complex impedances

Load impedances of complex slotline

terminations

J Machic, J Zehentner*

W Menzelo

*Faculty of Electrical Engineering Czech Technical University Technicki 2, 16627

Prague

6, Czech Republic

**University

of

Ulm Albert Einstein Allee 41, 89069 Ulm, Germany

Abstract

Single slotline has gained new interest due to application in uniplanar circuits, antennas or active radiators, allthese including slotline short circuits or even morecomplex terminations. Thispapergives a newapproachtocalculationof reflectioncoefficient and load impedances of such terminations using the spectral domain method. Furthermore, slotline resonators areanalysed in the similarway, and their complexresonantfrequencyis calculated. That is used in an alternative determination of the slotline load impedance by application of the transmission line theory. Computed results compare well with experiment. Closed-form formulaefor CAD purposes areproposed.

Introduction

Slotline plays an important role for antennas [1], radiating resonators[2], slotline fiRters apd couplers[3

-6],oruniplanar integratedcircuits[7].In^manyslotline components as well as in transitions to other planar waveguides like microstrip orcoplanar line, e.g. [8], [9],short-and open-circuitedslotlinesarerequired.To improve bandwidth, different slotline terminations as showninFig.^Iareused. While thesimpleslotline short circuithasbeenanalysedbydifferent authors[101,[11],

more complex terminations have been evaluated largely experimentally

112]

and theoretically by the methodoffinitedifferences [13]. Thispaper presents

a spectraldomain method (SDM) forthe analysis of generalslotline terminations. Incontrast to

[131,

SDM ismuchmoresuitable to evaluate the contnbutions of leakywaveexcitation andradiation,and ^itcaninclude easily dispersive effect of these terminations. Open resonant structures have beenanalysed in

[14].

^Now

thetechniqueused in[14] and[15] has been applied totheanalysisof slotlineresonators.

Analytical Approaches

Firstareflection coefficent of the slotline termination will be derived. Generally, theSDM is based onthe

procedure given in

1101

or

[11].

The slotlines in the structures as shown in

Fig.

I are divided using a

suitable

rectangularmesh.Thetangential

elctric

field along the slotisthendescribedbya summofpiecewise sirnsoidalfunctions(PWSFs). Oneof thefield elemerts isimpressedby agiven amplitude, in dependence on which the other amplitudes are computed. The reflection coefficient of the termination r is then calculated bymeans of the

standing

wave pattem in the

feeding

slotwhich is fittedtothefunction

E(z)=

A[exp(-jkz)

+rexp

(ksz)]

^I (1) by theleast squaresmethod. Aisan amplitde,

k.

is the

propagation

constant of thewave

guided

inthez direction by the

slotline

[161. r and A can be found

analytically minimizing

the leastsquares

discrepancy

between

(1)

and the

calculated standing

wavepattem.

Finally the reflectioncoeffkicentis

=-1

exp(2jk5yi) EexpQi(-jky3

^+P

EiexpikYi)

⁽²⁾

P £ iexp

(-jisyi)

^{f exp}

(-2Jksyi) £Eiexp(ksyi)

where

E1

are

calculated

values of the tangential field

along

theslotlinesampled atpoints

YV

numberofwhich is P.Load impedanceis foundbyasimplerecalculation

of known r.

A

modified

approach canbeusedfor

calculation

of the

slotline load

impedance. Theprocedure is based ontheanalysisoftheslotlineresonator

termninated

at both ends by the structure under

consideration.

The conceptofthe

complex

resonant

frequency [14], [151

is used.Inthesourcelesscase asetofequationssolved forthe unknown

PWSF's

amplitudes ishomogeneous.

Ithas

non-trivial

solutionwhen thedeterminantof the system matrix

equals

zero.

Accomplishment

of this

constraint

provides the complex resonantfrequency.

The

load

impedanceisthendetermined by meansof thecomplexresonantfrequency usingthetransmission linetheorycombinedwith the

lumped

element model ofthe

termination.

