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ANALYSIS OF A MODIFIED VARIABLE-LENGTH TUNING POST IN A RECTANGULAR WAVEGUIDE J. Roelvink and A.G. Williamson Department of Electrical and Computer Engineering The University of Auckland Private Bag 92019, Auckland, New Zealand [email protected] A modified variable-length post for use in high power waveguide tuning networks is analyzed to determine its reactance properties. Theoretical predictions are shown to be in good agreement with experimental results. An investigation of the reactance characteristics and the aperture voltage characteristics of the modified post, as a function of its dimensions, is also reported. Submission Date: 8 December 2006 Acceptance Date: 19 June 2007 Publication Date: 17 August 2007 INTRODUCTION Impedance matching and filter networks can be realized in a rectangular waveguide by introducing reactive discontinuities at certain points along the waveguide. Over the years, many waveguide discontinuities have been investigated and there exists much documented literature on their equivalent reactance, e.g. [Ragan, 1948; Montgomery et al., 1948; Marcuvitz, 1951; Lewin, 1951; Schwinger, 1968; Rizzi, 1988; Williamson, 1985, Williamson, 1986]. A commonly used waveguide discontinuity is the variable-length cylindrical post as it is easy to realize in practice and can be tuned over a relatively large capacitive range. The variable-length adjustment is usually achieved by using a threaded post to enter the waveguide through a threaded hole (often with a lock-nut to Keywords: Waveguides, waveguide discontinuities, impedance matching networks 41-2-12 prevent subsequent movement). A shortcoming of this approach is the limited metal-to-metal contact in the threaded section that results in a small but non-zero contact resistance. Whilst this form of construction may be satisfactory in many situations, for those where it is not, such as in high power applications, an alternative geometry is required. A technique to reduce the effect of the contact resistance associated with the threaded section is the introduction of a coaxial choke, shown in Figure 1(a), which presents a low impedance at the coax/waveguide aperture that is relatively insensitive to small changes in the resistance due to the screw thread (in practice there would be a tight fit between the post and the choke, which could easily be supplemented by an external clamp-ring after post adjustment). The post remains easily tunable by the threaded section beyond the choke and external to the waveguide. For a given frequency the effect of the coaxial choke on the fields within the Journal of Microwave Power & Electromagnetic Energy Vol. 41, No. 2, 2007 h z Coaxial choke section Screw thread 2a � y x 2b e (a) � OFF d (b) Figure 1. Modified variable-length tuning post (a) sectional view (b) simplified sectional view. waveguide can be modeled by the simplified arrangement shown in Figure 1(b) where there is a coaxial line section offset from the coax/waveguide aperture by a fixed length lOFF (more complicated choke arrangements can be modeled in an equivalent form of Figure 1(b)). In some practical implementations it has been noted that for some equivalent values of lOFF the reactive effect of the modified post can be rather insensitive to insertion depth, l. Of course, when lOFF is a multiple of λ/2, the reactive characteristic of the modified post should be the same as the standard variable-length post, that is with lOFF =0. Clearly, if this more expensive design to manufacture is to be used in a high power application to overcome the limitations of the standard variable-length post, a theoretical analysis or model that accounts for the effect of the coaxial choke would be invaluable to assist in determining appropriate dimensions. This is the motivation for the work reported here. In this paper an accurate theoretical analysis for the modified tuning post is presented. Theoretical and experimental results are reported that both confirm the accuracy of the model and illustrate examples of where the reactance can International Microwave Power Institute be quite insensitive to post depth as observed in practice. The reactance characteristics and the voltage in the coaxial aperture and coaxial line section are investigated and reported, illustrating what values of lOFF produce insensitive reactance characteristics and relatively large aperture voltages. THEORY The modified tuning post in Figure 1(a) is considered here as the simplified configuration shown in Figure 1(b) where there is a cylindrical post entering through a coaxial aperture that is shorted at an offset length, lOFF , and extends into the waveguide a length l (the coordinate system adopted here is that associated with the coaxial line). It is assumed that the waveguide and coax dimensions, and the operating frequency are such that only the TE10 mode can propagate in the waveguide, and only the TEM mode can propagate in the coaxial region. All metal surfaces are modeled to be perfectly conducting and to simplify the analysis it is assumed that the fields in the vicinity of the post are rotationally symmetric as in [Williamson, 1985]. 