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Design of couplers for traveling wave RF structures using 3D electromagnetic codes in the frequency domain
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 580 (2007) 1176–1183 www.elsevier.com/locate/nima
Design of couplers for traveling wave RF structures using 3D electromagnetic codes in the frequency domain David Alesinia,, Alessandro Galloa, Bruno Spataroa, Agostino Marinellib, Luigi Palumboa,b a
INFN-LNF, Via E. Fermi 40, I-00044 Frascati, Italy Universita` degli Studi di Roma ‘‘La Sapienza’’, Dip. Energetica, Via A. Scarpa 14 00161, Roma, Italy
b
Received 12 March 2007; received in revised form 21 June 2007; accepted 23 June 2007 Available online 28 June 2007
Abstract In this paper, a simple equivalent circuit model of a traveling wave RF structure with input and output couplers is introduced. A few properties related to evaluating the reflection coefficient of the single coupler are then obtained and discussed. From these properties, a general procedure to design couplers for traveling wave structures using 3D electromagnetic codes in the frequency domain is derived. Finally, an example of coupler design for a 2p/3 X band accelerating structure is illustrated. r 2007 Elsevier B.V. All rights reserved. PACS: 41.75.Lx; 29.17.+w; 41.20.Jb Keywords: RF structures design; Linear accelerator; Numerical codes
1. Introduction Traveling wave (TW) structures coupler design can be performed using 3D electromagnetic (e.m.) codes in the frequency domain. The technique can be applied to both accelerating [1] and deflecting [2] devices. In the first part of the paper, we introduce a simple equivalent circuit model of TW structures with input and output couplers and we discuss some important related properties. In the second part, we illustrate the procedure to design couplers with 3D e.m. codes. In particular, we will refer to the e.m. code HFSS [3] and we will illustrate the case of an X band 2p/3 mode accelerating structure. 2. Circuit model of TW structure The sketch of a disc-loaded TW structure is shown in Fig. 1. The cells of the TW section have to be designed in Corresponding author. Tel.: +39 06 94032639; fax: +39 06 94032256.
E-mail address: [email protected] (D. Alesini). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.06.045
order to have phase velocity of the working mode equal to the beam velocity [1]. The coupler cell dimensions have to be designed in order to minimize the reflected power at the waveguide input/output ports. Since, with e.m. codes, it is not possible to consider an infinite number of TW cells (or, equivalently, a boundary condition that perfectly matches the TW with a finite number of cells), one possible strategy to design the single couplers is to consider a TW structure with input and output couplers and a few cells (Fig. 1a). In this case it is possible to design the couplers by changing their dimensions (see Fig. 1b) minimizing the reflection coefficient at the waveguide input port. However, this does not necessarily correspond to a minimization of the coupler reflection coefficient. At a given frequency, in fact, one can have zero reflection coefficient at the waveguide input port because of cancellation between the reflected waves generated by the input and output couplers [4]. If this is the case, some backward wave is present in the structure, perturbing the cell-to-cell phase advance. Therefore, one also has to verify that the phase advance per cell in the TW structure is constant and equal to the nominal one (2p/3, as
ARTICLE IN PRESS D. Alesini et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 1176–1183
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Fig. 1. Sketch of a TW structure (a) and detail of the input coupler (b).
Fig. 2. Equivalent circuit model of the TW structure with couplers.
example). This procedure is, in general, very time consuming and we will not consider it. Other different techniques to find the correct coupler dimensions simulating a TW structure with input and output couplers and a few TW cells are discussed in Refs. [5,6]. In the following we propose a different design approach based on the simple circuit model of the structure with input and output couplers shown in Fig. 2. The two port networks with scattering matrix [S] correspond to the coupling cells regions that match the input/output waveguides to the disc loaded structure. Each cell of the TW structure is modeled by a two port networks, as discussed in Ref. [7]. If we consider the short-circuited structure without losses shown in Fig. 3(a), whose equivalent circuit is shown in
where Gs(n) is the reflection coefficient at the coupler waveguide (n is the position of the short circuited cell), S11 is the first element of the coupler scattering matrix and f is the phase advance per cell in the TW structure. Let us consider the two following quantities that represent the phase distortions at the waveguide input port due to the non-ideal coupler: s ðnÞ W10 ¼ ff½Gs ðnþ1Þ=G þf 2 s ðnþ1Þ þf W21 ¼ ff½Gs ðnþ2Þ=G 2
where +[x] indicates the phase of the complex number x. We have that (see Appendix B):
1 jS 11 j ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi sin f sin f 2 1 4 sin cos ð f W Þ d sin ð f W Þ sin d cos d sin 21 2 21 2 2 W21 sin W21 Fig. 3(b), we have (see Appendix A): Gs ðn þ 2Þ Gs ðn þ 1Þ ¼ ¼ ej2f Gs ðn þ 1Þ Gs ðnÞ ðwith Gs ðnÞ ¼ 1Þ
(2)
(3)
where
S 11 ¼ 03
ð1Þ
sin ð2f W10 W21 Þ ffi. sin d2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W10 sin W10 1 þ sin 2 cos ð 2f W W Þ 10 21 2 sin W21 sin W 21
(4)
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Fig. 3. (a) TW structure with short-circuited cells and (b) circuital model.
