Numerical Even- and Odd-Mode Analysis of Branch-Line Couplers

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1 Numerical Even- and Odd-Mode Analysis of Branch-Line Couplers Kazuhito MURAKAMI* and Hideaki FUJIMOTO** This paper presents that the numerical analysis using the central difference method is efficient for the evenand odd-mode analysis of the branch-line coupler. The boundary treatment which computes reflected and transmitted quantities at T-junction is carried out by signal flow graph. For 3-dB branch-line coupler, the electric and magnetic walls appeared by even-mode, odd-mode and all ports excitations are confirmed on the lines. It is shown that the simulated results for the equivalent circuits agree with those of the conventional analysis. Key words: microwave planar circuit, coupled transmission line, directional coupler, numerical analysis 1. Introduction Microwave simulators are actively used as a supporting tool of the microwave circuit design. The use is for the reduction of the development cost, shortening of the development period, and the reliability improvement of a system. In microwave planar circuits, the branch-line hybrid (branch-line coupler) is a passive element widely used as the directional coupler and the mixer, etc. Using such a discontinuity part for the design of microwave circuits is one of the important themes. Simulating the behavior and the characteristic of reflected and transmitted waves generated due to discontinuity is an important task. For the purpose of analyzing or designing microwave and millimeterwave circuits1>-3), it is also important to analyze directly the transmission line system corresponding to a physical structure given. For such a purpose there are a lot of electromagnetic field numerical analysis methods such as the finite element method, the moment method and the FDTD method with respect to the analysis of planar circuits. The use of threedimensional electromagnetic field analysis simulators have spread recently. Moreover, a circuit simulator 6) of SPICE system into which the transmission line model is worked have spread. In this paper, we present time solutions of voltage and current which occur by exciting a transmission line system with the sinusoidal signal. These solutions can be obtained by a numerical analysis based on the central difference method in which the boundary treatment is incorporated for T-junction. As for the boundary treatment, by introducing the theory of the signal flow graph we process the reflected waves and the transmitted waves, which are generated with the incident waves of the voltage and current, on each transmission line with respect to in- stantaneous values of time. The obtained time solutions express the voltage and current variations, and the behavior of the multiple reflection waves along the line is made visible. In order to verify the effectiveness of the proposed numerical analysis method we give a numeric calculation example. The example is about a 3dB branchline coupler, and gives solutions for the voltage and current. By using the obtained solutions, we show the standing wave and the isolation at a center frequency from the distributions of the steady-state voltage and current. Moreover, we verify that combined solutions, which are obtained from the numeric even- and odd-mode analysis, agree with original solutions about the corresponding equivalent circuits. When a circuit under consideration is excited by each of the even and odd modes, and by adding input signals at all ports, its equivalent circuit may be obtained from the electric and magnetic walls generated because of the symmetricalness of circuit structure. 2. Transmission line model and nu- merical analysis For the branch-line coupler, as shown in Fig.1(a), using a planar circuit, we consider a model consisting of transmission lines shown in Fig.l(b). The following is the well known telegraphers' equations : where V(x, t) and I(x, t) are the voltage and current, respectively, at the any point x, where is a distance from the input port to the point x, and at time t. Moreover, the constants R, L, G, and C are resistance, inductance, conductance, and capacitance, respectively, per unit length distributed along the line.

2 On the assumption of TEM mode propagation, the characteristic impedance Z0 and the phase velocity vp are where Ax i *i, At j r j for the incremental length Ox of the line and the incremental time At. From the above equations we can have the following equations for successive calculation. respectively. approximatione4l Applying the central to equation (1) gives difference where Yo = 1/Zo. Assume that each port of the branch-line coupler is terminated in a matched load. When exciting its Port 1 we set an initial value and a boundary value, as follows. Initial value: Here Vpk (t) (k = 1,., 4) is the voltage at Port k. From equation (3), obtain all solutions of voltage and current on the transmission lines. After that, extract the forward and backward waves due to the input and output responses obtained at each port of the coupler under consideration. As a result, for the input voltagevin (t) at Port 1 given by TT I \ 1 ITT 1L\ r7 r IL\\ we canget an output voltage Vpko(t) (k = 1,, 4) at each of other ports by the following: Fig.l Branch-line coupler and its equivalent circuits: (a) branch-line coupler; (b) transmission line model; (c) even mode equivalent circuit; (d) odd mode equivalent circuit.

