Different microstrip line discontinuities on a singlefield-based equivalent circuit model

W. Tang, Y.L. Chow and K.F. Tsang

Abstract: The method of moments (MoM) solves all microstrip line discontinuities by the joiningof segments of different sizes. A single equivalent circuit model may then be constructed to solvemany line discontinuities that are related to the joining of half-lines of different widths, providedthat the physics behind theMoM is well interpreted. From an existing open-circuit model the paperconstructs an alternative open-circuit model that applies to all such line joining discontinuities – forexample, the open-circuit itself, step, right-angle bend and T-junction. The maximum error is 1%without the use of any arbitrary constant.

1 Introduction

Analysis of the fundamental discontinuities in microstrips isof much importance since a complicated circuit is realisedby interconnecting circuit components with microstrip linesand their line discontinuities.

Various transmission-line discontinuities (e.g. open-cir-cuit with fringe, step, right-angle bend, T-junctions, bendswith mitre-cut, etc.) have been characterised by using theexcess inductance and excess capacitance; see [1], forexample. This characterisation is valid only in the quasi-static frequencies in which the dimensions of the disconti-nuity are much smaller than the wavelength. Beyond suchquasi-static frequencies, the full-wave moment method [2, 3]or FDTD [4] have to be used to analyse the behaviour ofmicrostrip discontinuities of arbitrary shape.

Equivalent circuit models [5] of microstrip discontinuitiesare widely used to enhance the speed of microwave circuitdesigns. The equivalent circuit parameters can be deter-mined directly from hardware experiments or from curve-fitted formulas [6], or extracted from software results [7–11];i.e. from the moment method as mentioned above.

Different circuit models have been developed for differentcircuit discontinuities. Such complexity makes interpreta-tion for improved design of a total circuit of manydiscontinuities difficult. However, only one moment methodprogram is needed to apply to all circuit discontinuities.Through a careful physical interpretation of the differentterms in the moment method [12], this paper succeeds inobtaining a single field-based equivalent circuit model formost commonly encountered microstrip discontinuities.

The single equivalent circuit model comes from thereinterpretation of the fringe capacitance of the existingopen circuit formula of a microstrip line [6]. This newly

interpreted circuit model added in tandem to the presentanalysis then gives the circuit models of different disconti-nuities, such as: step and right-angle bend (two opencircuits, in varied strip widths and directions, in tandem),T-junction (joining of three open circuits), and any otherjunctions created by joining different open circuits at theend of a half-line.

Compared with IE3D, the average errors of S-parametersare o1% within the quasi-static frequencies that arenumerically found to be below fh

ffiffiep

r ¼ 12GHzmm,where f and h are the frequency and the thickness,respectively, of the substrate er. This quasi-static limitcorresponds to h being at 4% of the wavelength in thesubstrate.

2 Basic circuit model of an open circuit

The equivalent circuit model is derived under the quasi-static condition, assuming that the metal is thin copper ofthickness 2mm. After the interpretation of the MoM,derivation of the equivalent circuits and the numericalexamples, the observed frequency limit of the quasi-staticcondition is given.

2.1 Usual equivalent circuit model of anopen circuitFigures 1a and 1b show, respectively, an open circuit ofmicrostrip and its usual equivalent circuit model [5, 6]. Inthe model in Fig. 1b, the open-circuit end at the T0 referenceplane is terminated with Cexcess, the excess capacitanceowing to the fringe field at the open circuit. To the left-handside of the T0 reference plane is an ideal transmission line.The capacitance Cexcess is taken directly from the highlyaccurate formula in [6], listed in the Appendix (Section 9).