Fig. 5. TW single cell dimensions of the structure working on the 2p/3 mode at 11.424 GHz.
Fig. 4. |S11| as a function of W10 for different values of W21 assuming f ¼ 2p/3.
The procedure gives two solutions for |S11|. As shown in Appendix B, to select the correct solution it is necessary, first of all, to calculate the following angle: W20 ¼ 2ff 1 þ jS11 jejðZ2 þ2ðnþ2ÞfÞ 2ff 1 þ jS 11 jejðZ2 þ2nfÞ (5) where
ð1=jS 11 j2 Þ þ 1 ð2 sin f sin d2 = sin W21 Þ2 Z2 ¼ arccos jS 11 j 2 þ 2f p.
ð6Þ
The correct solution is the one that satisfies the following equality1: W20 ¼ W21 þ W10 . (7) 1
It is easy to verify from Eqs. (3)–(6) that W20 has four possible solutions. The correct solution for |S11| is the one that allows satisfying Eq. (7) for one of the two values of Z2.
From previous formulae it is easy to verify that in the ‘‘standard’’ cases of TW structures (with, for example, f ¼ p/3, 2p/3, 5p/6), we have that: W10 þ W21 513jS 11 j51. f
(8)
To illustrate this, plots of |S11| as a function of W10 for different values of W21 are presented in Fig. 4 for f ¼ 2p/3. Under the condition of small |S11| we have (see Appendix C): S11 ffi
Gc ðn þ 1Þ Gc ðnÞ ej2f 1 ej2f
(9)
where Gc(n) is the reflection coefficient at the input port of a complete n-cell structure (with input and output couplers) and n is the number of TW cells. Under the condition of small |S11| we have that (see Appendix D): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðW21 þ W10 Þ2 jS 11 j ffi þ W210 W10 ðW21 þ W10 Þ. 2 2 2ðsin fÞ 4ðcos fÞ (10)
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3. Couplers design
Fig. 6. Phase and amplitude of the longitudinal electric field on the axis of the single cell obtained by HFSS (W is the stored energy per unit length).
Fig. 7. Short-circuited structures for coupler optimization.
We consider, as an example, the case of a TW accelerating structure working on the 2p/3 mode at 11.424 GHz. The dimensions of the single TW cell are given in Fig. 5, and they have been adjusted (using HFSS) in order to have the phase velocity of the fundamental harmonic equal to c at 11.424 GHz. The phase and amplitude of the longitudinal electric field on axis are reported in Fig. 6. Considering the properties discussed in the preceding section related to the short-circuited structure, it is possible to design the coupler according to the procedure discussed in detail in the following. Let us consider the short circuited structures shown in Fig. 7 that correspond to n ¼ 0, n ¼ 1 and n ¼ 2 shortcircuited cells. The optimum coupler dimensions are those satisfying Eq. (1). Starting from a non-optimized design it is possible to calculate the reflection coefficient of the coupler using Eq. (3) and change the coupler dimensions to minimize |S11| in a step-by-step iterative procedure. Since, as Eq. (3) shows, the coupler reflection coefficient depends on the two parameters W10 and W21, it is enough to vary only two of the input coupler dimensions until the residual |S11| value is within the specified range. Referring to Fig. 1(b), simulations have shown that the most sensitive parameters are w and Rc while the length Lc and the thickness tc can be kept fixed. This procedure can also be included in an optimization algorithm to find the optimum value in a faster way as illustrated in [4]. The values of W10 and W21 as functions of the geometrical parameter Rc near its optimum values (Rc ¼ 10.335 mm)
Fig. 8. (a) value of the phases W10, W10 as a function of the coupler parameters Rc (w ¼ 9.57 mm) and (b) amplitude of the reflection coefficient.
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are plotted in Fig. 8(a). The optimum value for w is 9.57 mm. The corresponding amplitudes of the reflection coefficient, calculated by Eqs. (3) and (10), are plotted in Fig. 8(b). From this plot it is easy to verify that Eq. (10) can be applied only under the condition expressed by Eq. (8). Moreover, from these calculations, it is possible to evaluate the sensitivity of the input coupler reflection coefficient to all coupler dimensions. The amplitude of the reflection coefficient as a function of frequency, assuming the optimum values of Rc and w, is plotted in Fig. 10(a). It has been calculated by Eq. (3) including the different single cell phase advances of the TW structure at the different frequencies given by the
dispersion curve of Fig. 9. The details of the amplitude of the reflection coefficients near the working frequency 11.424 GHz calculated by Eqs. (3) and (10) are plotted in Fig. 10(b). From the plot it is easy to verify that the input coupler has reflection coefficient below 0.05 in a bandwidth of 20 MHz near the working frequency. Outside this bandwidth the reflection coefficient grows and over a certain threshold Eq. (10) is no longer useable and we have to consider the more general expression (3). Finally, in Figs. 11 and 12 we report the simulation results of a 7-cell structure in terms of electric field profiles (amplitude and phase) and reflection coefficient at the input port in the case of no losses. The finite number of
Fig. 9. TW structure dispersion curve (HFSS results).
Fig. 11. Electric field on axis (amplitude and phase) as a function of the longitudinal coordinate in a 7-cell structure (HFSS results). PIN is the input power.
Fig. 10. Amplitude of the reflection coefficient as a function of frequency.
