on the thru-reflect-line (trl) numerical calibration and error analysis for parameter extraction of...

On the Thru-Reflect-Line (TRL) NumericalCalibration and Error Analysis for ParameterExtraction of Circuit Model

Lin Li,1 Ke Wu,1 Peter Russer2

1 Poly-Grames Research Center, Department of Electrical Engineering, Ecole Polytechnique deMontreal, Quebec, Canada H3C 3A72 Hochfrequenztechnik, Technische Universitaet Muenchen, Arcisstrasse 21, D-80333 Muenchen,Germany

Received 19 April 2005; accepted 14 October 2005

ABSTRACT: In this work, a resistor standard is introduced into our previously proposednumerical thru-reflect-line (TRL) calibration procedure in order to determine the character-istic impedance of the line standard of calibration on the basis of a deterministic method ofmoments (MoM) algorithm. A comprehensive analytical derivation is presented with regardsto electrical properties of such a resistor standard in comparison to other standards. Inaddition, an error analysis is detailed, which reveals correlations of characteristic parametersin connection with equivalent circuit model development from the conversions from field-based S-parameters to circuit-based Y- or Z-parameters. Interesting properties and criteriaare derived, allowing accurate parameter extractions. To validate the proposed numericalTRL calibration procedure with this new resistor standard concept and the developed erroranalysis, the characteristic impedance of a microstrip line is extracted from a commercialsoftware. In addition, a further example with microstrip discontinuity is shown and theeffectiveness of the proposed technique is verified. © 2006 Wiley Periodicals, Inc. Int J RF andMicrowave CAE 16: 470–482, 2006.

Keywords: TRL calibration technique; full-wave MoM simulator; parameter extractions; planardiscontinuity; equivalent-circuit model

I. INTRODUCTION

The deterministic method of moments (MoM) algo-rithms have been recognized as probably the mostpowerful candidates for accurate and efficient model-ing of multilayered planar or quasi-planar structures.However, approximate lumped current–voltagesources such as delta-voltage are generally used toexcite the structure, thereby allowing the determinis-

tic and efficient calculations of field parameters. Sincethe artificial sources can never describe the exact fieldprofile at the ports of excitation because of multilay-ered geometry and nonuniform field profile, the re-sulting “artificial” field discontinuities or differencesbetween the lumped sources and the “true” fields canbring errors or parasites to the calculated networkparameters or equivalent circuit models, which can besignificant for a planar structure.

To solve this problem of port discontinuities, sev-eral techniques have been presented [1–5]. But nofurther attempts have been made for a systematicunderstanding of this critical problem until the pro-posal of the short-open calibration (SOC) technique

Correspondence to: Professor Ke Wu; e-mail: [email protected]

DOI 10.1002/mmce.20167Published online 18 July 2006 in Wiley InterScience (www.

interscience.wiley.com).

© 2006 Wiley Periodicals, Inc.

470

[6–12]. Considering that nearly all commercial MoMsimulators can only provide the calculated networkparameters at a specified external location along thefeeding line and that the SOC need to calculate cur-rent and voltage at the middle of the line withoutresorting to an internal port, the SOC technique wasfound to be difficult in theory due to its compatibilityand integration with them. Therefore, it is not practi-cal for us to consider the SOC technique with thecommercial electromagnetic simulation and analysistools. To remove this hurdle, thru-reflect-line (TRL)and thru-resistor (TR) numerical calibration tech-niques were introduced that have successfully beencombined with such commercial EM software pack-ages as Agilent Momentum, Sonnet Lite EM simula-tors, and Zeland IE3D [13] to model the port discon-tinuities and extract the circuit models of planardiscontinuities [14–16] in a similar way as the SOCscheme. As such, an accurate equivalent circuit modelof the planar circuits can be generated from thisprocess.

However, the calibration reference impedance,which is equal to the characteristic impedance of theline standards, should be known a priori in the TRLtechnique. In general, the characteristic impedanceand propagation constant of the line standards arecalculated from a 2D modeling method, which as-sumes the lines to be infinitely long. Because of theinconsistency between the 2D and 3D impedancedefinitions [17] of transmission line, the characteristicimpedance of the line standards obtained from the 2Dmethod will yield some difficult-to-estimate error ef-fects [10, 17]. In the TRL calibration, the 3D defini-tion of characteristic impedance should be used.