41-2-13 The analysis involves finding the induced current distribution on the post surface resulting from the incident TE10 mode field, namely: E inc z ( π x − jk y x, y,z = Ei sin e 10 d ) (1) where k10 = k 2 − ( π d ) , k=2π/ λ and Ei is the magnitude of the incident TE10 mode field. Once the induced current has been determined, the scattered field and resulting reactive effect can be calculated by a similar method to that in [Williamson, 1986]. The calculation of the induced current is facilitated by the extension of two existing analyses, namely the induced current on a variable-length coaxially driven hollow post, IC (z) [Williamson, 1985] and the induced current by an incident TE10 field on a variable-length hollow post, IH (z) [Williamson, 1986]. These currents were obtained from the solutions of two integral equations and the details of their evaluation can be found in [Williamson, 1986]. Upon applying the principle of superposition, an expression for the total current induced on the post shown in Figure 1(b) can be given by: 2 () () () IT z = Ei I H z + V I C z (2) where V is the voltage in the coax/waveguide aperture at z=0. An obvious difficulty in the evaluation of (2) arises in determining V as the form of the electric field if the aperture is not known. However, if the aperture electric field is assumed to be of the same form as the TEM mode in the coaxial stub, the post is electrically thin (ka<<1) and kl>>ka. A good approximation for V can be given by: () V = − jX AP IT 0 (3) where IT(0) is the current flowing on the post at the coax/waveguide aperture and XAP is the 41-2-14 reactance of the shorted coaxial section referred to the aperture, namely: ( X AP = ZC tan klOFF ) (4) where ZC is the characteristic impedance of the coaxial region (note that for more complicated choke designs the expression for XAP will be different). By considering the total current at z=0, the voltage V can be found from (2) and (3). It is then a simple matter to calculate the total current from (2). The scattered TE10 mode field from the current on the post and the field in the aperture (using the TEM approximation [Williamson, 1985]) can be calculated using a similar approach to that adopted in [Williamson, 1986] and shown to be: π x − jk y Ezscat x , y , z = Er sin e 10 d ( ) (5) where e 2 jV Er = sin d h ln b a k10 d ( ) J kb − J ka − kη0 I J ka 0 0 k d 0 0 (6) 10 ( ) ( ) ( ) and I0 is the z-independent part of IT(z), η0 is the intrinsic impedance of free space while J0 is a Bessel function of the first kind. The first term of (6) represents the scattered TE10 mode field due to the coaxial line section while the second term represents the scattered TE10 mode field by the total current on the post. By comparing (1) and (5), it can be seen that the reflection coefficient is given by: E R= r (7) Ei The effect of the modified variable-length tuning post can be represented by the equivalent circuit shown in Figure 2 containing a single normalized Journal of Microwave Power & Electromagnetic Energy Vol. 41, No. 2, 2007 shunt reactance (a good approximation if the post is electrically thin), where R and X are related by: R=− 1 1 + 2 jX (8) COMPARISON OF THEORETICAL AND EXPERIMENTAL VALUES FOR REACTANCE The normalized reactance of a 6 mm diameter post (2a=6 mm) with a 15 mm diameter aperture (2b=15 mm) in a C-Band waveguide (d=47.55 d d=47.55 mm, h=22.15) was measured over the usual operating frequency range of the guide (3.955.85 GHz) for several fixed offset lengths, lOFF, using a similar approach to that outlined in [Roelvink and Williamson, 2005]1. Theoretical and experimental results for the normalized post reactance, X, are shown in Figure 3 for two situations lOFF/ λ=0.65 and lOFF/ λ=0.70. In both cases the agreement between theory and experiment is very satisfactory. However, the extensive range over which the post reactance is relatively unchanged is evident. CHARACTERISTICS OF THE MODIFIED TUNING POST In Figure 4, the normalized reactance, X , has been plotted as a function of l/h for a number of values of lOFF. It can be seen that when lOFF is in the range 0 < lOFF/λ < 0.25, the reactance is greater than that of the standard tuning post (lOFF=0) while in the range 0.25≤ lOFF/λ< 0.5 the reactance is of a lesser value. Note that the reactance changes widely when lOFF/λ is close to 0.25 as in this case XAP is large. An important parameter to consider, Waveguide Port 1 jX Waveguide Port 2 Figure 2. Modified variable-length tuning post equivalent circuit. particularly in high power applications, is the induced voltage in the aperture, V . Once V has been determined, all voltages in the coaxial choke can be determined. Figure 5 gives results for V V, calculated from (2) and (3) (with Ei=1), as a function of l/h for several values of