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with the electromagnetic code HFSS, has been finally illustrated. Acknowledgment The authors would like to thank Dr. Erk Jensen for helpful discussions and suggestions. Appendix A Let us consider the first implication: S11 ¼ 0 )
Gs ðn þ 2Þ Gs ðn þ 1Þ ¼ ¼ ej2f . Gs ðn þ 1Þ Gs ðnÞ
(A.1)
The reflection coefficient at the input port is simply given by: Gs ðnÞ ¼ S 11 Fig. 12. Reflection coefficient at the coupler waveguide as a function of frequency in the case of a 7-cell structure (HFSS results).
reflection coefficient minima is given by the resonant SW patterns generated in the structure by the reflections at the input and output couplers. In fact, as previously observed, the coupler has a finite bandwidth, and over this, there are reflections of the TW. The minima are located in the passband of the periodic structure shown in Fig. 9 and their number is equal to the number of cells. Increasing the number of cells, we progressively increase the number of minima in the pass-band. This design procedure is generally applicable even to the cases of structures with losses and/or constant gradient instead of constant impedance. In the first case, the power losses in the first cells of the structure are, in general, so small [1] that it is possible to design the couplers considering the ideal structure without losses. It is easy to verify with simulations of the real structure (still considering few cells) that the results are correct. In the second case, the cell dimension variations after the coupler are, in general, so small [1] that it is possible to design the couplers considering a constant impedance structure with few equal cells. In this case input and output couplers have to be designed separately to match the initial and final values of the cell irises.
(A.2)
If we suppose that the structure has no losses, we have [7]: 8 jS 11 j ¼ jS 22 j ¼ a > > pffiffiffiffiffiffiffiffiffiffiffiffiffi < 1 a2 ejZ12 S 12 ¼ S21 ¼ (A.3) > > : Z þ Z ¼ 2Z p 1
2
12
where a is a real number p1 and Z1, Z2 and Z12 are the phases of the scattering coefficients S11, S22 and S12, respectively. Substituting Eq. (A.2) in Eq. (A.1) we have Gs ðn þ 2Þ Gs ðn þ 1Þ ¼ ¼ ej2f . Gs ðn þ 1Þ Gs ðnÞ Let us now consider the reverse implication: Gs ðn þ 2Þ Gs ðn þ 1Þ ¼ ¼ ej2f ) S 11 ¼ 0. Gs ðn þ 1Þ Gs ðnÞ Let us suppose S116¼0. From Eqs. (A.2) and (A.3), we have: ð1 a2 Þej2Z12 ej2nf 1 þ aejZ2 ej2nf aejZ1 þ ejðZ1 þZ2 2nfÞ ¼ 1 þ aejZ2 ej2nf 1 þ aejðZ2 þ2nfÞ ¼ ejðZ1 þZ2 2nfÞ , 1 þ aejðZ2 2nfÞ
Gs ðnÞ ¼ aejZ1
4. Conclusions We have illustrated the procedure to design couplers of TW structures with 3D electromagnetic codes working in frequency domain. The procedure is based on a proper analysis of the phase of the reflection coefficient for different length short-circuited structures. This design procedure is based on a simple circuit model that has been presented and solved. An example of input coupler design of an X band structure, using this technique implemented
S 12 S 21 ej2nf . 1 þ S 22 ej2nf
ðA:4Þ
the magnitude of the reflection coefficient is always equal to 1. Concerning the phase2 it is given by: (A.5) ffGs ðnÞ ¼ Z1 þ Z2 2nf þ 2ff 1 þ aejðZ2 þ2nfÞ , 2
The symbol +[x] means the phase of the complex number x.
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Considering the polygon ABCO we have: p 2f þ W10 þ W21 þ d2 þ d1 ¼ 2p ) d1 2 2 ¼ p þ 2f W10 W21 d2 .
ðB:3Þ
From the two Eqs. (B.2) and (B.3), it is easy to derive sin ð2f W10 W21 Þ ffi. sin d2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W10 sin W10 1 þ sin 2 cos ð2f W W Þ 10 21 2 sin W21 sin W 21
(B.4) By applying the law of cosines to the triangle BOW, we have: 1 ¼ jc1 j2 þ a2 2ajc1 j cos ðf W21 d2 þ p=2Þ 2 jc1 j jc1 j 1 cos ðf W21 d2 þ p=2Þ. ) 21¼ 2 a a a
Fig. A1. Complex numbers ch of Eq. (A.6) for h ¼ 0, 1, 2.