The SOC has successfully been used to determinethe 3D characteristic impedance of a transmission line[10]. Nevertheless, the SOC technique is not directlycompatible with the existing commercial EM soft-ware, as it needs to calculate current and voltage at themiddle of the transmission line in the program basedon the MOM algorithm. In order to determine the 3Dcharacteristic impedance of the line standards for im-proving accuracy in the parameter extraction of planarcircuits, we have introduced an alternative standardcalled the “resistor standard” in the numerical TRLcalibration [16]. The resistor standard is similar to thatused in [15], but does not involve the same applica-tion concept. In [16], the resistor standard was used asa supplement of the TRL method to determine thecharacteristic impedance of the transmission line. Inthis work, we provide a further discussion of thisadditional standard and its related analytical deriva-tions.

The problem of port discontinuity involved in thedeterministic MOM algorithms has been studied in [3,18]. However, only lumped-circuit models of the portdiscontinuity have been proposed and developed. TheTRL calibration procedure works with the T- andS-parameters. However, the extraction of lumped-circuit models of microwave structures will be moreconvenient and can be clearly understood if we use Y-or Z-parameters. Absolute and relative errors in con-nection with the transformation from the S-parametersto the Z- or Y-parameters, which are critical in theunderstanding of accuracy of circuit models, are notknown at all thus far. This article also presents adetailed analysis of such errors, showing what degreeof error impact the port discontinuity brings to theequivalent-circuit models.

II. THEORY

In the TRL calibration, three calibration standards(through, reflect, and line) are used to extract theembedded error terms [19]. Compared to other tech-niques such as short-open-load-through (SOLT) cali-bration, the TRL standards are more easily realizable.The only critical parameter is the characteristic im-pedance of line standards. For the TRL calibration,the characteristic impedance of the line standardshould be known exactly at the very beginning, theTRL itself could not determine such characteristicimpedance. As in [16], to determine this importantparameter, we propose a new resistor standard, asshown in Figure 1(f). The connection of this resistorstandard is the same as that of the reflect standard. In[16], we proved that such a resistor standard can beused to determine the characteristic impedance of theline standard. But no clear analytical scheme wasgiven in [16], which is critical for the understandingand applications of the proposed technique. Next wediscuss how the resistor standard works. The imped-ance of the resistor standard is chosen to be thereference impedance used in the calibration proce-dure, and the reason for this is discussed below. Sucha resistor standard is very easy to implement in com-mercial full-wave EM software.

With the same definition as in [20, 21], we candefine

Mx � ANxB�1 (1)

where A and B�1 are the cascading matrices of theerror boxes A and B; Nx is the cascading matrices ofthe standards and Mx is the cascading matrices ob-tained at reference planes P1 and P2.

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

Numerical TRL Calibration and Error Analysis 471

From the through connection, we obtain

M1 � AN1B�1, (2)

N1 � � I 00 I� . (3)

From the line connection, we can obtain

M2 � AN2B�1. (4)

If the characteristic impedance of the line standardis known and equal to the reference impedance in thecalibration procedure, we can write

N2 � �e��l 00 e��l� . (5)

As stated in [20, 21], error parameters in the errorbox can be determined by measurements of the three

TRL standards connections and the true parameters ofthe device under test (DUT) can be extracted cor-rectly. If the characteristic impedance of the line stan-dard is not known exactly, however, eq. (5) cannot beused. The TRL calibration procedure cannot be com-pleted. Or if there is error in the characteristic imped-ance of the line standard, it will cause error in theextracted parameters of DUT. So we need anotherstandard to generate the characteristic impedance ofthe line standard. As in [21], we define two similarmatrices P and Q as follows:

Q � M2M1�1 and P � N2N1

�1 with

P � A�1QA. (6)

Because P and Q are similar, we can define aneigenvalue matrix �, which gives

P � N2N1�1 � N2 � X�X�1, (7)

Figure 1. Block diagrams of the proposed improved numerical TRL calibration procedure: (a)two-port core circuit under modeling; (b) error model for numerical TRL calibration; (c) throughstandard connection; (d) line standard connection; (e) reflect standard connection; (f) resistorstandard connection.


