Load impedances of complex slotlineterminations

J Machic, J Zehentner*W Menzelo*Faculty of Electrical EngineeringCzech Technical UniversityTechnicki 2, 16627 Prague 6,Czech Republic**University of UlmAlbert Einstein Allee 41,89069 Ulm, Germany

AbstractSingle slotline has gained new interest due to

application in uniplanar circuits, antennas or activeradiators, all these including slotline short circuits oreven more complex terminations. This paper gives anew approach to calculation of reflection coefficient andload impedances of such terminations using thespectral domain method. Furthermore, slotlineresonators are analysed in the similar way, and theircomplex resonant frequency is calculated. That is usedin an alternative determination of the slotline loadimpedance by application of the transmission linetheory. Computed results compare well withexperiment. Closed-form formulae for CAD purposesare proposed.

IntroductionSlotline plays an important role for antennas [1],

radiating resonators [2], slotline fiRters apd couplers [3- 6], or uniplanar integrated circuits [7]. In many slotlinecomponents as well as in transitions to other planarwaveguides like microstrip or coplanar line, e.g. [8],[9], short- and open-circuited slotlines are required. Toimprove bandwidth, different slotline terminations asshown in Fig. I are used. While the simple slotline shortcircuit has been analysed by different authors [101, [11],more complex terminations have been evaluatedlargely experimentally 112] and theoretically by themethod of finite differences [13]. This paper presentsa spectral domain method (SDM) for the analysis ofgeneral slotline terminations. In contrast to [131, SDMis much more suitable to evaluate the contnbutions ofleaky wave excitation and radiation, and it can includeeasily dispersive effect of these terminations. Openresonant structures have been analysed in [14]. Nowthe technique used in [14] and [15] has been appliedto the analysis of slotline resonators.

Analytical ApproachesFirst a reflection coefficent of the slotline termination

will be derived. Generally, the SDM is based on the

procedure given in 1101 or [11]. The slotlines in thestructures as shown in Fig. I are divided using asuitable rectangular mesh. The tangential elctric fieldalong the slot isthen described by a summ of piecewisesirnsoidal functions (PWSFs). One of the field elemertsis impressed by a given amplitude, in dependence onwhich the other amplitudes are computed. Thereflection coefficient of the termination r is thencalculated by means of the standing wave pattem inthe feeding slot which is fitted to the function

E(z) = A[exp(-jkz) + r exp (ksz)] I (1)by the least squares method. A is an amplitde, k. isthe propagation constant of the wave guided in the zdirection by the slotline [161. r and A can be foundanalytically minimizing the least squares discrepancybetween (1) and the calculated standing wave pattem.Finally the reflection coeffkicent is=-1 exp(2jk5yi) EexpQi(-jky3 + P EiexpikYi) (2)

P £ iexp (-jisyi) f exp (-2Jksyi) £Eiexp(ksyi)where E1 are calculated values of the tangential fieldalong the slotline sampled at points YV number of whichis P. Load impedance is found by a simple recalculationof known r.A modified approach can be used for calculation of

the slotline load impedance. The procedure is basedon the analysis of the slotline resonator termninated atboth ends by the structure under consideration. Theconcept of the complex resonant frequency [14], [151is used. In the sourceless case a set of equations solvedfor the unknown PWSF's amplitudes is homogeneous.It has non-trivial solution when the determinant of thesystem matrix equals zero. Accomplishment of thisconstraint provides the complex resonant frequency.The load impedance is then determined by means ofthe complex resonant frequency using the transmissionline theory combined with the lumped element modelof the termination. The slotline resonator is treated asa section of the slotline terminated at the distance L byimpedances normalized values of which are Z = R+ jXTerminating impedances are normalized to thecharacteristics impedance of the slotline. After settingthe known complex resonant frequency f= f, + jf, andthe length of the resonator L into the resonant conditionZ can be find from

F Z+itid1cL) 1 (3)Im[Z 1+jZtg(ksL) =

Q = fr/(2fi) = ksL(1 - IZl 2)-i(4)

where 0 is the resonator qualiy factor. This approachis valid as far the equivalent circuit of the resonatorholds, i.e. the width of the slotline is negligible incomparison with the resonator's length.

