# Equivalent circuit analysis of ridge-loaded folded-waveguide slow-wave structures for millimeter-wave traveling-wave tubes.

1. INTRODUCTIONThe folded waveguide (FWG) slow-wave structure (SWS) is one of the most important millimeter and terahertz-wave amplifiers which are suitable for traveling-wave tubes (TWTs) [1-7]. As a full-metal structure, the FWG circuit has been shown to have excellent performances in lower RF loss, higher power handing capability and simple input and output circuit transitions. However, the previous investigations on FWG-TWTs reveal that their electron efficiency are relatively low, which are mainly limited by the beam-wave interaction impedance [8-13]. In order to increase the electron efficiency of this kind of structure, a novel structure, ridge-loaded folded-waveguide slow-wave structure (RLFWG-SWS), has been presented by our research group [14]. Although the boundary conditions are very complicated, we still made some rough analysis of this structure according to the field-matching method under the following conditions:

1) The reflections from the waveguide bend and the beam tunnel are negligible.

2) The RLFWG-SWS was straightened and an equivalent analytical model was adapted.

All these assumptions will bring some errors, which will further impact on the accuracy of the design of TWTs, especially influence on the dispersion characteristics which is more sensitive to predict the performance of TWTs. For the sake of studying this structure as accurate as possible, we seek to use other method to analyze the RLFWG-SWS. Just because of this reason, equivalent circuit model based on transmission line cascade network, which can be used to analyze the high frequency characteristics of RLFWG-SWS, has been proposed.

With respect to the equivalent circuit, some scholars have paid attention to it, such as Booske et al. [12], Liu [15], Na et al. [16], Han et al. [17] and Sumathy et al. [18]. Na et al. [16] and Han et al. [17] exclude the effect of the beam hole. A thorough equivalent circuit analysis of folded-waveguide slow-wave structure including the effect of the beam hole was proposed by Booske et al. [12]. By means of the high pass filter model, the analysis of the high-frequency characteristic of FWG-SWS has been developed by Sumathy et al. [18].

In this paper, a complete equivalent circuit model is developed for studying V-band RLFWG-SWS. As an import atmospheric attenuation window, the V-band has been shown great potential on both military and civil areas, especially for complex military communication and information countermeasure [19-21]. The frequency range of 60-65 GHz is a region of the millimeter-wave spectrum that has been designated for inter-satellite communications.

The dispersion properties and the beam-wave interaction impedance are calculated using equivalent circuit model and three-dimensional (3-D) high-frequency electromagnetic simulation software (HFSS) [22]. The effects of the beam hole, the discontinuity of waveguide bend and the change of the narrow side dimension are considered in our model. The analytical results agree well with those obtained by HFSS.

This paper is organized as follows. A brief introduction is presented in Section 1. The analysis of the equivalent circuit of RLFWG-SWS is given in Section 2. The FWG-SWS, as a special case of RLFWG-SWS, which can be formed by changing the height of the ridge of RLFWG-SWS, is also discussed in this Section. The numerical analysis of high-frequency characteristics of RLFWG-SWS and FWG-SWS are given in Section 3. A brief summary is given in Section 4.

2. EQUIVALENT CIRCUIT OF RLFWG-SWS

The circuit is formed by the E-plane bend of a folded rectangular waveguide. The ridges are loaded on the straight portions of FWGSWS to increase the longitudinal electric field in the beam tunnel. It is assumed that the lowest transverse electric mode, [TE.sub.10,] propagatesalong the fold path shown by dotted line in Figure 1(a). An electronbeam passes through a small hole on the broad wall of the serpentine waveguide where the electric field intensity peaks. The beam-wave interaction in RLFWG-SWS occurs when a beam phase velocity is synchronized with the effective slow-down wave phase velocity. Figure 1(b) shows the transverse dimensions of the RLFWG-SWS. According to transmission line theory, it is feasible that the ridge-loaded folded waveguide can be modeled with an equivalent circuit, which is shown in Figure 2.