and therefore: 8 G ðnþ1Þ ff Gs s ðnÞ ¼ 2f þ 2ff 1 þ aejðZ2 þ2ðnþ1ÞfÞ > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > c1 > > > > jðZ þ2nfÞ 2 > 2ff 1 þ ae > > > |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} > < c0 G ðnþ2Þ s > ff Gs ðnþ1Þ ¼ 2f þ 2ff 1 þ aejðZ2 þ2ðnþ2ÞfÞ > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > c2 > > > > jðZ2 þ2ðnþ1Þf > > 2ff 1 þ ae > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > : c
ðB:5Þ Moreover since from (B.2) we have: jc1 j 2 sin f sin d2 ¼ a sin W21
(A.6)
1
and the complex numbers ch ¼ 1 þ aejðZ2 þ2ðnþhÞfÞ can be easily plotted for h ¼ 0,1,2 and are shown in Fig. A1. It is straightforward to verify that, except in the singular case of f ¼ p/2 and Z2 ¼ 0, p, the complex numbers c0, c1, c2 cannot all have the same phases if a6¼0 and therefore: Gs ðn þ 1Þ aej2f Gs ðnÞ
or
Gs ðn þ 2Þ aej2f . Gs ðn þ 1Þ
This result contradicts the hypothesis. Therefore S 11 has to be necessarily equal to 0. Appendix B In Fig. A1 we have: ff ðGs ðn þ 1ÞÞ=ðGs ðnÞÞ W10 ¼ þf 2 ff ðGs ðn þ 2ÞÞ=ðGs ðn þ 1ÞÞ W21 ¼ þ f. ðB:1Þ 2 By applying the sine theorem to the triangle ABO and BCO we obtain: 9 sin W10 sin d1 = ¼ 2a sin f jc1 j sin W10 ) sin d1 ¼ sin d2 . (B.2) sin W21 sin d2 ; sin W21 ¼ 2a sin f jc1 j
(B.6)
we obtain: 1 2 sin f sin d2 2 4 sin f sin d2 1¼ a2 sin W21 sin W21 cos ðf W21 d2 þ p=2Þ.
ðB:7Þ
From this equation it is easy to obtain Eq. (3). Concerning the choice between the two possible solutions we can apply the law of cosines to the triangle BOW obtaining the value of Z2: jc1 j2 ¼ 1 þ a2 2a cos ðp þ Z2 2fÞ 1 þ a2 jc1 j2 ) Z2 ¼ arccos þ 2f p ) 2a 0 2 1 2 sin f sin d2 1 þ 1 2 sin W21 B a C ) Z2 ¼ arccos @a A |{z} 2 Eq:ðB:6Þ
þ 2f p
ðB:8Þ
with these values of Z2 it is possible to calculate the phase distortions at the waveguide input port due to the non-ideal coupler using Eq. (A.6). The correct solution for a is the one that reproduces for one of the two values of Z2 the initial phase distortions or its sum as expressed in Eq. (7). Appendix C If jS 11 j 1 it follows that: Gc ðnÞ ¼ S 11 þ ¼ aejZ1 þ
S212 S 22 ej2nf 1 S 222 ej2nf
ð1 a2 Þaejð2Z12 þZ2 2nfÞ ffi a½ejZ1 þ ejð2Z12 þZ2 2nfÞ , 1 a2 ejð2Z2 2nfÞ
substituting this expression in Eq. (9) we have immediately that the equality is satisfied for Gc.
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Appendix D
ffGs ðnÞ ¼ Z1 þ Z2 2nf a sin ð2nf Z2 Þ þ 2arctan ffi Z1 1 þ a cos ð2nf Z2 Þ þ Z2 2nf þ 2a sin ð2nf Z2 Þ and therefore: 8 W10 ¼ a½sin ð2ðn þ 1Þf Z2 Þ sin ð2nf Z2 Þ > > < ¼ 2a sin ðfÞ cos ðf Z2 2nfÞ > > : W ¼ W þ W ¼ 2a sin ð2fÞ cos ð2f Z 2nfÞ 20
21
10
2
W20 cos ðfÞ cos ðf Z2 2nfÞ 2a sin ð2fÞ ¼ sin ðfÞ sin ðf Z2 2nfÞ 2 W20 ) þ cos2 ðfÞ cos2 ðf Z2 2nfÞ 2a sin ð2fÞ W20 cos ðf Z2 2nfÞ ¼ sin2 ðfÞ sin2 ðf Z2 2nfÞ 2a sin ðfÞ 2 2 W20 W10 W10 W20 ) þ cos2 ðfÞ 2a sin ð2fÞ 2a sin ðfÞ ½2a sin ðfÞ2 )
2 # W10 ¼ sin ðfÞ 1 2a sin ðfÞ 2 W20 þ ½2W10 cos ðfÞ2 4W10 W20 ) cos ðfÞ ¼ sin2 ðfÞ 16a2 sin2 ðfÞ 4W210 2 W20 þ 4W210 4W10 W20 ¼ 16a2 sin4 ðfÞ ) cos ðfÞ 2
Considering the expression (A.5) if jS11 j ¼ a51 we have:
1183
From this last equality it is easy to derive Eq. (10). References [1] P.M. Lapostolle, A.L. Septier (Eds.), Linear Accelerators, NorthHolland, Amsterdam, 1970. [2] P. Bernard, H. Lengeler, On the design of disk-loaded waveguides for the RF separator. CERN 68-30, Geneva, 1968. [3] /www.ansoft.comS. [4] A. Marinelli, Progetto di un accoppiatore in guida d’onda per una struttura accelerante ad onda viaggiante in banda X, Tesi di Laurea, University of Rome ‘‘La Sapienza,’’ 2004. [5] N.M. Kroll, et al., Applications of Time Domain Simulation to Coupler Design for Periodic Structures, in: Proceedings of the 20th International Linac Conference (LINAC 2000), August 21–25, 2000, Monterey, CA, SLAC-PUB-8614. [6] C. Nantista, et al., Phys. Rev. ST Accel. Beams 7 (2004) 072001. [7] R.E. Collin, Foundations for Microwave Engineering, McGraw-Hill, Inc., New York, 1992.