2.2 Interpretation of an open circuit fromfield based moment methodThe usual model of an excess capacitance Cexcess gives theequivalent effect of the open circuit but is not the true circuitmodel based on physics. As a result, the Cexcess modelcannot be extended for use in other discontinuities likebends and T-junctions. The true circuit model of an opencircuit is derived through the physics contained in themoment method as given below:

W. Tang is with the Department of Electronic Engineering, Nanjing Universityof Science & Technology, Nanjing 210094, China

Y.L. Chow is with the Department of Electrical and Computer Engineering,University of Waterloo, Waterloo ON N2L 3G1, Canada

K.F. Tsang is with the Department of Electronic Engineering, City Universityof Hong Kong, Hong Kong SAR, China

r IEE, 2004

IEE Proceedings online no. 20040268

doi:10.1049/ip-map:20040268

Paper first received 8th May 2003 and in revised form 13th January 2004

256 IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 3, June 2004

(i) We start by studying the importance of the mutualinductance and capacitance in a continuous line. Acontinuous line is characterised by the distributed induc-tance; for example, l1 in H/m and capacitance c1 in F/m. Ifthe line is finite, it may be solved by the moment methodwith segmentation along the line of length Dz. In a momentmatrix, it is easy to see that the ‘distributed inductanceintegrated’ over a segment should be much larger than the‘self-inductance’ of the same segment in isolation. Thereason is that the ‘distributed inductance integrated’ is theself-inductance plus the mutual inductances from allneighbouring segments. The ‘distributed capacitance inte-grated’ over the same segment, however, may be muchsmaller than the ‘self-capacitance’ of the segment inisolation. The reason is that whilst the mutual voltages onthe capacitor are identical in form to those on the induct thevoltages are related to the reciprocal of capacitance. Thereciprocal causes the ‘distributed capacitance integrated’ tobe smaller.

(ii) If the line is now cut in the middle and the right-handhalf-line (branch 2) deleted the remaining left-hand half-line(branch 1) ends in an open circuit. Those segments to theleft and near the open-circuit end are now deprived of thesubstantial mutual inductance and mutual voltage from theright. This means that, at these segments, the distributedinductance is effectively decreased and the distributedcapacitance is effectively increased. The increased distrib-uted capacitance is easily noticed, and modelled as a Cexcess

but the decreased distributed inductance is not noticed asthere is little current flowing through the segments near theopen circuit.

In standard practice, the decrease in voltage, which maybe called fringe voltage (�DV1), is converted to the excess,or the fringe, capacitance Cexcess in parallel, as in Fig. 1b. InLaplace’s equation, however, the voltages are additive as�DV1+V1 (fringe and self). The physics therefore dictatethat there should be a corresponding fringe capacitanceDC1, in series with the distributed capacitance C1 (integrat-ing the last MoM segment Dz by the open end) in Fig. 1c.

Following the fringe voltage, the series fringe capacitance isalso negative, i.e. �DC. This also ensures that the totalcapacitance in Fig. 1c (C1 in series with �DC1) is increasedto agree with the corresponding total capacitance in Fig. 1b(C1 in parallel with a positive Cexcess). The formulas of�DC1 and C1 of the open circuit are given in the followingSection.

(iii) We now continue with the physics interpretationbeyond the open circuit. If the truncated right-hand half-line (branch 2) is now restored (equivalent to a stepdiscontinuity in Section 4 but with no change in stripwidth), the mutual voltage DV12 from the right-hand half-line must cancel the fringe voltage �DV1 of the open-circuitline. As the voltage reciprocal Cexcess is known from [6], thefringe voltage �DV1 and therefore the mutual voltage DV12

from the branch 2 half-line are known. As discussed indetail later, the mutual voltage corresponds to a mutualcapacitor DC12 (positive) in series, thus cancelling �DC1 inFig. 2. Figure 2 represents a T-junction; for it to representthe restored line (a step with no change in line width), thethird branch should be deleted, along with all its mutuals –DC3n as well as DL3n – where n¼ 1 or 2.

(iv) The vector potential has the same form as the scalarpotential (voltage) along a microstrip line, cut and restoredas before. Therefore the same arguments can be applied toobtain the mutual vector potential leading to the mutualinductance DL12 (positive) from the right-hand half-line.