The RLFWG-SWS can be represented by different circuit components, which have their own transfer matrix [23]. Since folded waveguide is a periodic structure, the circuit parameters can be treated only with one pitch, the analytical model of the ridge-loaded folded waveguide is divided into six parts.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[A.sub.2]) a uniform circular E-plane bending of rectangular waveguide;

[C.sub.2]) straight rectangular waveguide;

[B.sub.2]) an E-plane junction of the straight rectangular waveguide and a uniform circular bend of rectangular waveguide;

[E.sub.2]) another straight rectangular waveguide, which change the narrow dimension basis of the former straight rectangular waveguide;

[D.sub.2]) the junction of the two straight rectangular;

[F.sub.2]) electron beam-hole;

Each part has their own transfer matrix by use of the knowledge of microwave networks. It can be written as follows [23]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[B.sub.2] = (1/ j[B.sub.2]/1) (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[D.sub.2] = (1/0) ([jX.sub.2] /1) (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[F.sub.2] = (1/0 -j[X.sub.2] 0/1) (6)

where, [[beta].sub.0] is the axial propagation constant of the straight waveguide ([C.sub.2]) and ([E.sub.2]) and [[beta].sub.1] the axial propagation constant of the uniform circular bend of rectangular cross section waveguide ([A.sub.2]). [L.sub.1], [L.sub.2], [L.sub.3] are the path length for the electromagnetic wave in the waveguide ([A.sub.2]), ([C.sub.2]) and ([E.sub.2]), respectively. The straight waveguide ([C.sub.2]) is treated as uniform transmission line with characteristics impedance of [Z.sub.o] and wave admittance of [Y.sub.o]. While the serpentine E-plane bend waveguide ([A.sub.2]), is also considered to uniform transmission line and has the characteristics impedance [Z.sub.1] and wave admittance [Y.sub.12]. Meanwhile, [E.sub.2] is treated as uniform transmission line with characteristics impedance of [Z.sub.01] and wave admittance of [Y.sub.01]. The discontinuity ([B.sub.2]) between serpentine E-plane bending waveguide ([A.sub.2]) and straight waveguide ([C.sub.2]) is represented by series reactance [X.sub.1].The discontinuity ([D.sub.2]) between straight waveguide ([C.sub.2]) and straight waveguide ([E.sub.2]) is represented by shunt admittance [B.sub.2] and the beam-hole on the straight waveguide is represented by series reactance [X.sub.2].

Expressions for the parameters in the above matrixes are given as [23,24]

[Z.sub.1] [Z.sub.0] = 1 + 1/24[(2[b.sub.1]/p).sup.2] - 1/60 [(2[b.sub.1]/p).sup.2] (2[pi][b.sub.1]/[[lambda].sub.g]).sup.2] (7)

[X.sub.1]/[Z.sub.0] = 32/([pi].sup.2) (2[pi][b.sub.1]/[[lambda].sub.g]).sup.3] [(2[b.sub.1]/p).sup.2] [infinity][summation over (n=1,3,...) 1/n [square root of 1 -([(2[b.sub.1)/n[lambda]g.sup.2)] (8)

[X.sub.2]/[Z.sub.01] =1/(4[pi].sup.2)[[beta].sub.0][b.sub.2][(2[r.sub.c /a.sup.3) (9)

[[lambda]'.sub.g] = [[lambda].sub.g]{1 + 1/12[(2[b.sub.1]/p).sup.2][1/2 - 1/5 (2[pi][b.sub.1]/[[lambda].sub.g]).sup.2]} (10)

[Z.sub.01] = [b.sub.2]/ [b.sub.1][Z.sub.0] (11)

[B.sub.2]/[Y.sub.0] =(2[b.sub.1]/[[lambda].sub.g])In csc ([pi][b.sub.2]/2[b.sub.1]) (12)

Here, p is the pitch of the RLFWG-SWS, [b.sub.2] the narrow dimension of the waveguide ([E.sub.2]), and [[lambda]'.sub.g] the guide-wavelength of the circular bend of rectangular cross section waveguide ([A.sub.2]). a is waveguide broad wall dimension, [b.sub.1] the narrow dimension of the waveguide ([C.sub.2]),[lambda] the free space wavelength, [[lambda].sub.g] the guide-wavelength of straight waveguide ([C.sub.2]) and ([E.sub.2]), and [r.sub.c] the tunnel radius.

3. ANALYSIS OF HIGH-FREQUENCY

CHARACTERISTICS OF RLFWG-SWS

3.1. Dispersion Relation

Treating the entire folded waveguide as a single transmission line, the transfer matrix H for one pitch of the circuit can be obtained by serial multiplication of the individual transfer matrix. Since the cascaded matrix is solvable, the high frequency characteristics of the RLFWG-SWS can be determined.