3 3. Boundary treatment In discontinuities within the branch-line coupler, the boundary treatment for the reflected and transmitted waves is required about the incident wave that arrives from each line. The boundary treatment of T-junction is done by processing the reflected and transmitted waves, and this processing is especially performed by applying the theory of the signal flow graph, as shown in Fig. 2. The reflection coefficient Fk (k = 1, 2, 3) of each port of T-junction can be described by the following equations. Fig.3 Steady-state voltage and current variations. For the incident voltage and current on each transmission line, the corresponding reflected and transmitted quantities can be written in the forms: Here, superscripts "+" and "-" mean the incident wave and the transmitted wave (the reflected wave is included), respectively. The boundary treatment at each of discontinuities is done by calculating the reflected and transmitted quantities with respect to the incident voltage and current on each line at each time. 4. Simulation results As an example of the numerical analysis of the coupler under consideration, in Fig.3 we show variations of the steady-state voltage and current, when exciting its Port 1 with a voltage generator e(t) = sin (2ir fat) (10 Center frequency). (11) From the result, Port 3 is recognized as the isolated port, and moreover -3dB is then observed for the input signal to be output to Port 2 and Port 4, respectively. From this figure we can see these voltage and current variations, respectively, are almost corresponding to the phenomenon of electric field distribution measured and the current density distribution.5) Fig.2 Signal flow graph of T-junction. I r ) Even- and odd-mode excitations We shall apply the proposed numerical analysis ethod to the even- and odd-mode analysis. When

4 two ports, Port 1 and Port 3, are simultaneously and sinusoidally excited, i.e., for the steady-state when the coupler is excited by each of even mode and odd mode, Figs. 4 and 5 show variations of voltage and current, respectively. From each figure, we can confirm that there are each of the electric and magnetic walls at the center part of coupler. When these two modes with respect to voltage are superposed, its result is in good agreement with the result shown in Fig.3(a). Of course, as shown in Fig.3(b), about the current also holds similarly. 1) Even mode Fig.4 illustrates the voltage and current variations under conditions V1 = V (t), V3 = V(t), V2 = 0, V4 = 0, i.e. under the even mode excitation. 2) Odd mode Fig.5 illustrates the voltage and current variations under conditions V1 = V (t), V3 = -V (t), V2 = 0, V4 = 0, i.e. under the odd mode excitation. 3) Equivalent circuits Fig.6 shows two equivalent circuits corresponding to the even mode and odd mode excitations, respectively. We have applied these circuits to obtain the variations shown in the above. The conventional analytical result for voltage that is obtained by using the even- and odd-mode equivalent circuits agrees with the lower half of Fig.3(a). On two voltages with respect to two modes, it should be noted that the superposition of their sum and difference agrees with the result shown in Fig.3(a) completely. Similarly, on two currents with respect to two modes the corresponding superposition agrees with that shown in Fig.3(b). B) Simultaneous all ports excitations Fig.7 illustrates the voltage and current variations in a steady-state response with respect to when each port of branch-line coupler is excited by four patterns simultaneously. It is observed respectively that, at a center of the main line and of branches, there exist the electric and magnetic walls that arise due to the symmetricalness of the circuit structure. Moreover, it agrees to the results, shown in Fig.3, when four kinds of all voltage solutions are superposed. Of course, the above about the current solutions is also true. 5. Conclusions We have shown the steady-state solutions with respect to voltages and currents on all transmission lines within the branch-line coupler. These solutions have been obtained by applying the time domain analysis based on a numerical analysis using the center difference method. Fig.4 Variations for the even mode excitaion. (a) Even mode equivalent circuit Fig.5 Variations for the odd mode excitaion. Fig.6 (b) Odd mode equivalent circuit Equivalent circuits for two modes.

5 1) V1=v0, V34/0, 1124/0, V4=v(t) Moreover, we have presented the usefulness of the numerical analysis method proposed in this paper by showing the following: (1) the presentation for voltage and current variations on the transmission lines about the even and odd mode excitations, and about the excitation at all ports where are terminated by matched loads, (2) verification for the existence of the electric wall and the magnetic wall, and of the existence of equivalent circuits because of these walls, (3) numeric even-mode and odd-mode analysis. The presented numerical analysis method can be used as a simple simulation tool for two purposes mainly: (1) for understanding characteristics of circuits, (2) for visualizing the operation phenomenon of planar circuits. References 2 ) V1=V(t), V3= 3/(t), V2.V(t), V4=V(t) 1) R.E.Collin, "Foundations for microwave engineering, 2nd ed.," McGraw-HILL, (1992). 2) K.C.Gupta, et al, "Microstrip Lines and Slotlines, 2nd ed.", ARTECH HOUSE, Norwood, (1996). 3) G.L.Matthaei, L. Young and E.M.T. Jones, "Microwave filters, impedance-matching networks, and coupling structures," ARTECH HOUSE, Norwood, MA, (1980). 4) K.Murakami, IEICE General Conf. 2004, Mar, (2004) A ) G.Fred, "Microstrip circuits," John Wiley & Sons.Inc., NewYork, NY, (1994). 6) Y.Yamashita, "Foundation of microwave simula tors," Publishing of IEICE, Tokyo, (2004). 3) V1=v(t), V3=-V0, V2=v(t), V4=.1/(t) 4) V1=V(t),V3=v(t),V2=-v(t),V4=-v(t)j I Fig.7 Voltage and current variations for when all ports is simultaneously excited. 5