3 Formulas for equivalent circuit of open circuit

To establish quantitatively the equivalent circuit of a linediscontinuity (for example, for an open-circuit or a stepdiscontinuity), the first step is to establish the area boundaryof the discontinuity. For a step discontinuity, it is twosegments, each of length Dz, from the step junction in eitherdirection. We shall assume beyond the boundary that onlythe fundamental mode of the transmission line propagates.For this requirement, following the moment segment in [13,14], we take the segment length as Dz¼ 2h, where h is thesubstrate thickness. The classical ‘addition theorem’ [15], ofa point charge displaced from the origin, ensures that good

∆zp∆z

T1 T0

a

T0

Cexcess

b

T1 T0

−∆C1 from −∆V1

C1

−∆L1L1/2 L1/2

c

Fig. 1 Open and equivalent circuit models of microstripa Layout of open circuit, at T0 end of a half-line;b Usual equivalent circuit;c Our equivalent circuit with discontinuity boundary at Dz(¼ 2h) forconnecting transmission line. Measuring port is at Dzp¼ 2000mm fornumerical results in Fig. 3

C1

T2

C2

∆C21

L2/2 L2/2L1/2L1/2

∆C23

C3

−∆C3

−∆C2−∆C1

∆C31

−∆L3

−∆L2−∆L1 ∆L21∆L13

∆C13

∆C12

∆L12 ∆L23

L3/2 L3/2 T1

∆L32 ∆L31

∆C32

T3

Fig. 2 General field-based circuit model of a T-junction withdashed boxes of mutual effectsIn the boxes, DLmn and DCmn are the mutual inductance and mutualcapacitance on line m from an adjacent line n

IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 3, June 2004 257

attenuation of multipole effects gives a good TEM modebeyond the boundary of the equivalent circuit andindependent of the strip width W/h.

For the open circuit of Fig. 1a, the one segment Dz isbetween the reference plane T1 and T0 at the open-circuitend. This segment may be modelled by a T equivalentcircuit, with L1/2 as horizontal branches and C1 as thevertical branch, with

L1 ¼ l1Dz and C1 ¼ c1Dz ð1Þwhere l1 and c1 are the known distributed inductance andcapacitance of the microstrip line [16]. The vertical branchC1 is in series with fringe capacitance �DC1 in theequivalent circuit of the open circuit in Fig. 1c. Thedistributed fringe inductance �DL1 is also shown forcompleteness, although there is actually little current flowtoward the open-circuit end. The C1 and �DC1 in series inFig. 1c must be equal to the Cexcess and the fringe C1 inshunt in Fig. 1b. That is,

1

1=C1 � 1=DC1¼ C1 þ Cexcess ð2Þ

The fringe �DC1, found as Cexcess, is known from [6] andincluded in the Appendix (Section 9). For convenience, wecan write

DC1 ¼C1ðC1 þ CexcessÞ

Cexcessð3Þ

We can obtain the corresponding fringe DL1 from the L-Cduality of the transmission line theory in air. The duality inair with a series C1 and �DC1 capacitors may appearcomplicated. It has the form

1

1=C0 � 1=DC0

� �ðL1 � DL1Þ ¼ m0e0ðDzÞ2

where DC0 is DC1 of (3) in air and the similar is for C0.However, it is straightforward to reduce the duality to themore convenient form of

�DL1 ¼ �m0e0DC0ðDzÞ2 ð4Þ

We may note that both the fringe inductance andcapacitance have attached negative signs, giving negativenumerical values.

Fig. 1c shows the basic equivalent circuit model of anopen circuit at a half-line from which other discontinuitiesare constructed. No arbitrary constants were required inconstructing the associated equations.

4 Addition of lines to form other discontinuities

In general, a line discontinuity (i.e. a junction) occurs whenthe equivalent open circuits of two or more half-lines aresimply connected.

Figure 2 shows the equivalent circuit of the T-junction(three open circuits joined). The main differencebetween the open circuit in Fig. 1c and the junction inFig. 2 is that the latter has mutual inductances DLmn

and mutual capacitances DCmn on the mth half-line andfrom the nth half-line. In the step or bend discontinuity,where two open circuits join, the third branch and itsassociated mutual inductances and capacitances are to bedeleted.

4.1 The StepTwo open circuits are now joined in a straight line withdifferent widths W1 and W2 for their half-lines. Across thejunction of W1 and W2, there are neither voltage changesnor current changes. The latter means that there is no

change in the mutual inductance, i.e. DL12 still cancels�DL1 in Fig. 2. Similarly DL21 cancels DL2. Note that thedifferent line widths mean that DL21aDL21. However, thechange in widths causes a change in the distributedcapacitance from c1 to c2. The ‘no-change in voltage’ thenresults in a new mutual capacitance pair of

DC12 ¼c1c2

� �DC1 and DC21 ¼

c2c1

� �DC2 ð5Þ

which only cancel DC1 and DC2 with c1¼ c2, i.e. when theline widths do not change.

