Abstractâ In this paper, an equivalent circuit model of square waveguide T-junction is derived from numerical results obtained by mode-matching analysis.
Equivalent Circuit Model of Square Waveguide T-junction for Ortho-Mode Transducers Yun Tao # , Zhongxiang Shen # , and Gang Liu ∗ #
School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798 ∗ ST Electronics (Satcom & Sensor Systems) Pte Ltd 100 Jurong East Street 21, ST Electronics Jurong East Building, Singapore 609602
Abstract— In this paper, an equivalent circuit model of square waveguide T-junction is derived from numerical results obtained by mode-matching analysis. The resultant 5-port network is simplified as a combination of a 3-port network and a 2-port network. A short-circuited branch ortho-mode transducer (OMT) is then designed in circuit simulator using this equivalent circuit model. Measured results exhibit good agreement with those from circuit simulation. Index Terms— Equivalent circuit, multi-port, ortho-mode transducer, T-junction, square waveguide.
T0'
(a) 3-D view
I. I NTRODUCTION
Fig. 1.
Waveguide T-junctions have been studied intensively for a long time because it is an essential component in numerous microwave applications [1], [2]. Equivalent circuits of a rectangular waveguide T-junction were initially derived based on the electrostatic approximations [1]. Later on, some improved theory and methods were developed to provide more accurate models [3]–[6]. In the design of ortho-mode transducers (OMTs), the square waveguide T-junction is of most interest. Although it can be analyzed with electromagnetic (EM) field simulators, its equivalent circuit cannot be directly obtained from the rectangular E-plane or H-plane ones. This is because two degenerated TE10 and TE01 modes are propagating simultaneously in the square waveguide, which results in an electrical 5-port network, rather than a conventional 3-port one. The synthesis method for this multi-port network is still not fully established [7], [8]. Therefore, although many OMTs have been designed in different antenna feed systems, the design of OMTs still relies mostly on experience [9], cut and try or full-wave optimization [10]. For the purpose of efficient design of OMTs, a good equivalent circuit model of this multi-port network is most desired. In this paper, the square waveguide T-junction is firstly analyzed by the mode matching (MM) method to obtain the generalized scattering matrix (GSM). The resultant scattering matrix of propagating modes is then converted into a normalized Y-matrix, which can be readily transferred to a 5port equivalent circuit model [11]. Since the cross-polarization coupling is found to be negligible, this 5-port model is, therefore, simplified as two individual 3-port and 2-port networks. Based on this equivalent circuit model, an X band short-
978-1-4244-2804-5/09/$25.00 © 2009 IEEE
T0
T0
(b) Top view
Square waveguide T-junction.
circuited branch OMT can be designed with circuit simulator, such as Agilent’s Advanced Design System (ADS), which is much faster than the original electromagnetic field simulator. A designed OMT is then fabricated and tested to verify the validity of our equivalent circuit model. II. D ERIVATION OF E QUIVALENT C IRCUIT A square waveguide T-junction is shown in Fig. 1, where a rectangular waveguide is longitudinally connected to the side wall of a square waveguide. Because the square waveguide supports two degenerated TE10 and TE01 modes simultaneously, this T-junction can be electrically represented by a 5-port network. The general equivalent circuit of a 5-port network is shown in Fig. 2. The series and shunt admittances can be calculated from its Y-matrix [11]. ⎧ ⎨ Ysij = −Yij 5 (1) Yij ⎩ Ypi = j=1
where Yij is the element value of Y-matrix, i, j = 1, 2, ..., 5. It should be noted that the value of calculated circuit elements may be different due to the different definitions of the equivalent current’s direction. For the lossless network, the normalized Y-matrix can be obtained by [12] Y = −jS−1 i (I − Sr )
(2)
where Sr and Si are the real and imaginary parts of its Smatrix respectively, I is the identical matrix. The S-matrix
1013
IMS 2009
s2 3
T0 Port 5
Ys34
s1 4
Y 5 s2
Y
s1 5
5 s3
Y
15
Y
Yp5
Yp1
T0
Port 5 Ys12
15
Yp1
Yp3
Port 3
Ys13
Ys
Ys
Yp1
Port 1
Port 3
Ys13
Y
Port 1
Yp5
Y
T0'
Port 2
s4 5
Port 4 Ys24
Port 2 Yp2
Yp2
Port 4
Ys24
T0
Yp4
Yp2 T0
Fig. 3. Simplified equivalent circuit of a square waveguide Tjunction. Fig. 2.