For the single equivalent matrix, the following expressions can be obtained according to [24]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Here, [V.sub.1] and [V.sub.2] are the voltages at the output and input of the equivalent transmission line, respectively. k is the phase shift. Y and Z are the admittance and impedance of the FWG equivalent circuit, respectively.

As discussed above, the cascaded product of these individual circuit can be equated to the single equivalent transmission line matrix [H].

H = [[A.sub.2] ] ][B.sub.2]] [[C.sub.2]] [[D.sub.2]] [[E.sub.2]] [[F.sub.2]] [[E.sub.2]] [[D.sub.2]] [[C.sub.2]] [[B.sub.2]] [[A.sub.2]] (15)

Substituting (1)-(6) into (15), the dispersion relation can be obtained, which can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

We can obtain the dispersion relation by solving the Equations (16)-(18). The effective phase shift per period of the structure, as seen by the electron beam, is given by

[beta] = k+[pi]/p = arccos(cos[k.sub.1]+cos[k.sub.2]+[pi]/p p (19)

Here, the additional [pi] term accounts for the inherent 180[degrees] phase shift of the electric field with respect to the electrons in one period, due to

the folded nature of the RLFWG-SWS.

The phase velocity as seen by electrons can be obtained as follows: [v.sub.p] = w/[beta].

3.2. Interaction Impedance

The on-axis interaction impedance of the RLFWG-SWS can be obtained as following expressions which stem from R. G. E. Hutter 25].

[k.sub.c]=[Z.sub.01][1/[beta][0.sub.p].sup.2][sin/[[beta].sub.0][b.sub.2]/2.sup.2] 1/[I.sup.2.bub.0] (20)

[Z.sub.01] = 120[pi] 2[b.sub.2]/a [[lambda].sub.g]/[lambda] (21)

[T.sup.2] = [[beta].sup.2.sub.0] - (w/c).sup) (22)

where w is the angular frequency of the wave (rad/s), c the speed of light, and [I.sub.o] a modified Bessel function.

3.3. Analysis of FWG-SWS

The FWG-SWS is a special case of RLFWG-SWS, which can be formed if [L.sub.3] is equal to zero. In the particular case, both [D.sub.2] and [E.sub.2] circuits vanished. A periodic equivalent circuit model of FWG-SWS is shown as Figure 3. It is particular note that the discontinuity of the beam-hole on the straight waveguide is represented by series reactance [X.sub.2], which is different from the beam-hole having been represented by n type network 12].

The high-frequency characteristics of SFWG-SWS are analyzed by the means of the same approach as the RLFWG-SWS.

Based on the above analysis of RLFWG-SWS, the dispersion relation and interaction impedance can be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

[k.sub.c] =[Z.sub.0][1/[[beta].sub.p.].sup.2][sin[[beta].sub.b]/2[[beta].sub.b]/2].sup2] 1/[I.sub.2.sup.0]([Tr.sub.c]) (24)

where

[T.sub.1] = cos2([[beta].sub.o][L.sub.o)cos2 [[beta].sub.1][L.sub.1)

[T.sub.2] = sin2([[beta].sub.o][L.sub.o)sin2 [[beta].sub.1][L.sub.1)

[T.sub.3] = sin2([[beta].sub.o][L.sub.o)cos2 [[beta].sub.1][L.sub.1)

[T.sub.4] = cos2([[beta].sub.o][L.sub.o)sin2 [[beta].sub.1][L.sub.1)

[T.sub.5] = sin2([[beta].sub.1][L.sub.1)+ [T.sub.4]

[T.sub.6] = cos2([[beta].sub.1][L.sub.1)- [T.sub.1]

[T.sub.7] = sin2([[beta].sub.1][L.sub.1)- [T.sub.4]

[FIGURE 3 OMITTED]

4. RESULTS OF NUMERICAL CALCULATIONS

The dispersion relation (19) and the interaction impedance (20) in the previous section can be solved numerically. For comparison, we also simulated the structure using 3-D electromagnetic code HFSS where the circuit has been considered to be lossless.