4.2 The bendTwo open circuits are now joined at a angle y with differentW1 and W2. To avoid ambiguity, we may note that the no-bend case is at y¼ 00. We then get

DL12 ¼ DL1 cos y and DL21 ¼ DL2 cos y ð6ÞAdditionally, through trigonometry and by applyingCoulomb’s Law to (3), we get

DC12 ¼ 1= cosy2

� �� �c1c2

� �DC1 and

DC21 ¼ 1= cosy2

� �� �c2c1

� �DC2

ð7Þ

The right-angle bend is simply the case with y¼ 901.The T� (or Y�) junction joining three open circuits has

an equivalent circuit that is shown in Fig. 2 without branchdeletion. Any such junction with different line widths can beconsidered to be a combination of the step and the bend.Therefore no further formula adjustments beyond (5)–(7)are necessary.

Formulas (1)–(7) have been derived for the discontinuitiesof a microstrip line. However, the general form of theformulas and their interpretation in physics indicate thatthey are applicable to discontinuities of other lines, e.g. striplines, coaxial lines, coplanar lines and likely coplanarwaveguides. This is provided, of course, that an accurateformula of the corresponding Cexcess in (2), for the opencircuit, is available.

5 Numerical results

We have assumed in Section 3 that, beyond the boundaryarea of the discontinuity, only the fundamental mode of thetransmission line propagates. For the IE3D computation, toavoid any remnant higher order modes, each measuringport at the discontinuity is assumed, not at the equivalentcircuit boundary at length Dz¼ 2h from the discontinuityjunction, but further along the connecting microstrip linesto a port distance Dzp. The Dzp value of each example willbe listed in the text. To simulate for the ohmic loss inhardware, the connecting microstrip is assumed to becopper at a thickness 2mm. The load connected to each portis taken to be 50O.

5.1 Open circuitFor an open circuit, the equivalent circuit model is that ofFig. 1c. Figure 3 shows and compares the results by thefield-based equivalent circuit model and IE3D for er¼ 9.6and W¼ h¼ 100mm (49.9O). The port distanceDzp¼ 2000mm. The error of 7S117 of our equivalent circuitfrom IE3D is o1%. As observed in Fig. 3, the error in itsphase is even smaller. This small phase error is expected asthe open-circuit model is converted directly from theaccurate formula of Cexcess [6].

258 IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 3, June 2004

For the multi-port examples below, the errors ofamplitudes (i.e. various 7S117 and 7S217 from our circuitmodel against those from the numerical IE3D) are of moreinterest than the errors of phases. Further, the phase errorsall turn out to be just as small as those of the one-portexample above. Therefore, only the amplitudes are plottedand not the phases.

5.2 Step discontinuitySteps in width of a microstrip line are used in lowpass filters,and in impedance and general transformation circuits. Thelayout of a step is shown in Fig. 4a. Figure 4b shows the

comparison results of 7S117, by the field-based equivalentcircuit model and IE3D for er¼ 2.55, h¼ 200mm,W1¼ 600mm (48.1O) and W2¼ 200mm (89.6O). The portdistances are Dzp¼ 2000mm. The result (dashed line) by thetransmission line theory (without fringe field) in the stepdiscontinuity is also shown in Fig. 4b.

The average errors of 7S117 and 7S217 of the equivalentcircuit and from those of IE3D are o1%, until the quasi-static limit at fh

ffiffiffiffierp ¼ 12GHzmm is exceeded. Translated,

the limit means a 4% substrate wavelength in substratethickness. For a number of examples of other dielectricconstants and different geometry, the average errors of7S117 and 7S217 are also found to be o1%.

To give a feeling on the values of the components,Table 1 lists the values for the step discontinuity example of

Fig. 4. They correspond to Fig. 2 with the third branchdeleted. It is to be noted in the table that: (i) as expected, thesegmental (e.g. the distributed l times Dz) and mutualparameters are of similar orders of magnitude (e.g. betweenL1/2 and DL12); (ii) the mutual parameters do not havereciprocity (e.g. DC21 and DC12 are not equal). The reason isthat a step has unequal line widths across the junction, andtherefore unequal characteristic impedances and nonreci-procate mutual parameters.