Theslotline resonatoris treatedas asection ofthe

slotline

terminatedatthedistanceLby impedances

normalized

valuesofwhich are Z=R+jX Terminating impedances are normalized to the

characteristics impedance

of the

slotline.

After setting the known

complex

^resonant

frequency

f= f,+

jf,

^and

thelengthof theresonatorLinto the resonant

condition

Z can be findfrom

F Z+itid1cL) 1 (3)

Im[Z

1+jZtg(ksL)

⁼

Q

=

fr/(2fi)

=

ksL(1

^-

IZl

^2)-i

⁽⁴⁾

where

0

is theresonator

qualiy factor. This approach

is

valid

^asfar the equivalent

circuit

of the resonator holds, i.e. the width of the slotline is negligible in comparison

with

the resonator's

length.

Results Evaluation

A number of different slotline terminations on the substrate 1.27 mmthickwithdielectric constant of11 wereinvestigated both theoreticallyandexperimentally.

To measure thecircuit, excitation of the slot modewith low losses at respective transitions is important.

Therefore, the wave was transmitted from the waveguide to the finline and further from the finline to the openslotline.Nevertheless, radiation andstanding wavesoccured at thesetransitions and resultedinthe strong ripple of the measured characteristics.

Behaviourof theslotlineresonatorswastestedusing the structureshown in Fig.2.

Figs. 3, 4, 5 showcalculated and measured return lossesof threeslotlineterminationsgiveninFig.1a,b, c. Although in all cases the measured patternshow strong ripple, the general behaviour of thecurvesfits verywell.

Calculated normalized terminal resistance and reactanceof the short-circuited slotline in the span of widthfrom 0.1 to3.25 mmforH/X0inthe range from 0.00425 to0.0845 aregiveninFig. 6 where

permittivity

of thesubstrate is11 and H denotes itsthickness.

Fig. 7 shows calculated normalized terminating impedance of theshort-circuited slotline obtained by thepresentmethod with thatpublishedin

(10]

andwith measured values

given

in

[17].

Agood agreement is evident. Similar comparison is made in Fig. 8. The agreementof normalized terminal reactance with data obtained in

[18]

by the integral equation technique is quite good. Resistances can be compared at low frequencies only

with

thediscrepancy growingforthe frequencyraise.

As to the values shown in [13]: leaky waves are excitedbythe slotline (£r=9.8, H= 1.5mm, W =0.75 mm) from app. 28 GHz upward as follows from calculationaccordingto

[16].

Above thisfrequency the slotline does not transmit bound wave and consequently definition of the terminal impedance is sensless.

The main contribution to the losses at the slotline

terminations

is caused byexcitation of surface waves and byradiation. Losses in

dielectric

playaminor role onlyas it has beenproved byaccounting itscomplex permittivity.

Tomeasurethe resonant frequencyf,and the

quality

factor 0 of the

short-circuited

slotline resonator the circuit shown in Fig. 2 was manufactured on the substratewith permittivity 11 and 1.27mmthick.The resonatorislooselycoupled to

microstrip

feeding lines onthe

opposite side

of the substrate.Measureddata andcorresponding

values

resulting from theresonator

analysis

mentioned aboveagree well. Forillustration Tab. 1

gives

calculated f=

fr

+

jfi

andmeasuredfm-

f=rm

+

jfi

complexresonantfrequenciesalongwith relevant O and

Cm.

respectivelyforshort-circuitedresonatorsof unequal lengths.

Relatively

highererrorsAfifollow from measurementaccuracy ofOm.

Comparison of normalized short-circuited slotline terminating impedancecalculated directly bySDM

with

valuesprovided bythe transmission line theory

utilizing

calculatedcomplexresonantfrequency of the slotline resonatorisshownin Fig.9.

