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Tel: +49 40 42878 2758, Fax: +49 40 42878 2755, email: [email protected]. Abstract- An efficient ... plers based on the effective even and odd mode characteristic ... useful, for instance, when designing compensated high directivity.
Advanced Characterization and Design of Compensated High Directivity Quadrature Coupler Johannes Muller and Arne F. Jacob Institut fUr Hochfrequenztechnik, Techn. Univ. Hamburg-Harburg, 21073 Hamburg, Germany Tel: +49 40 42878 2758, Fax: +49 40 42878 2755, email: [email protected]

Abstract- An efficient design procedure for directional cou­ plers based on the effective even and odd mode characteristic

A

impedances and electrical lengths is presented. These quantities which are extracted from the scattering parameters of the overall

A'

coupler structure allow for a systematic optimization. This is useful, for instance, when designing compensated high directivity couplers or when large bandwidth performance is desired. The method is applied to optimize a 2 GHz wiggly line coupler. Measurements yield a directivity of better than 30 dB from DC

Zo,eff = fct(ri,i' Zret) Ze,eff fct(ri,i Zret) E\,eff= fct(ri) Ge,eff= fct(ri)

to 3.5 GHz, confirming the simulation results.

=

I.

INTRODUCTION

Directional couplers are widely used in microwave circuits. They commonly serve as power splitters in measurement systems or as key components in filters, matching networks, or Marchand baluns. In case of a two fold symmetry they are commonly known as quadrature couplers because the phase difference between the coupled and the through port is 90°. This class of couplers is especially attractive because the synthesis of the four-port can be reduced to the synthesis of four one-ports [1]-[3]. The different even and odd mode phase velocities of cou­ plers in inhomogeneous media impair directivity and match­ ing. Many compensation schemes have been devised in the past. They usually aim at equalizing the phase velocities at some given frequency (usually the center frequency). This can be done by analytically calculating the value of the compen­ sating elements as in the external methods [2], [4], [5], or iteratively as in the case of the internal compensation methods which use dielectric layer [6], quasi suspended substrates [7] or the well known wiggly line technique [8], to name only a few. In a lossless environment, perfect compensation is easily achieved at the design frequency. However, because of dispersion, the bandwidth is limited. There is no systematic solution to this problem up to now. In the following we propose to characterize a given coupler over a wide band by means of effective even and odd mode electrical lengths as well as effective impedances as shown in Fig. 1. The remaining error can then be expressed as the deviation of these parameters from the ideal values. Similar approaches are known from classical filter and matching theory as 'image impedance' and 'image propaga­ tion function' [9]. The 'image parameter' theory has, to the authors' knowledge, not yet been used for the characterization of compensated directional couplers.

978-1-4244-7732-6/10/$26.00 ©2010 IEEE

724

'

Zoeff' Zeeff

Fig. 1.

Symmetry planes of a quadrature coupler and effective

parameter representation.

II. THEORY When reducing a coupler to a pair of coupled lines It IS best described in terms of even and odd mode parameters. To provide perfect matching at all ports and ideal isolation between two pairs of ports, the quadrature coupler must fulfill two conditions: • •

the two Eigenmodes of the coupler must be degenerated, the coupler impedance Zc ';ZeZo must match the system (reference) impedance Zref (e.g. 500), =

Ze and Zo being the even and odd mode impedance, re­ spectively. Actual couplers exhibit transitions (usually mitered bends) to the connecting port lines and often also some kind of compensation. To optimize the coupler as a whole these elements have then to be included in the design procedure. We thus introduce a set of effective even and odd mode parameters that take into account the whole structure and require them to satisfy the above conditions. Obviously, these parameters become the actual even and odd mode quantities if only coupled lines are considered.

IMS 2010

The reference planes should thus be placed on the con­ necting lines ahead of the transition as shown in Fig. 1. As mentioned earlier, 4-port directional couplers with two fold symmetry can be decomposed into four I-ports [1]. From the scattering parameters one obtaines the eigenreftections rij of the decomposed I-port as indicated in Fig. 1:

ree roe reo roo

=

=

=

=

( 811 + 821 + 831 + 841 ) , ( 811 + 821 - 831 - 84 d , ( 811 - 821 - 831 + 84 d , ( 811 - 821 + 831 - 84 d ,

(la) (lb)

Fig. 2.