5.3 Right-angle bendFor a right-angle bend of the same line width, Figs. 5a–5cshow the layout and the comparison of S11 and S21,respectively, by the field-based equivalent circuit model andIE3D for er¼ 4.5, h¼ 250mm and W1¼W2¼ 500mm(48.2O). The port distances Dzp¼ 1000mm. The averageerrors of 7S117 and 7S217 of our model are small,o1%. Thesmall size of the errors indicates that our equivalent circuitmodel is correct. In turn, it means that the reflection of thebend is a result of the simple loss in the mutual inductanceand a slight gain in mutual voltage across the right anglebend.

5.4 T-junctionThe T-junction, a joining of three open circuits, is shown inFig. 6a. All three ports are loaded with 50O. Fig. 6b showsthe comparison results of S-parameters by the field-based

0 5 10 15 20 25 30 35 40−2.0

−1.5

−1.0

−0.5

0.0

IE3Dour equivalentcircuit model

frequency, GHz

|S11

|, dB

−200

−150

−100

−50

0

50

100

150

200

Q.S.

phase of S11

|S11|

W = 100µmh = 100µm�r = 9.6

phas

e of

S11

, deg

.

Fig. 3 Comparison of S11 of open circuit in Fig. 1 with measuringport at Dzp¼ 2000mm, along a microstrip line that is 2mm thick withcopper lossQS means the quasi-static limit given by fh

ffiffiffiffierp ¼ 12GHzmm

∆zp∆zp

∆z

∆z

T2T1

W2W1

0 10 20 30

−50

−40

−30

−20

−10

0

40

QS

W1 = 600µmW2 = 200µmh = 200µm

�r = 2.55

|S11

|, dB

frequency, GHz

IE3Dour equivalent circuittransmission line theory

a

b

Fig. 4 Step layout and S1 comparison resultsa Layout of a step discontinuityb Comparison of S11 with measuring ports at Dzp¼ 2000mm

Table 1: Values of components in equivalent circuit of stepstructure in Fig. 4 below quasi-static limit

Components in equivalentcircuit of step discontinuity

Values

L1/2 4.7978�10�11 H

�DL1 �3.6641�10�11 H

DL12 3.6641�10�11 H

C1 4.0894�10�14 F

�DC1 �1.8682�10�13 F

DC12 4.5227�10�13 F

L2/2 8.5175�10�11 H

�DL2 �4.7730�10�11 H

DL21 4.7730�10�11 H

C2 2.1116�10�14 F

�DC2 �1.3073�10�13 F

�DC21 8.4377�10�14 F

Values are based on evaluation of Cexcess of [6] at 10GHz. Stepparameters are er¼ 2.55, h¼ 200mm, W1¼ 600mm (48.1O) andW2¼200mm (89.5O)

IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 3, June 2004 259

equivalent circuit model and IE3D, respectively. Theequivalent circuit results agree very well with the IE3Dresults, o1% maximum below the quasi-static limit of fhffiffiffiffierp ¼ 12 GHzmm, i.e. at 19GHz in Fig. 6b. Thetransmission line result with no mutual effects has a muchlarger error, of course. The structure parameters of theT-junction are: er¼ 9.6, h¼ 100mm, W1¼ 100mm (49.9O)and W2¼W3¼ 15mm (98.6O).

The port distances, Dzp¼ 300mm, chosen in this exampleare much shorter than those in the previous examples. Thereason is that the T-junction has more branchesand therefore more radiative attenuation to appear inS-parameters from the IE3D computation. For conveniencein circuit analysis, we may want to assume that theT-junction does not radiate (as the equivalent circuit inFig. 2), and all radiation is relegated to the connectingmicrostrip lines of finite length. In IE3D computations,radiation is unavoidable. It can be reduced, however, for theT-junction computation by having as short a port distanceas possible, i.e. before reaching the region of higher ordermodes in the T-junction.