Slotline resonators with terminations shaped in accordancewith

Fig.1

b,chavebeen manufacturedand

their resonat frequencies were measured. The rectangularpatch had 2x2 mmsize and the fan patch with 90°anglehad 2 mmradius. Bothresonators were 14 mm long with 0.15mm wide slot. They were fed according to Fig. 2. Their characteristics are nearly identical.

Closed-Form Formulae of Short- Circuited Slotline

An exact model of the

short-circuited

slotline based onSDM can not beimplanted intoCAD packages. For that a set of impedances Z =R+jX normalizedtothe characteristicimpedanceoftheslotlinecalculated by the present technique quoted above has beenfitted by the least squares method.The resultantclosed-form formulae are

R(x,y)

⁼ Al.

exp(A2y)- A3. exp(-A4y- AsA6)

X(x,y) =

(5)

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

(6) where x⁼W/H, y=

FVX0,

^Wiswidth of the slotline and H 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-08111l)

-o.oo(o.

^{os +soi-i.ss5)-+}

+0.O000x&3

B4=

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

(15)

(16) X 0.053+0

0025.ap(Aft-x2.1914)

-o.o01.eV(-14lx-1.nl3)-o.0024c(eV

2.2.11)

^-

-20(x+(1.7r0.x+io)4sL- .)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⁾

1-1

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

xas

⁺

+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 range froml/2 to 20 GHz for a slot line with the slot width from 0.1 to3.0/1.5mm onthesubstrates1.27/0.635mrr thickhavingpermittivitySr=11. Intheother words the) hold for 4r=11 and y E(0.00425,0.0846), x e (0.0787,2.362)when H =1.27mm andx e

(0.157.2.362

when H=0.635mm.

Conclusions

Two different techniques based on the spectra domain method for calculating the load impedance o thecomplex slotlineterminationshavebeenpresentec and verified by experiments with a good agreement Influence ofthe surface waveleakage and radiationis clearly seen at higherfrequencieswhen the reflectior coefficient isgoingdown.Dielectric losses mostly

play

minorrole. For CADpurposes, terminating impedance of the short-circuited slotline is approximated by the closed-form formulae with sufficient accuracy.

Acknowledgement

This work has been

supported

in partbythegrants given by the Commision of European Communities under the scheme of Cooperation in Science anc Technology with Central and Eastern Europear Countries, Research Fellowships No CIPA351OCT920643and

CIPA3510CT923158

andb5 the CTU ofPragueinternalgrant No. 1003802.

References

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IEEE Trans. MicrowaveTheoryTechn., vol. MTT-20, 1972 No.11, pp. 760-762.

[2] Luy J.F., Buechler J., Thieme M., Biebi E.M.

Matching an active mm-wave slotline antennas Electron Lett.,vol. 29, Sept.1993, pp. 1772-1774.

[3] Mariani E.A., Agrios J.P.: Slot-line filters anc couplers, IEEE Trans.

Microwave

TheoryTechn., vol

MU-18,

1970, No. 12,pp. 1089-1095.

[4]

Schieck B., Kohler J.: Improving the isolation o 3-dB couplers in microstrip-slotline technique, IEEE Trans. Microwave

Theory

Techn., vol. MTT-26, 1978

No. 1, pp. 5-7.

[5] Ho C.H., Fan L., Chang K.: A broad-band

uniplanar

slotline

hybrid ring

coupler with over one octave

bandwidth,

IEEEMTT-S Digest, 1993, pp.585- 8) 588.

[6]

Ho

C.H.,

Fan

L., Chang

^K. :Broad-band

uniplanar hybrid-ring

and branch-line

couplers, IEEE

Trans. MicrowaveTheory

Techn.,

vol. MTT-41, 1993, No. 12, pp. 2116-2125.

[7]

Ogava

H., Minagawa A.:

Uniplanar

MIC balanced multiplier - a proposed new structure for MIC's, IEEE Trans. Microwave Theory Techn., vol.

MTT-35,

1987, No. 12, pp. 1363-1368.

[8] Schuppert B.: Microstrip/slotline transitions:

modeling

and

experimental investigation,

IEEE

Trans.