Nuclear Instruments and Methods in Physics Research A 580 (2007) 1176–1183 www.elsevier.com/locate/nima
Design of couplers for traveling wave RF structures using 3D electromagnetic codes in the frequency domain David Alesinia,, Alessandro Galloa, Bruno Spataroa, Agostino Marinellib, Luigi Palumboa,b a
INFN-LNF, Via E. Fermi 40, I-00044 Frascati, Italy Universita` degli Studi di Roma ‘‘La Sapienza’’, Dip. Energetica, Via A. Scarpa 14 00161, Roma, Italy
b
Received 12 March 2007; received in revised form 21 June 2007; accepted 23 June 2007 Available online 28 June 2007
Abstract In this paper, a simple equivalent circuit model of a traveling wave RF structure with input and output couplers is introduced. A few properties related to evaluating the reflection coefficient of the single coupler are then obtained and discussed. From these properties, a general procedure to design couplers for traveling wave structures using 3D electromagnetic codes in the frequency domain is derived. Finally, an example of coupler design for a 2p/3 X band accelerating structure is illustrated. r 2007 Elsevier B.V. All rights reserved. PACS: 41.75.Lx; 29.17.+w; 41.20.Jb Keywords: RF structures design; Linear accelerator; Numerical codes
1. Introduction Traveling wave (TW) structures coupler design can be performed using 3D electromagnetic (e.m.) codes in the frequency domain. The technique can be applied to both accelerating [1] and deflecting [2] devices. In the first part of the paper, we introduce a simple equivalent circuit model of TW structures with input and output couplers and we discuss some important related properties. In the second part, we illustrate the procedure to design couplers with 3D e.m. codes. In particular, we will refer to the e.m. code HFSS [3] and we will illustrate the case of an X band 2p/3 mode accelerating structure. 2. Circuit model of TW structure The sketch of a disc-loaded TW structure is shown in Fig. 1. The cells of the TW section have to be designed in Corresponding author. Tel.: +39 06 94032639; fax: +39 06 94032256.
E-mail address: [email protected] (D. Alesini). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.06.045
order to have phase velocity of the working mode equal to the beam velocity [1]. The coupler cell dimensions have to be designed in order to minimize the reflected power at the waveguide input/output ports. Since, with e.m. codes, it is not possible to consider an infinite number of TW cells (or, equivalently, a boundary condition that perfectly matches the TW with a finite number of cells), one possible strategy to design the single couplers is to consider a TW structure with input and output couplers and a few cells (Fig. 1a). In this case it is possible to design the couplers by changing their dimensions (see Fig. 1b) minimizing the reflection coefficient at the waveguide input port. However, this does not necessarily correspond to a minimization of the coupler reflection coefficient. At a given frequency, in fact, one can have zero reflection coefficient at the waveguide input port because of cancellation between the reflected waves generated by the input and output couplers [4]. If this is the case, some backward wave is present in the structure, perturbing the cell-to-cell phase advance. Therefore, one also has to verify that the phase advance per cell in the TW structure is constant and equal to the nominal one (2p/3, as
ARTICLE IN PRESS D. Alesini et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 1176–1183
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Fig. 1. Sketch of a TW structure (a) and detail of the input coupler (b).
Fig. 2. Equivalent circuit model of the TW structure with couplers.
example). This procedure is, in general, very time consuming and we will not consider it. Other different techniques to find the correct coupler dimensions simulating a TW structure with input and output couplers and a few TW cells are discussed in Refs. [5,6]. In the following we propose a different design approach based on the simple circuit model of the structure with input and output couplers shown in Fig. 2. The two port networks with scattering matrix [S] correspond to the coupling cells regions that match the input/output waveguides to the disc loaded structure. Each cell of the TW structure is modeled by a two port networks, as discussed in Ref. [7]. If we consider the short-circuited structure without losses shown in Fig. 3(a), whose equivalent circuit is shown in
where Gs(n) is the reflection coefficient at the coupler waveguide (n is the position of the short circuited cell), S11 is the first element of the coupler scattering matrix and f is the phase advance per cell in the TW structure. Let us consider the two following quantities that represent the phase distortions at the waveguide input port due to the non-ideal coupler: s ðnÞ W10 ¼ ff½Gs ðnþ1Þ=G þf 2 s ðnþ1Þ þf W21 ¼ ff½Gs ðnþ2Þ=G 2
where +[x] indicates the phase of the complex number x. We have that (see Appendix B):
1 jS 11 j ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi sin f sin f 2 1 4 sin cos ð f W Þ d sin ð f W Þ sin d cos d sin 21 2 21 2 2 W21 sin W21 Fig. 3(b), we have (see Appendix A): Gs ðn þ 2Þ Gs ðn þ 1Þ ¼ ¼ ej2f Gs ðn þ 1Þ Gs ðnÞ ðwith Gs ðnÞ ¼ 1Þ
(2)
(3)
where
S 11 ¼ 03
ð1Þ
sin ð2f W10 W21 Þ ffi. sin d2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W10 sin W10 1 þ sin 2 cos ð 2f W W Þ 10 21 2 sin W21 sin W 21
(4)
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Fig. 3. (a) TW structure with short-circuited cells and (b) circuital model.
Fig. 5. TW single cell dimensions of the structure working on the 2p/3 mode at 11.424 GHz.