General equivalent circuit model of a 5-port network.
can be obtained from numerical methods, such as the MM method, the finite-element method (FEM) and the method of moments (MoM). In this paper, the mode-matching method is employed. Although the GSM can be obtained by the modematching method, only the S-parameters of propagating modes are of interest. Although the general equivalent circuit model of a 5port network is rather complicated, the one for our square waveguide T-junction may be simplified by simple physical reasoning. The TE10 and TE01 modes are intrinsically orthogonal in the square waveguide, and the obstruction of the rectangular branch waveguide does not contribute much to the cross-polarization. Therefore, it is possible to separate the whole 5-port network into two individual orthogonal networks. For TE10 mode, it is a 3-port network because it can propagate among Ports 1, 3 and 5. For TE01 mode, it is a 2-port network corresponding to Ports 2 and 4 because it is evanescent in the rectangular side arm, which can be considered as a reactive load rather than an electrical port. This understanding is verified from the calculated values of circuit elements. The admittances Ys12 , Ys14 , Ys25 are almost zero, which are corresponding to open circuit. This shows that crosspolarization coupling between the orthogonal ports is really very small, as we expected. Meanwhile, due to the symmetry, Yp1 = Yp3 , Yp2 = Yp4 , Ys15 = Ys35 . The equivalent circuit of a square waveguide T-junction, therefore, is simplified as two individual 3-port and 2-port networks, as shown in Fig. 3. This equivalent circuit model can be then loaded into circuit simulator, such as Agilent’s Advanced Design System (ADS), for efficient analysis and design of OMTs. As an example, we consider an X band square waveguide T-junction here. Although the slot-coupled T-junction can be modeled in the same manner, the directly mounted case is considered here for simplicity. The dimension of the square
waveguide is s = 18mm, while the rectangular waveguide is WR90 (a = 22.86mm, b = 10.16mm). The S-parameters calculated by circuit simulator are compared with those obtained by the mode-matching method, as shown in Fig. 4. Good agreement is observed from 8.3GHz to 11.7GHz, which covers most of the entire X band. The discrepancy in the low and high frequencies is due to the fact that only the propagating modes are considered in this equivalent circuit model: all modes are cutoff below 8.3GHz; higher-order modes start to become propagating near 11.7GHz. III. D ESIGN OF S HORT-C IRCUITED B RANCH OMT A short-circuited branch OMT can be implemented by orthogonally cascading two square waveguide T-junctions with one end terminated by a short-circuit, as shown in Fig. 5. With the equivalent circuit model obtained above, this structure can be efficiently analyzed in circuit simulation rather than the full-wave one [13], as shown in Fig. 6. The lengths of the two transmission lines l1 and l2 can be easily optimized in the circuit simulator to obtain the desired S-parameters. It is noticed that in this circuit model, the cross-polarization between the orthogonal port is ignored because this structure can intrinsically suppress it [13]. When l1 = 1.07mm and l2 = 5.64mm, the S-parameters of designed OMT are shown in Fig. 7. Results calculated by the equivalent circuit model are still in good agreement with those obtained by the mode-matching method. The center frequency is located at about 9.75GHz. For the vertically polarized Port 1, the 15dB return loss bandwidth is about 10%, while for the horizontally polarized Port 2, it is about 5%. To verify our design derived from the proposed equivalent circuit model, two identically designed OMTs are fabricated and connected in the back-to-back configuration, as shown in Fig. 8. The port numbers of the back-to-back configuration are redefined accordingly. Figure 9 presents measured and
1014
1.2 Mode Matching Method Equivalent Circuit
1
Magnitude
0.8
S11
0.6 S
15
0.4
0.2 S22 0
8
8.5
9
9.5 10 10.5 Frequency (GHz)
11
11.5
12
(a) Magnitude
200
Fig. 5.
Mode Matching Method Equivalent Circuit
150
Short-circuited branch OMT.
100 Phase (degree)
S15 50
l1
l2
0 S11
Port 1
−50
Port 4
−100
−200
Port 3
S22
−150
Port 2 8
8.5
9
9.5 10 10.5 Frequency (GHz)
11
11.5
12
(b) Phase Fig. 4. S-parameters of a square waveguide T-junction (s = 18mm, a = 22.86mm, b = 10.16mm).
simulated S-parameters of the fabricated structure shown in Fig. 8. It is seen that measured results are in good agreement with those from our proposed circuit model over the interested band, and more resonant frequency points are observed due to the connection section between two OMTs, as shown in Fig. 9. Measured Isolation between Ports 1 and 2 is better than 30dB over the entire X band, which indicates the correctness of our assumption. These results verify the validity of our equivalent circuit model. It should be mentioned that we only consider the propagating modes in the square waveguide. IV. C ONCLUSION An equivalent circuit model for the square waveguide Tjunction has been derived from numerical results obtained
Fig. 6.
