Altman 19/10/017 "Waveguide systems", S. Kazakov, JAS