Fig. 4. |S11| as a function of W10 for different values of W21 assuming f ¼ 2p/3.
The procedure gives two solutions for |S11|. As shown in Appendix B, to select the correct solution it is necessary, first of all, to calculate the following angle: W20 ¼ 2ff 1 þ jS11 jejðZ2 þ2ðnþ2ÞfÞ 2ff 1 þ jS 11 jejðZ2 þ2nfÞ (5) where
ð1=jS 11 j2 Þ þ 1 ð2 sin f sin d2 = sin W21 Þ2 Z2 ¼ arccos jS 11 j 2 þ 2f p.
ð6Þ
The correct solution is the one that satisfies the following equality1: W20 ¼ W21 þ W10 . (7) 1
It is easy to verify from Eqs. (3)–(6) that W20 has four possible solutions. The correct solution for |S11| is the one that allows satisfying Eq. (7) for one of the two values of Z2.
From previous formulae it is easy to verify that in the ‘‘standard’’ cases of TW structures (with, for example, f ¼ p/3, 2p/3, 5p/6), we have that: W10 þ W21 513jS 11 j51. f
(8)
To illustrate this, plots of |S11| as a function of W10 for different values of W21 are presented in Fig. 4 for f ¼ 2p/3. Under the condition of small |S11| we have (see Appendix C): S11 ffi
Gc ðn þ 1Þ Gc ðnÞ ej2f 1 ej2f
(9)
where Gc(n) is the reflection coefficient at the input port of a complete n-cell structure (with input and output couplers) and n is the number of TW cells. Under the condition of small |S11| we have that (see Appendix D): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðW21 þ W10 Þ2 jS 11 j ffi þ W210 W10 ðW21 þ W10 Þ. 2 2 2ðsin fÞ 4ðcos fÞ (10)
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3. Couplers design
Fig. 6. Phase and amplitude of the longitudinal electric field on the axis of the single cell obtained by HFSS (W is the stored energy per unit length).
Fig. 7. Short-circuited structures for coupler optimization.
We consider, as an example, the case of a TW accelerating structure working on the 2p/3 mode at 11.424 GHz. The dimensions of the single TW cell are given in Fig. 5, and they have been adjusted (using HFSS) in order to have the phase velocity of the fundamental harmonic equal to c at 11.424 GHz. The phase and amplitude of the longitudinal electric field on axis are reported in Fig. 6. Considering the properties discussed in the preceding section related to the short-circuited structure, it is possible to design the coupler according to the procedure discussed in detail in the following. Let us consider the short circuited structures shown in Fig. 7 that correspond to n ¼ 0, n ¼ 1 and n ¼ 2 shortcircuited cells. The optimum coupler dimensions are those satisfying Eq. (1). Starting from a non-optimized design it is possible to calculate the reflection coefficient of the coupler using Eq. (3) and change the coupler dimensions to minimize |S11| in a step-by-step iterative procedure. Since, as Eq. (3) shows, the coupler reflection coefficient depends on the two parameters W10 and W21, it is enough to vary only two of the input coupler dimensions until the residual |S11| value is within the specified range. Referring to Fig. 1(b), simulations have shown that the most sensitive parameters are w and Rc while the length Lc and the thickness tc can be kept fixed. This procedure can also be included in an optimization algorithm to find the optimum value in a faster way as illustrated in [4]. The values of W10 and W21 as functions of the geometrical parameter Rc near its optimum values (Rc ¼ 10.335 mm)
Fig. 8. (a) value of the phases W10, W10 as a function of the coupler parameters Rc (w ¼ 9.57 mm) and (b) amplitude of the reflection coefficient.
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are plotted in Fig. 8(a). The optimum value for w is 9.57 mm. The corresponding amplitudes of the reflection coefficient, calculated by Eqs. (3) and (10), are plotted in Fig. 8(b). From this plot it is easy to verify that Eq. (10) can be applied only under the condition expressed by Eq. (8). Moreover, from these calculations, it is possible to evaluate the sensitivity of the input coupler reflection coefficient to all coupler dimensions. The amplitude of the reflection coefficient as a function of frequency, assuming the optimum values of Rc and w, is plotted in Fig. 10(a). It has been calculated by Eq. (3) including the different single cell phase advances of the TW structure at the different frequencies given by the
dispersion curve of Fig. 9. The details of the amplitude of the reflection coefficients near the working frequency 11.424 GHz calculated by Eqs. (3) and (10) are plotted in Fig. 10(b). From the plot it is easy to verify that the input coupler has reflection coefficient below 0.05 in a bandwidth of 20 MHz near the working frequency. Outside this bandwidth the reflection coefficient grows and over a certain threshold Eq. (10) is no longer useable and we have to consider the more general expression (3). Finally, in Figs. 11 and 12 we report the simulation results of a 7-cell structure in terms of electric field profiles (amplitude and phase) and reflection coefficient at the input port in the case of no losses. The finite number of
Fig. 9. TW structure dispersion curve (HFSS results).
Fig. 11. Electric field on axis (amplitude and phase) as a function of the longitudinal coordinate in a 7-cell structure (HFSS results). PIN is the input power.
Fig. 10. Amplitude of the reflection coefficient as a function of frequency.