lOFF. It can be seen that as lOFF increases from zero, the voltage in the aperture increases. When lOFF/ λ=0.2 there is a significant voltage increase around l/h=0.10. In general, the trend is for the voltage to be largest around lOFF/λ=0.25 when XAP is large (for this case this is also the location of the maximum voltage in the coaxial line region). As lOFF/λ is increased beyond 0.25 towards 0.5, the aperture voltage decreases (the maximum voltage now occurs in the coaxial line section). The aperture voltage is of interest as the corners associated with the transition from coaxial line to rectangular waveguide give rise to relatively high electric fields and so arcing commonly occurs first in this region. Consequently, values of lOFF resulting in relatively high aperture voltages should be avoided. In Figure 6 the reactance is shown for three frequencies when lOFF/λ= 0.2. It was found that the difference in reactance between each of the frequencies was relatively larger for values of lOFF/λ that are close to 0.25 than for values around 0.5. The above results illustrate that while it may be possible to increase the reactance tuning range for some values of lOFF, the disadvantages of 1 In these measurements a solid post was used, whereas the theory presented here is for hollow ended posts. The difference in reactance of each form has been investigated elsewhere [Roelvink and Williamson, 2005]. In order to invoke a more meaningful comparison, the effective hollow length, ls, was taken to be ls=l + 0.12a [Roelvink and Williamson, 2005]. International Microwave Power Institute 41-2-15 4 �OFF = 0.70� X 2 0.65� 0 -2 -4 0 0.2 0.4 �/h 0.6 0.8 1 Figure 3. Theoretical and experimental results for the normalized reactance, X, for several values of lOFF as a function of l/h for the case h=22.15 mm , d=47.55 mm , e/d=0.5, 2a=6 mm, 2b=15 mm and f=4.9 GHz. — Theory, ••• Experimental results. 4 �OFF = 0.2� 0.1� 0 X 0� 0.4� -4 0.3� 0.25� -8 0 0.2 0.4 �/h 0.6 0.8 1 Figure 4. Theoretical results for the normalized reactance, X, for several values of lOFF as a function of l/h for the case h=22.15 mm , d=47.55 mm , e/d=0.5, 2a=6 mm, 2b=10 mm and f=4.9 GHz. insertion insensitivity, increased aperture voltage and frequency sensitivity are significant. A value of lOFF somewhere in the region 0.35λ−0.6λ produces a reactance range similar to that of the standard variable-length tuning post that is relatively sensitive to insertion without greatly suffering from the above disadvantages. 41-2-16 CONCLUSION A modified variable-length tuning post for use in high power impedance matching applications has been presented and an accurate analysis for predicting it’s reactance characteristics has been developed that has been shown to be in good agreement with experimental measurements. Journal of Microwave Power & Electromagnetic Energy Vol. 41, No. 2, 2007 10 0.015 5.1GHz 5 f = 4.9GHz V X �OFF = 0.2� 0.01 0.1� 0.3� 0.005 -5 0.4� 0 0 4.7GHz 0 0.2 0.4 �/h 0.6 0.8 1 -10 0 0.2 0.4 �/h 0.6 0.8 1 Figure 5. Theoretical results for the aperture voltage,V, for several values of lOFF as a function of l/h for the case h=22.15 mm , d=47.55 mm , e/d=0.5, 2a=6 mm, 2b=10 mm and f=4.9 GHz. Figure 6. Theoretical results for the normalized reactance, X, for several frequencies as a function of l/h for the case h=22.15 mm , d=47.55 mm, e/d=0.5, 2a=6 mm, 2b=10 mm and lOFF=0.2λ. This model has permitted an investigation into the reactance and voltage characteristics of the modified tuning post. A choice of 0.35<lOFF/ λ<0.6 would produce a tuning post with a useful tuning range. Developing a simple tuning structure that can be tuned through both a capacitive and inductive range is the focus of ongoing research by the authors. Lewin, L. (1951). Advanced Theory of Waveguides. Waveguides Iliffe, London. Schwinger, J.S., and Saxon, D.S. (1968). Discontinuities in Waveguides: Notes on Lectures. Lectures Gordon and Breach, New York. Rizzi, P.A. (1988). Microwave Engineering Passive Circuits. Prentice-Hall, New Jersey. Williamson, A.G. (1985). “Coaxially fed, hollow probe in a rectangular waveguidE.” IEE Proc. H, Microwaves, Antennas & Propag. 132, pp.273-285. Williamson, A.G. (1986). “Variable-length cylindrical post in a rectangular waveguide.” IEE Proc. H, Microwaves, Antennas & Propag. 133, pp.1-9. Roelvink, J., and Williamson, A.G. (2005). “Reactance of hollow solid and hemispherical-cap cylindrical posts in rectangular waveguide.” IEEE Trans. Microwave Theory Tech. MTT-53, pp.3156-3160. ACKNOWLEDGMENTS The authors wish to thank Keam Holdem Associates, Auckland, New Zealand, for being the industrial partner of a Technology New Zealand TIF Award, and Dr. R. Keam for several helpful discussions. REFERENCES Ragan, G.L. (1948). Microwave Transmission Circuits. McGraw-Hill. Montgomery, C.G., Dicke, R.H., and Purcell, E.M. (1948). Principles of Microwave Circuits. McGraw-Hill. Marcuvitz, N. (1951). Waveguide Handbook . McGrawHill, New York. International Microwave Power Institute 41-2-17