5.5 Interpretation of mitre-cut on a bend orT-junctionA mitre-cut, at a right-angle or other bend, should increasethe self-inductance and decrease the self-capacitance withinthe boundary of the bend, and compensate the mutualeffects, inductance loss and voltage gain discussed in Section5.3. In other words, the mitre would: (i) restore the total

distributed inductance and capacitance along the bend toapproximate the value of an unbent line; (ii) as aconsequence, restore the characteristic impedance; and (iii)at the end, reduce the reflection S11. The formula of a mitre-cut to the changes in the self-inductance and the self-capacitance of the bend is to be derived in a separate paper.

Across a bend, both the current and voltage continue,and therefore the reflection from the mutual effects cansimply be reduced by a mitre-cut. Across a T-junction,however, the voltage still continues but the current splits,and splits differently each time the load changes at the endof a branch of the T. Therefore its reflection, caused by themutual effects across the T with possible changes over time,cannot be reduced by an unchanging mitre-cut.

5.6 Results beyond the frequency of thequasi-static limitThe quasi-static (QS) limit has been defined, at the end ofthe Introduction, as the frequency f at which the substratethickness h is 4% of the wavelength in the substrate.Beyond this frequency, the above examples indicate that theS-parameter errors from the quasi-static circuit model canbe 41% from the full wave results, such as those from thecommercial software IE3D.

It is now interesting to see what the errors of the model atthe higher frequencies actually are, from not only IE3D, butalso from other full wave solutions in the literature.Different variations of the four basic structures of theabove have been tested. The observation is that the errorsseldom get larger than 3%, even up to three times thefrequency limit of quasi-statics, provided that radiation

∆zp

∆zp

∆z

∆z

T2

T1

W2

W1

0 10 20 30−70

−60

−50

−40

−30

−20

−10

0

40

QSW1 = W2 = 500µmh = 250µm�r = 4.5

|S21|

|S11|

|S11

| and

|S21

|, dB

frequency, GHz

IE3D our equivalent circuit

b

a

Fig. 5 Bend layout and S11, S21 Comparisonsa Layout of right-angle bendb Comparison of S11 and S21 with measuring ports at Dzp¼ 1000mm.Dzp and Dz measured from centre of bend. Trivial results fromtransmission line theory are not plotted in this example

∆zp∆zp

∆z∆z

T1

port 3

port 1

W1

W3 Wa

∆zT3 T2

∆zp

W2 port 2

0 5 10 15 20 25 30 35 40

−12

−10

−8

−6

−4

−2

QS

W1 = 100µmW2 = W3 = 15µmh = 100µm�r = 9.6

|S21|

|S11||S11

| and

|S21

|, dB

frequency, GHz

IE3D our equivalent circuit transmission line theory

a

b

Fig. 6 T-junction layout and S11, S21 comparisonsa Layout of a T-junctionb Comparison of S11 and S21 of T-junction with measuring ports atDzp¼ 300mm Dzp and Dz are measured from centre of T-junction

260 IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 3, June 2004

from the junction is not a problem. Radiation can be aproblem if the reflection from line discontinuity is high.Typical discontinuity examples are an open circuit, or aT-junction with an effective open circuit appearing in one ofits ports (e.g. owing to a stub beyond the port).

We have selected one of the variations to demonstrate theabove observation beyond the QS limit. It is the sameT-junction of Fig. 6a but, unlike those for Fig. 6b, forFig. 7, the widths of the arms are now equal. That is:W1¼W2¼W3¼ 0.56mm, with er¼ 9.7 and h¼ 0.635mm.This is the example from Mehran [17], compared andverified in [18]. Fig. 7 shows comparison of four responses:our equivalent circuit model, IE3D (MoM), Mehran(waveguide model) and transmission line theory. Plottedin linear scale of magnitude Smn for easier error checking,Fig. 7 shows that the agreement is excellent between the firstthree responses, i.e. at a deviation of r3% (from amaximum of Smn¼ 1) up to 20GHz (i.e. 3.3 times thequasi-static limit) between all three. In other words, untilradiation effects set in, our simple circuit model is just asaccurate as the full wave solutions, e.g. [17–19].