Figure 4 shows the dispersion relation and interaction impedance of RLFWG-SWS. In Figure 4(a), the values of normalized phase velocity calculated in each model differ from each other by an average of 0.79%. As can be seen in Figure 4(b), there is fairly good agreement between two models. The greatest variation in estimated impedance seems to be at the lower band edge, which is far from the center frequency. Both dispersion relation and interaction impedance obtained by equivalent circuit have some difference from HFSS. This is because that all the discontinuities in the RLFWG-SWS are represented by various lumped parameters and the accuracy rely on the ratio of tunnel radius to narrow dimension of waveguide.

In order to study the high-frequency characteristics of RLFWG-SWS more specifically, the influences of the ridge dimensions on dispersion relation and interaction impedance are investigated. It is shown in Figure 5 that, with the ridge thickness increases, the relative passband decreases rapidly and the interaction impedance increases noticeably. But we should choose the properly thickness of the ridge considering both passband and interaction impedance. Figure 6 shows that the effect on high-frequency characteristics by changing the height of the ridge. It can be concluded that, with the increase of [L.sub.3,] the normalized phase velocity decreases, the relative passband becomes narrow and the interaction impedance increases. It is important to note that the impedance increases slowly particularly as [L.sub.3]/L is greater than 0.4.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The dispersion relation (23) and the interaction impedance (24) of FWG-SWS also can be numerical calculated. We used four kinds of method to analyze the FWG-SWS, including the equivalent circuit approaches reported in references [12] and [16], HFSS, and the present method. The numerical calculation results are shown in Figure 7. It is indicated that the other three calculation data agree well with the numerical calculation data from the present method, which support our theory and show that the present method is reliable. On the other hand, since the present method is analytic, it is easier to work into an optimization algorithm and is faster to compute as long as given the parameters of FWG-SWS without setting a model. This means that

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

the present method is high efficiency.

Figure 8 shows that the effect of ridge-loaded on the dispersion relation and interaction impedance. It is observed that the RLFWG-SWS have lower normalized phase velocity and higher interaction impedance than FWG-SWS.

5. CONCLUSION

In this paper, an equivalent circuit approach of RLFWG-SWS is proposed. Unlike time-consuming 3-D electromagnetic simulation software, the present analysis is simple and provides closed-form expressions for obtaining the dispersion and the interaction impedance characteristics of the structure. It is found that the analysis results based on the equivalent circuit fairly agree with the results obtained by HFSS.

Such a circuit analysis is quite useful as a design tool which is easier to carry out an optimization algorithm with high efficiency, particularly for rapid parametric studies in the early phase of the design. The explicit relations between the design parameters provide physical insights into the dependence of the device performance on the structures parameters. Moreover, this method has several merits such as simple, fast, sufficient accurate and etc. In the future, in order to increase the electron efficiency of the interaction, it is feasible to search other novel slow-wave structures which are related to SFWG-SWS. By means of the equivalent circuit approach, we also can analyze these novel slow-wave structures which have more complicated boundary conditions than conventional SFWG-SWS. Subsequently, we plan to analyze the ridge-vane loaded folded waveguide slow-wave structure and the related analysis is in progress.

Received 26 April 2012, Accepted 14 June 2012, Scheduled 21 June 2012

ACKNOWLEDGMENT

This work was supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 61125103), the Vacuum Electronics National Lab Foundation, China (Grant No. 9140C050101110C0501), the National Natural Science Foundation of China (Grant Nos. 60971038 and 60971031) and the Fundamental Research Funds for the Central Universities, China (Grant Nos. ZYGX2010J1054 and ZYGX2011J034).

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Y. Hou *, J. Xu, H.-R. Yin, Y.-Y. Wei, L.-N. Yue, G.-Q. Zhao, Y.-B. Gong

National Key Laboratory of Science and Technology on Vacuum Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China

* Corresponding author: Yan Hou (houyano3o8@i63.com).

Table 1. Parameters of the RLFWG-SWS. Parameter Value (mm) waveguide broad wall dimension a 2.96 narrow dimension of the waveguide b 0.65 the length of the straight waveguide L 0.8 the length of the pitch of the RLFWG-SWS p 0.88 the radius of the beam tunnel [r.sub.c] 0.25 thickness of the ridge [b.sub.3] 0.2b height of the ridge 2[L.sub.3] L

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Author: | Hou, Y.; Xu, J.; Yin, H.-R.; Wei, Y.-Y.; Yue, L.-N.; Zhao, G.-Q.; Gong, Y.-B. |
---|---|

Publication: | Progress In Electromagnetics Research |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | Jul 1, 2012 |

Words: | 3636 |

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