(Id)

0.7

where the subscripts i and j stand for the even Ce') or odd Co') mode with respect to the symmetry planes X-X' and Y­ Y', respectively. The input impedance of each I-port can be written as: 1 + rik (2) Zik, in Zref' ---' 1 - rik

o. 6

E

� 0.5

Zoe, in Zeo, in Zoo, in

=

=

=

Ze,eff . cotanh(j8e,eff/2), Zo,eff . cotanh(j8o,eft/2), Ze,eff' tanh(j8e,eff/2),

(3d)

In principle, 8i,eff is complex, imag( 8i,eff) being the loss term. Equations (3a) and (3b) correspond to an open circuit in the Y-Y' symmetry plane, (3c) and (3d) to a short circuit. Multiplying (3a) by (3c) and (3b) by (3d) eliminates the tanh and cotanh terms. By further using (2), the effective characteristic impedance of even and odd mode can be written as

Ze,eff

Zo,eff

=

=

JZee, in' Zeo, in

JZoe, in . Zoo, in

=

=

Zref

Zref

1 + ree 1 + reo . 1 - ree 1 - reo' (4a) 1 + roe 1 - roe

1 + roo 1 - roo (4b)

Consequently, the effective coupler impedance is

ZC,eff

=

JZe,eff . Zo,eff'

(5)

Finally, inserting (3a) or (3c) in (4a) and (3b) or (3d) in (4b) and combining with (2) the effective electrical lengths 8�,eff real(8e,eff) can be determined: =

8�,eff 8�,eff

=

=

( (

2· imag atanh

( 1 + reo) . ( 1- ree) (1 - reo) . ( 1 + ree)

2 . imag atanh

( 1 + roo) . ( 1 - roe) ( 1 - roo) . ( 1 + roe)

978-1-4244-7732-6/10/$26.00 ©2010 IEEE

--------------

4-

� -

� :�::;�=;): �-':.__,:::. .

�e eff [0]

-3 ----------- -3-�----------- -3---y-------------- -2----- ---- 1----------- �1--------------- -'\---

-- - -----

----- --_____

C)

@) ZC,eff[Ohm]

� -� ---------------

---

�::::::::7�:::::::::: :�L:::: -

c::>:J�

�------ �-------- 2 ------------- --·2-----------------'---O. 3 '---� "---�-�--'---"--'----'---------' 1.09 1.06 1.07 1.08 1.10 width we [mm]

(3a)

Zo,eff' tanh(j8o,eff/2).

-

! 0.4 1

(3b) (3c)

-6 SO. ----5 ---- ------

-

It can also be expressed in terms of the effective characteristic impedance Zi,eff of the considered mode as well as its effective electrical length 8i,ef r

=

-------

----

E

=

Zee, in

Parameters of the wiggly line coupler.

(1c)

Fig. 3.

Contour plot of the effective coupler impedance and the

effective length difference as a function of wiggle depth d and line width

We

at f

=

2 GHz.

The residual errors with respect to the impedance and phase velocity condition can now be expressed as follows:

I:1ZC,eff 1:18eff

=

=

ZC,eff - Zref, 8�,eff - 8�,eff'

(7a) (7b)

The scattering parameters of the coupler only indicate whether the design goal has been reached and contain limited information on the error sources. In contrast, the effective coupler impedance and effective electrical lengths not only allow to distinguish the two sources of error, but also reveal the algebraic sign and absolute value of each error. This information enables a more systematic design optimization as well as a quicker redesign if required. It should be mentioned that propagation losses are in principle included in (6a) and (6b). In this paper, however, losses are neglected. They will be accounted for in future work. In the following section this approach is applied to the design of a 20 dB coupler with wiggly-line compensation. III. ANALYSIS OF WIGGLY LINE COUPLER

) )

,

(6a)

.

(6b)

A detail of the wiggly line coupler model is depicted in Fig. 2. The critical dimensions are the coupling gap g, the line width We, the wiggle depth d, and the wiggle period 5. The substrate with thickness h 0. 508 mm, metalization thickness t 17/Lm, and permittivity Er 3. 5 is Rogers' R04003c [10]. =

=

725

=

IMS 2010

a)

10 8 6 4 � :0: 2 Q)

CD