We may notice that, as frequency increases towards thequasi-static limit, there is a general downward trend in thefirst 2 S11 values in Fig. 6b as opposed to the generalupward trend in the S11 values shown in Fig. 7. This isbecause the widths Wn of the arms are different in theT-junctions of Figs. 6 and 7. From the equations and thecircuit models in this paper, it is easy to see that thedifferences in widths must cause different levels of self- andmutual inductance and capacitance (across the junction),and different pseudo-quarter-wave transformations (alongeach arm). These differences therefore cause the differencein trends.

6 Conclusions

The accurate results of the different microstrip disconti-nuities (open-circuit, step, right-angle bend and T-junction)show that the general field-based equivalent circuit model isindeed very useful for the analysis of microwave compo-nents and circuits. The comparison with IE3D shows that

the maximum errors of S-parameters are o1% below thequasi-static limit fh

ffiffiep

r ¼ 12 GHzmm.The usual equivalent circuits for different discontinuities

are different and need many (frequently 410) arbitraryconstants to be curve-fitted. However, in this paper, noarbitrary constant is needed other than those in the initialformula of Cexcess of the open circuit from [6], in theAppendix (Section 9). The general field-based equivalentcircuit model (Fig. 2) is accurate. Having no arbitraryconstant, it is simple and gives very good physical insightsfor various design adjustments in the microstrip linediscontinuities.

It is possible to improve the accuracy further to below themaximum of 1% error by using arbitrary constants.Arbitrary constants invariably obscure the physical insightneeded for versatility in designs and design types, especiallyfor the initial designs. As a rule, therefore, no arbitraryconstants are added to the formulas of the components ofthe equivalent circuit whenever possible.

It is interesting to point out that the above equivalentcircuits of step, bend and T-junction are ‘bootstrapped’ in afuzzy EM [16] manner from the single equivalent circuit ofan open circuit of a microstrip. Unlike before, however, thebootstrapping starts not from one of our own fuzzy EMresults or from a classical solution such as a conformal map,but from reinterpreting an existing approximate formula ofothers, e.g. the open circuit in [6]. This implies that existingdesign formulas in the literature can be used not only to givenumerical results to their specific structures, but also togenerate simple design formulas for some related structures,which may look very different at first glance, e.g. microstripline discontinuities of crosses, Y-junctions, notches andbends with mitre-cuts.

This paper reduces the line discontinuities to one simplelumped circuit model as in Figs. 1 and 2, and broadens thebandwidth to 3.3 times the quasi-static limit as in Fig. 7, i.e.substrate h may become close to a radian where significantradiation and surface waves occur. These two points meanthat the responses of line discontinuities in a digital circuitmay now be easily solved in the time domain with circuitsoftware like SPICE, without any Fourier transform fromthe frequency domain. An early work on this has beenreported in [20].

7 Acknowledgments

This paper was supported by a CERG grant (no. 9040522)of Hong Kong at City University, and by an NSERCDiscovery Grant (no. 080-6166) of Canada at the Universityof Waterloo.

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5 Easter, B.: ‘The equivalent circuit of some microstrip discontinuities’,IEEE Trans. Microwave Theory Tech., 1975, 23, pp. 655–660

6 Kirschning, M., Jansen, R.H., and Koster, N.H.L.: ‘Accurate modelfor open end effect of microstrip lines’, Electron. Lett., 1981, 17,pp. 123–125

7 Garg, R., and Bahl, I.J.: ‘Microstrip discontinuities’, Int. J. Electron.,1978, 45, (1), pp. 81–87

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1.0

QS

|S11|

|S21|

mag

nitu

de

frequency, GHz

our equivalent circuit IE3D transmission line theory R. Mehran

Fig. 7 Comparison (in magnitude scale) of S11 and S21 of T-junction between four sets of results: our circuit model, IE3D,transmission line theory and Mehran [17, 18] Measuring ports areat Dzp¼ 20 mm from enter of T-junction

IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 3, June 2004 261

8 Benedek, P., and Silvester, P.: ‘Equivalent capacitances for microstripgaps and steps’, IEEE Trans. Microw. Theory Tech., 1972, 20, (11),pp. 729–733

9 Farrar, A., and Adams, A.T.: ‘Matrix methods for microstrip three-dimensional problems’, IEEE Trans. Microw. Theory Tech., 1972, 20,(8), pp. 497–504