610

Results EvaluationA number of different slotline terminations on the

substrate 1.27 mm thick with dielectric constant of 11were investigated both theoretically and experimentally.To measure the circuit, excitation of the slot mode withlow losses at respective transitions is important.Therefore, the wave was transmitted from thewaveguide to the finline and further from the finline tothe open slotline. Nevertheless, radiation and standingwaves occured at these transitions and resulted in thestrong ripple of the measured characteristics.Behaviour of the slotline resonators was tested usingthe structure shown in Fig. 2.

Figs. 3, 4, 5 show calculated and measured returnlosses of three slotline terminations given in Fig. 1a, b,c. Although in all cases the measured pattern showstrong ripple, the general behaviour of the curves fitsvery well.Calculated normalized terminal resistance and

reactance of the short-circuited slotline in the span ofwidth from 0.1 to 3.25 mm for H/X0 in the range from0.00425 to 0.0845 are given in Fig. 6 where permittivityof the substrate is 11 and H denotes its thickness.

Fig. 7 shows calculated normalized terminatingimpedance of the short-circuited slotline obtained bythe present method with that published in (10] and withmeasured values given in [17]. A good agreement isevident. Similar comparison is made in Fig. 8. Theagreement of normalized terminal reactance with dataobtained in [18] by the integral equation technique isquite good. Resistances can be compared at lowfrequencies only with the discrepancy growing for thefrequency raise.As to the values shown in [13]: leaky waves are

excited by the slotline (£r = 9.8, H= 1.5 mm, W = 0.75mm) from app. 28 GHz upward as follows fromcalculation according to [16]. Above this frequency theslotline does not transmit bound wave andconsequently definition of the terminal impedance issensless.The main contribution to the losses at the slotline

terminations is caused by excitation of surface wavesand by radiation. Losses in dielectric play a minor roleonly as it has been proved by accounting its complexpermittivity.To measure the resonant frequency f, and the quality

factor 0 of the short-circuited slotline resonator thecircuit shown in Fig. 2 was manufactured on thesubstrate with permittivity 11 and 1.27 mm thick. Theresonator is loosely coupled to microstrip feeding lineson the opposite side of the substrate. Measured dataand corresponding values resulting from the resonatoranalysis mentioned above agree well. For illustrationTab. 1 gives calculated f= fr + jfi and measured fm-f=rm+ jfi complex resonant frequencies along with relevantO and Cm. respectively for short-circuited resonators ofunequal lengths. Relatively higher errorsAfi follow frommeasurement accuracy of Om.Comparison of normalized short-circuited slotline

terminating impedance calculated directly by SDM withvalues provided by the transmission line theory utilizingcalculated complex resonant frequency of the slotlineresonator is shown in Fig. 9.Slotline resonators with terminations shaped in

accordance with Fig.1 b, c have been manufactured and

their resonat frequencies were measured. Therectangular patch had 2x2 mm size and the fan patchwith 90° angle had 2 mm radius. Both resonators were14 mm long with 0.15mm wide slot. They were fedaccording to Fig. 2. Their characteristics are nearlyidentical.

Closed-Form Formulae of Short-Circuited SlotlineAn exact model of the short-circuited slotline based

on SDM can not be implanted into CAD packages. Forthat a set of impedances Z = R+ jX normalized to thecharacteristic impedance of the slotline calculated bythe present technique quoted above has been fittedby the least squares method. The resultant closed-formformulae are

R(x,y) = Al. exp(A2y)- A3. exp(-A4y- AsA6)

X(x,y) =

(5)

(6)+B5,.exP 61BIY-B'7I1

where x = W/H, y = FVX0, W is width of the slotline andH is thickness of the substrate. Further