Equivalent circuit model of short-circuited branch OMT.
by the mode-matching analysis. The original 5-port network is separated into two individual 3-port and 2-port networks. With this model, it is possible to analyze the ortho-mode transducers in a very efficient circuit simulator and can avoid the complex treatment of multi-port network. A short-circuited branch OMT has been designed and fabricated as an example. Measured results agree very well with those from circuit simulation, which verifies our circuit model approach. ACKNOWLEDGEMENT This project is supported by Singapore Technologies Electronics Limited (ST Electronics), Singapore. The authors would like to thank Shishan Qi for his assistance of measurement.
1015
5 0
0 −5
−10
Magnitude (dB)
Magnitude (dB)
−10 −15 −20 −25 S22 Mode Matching Method
−35
S11 Equivalent Circuit
−30
−40
S11 Mode Matching Method −30
−20
S
11
−50
S
13
S22 Equivalent Circuit −40
8
8.5
9
9.5 10 10.5 Frequency (GHz)
11
11.5
−60
12
Fig. 7. Calculated S-parameters of a short-circuited branch OMT (s = 18mm, a = 22.86mm, b = 10.16mm, l1 = 1.07mm, l2 = 5.64mm).
Measured
S11 Equivalent Circuit Measured
S13 Equivalent Circuit 8
8.5
9
9.5 10 10.5 Frequency (GHz)
11
11.5
12
0
Magnitude (dB)
−10
−20
−30
−40 S
22
−50
S
24
−60
Fig. 8.
Measured
S22 Equivalent Circuit Measured
S24 Equivalent Circuit 8
8.5
9
9.5 10 10.5 Frequency (GHz)
11
11.5
12
Back-to-back configuration of short-circuited branch OMT. Fig. 9. Calculated and measured S-parameters of the designed OMT in the back-to-back configuration (s = 18mm, a = 22.86mm, b = 10.16mm, l1 = 1.07mm, l2 = 5.64mm).
R EFERENCES [1] N. Marcuvitz, Waveguide Handbook. London: Peter Peregrinus Ltd., 1986. [2] E. D. Sharp, “An exact calculation for a T-junction of rectangular waveguides having arbitrary cross sections,” IEEE Trans. Microwave Theory Tech., vol. MTT-15, no. 2, pp. 109–116, Feb. 1967. [3] T. Obata and J. Chiba, “Improved theory for E-plane symmetrical Tee junctions,” IEEE Trans. Microwave Theory Tech., vol. 37, no. 3, pp. 624–627, Mar. 1989. [4] B. N. Das, A. Chakraborty, and N. V. S. N. Sarma, “S marix of slotcoupled H-plane Tee junction using rectangular waveguides,” IEEE Trans. Microwave Theory Tech., vol. 38, no. 6, pp. 779–781, June 1990. [5] X.-P. Liang, K. Zaki, and A. Atia, “A rigorous three plane modematching technique for characterizing waveguide T-junctions, and its application in multiplexer design,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 12, pp. 2138–2147, Dec. 1991. [6] P. Lampariello and A. A. Oliner, “New equivalent networks with simple closed-form expressions for open and slit-coupled E-plane Tee junctions,” IEEE Trans. Microwave Theory Tech., vol. 41, no. 5, pp. 839–847, May 1993. [7] J. Uher, J. Bornemann, and U. Rosenberg, Waveguide Components for Antenna Feed Systems: Theory and CAD. Norwood, MA: Artech House, 1993.
[8] A. Morini, T. Rozzi, and G. Venanzoni, “Analytical formulas for the design of orthomode transducers,” in 34th European Microwave Conference, vol. 1, Oct. 2004, pp. 273–276. [9] S. Skinner and G. James, “Wide-band orthomode transducers,” IEEE Trans. Microwave Theory Tech., vol. 39, no. 2, pp. 294–300, feb 1991. [10] B. Piovano, C. Zarba, G. Bertin, L. Accatino, and M. Mongiardo, “CAD and optimization of compact ortho-mode transducers,” in IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, 1999, pp. 257–260. [11] J. Kim, H.-T. Kim, K. Kim, J.-S. Lim, and D. Ahn, “An equivalent circuit model for multi-port networks,” in 37th European Microwave Conference, Oct. 2007, pp. 901–904. [12] R. Levy, “Derivation of equivalent circuits of microwave structures using numerical techniques,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 9, pp. 1688–1695, Sept. 1999. [13] N. Yoneda, M. Miyazaki, M. Tanaka, and H. Nakaguro, “Design of compact-size high isolation branching OMT by the mode-matching technique,” in 26th European Microwave Conference, vol. 2, Oct. 1996, pp. 848–851.
1016