10 Gopinath, A., Thomson, A.F., and Stephenson, I.M.: ‘Equivalentcircuit parameters of microstrip step change in width and crossjunctions’, IEEE Trans. Microw. Theory Tech., 1976, 24, (3),pp. 142–144

11 Itoh, T., Mittra, R., and Ward, R.D.: ‘A method for computing edgecapacitance of finite and semi-infinite microstrip lines’, IEEE Trans.Microw. Theory Tech., 1972, 20, (12), pp. 847–849

12 Harrington, R.F.: ‘Field computation by moment method’ (IEEE,1991, 2nd edn.)

13 Chow, Y.L., and El-behery, I.N.: ‘An approximate dynamic spatialGreen’s function for microstriplines’, IEEE Trans. Microw. TheoryTech., 1978, 26, pp. 978–983

14 Chow, Y.L.: ‘An approximate dynamic Greens function in threedimensions for finite length microstripline’, IEEE Trans. Microw.Theory Tech., 1980, 28, pp. 393–397

15 Smythe, W.R.: ‘Static and dynamic electricity’ (Rev. printing Hemi-sphere, New York, 1989, 3rd edn.)

16 Chow, Y.L., and Tang, W.C.: ‘Development of CAD formulas ofintegrated circuit components – fuzzy EM formulation followed byrigorous derivation’, J. Electromagn. Waves Appl., 2001, 15, (8),pp. 1097–1119

17 Mehran, R.: ‘The frequency dependent scattering matrix of microstripright angle bends, T-junctions and crossings’, Arch. Electr. Uebertrag,1975, 29, pp. 454–460

18 Yakovlev, A., and Gnilenko, A.: ‘Analysis of microstrip disconti-nuities using the method of integral equations for overlapping regions’,IEE Proc. –Microw. Antennas Propag., 1997, 144, pp. 449–457

19 Wu, S.-C., Yang, H.-Y., Alexopoulos, N.-G., and Wolff, I.: ‘A rigo-rous dispersive characterization of microstrip cross and T-junc-tions’, IEEE Trans. Microw. Theory Tech., 1990, 38, pp. 1837–1844

20 Harms, P.H., and Mittra, R.: ‘Equivalent circuits for multiconductormicrostrip bend discontinuities’, IEEE Trans Microw. Theory Tech.,1993, 41, pp. 62–69

9 Appendix

An accurate and popular formula for the excess capacitanceat an open ended line is from Kirschning et al. [6]. Based on

functional approximations of the low frequency calculationsfrom a rigorous hybrid-wave analysis [11], it is:

DlL=h ¼ ACE=D ð8Þ

A ¼ 0:434907ðe0:81r;eff þ 0:26Þ½ðW =hÞ0:8544 þ 0:236�ðe0:81r;eff þ 0:189Þ½ðW =hÞ0:8544 þ 0:87�

ð9Þ

B ¼ 1þ ½ðW =hÞ0:371=ð2:358er þ 1Þ� ð10Þ

C ¼ 1þ ð0:5274=e0:9236r;eff Þ tan�1f0:084ðW =hÞ19413=B ð11Þ

D ¼ 1þ 0:0377½6� 5 expf0:036ð1

� erÞg� tan�1f0:067ðW =hÞ1:456g ð12Þ

E ¼ 1� 0:218 expð�7:5W =hÞ ð13ÞHere, DlL is the excess length of the excess capacitor, andthe excess capacitance is therefore

Cexcess ¼ 2pfZL cotð2pfDlLffiffiffiffiffiffiffiffiffiffier;effp

=v0Þ� ��1 ð14Þ

v0 is the speed of light in free space, f is the frequency, andZL and er,eff are the characteristic impedance and effectivedielectric constant of microstrip, given in [6] or in a simplerform in [16]. The maximum error of the formulas iso2.5%in the capacitance Cexcess for 0.01rW/hr100 and1rerr50. This maximum error in Cexcess, together withthe possible 1% maximum errors of the distributed l andc of the microstrip lines of the multi-port junctions,apparently translates to the 1% maximum errors in theS-parameters in all discontinuities in the paper.

262 IEE Proc.-Microw. Antennas Propag., Vol. 151, No. 3, June 2004