A1l = 03199,sx(0.26107x)+0.003911.apf3.70742(x-1.46771)3-0.310381-0.05M36SX

A2 = 737.873421wv(-02H x) 6959f.-1M.70971l-22.28357x2

Al = L.0O9.y(0.&176S%x)+0.0 [p.42346x-41.11394)1-0.94566-.0l1x

A, = 200.405Stk (-0M.U7945x)-492144+64.64021x-16.7656lxt

AS =-.0653+0.05357x-0.05139(x+0.0219I)1"

A6 = (o.u6+0.01 zuxsY1

a, = 9xrl.)3r)t+035 (-o.8lx-0.312) +o.sx+3.s-5+l.s+>-hx+0S)]f

(7)

Cs)

(9)

(10)

(11)

(12)

(13)

B2 = Meg..(-0.S10179) +25+Sx+0.U)--20.7x-l0.P(-31-tUl) +

(14)

B3 = 0t0903.c(4076x)-.O.00 (0.OM+ SOk-0 8111l) -o.oo(o.os +soi- i.ss5) -+

+ 0.O000x&3

B4 = 0.m3."(-3.lk-O.S5J)+1.5

(15)

(16)

X 0.053+00025.ap(Aft-x2.1914) -o.o01.eV(-14lx-1.nl3) -o.0024c(eV 2.2.11) -

-20(x+ 1.7r( 0.x+io)4s-L .)o42(0.0+Ox-0.1D(5) (

+D.oowo.1t6+ 10*- I. 0.05[)-.l(-10dk- 1.278D- IIx- 1.278).i-1.281-' (17)

611

1-1

6 = [850 -9oo(1 + 5oIx - 1.278Ia) (x-1.278). ix - 1.2781 + 1150 + 8xas +

+121(x+150Ix- .L133) +3O.exp(15lx-0.312) -0.0016(x-0.07)9

7= 0.0179+0.005x048 -2.9(x+ 1.24)- -0.0014.ex(-4Ix-0.9812) +0.0016.exp(-31x- 1.9ts5)

Eqns. (5) and (6) are valid in the frequency rangefroml/2 to 20 GHz for a slot line with the slot widthfrom 0.1 to 3.0/1.5 mmon the substrates 1 .27/0.635mrrthick having permittivity Sr=1 1. In the other words the)hold for 4r=11 and y E(0.00425,0.0846), x e(0.0787,2.362) when H =1.27mm and x e (0.157.2.362when H=0.635mm.

ConclusionsTwo different techniques based on the spectra

domain method for calculating the load impedance othe complex slotline terminations have been presentecand verified by experiments with a good agreementInfluence of the surface wave leakage and radiation isclearly seen at higher frequencies when the reflectiorcoefficient is going down. Dielectric losses mostly playminor role. For CAD purposes, terminating impedanceof the short-circuited slotline is approximated by theclosed-form formulae with sufficient accuracy.

AcknowledgementThis work has been supported in part by the grants

given by the Commision of European Communitiesunder the scheme of Cooperation in Science ancTechnology with Central and Eastern EuropearCountries, Research Fellowships NoCIPA351OCT920643 and CIPA3510CT923158 and b5the CTU of Prague internal grant No. 1003802.

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Trans. Microwave Theory Techn., vol. MTT-20, 1972No. 11, pp. 760-762.

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No. 1, pp. 5-7.[5] Ho C.H., Fan L., Chang K.: A broad-band

uniplanar slotline hybrid ring coupler with over oneoctave bandwidth, IEEE MTT-S Digest, 1993, pp. 585-

8) 588.[6] Ho C.H., Fan L., Chang K. :Broad-band

uniplanar hybrid-ring and branch-line couplers, IEEETrans. Microwave Theory Techn., vol. MTT-41, 1993,No. 12, pp. 2116-2125.

[7] Ogava H., Minagawa A.: Uniplanar MICbalanced multiplier - a proposed new structure forMIC's, IEEE Trans. Microwave Theory Techn., vol.MTT-35, 1987, No. 12, pp. 1363-1368.

