Design and Analysis of a Novel Broadband Tweaked T-shaped Stub-loaded Quadrature Power-Splitter

Abstract

In this paper analytical approach to design a broadband microstrip quadrature 3-dB power splitter is proposed and validated. The proposed quadrature splitter in this paper utilizes a double-stage coupled line Wilkinson power divider in conjunction with a modified phase shifter to induce phase delay at one output port. Compared with the conventional T- shaped stub, the proposed configuration of Tweaked T-shaped stubs exhibits better performance without adding any fabrication complexity, which in turn improves the performance of the quadrature power splitter. In testing, the proposed quadrature Wilkinson power divider covers 2–4.5 GHz (fractional bandwidth FBW = 76.9%) working frequency with 10-dB impedance and 15-dB isolation bandwidth. Furthermore, this design has a very good phase and amplitude balance, with a 7° phase and 1 dB amplitude imbalance.

Appendix

$$2Z_{0} \, = \, Z_{1e} \frac{{Z_{in1} + jZ_{1e} \tan \theta }}{{Z_{1e} + jZ_{in1} \tan \theta }}$$

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$$Z_{in1 \, } = \, Z_{2e} \frac{{Z_{0} + jZ_{2e} \tan \theta }}{{Z_{2e} + jZ_{0} \tan \theta }}$$

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$$\left\{ \begin{gathered} Z_{2e} = Z_{0} \sqrt {\frac{{\sqrt {1 + 8\tan^{4} \theta \, } + 1}}{{\tan^{2} \theta }}} \hfill \\ Z{}_{1e} = Z_{0} \sqrt {\frac{{\sqrt {1 + 8\tan^{4} \theta \, } - 1}}{{\tan^{2} \theta }}} \, \hfill \\ \end{gathered} \right.$$

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$$Z_{in2} \, = \frac{{jR_{1} Z_{1o} \tan \theta }}{{R_{1} + 2jZ_{1o} \tan \theta }}$$

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$$Z_{in3} \, = Z_{2o} \frac{{Z_{in2} + jZ_{2o} \tan \theta }}{{Z_{2o} + jZ_{in2} \tan \theta }}$$

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$$Z_{0} = \frac{{R_{2} Z_{in3} }}{{R_{2} + 2Z_{in3} }}$$

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$$\left\{ \begin{gathered} R_{1} = \frac{{2Z_{1O} Z_{2O} \tan^{2} \theta }}{{\sqrt {(Z_{1O} + Z_{2O} )\tan^{2} \theta [Z_{1O} \tan^{2} \theta - Z_{2O} ]} }} \hfill \\ R_{2} = \frac{{2Z_{0} (Z_{1O} + Z_{2O} )Z_{{^{{_{2O} }} }}^{2} \tan^{2} \theta }}{{(Z_{1O} Z_{{^{{_{2O} }} }}^{2} - Z_{0}^{2} Z_{1O} + Z_{{^{{_{2O} }} }}^{3} )\tan^{2} \theta + Z_{0}^{2} Z_{2O} }} + \frac{{2Z_{0}^{2} Z_{2O} \sqrt {(Z_{1O} + Z_{2O} )} \tan^{2} \theta [Z_{1O} \tan^{2} \theta - Z_{2O} ]}}{{(Z_{2O}^{2} Z_{1O} - Z_{0}^{2} Z_{1O} + Z_{{^{{_{2O} }} }}^{3} )\tan^{2} \theta + Z_{0}^{2} Z_{2O} }} \hfill \\ \end{gathered} \right.$$

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$$S_{11e} = \, \frac{{\left[ {Z_{0} (2Z_{2e}^{2} - Z_{1e}^{2} )\sin^{2} \theta - Z_{0} Z_{1e} Z_{2e} \cos^{2} \theta + j(Z_{1e} + Z_{2e} )(Z_{1e} Z_{2e} - 2Z_{0}^{2} )\sin \theta \cos \theta } \right]}}{{\left[ {(3Z_{0} Z_{1e} Z_{2e} \cos^{2} \theta ) - Z_{0} (2Z_{2e}^{2} + Z_{1e}^{2} )\sin^{2} \theta + j(Z_{1e} + Z_{2e} )(Z_{1e} Z_{2e} + 2Z_{0}^{2} )\sin \theta \cos \theta } \right]}}$$

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$$S_{12e} = \, \frac{{\left[ {2\sqrt 2 Z_{0} Z_{1e} Z_{2e} } \right]}}{{\left[ {(3Z_{0} Z_{1e} Z_{2e} \cos^{2} \theta ) - Z_{0} (2Z_{2e}^{2} + Z_{1e}^{2} )\sin^{2} \theta + j(Z_{1e} + Z_{2e} )(Z_{1e} Z_{2e} + 2Z_{0}^{2} )\sin \theta \cos \theta } \right]}}$$

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$$S_{22e} = \, \frac{{\left[ {Z_{0} (Z_{1e}^{2} - 2Z_{2e}^{2} )\sin^{2} \theta + Z_{0} Z_{1e} Z_{2e} \cos^{2} \theta + j(Z_{1e} + Z_{2e} )(Z_{1e} Z_{2e} - 2Z_{0}^{2} )\sin \theta \cos \theta } \right]}}{{\left[ {(3Z_{0} Z_{1e} Z_{2e} \cos^{2} \theta ) - Z_{0} (2Z_{2e}^{2} + Z_{1e}^{2} )\sin^{2} \theta + j(Z_{1e} + Z_{2e} )(Z_{1e} Z_{2e} + 2Z_{0}^{2} )\sin \theta \cos \theta } \right]}}$$

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$$S_{22O} = \, \frac{{\left[ \begin{gathered} (R_{1} R_{2} Z_{0} Z_{1O} - 2R_{2} Z_{1O} Z_{2O}^{2} + 4Z_{0} Z_{1O} Z_{2O}^{2} )\sin^{2} \theta - Z_{0} Z_{2O} R_{1} R_{2} \cos^{2} \theta + j(R_{1} R_{2} - 2R_{1} Z_{0} )Z_{2O}^{2} \sin \theta \cos \theta \hfill \\ + jZ_{1O} Z_{2O} (R_{1} R_{2} - 2R_{1} Z_{0} - 2R_{2} Z_{0} )\sin \theta \cos \theta \hfill \\ \end{gathered} \right]}}{{\left[ \begin{gathered} - (R_{1} R_{2} Z_{0} Z_{1O} + 2R_{2} Z_{1O} Z_{2O}^{2} + 4Z_{0} Z_{1O} Z_{2O}^{2} )\sin^{2} \theta + Z_{0} Z_{2O} R_{1} R_{2} \cos^{2} \theta + j(R_{1} R_{2} + 2R_{1} Z_{0} )Z_{2O}^{2} \sin \theta \cos \theta \hfill \\ + jZ_{1O} Z_{2O} (2R_{1} R_{2} + 2R_{1} Z_{0} + 2R_{2} Z_{0} )\sin \theta \cos \theta \hfill \\ \end{gathered} \right]}}$$

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