Microwave TheoryTechn., vol. MTT-36, 1988, No. 8, pp. 1272-1282.

[9] Ho C.H., Fan L., Chang K.: Experimental investigationsofCPW-slotline transitions of uniplanar microwave integrated circuits, IEEE MTT-S

Digest,

1993, pp. 877-880.

[10] Yang H.Y., Alexopoulos N.G.: A dynamic model formicrostrip-slotline transitionandrelated

structures,

IEEE Trans. Microwave

Theory

Techn., vol. MTT-36, 1988, No.2, pp. 286-293.

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circuits,

IEEE Trans. Microwave Theory Techn., vol. MTT-39, 1991, No. 12. pp. 2148-2153.

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MicrowaveTheory

Techn.,vol.

MTT-37, 1989, No. 10, pp. 1639-1641.

[13] Fusco

V.F.,

Chen Q: Slotline short. and open ..circuit analysis by the finite-difference time-domain method, Proc. of 24thEuMC, Cannes, France, Sept.

1994,

vol. 2, pp.1 720-1726.

[141Itoh

T.,

MenzelW.: Afull-wave analysismethod for open microstrip structures, IEEE TTrans.

Antennas

Propag., vol.

AP-29,1981,

No.1, pp. 63-68.

[1-5] Bhat B., Koul S.K.: Analysis, design and applications of fin lines,ArtechHouse,Norwood,1987.

[16] Machac

J.,

Zehentner J.:Leaky waves on a slot line, Proc. of 1995 lnternational S9mposium on Electromagnetic Theory, URSI, St.Petersburg,Russia, May

1995,

pp. 761-763.

[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 coplanar

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MTT-39,1991,

No. 1, pp. 112-116.

Tab. 1 Calculatedf,Qand measured

fm,

Qm complex

resonnt

f nciesand

quality

factors of the short-circuitedslotlineresonators

(w=O.15

mm, H=1.27mm,

.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

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

20

Frequew (GH4

Fig3. Retunloss ofthe

short-cruted

slodine

0

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

-10

-15 5 10 ¹⁵ 20

Freqlun

(GHz)

Fig. 5 Retunlossofthe open-circuitedslotine

* S

I 1

-I £

t * I

II~~~~~~~~~~~~~~

r---

-

|

topfild (dot)

---- back side (rrdoslps) Fig.

2

Slotline reisoator fed by microstrip lines

-5

&-10

-15 -20

5 10 ¹⁵ 20

FrequenWc (GHz) Fig

4 Retum lossoftheopen-circitedsotline

1.00

0.75

0.50

0.25

0.000.000

0.025 0.050

0.075

W;L

Fig 6 Normalizedresistan andreaaneofthe

short-circuited slodine

n

8 5

icu

-o

II

prownt

thway // / Q0.8 -

presnt thery

2.74

mn-

prSDn

---SDMIn10]

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 7

a

Frequency (GHz)

b

Frequency (GHz

Fig.7 Comparisonofnormalizedterminatingimdance of theshort-cicted slotlinecalculatedbythepresent method andthatgivenin[10]and(17]

0.5

1

^0.3

10.2

0.1

5 10 ¹⁵ 20 25 30 0.0

Frequency (CHz)

a

-

present theory

*

Integr. equ.

In

[1 8]

e.S.7

HO.635mmWe-0-25

/

er8*7

1-10.635mm W-0.254mm

5 10 ¹⁵ 20 25 30

Frequency (CGz)

b

Fig. 8 Comparison^ofnormalizedterminatingimpedance of theshort-circuitedslotline calculatedbythepresent

method andthat given in[18]

80.4

J0.2 0.1

0.0 5 10 15

Frequeny ((H

²⁰

Fig,9 Loadimpedanceoftheshort-circuited slotlinecalated directlybySDM,

AAA mthecomplexresonant frequencyofthesloteresnator

-8

0.08

0.02

0.00