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[9] Ho C.H., Fan L., Chang K.: Experimentalinvestigations of CPW-slotline transitions of uniplanarmicrowave integrated circuits, IEEE MTT-S Digest,1993, pp. 877-880.[10] Yang H.Y., Alexopoulos N.G.: A dynamic model

for microstrip-slotline transition and related structures,IEEE Trans. Microwave Theory Techn., vol. MTT-36,1988, No. 2, pp. 286-293.

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[13] Fusco V.F., Chen Q: Slotline short. and open..circuit analysis by the finite-difference time-domainmethod, Proc. of 24thEuMC, Cannes, France, Sept.1994, vol. 2, pp.1 720-1726.[141 Itoh T., Menzel W.: A full-wave analysis method

for open microstrip structures, IEEE TTrans. AntennasPropag., vol. AP-29,1981, No.1, pp. 63-68.

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[17] KnorrJ.B., Saenz J.:,End effect ina shortedslot,IEEE Trans. Microwave Theory -Techn., vol. MTT-21,1973, No. 9, pp. 579-580.[18] Drissi M., Hanna V.F., Citerne J.: Analysis of

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Tab. 1 Calculated f, Q and measured fm, Qm complex resonnt f ncies and quality factorsof the short-circuited slotline resonators (w=O.15 mm, H=1.27 mm,.r = 11)

L (mm) f fr+Jfi (GHz) Q fm =f +Jfim (GHz) Qi( Afr (%) Ar i (%) AQ (/)14.95 4.308+jO.0352 42.11 4.353+j0.0324 44.82 1.04 |795 6.46

3.75 14.146+jO.441 15.35 14.340+jO.4813 13.81 -1.37 9.14 10.03

612

oY{Y

a b c d

Fig. 1 Short- andopnuited slodine

0

------ -thecly-15-10 e~~~xpeim.

i f'sI H-1.27mmy W-1.25mm

0 5 10 15 20Frequew (GH4

Fig 3. Retun loss ofthe short-cruted slodine

0I V w --b"s-' _S=_ [L * | w * w | n

-10

-155 10 15 20

Freqlun (GHz)Fig. 5 Retun loss of the open-circuited slotine

* S

I

1

-I £

t* I

r---II~~~~~~~~~~~~~~

- | topfild (dot)---- back side (rrdoslps)

Fig. 2 Slotline reisoator fed by microstrip lines

-5

&-10

-15

-20

5 10 15 20FrequenWc (GHz)

Fig 4 Retum loss of the open-circited sotline

1.00

0.75

0.50

0.25

0.000.000 0.025 0.050 0.075

W;L

Fig 6 Normalized resistan andreaaneoftheshort-circuited slodine

613

n

8

5icu-o

II

prowntthway // / Q0.8 - presnt thery 2.74mn- prSDn -----SDM In10]

e~-121-1-3.073mm 1.7427 0. ao exper. In [17] c

7

). 1

2 3 4 5 6 7 8 2 3 4 5 8 7a Frequency (GHz) b Frequency (GHz

Fig. 7 Comparison of normalized terminating imdance of the short-cicted slotline calculated by the presentmethod and that given in [10] and (17]

0.5

1 0.3

10.20.1

0.05 10 15 20 25 30

Frequency (CHz)a

- present theory* Integr. equ. In [18]

e.S.7HO.635mmWe-0-25 /

er8*7 1-10.635mm W-0.254mm

5 10 15 20 25 30Frequency (CGz)

bFig. 8 Comparison of normalized terminating impedance of the short-circuited slotline calculated by the present

method and that given in [18]

80.4

J0.2

0.1

0.05 10 15

Frequeny20

((HFig, 9 Load impedance ofthe short-circuited slotline calated directly by SDM,

AAA mthe complex resonant frequency of the slote resnator

614

-8

0.08

0.02

0.00