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with the electromagnetic code HFSS, has been finally illustrated. Acknowledgment The authors would like to thank Dr. Erk Jensen for helpful discussions and suggestions. Appendix A Let us consider the first implication: S11 ¼ 0 )
Gs ðn þ 2Þ Gs ðn þ 1Þ ¼ ¼ ej2f . Gs ðn þ 1Þ Gs ðnÞ
(A.1)
The reflection coefficient at the input port is simply given by: Gs ðnÞ ¼ S 11 Fig. 12. Reflection coefficient at the coupler waveguide as a function of frequency in the case of a 7-cell structure (HFSS results).
reflection coefficient minima is given by the resonant SW patterns generated in the structure by the reflections at the input and output couplers. In fact, as previously observed, the coupler has a finite bandwidth, and over this, there are reflections of the TW. The minima are located in the passband of the periodic structure shown in Fig. 9 and their number is equal to the number of cells. Increasing the number of cells, we progressively increase the number of minima in the pass-band. This design procedure is generally applicable even to the cases of structures with losses and/or constant gradient instead of constant impedance. In the first case, the power losses in the first cells of the structure are, in general, so small [1] that it is possible to design the couplers considering the ideal structure without losses. It is easy to verify with simulations of the real structure (still considering few cells) that the results are correct. In the second case, the cell dimension variations after the coupler are, in general, so small [1] that it is possible to design the couplers considering a constant impedance structure with few equal cells. In this case input and output couplers have to be designed separately to match the initial and final values of the cell irises.
(A.2)
If we suppose that the structure has no losses, we have [7]: 8 jS 11 j ¼ jS 22 j ¼ a > > pffiffiffiffiffiffiffiffiffiffiffiffiffi < 1 a2 ejZ12 S 12 ¼ S21 ¼ (A.3) > > : Z þ Z ¼ 2Z p 1
2
12
where a is a real number p1 and Z1, Z2 and Z12 are the phases of the scattering coefficients S11, S22 and S12, respectively. Substituting Eq. (A.2) in Eq. (A.1) we have Gs ðn þ 2Þ Gs ðn þ 1Þ ¼ ¼ ej2f . Gs ðn þ 1Þ Gs ðnÞ Let us now consider the reverse implication: Gs ðn þ 2Þ Gs ðn þ 1Þ ¼ ¼ ej2f ) S 11 ¼ 0. Gs ðn þ 1Þ Gs ðnÞ Let us suppose S116¼0. From Eqs. (A.2) and (A.3), we have: ð1 a2 Þej2Z12 ej2nf 1 þ aejZ2 ej2nf aejZ1 þ ejðZ1 þZ2 2nfÞ ¼ 1 þ aejZ2 ej2nf 1 þ aejðZ2 þ2nfÞ ¼ ejðZ1 þZ2 2nfÞ , 1 þ aejðZ2 2nfÞ
Gs ðnÞ ¼ aejZ1
4. Conclusions We have illustrated the procedure to design couplers of TW structures with 3D electromagnetic codes working in frequency domain. The procedure is based on a proper analysis of the phase of the reflection coefficient for different length short-circuited structures. This design procedure is based on a simple circuit model that has been presented and solved. An example of input coupler design of an X band structure, using this technique implemented
S 12 S 21 ej2nf . 1 þ S 22 ej2nf
ðA:4Þ
the magnitude of the reflection coefficient is always equal to 1. Concerning the phase2 it is given by: (A.5) ffGs ðnÞ ¼ Z1 þ Z2 2nf þ 2ff 1 þ aejðZ2 þ2nfÞ , 2
The symbol +[x] means the phase of the complex number x.
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Considering the polygon ABCO we have: p 2f þ W10 þ W21 þ d2 þ d1 ¼ 2p ) d1 2 2 ¼ p þ 2f W10 W21 d2 .
ðB:3Þ
From the two Eqs. (B.2) and (B.3), it is easy to derive sin ð2f W10 W21 Þ ffi. sin d2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 W10 sin W10 1 þ sin 2 cos ð2f W W Þ 10 21 2 sin W21 sin W 21
(B.4) By applying the law of cosines to the triangle BOW, we have: 1 ¼ jc1 j2 þ a2 2ajc1 j cos ðf W21 d2 þ p=2Þ 2 jc1 j jc1 j 1 cos ðf W21 d2 þ p=2Þ. ) 21¼ 2 a a a
Fig. A1. Complex numbers ch of Eq. (A.6) for h ¼ 0, 1, 2.
and therefore: 8 G ðnþ1Þ ff Gs s ðnÞ ¼ 2f þ 2ff 1 þ aejðZ2 þ2ðnþ1ÞfÞ > > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > c1 > > > > jðZ þ2nfÞ 2 > 2ff 1 þ ae > > > |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} > < c0 G ðnþ2Þ s > ff Gs ðnþ1Þ ¼ 2f þ 2ff 1 þ aejðZ2 þ2ðnþ2ÞfÞ > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > > > c2 > > > > jðZ2 þ2ðnþ1Þf > > 2ff 1 þ ae > > |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > : c
ðB:5Þ Moreover since from (B.2) we have: jc1 j 2 sin f sin d2 ¼ a sin W21
(A.6)
1
and the complex numbers ch ¼ 1 þ aejðZ2 þ2ðnþhÞfÞ can be easily plotted for h ¼ 0,1,2 and are shown in Fig. A1. It is straightforward to verify that, except in the singular case of f ¼ p/2 and Z2 ¼ 0, p, the complex numbers c0, c1, c2 cannot all have the same phases if a6¼0 and therefore: Gs ðn þ 1Þ aej2f Gs ðnÞ
or
Gs ðn þ 2Þ aej2f . Gs ðn þ 1Þ
This result contradicts the hypothesis. Therefore S 11 has to be necessarily equal to 0. Appendix B In Fig. A1 we have: ff ðGs ðn þ 1ÞÞ=ðGs ðnÞÞ W10 ¼ þf 2 ff ðGs ðn þ 2ÞÞ=ðGs ðn þ 1ÞÞ W21 ¼ þ f. ðB:1Þ 2 By applying the sine theorem to the triangle ABO and BCO we obtain: 9 sin W10 sin d1 = ¼ 2a sin f jc1 j sin W10 ) sin d1 ¼ sin d2 . (B.2) sin W21 sin d2 ; sin W21 ¼ 2a sin f jc1 j
(B.6)
we obtain: 1 2 sin f sin d2 2 4 sin f sin d2 1¼ a2 sin W21 sin W21 cos ðf W21 d2 þ p=2Þ.
ðB:7Þ
From this equation it is easy to obtain Eq. (3). Concerning the choice between the two possible solutions we can apply the law of cosines to the triangle BOW obtaining the value of Z2: jc1 j2 ¼ 1 þ a2 2a cos ðp þ Z2 2fÞ 1 þ a2 jc1 j2 ) Z2 ¼ arccos þ 2f p ) 2a 0 2 1 2 sin f sin d2 1 þ 1 2 sin W21 B a C ) Z2 ¼ arccos @a A |{z} 2 Eq:ðB:6Þ
þ 2f p
ðB:8Þ
with these values of Z2 it is possible to calculate the phase distortions at the waveguide input port due to the non-ideal coupler using Eq. (A.6). The correct solution for a is the one that reproduces for one of the two values of Z2 the initial phase distortions or its sum as expressed in Eq. (7). Appendix C If jS 11 j 1 it follows that: Gc ðnÞ ¼ S 11 þ ¼ aejZ1 þ
S212 S 22 ej2nf 1 S 222 ej2nf
ð1 a2 Þaejð2Z12 þZ2 2nfÞ ffi a½ejZ1 þ ejð2Z12 þZ2 2nfÞ , 1 a2 ejð2Z2 2nfÞ
substituting this expression in Eq. (9) we have immediately that the equality is satisfied for Gc.
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Appendix D
ffGs ðnÞ ¼ Z1 þ Z2 2nf a sin ð2nf Z2 Þ þ 2arctan ffi Z1 1 þ a cos ð2nf Z2 Þ þ Z2 2nf þ 2a sin ð2nf Z2 Þ and therefore: 8 W10 ¼ a½sin ð2ðn þ 1Þf Z2 Þ sin ð2nf Z2 Þ > > < ¼ 2a sin ðfÞ cos ðf Z2 2nfÞ > > : W ¼ W þ W ¼ 2a sin ð2fÞ cos ð2f Z 2nfÞ 20
21
10
2
W20 cos ðfÞ cos ðf Z2 2nfÞ 2a sin ð2fÞ ¼ sin ðfÞ sin ðf Z2 2nfÞ 2 W20 ) þ cos2 ðfÞ cos2 ðf Z2 2nfÞ 2a sin ð2fÞ W20 cos ðf Z2 2nfÞ ¼ sin2 ðfÞ sin2 ðf Z2 2nfÞ 2a sin ðfÞ 2 2 W20 W10 W10 W20 ) þ cos2 ðfÞ 2a sin ð2fÞ 2a sin ðfÞ ½2a sin ðfÞ2 )
2 # W10 ¼ sin ðfÞ 1 2a sin ðfÞ 2 W20 þ ½2W10 cos ðfÞ2 4W10 W20 ) cos ðfÞ ¼ sin2 ðfÞ 16a2 sin2 ðfÞ 4W210 2 W20 þ 4W210 4W10 W20 ¼ 16a2 sin4 ðfÞ ) cos ðfÞ 2
Considering the expression (A.5) if jS11 j ¼ a51 we have:
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From this last equality it is easy to derive Eq. (10). References [1] P.M. Lapostolle, A.L. Septier (Eds.), Linear Accelerators, NorthHolland, Amsterdam, 1970. [2] P. Bernard, H. Lengeler, On the design of disk-loaded waveguides for the RF separator. CERN 68-30, Geneva, 1968. [3] /www.ansoft.comS. [4] A. Marinelli, Progetto di un accoppiatore in guida d’onda per una struttura accelerante ad onda viaggiante in banda X, Tesi di Laurea, University of Rome ‘‘La Sapienza,’’ 2004. [5] N.M. Kroll, et al., Applications of Time Domain Simulation to Coupler Design for Periodic Structures, in: Proceedings of the 20th International Linac Conference (LINAC 2000), August 21–25, 2000, Monterey, CA, SLAC-PUB-8614. [6] C. Nantista, et al., Phys. Rev. ST Accel. Beams 7 (2004) 072001. [7] R.E. Collin, Foundations for Microwave Engineering, McGraw-Hill, Inc., New York, 1992.