472 Li, Wu, and Russer

Q � Y�Y�1, (8)

P � XY�1QYX�1, (9)

A � YX�1. (10)

We can write the characteristic impedance of theline standard as Z, where Z is an unknown. Eq. (5)then becomes

N2

� �ch(�l) �Z2 � Z0

2

2ZZ0sh(�l)

Z2 � Z02

2ZZ0sh(�l)

�Z2 � Z0

2

2ZZ0sh(�l) ch(�l) �

Z2 � Z02

2ZZ0sh(�l)�.

(11)

From the above matrix calculation, the eigenvalueof N2 is given by

� � �e��l 00 e��l� . (12)

So, N2 can be expressed by

� � X�1N2X � X�1PX, (13)

where the columns of X are composed of eigenvectorsof N2. The eigenvectors can be written as

X � X0�, (14)

� � diag��1, �2�, (15)

where � are arbitrary constants.One possible X0 is

X0 � �1 aa 1� , a �

Z � Z0

Z � Z0. (16)

In the same way, we can write Y in eq. (8) as

Y � Y0�, � � diag��1, �2�. (17)

Because Q is the measured data at reference planesP1 and P2, so we can obtain Y0, except for an arbi-trary constant �. From eqs. (10), (14), and (17), wecan obtain

A � YX�1 � Y0��1�1 aa 1��1

� Y0��11

1 � a2 � 1 �a�a 1 � � A0K� 1 �a

�a 1 �� A0K�1 b

b 1�, (18)

where A0 � Y0, b � �a, and K � diag((�1/�1(1 �

a2)), (�2/�2(1 � a2))) � �k1 00 k2

�, where k1 and k2

are arbitrary constants.Eq. (18) can be written as

A � A0K�1 bb 1�

� �A0,11k1 � A0,12k2b A0,11k1b � A0,12k2

A0,21k1 � A0,22k2b A0,21k1b � A0,22k2�

� �A11 A12

A21 A22� . (19)

From eq. (2), we obtain

B � M1�1AN1 � M1

�1A � M1�1A0K�1 b

b 1�� B0K�1 b

b 1�� B0,11k1 � B0,12k2b B0,11k1b � B0,12k2

B0,21k1 � B0,22k2b B0,21k1b � B0,22k2�

� �B11 B12

B21 B22� , (20)

where B0 � M1�1A0.

So far, there are three unknowns to be determinedin A and B: b(b � �a), k1, and k2. The methodstated in [21] cannot be applied here because there arethree unknowns instead of two unknowns. That iswhy we have to introduce another resistor standard tocomplete the calibration procedure. As in [21], usinga known value resistor connection yields

�Xa � � A11� � A12�� A21� � A22��1, (21a)

�Xb � �B22� � B21��B12� � B11��1. (21b)

At first, we use two match standards with � � 0.From eq. (21a), we have

A12 � �XaA22. (22)

Combining eqs. (19) and (22) leads to



A0,11k1b � A0,12k2 � �XaA0,21k1b � �XaA0,22k2,

(23a)

k1b � � A0,11 � �XaA0,21��1��XaA0,22 � A0,12�k2.

(23b)

We can write eq. (23b) as

k1b � X1k2, (24)

where X1 � ( A0,11 � �XaA0,21)�1(�XaA0,22 �A0,12).

In the same way, from eqs. (20) and (21b), we canobtain

B21 � �XbB11, (25a)

B0,21k1 � B0,22k2b � �XbB0,11k1 � �XbB0,12k2b.

(25b)

Eq. (25b) can be written as

k2b � X2k1, (26)

where X2 � (B0,22 � �XbB0,12)�1(�XbB0,11 �B0,21).

Solving eqs. (24) and (26) generates a condition asfollows

b2 � X1X2, b � ��X1X2. (27)

So the characteristic impedance of the line stan-dard is given by

Z � Z0

1 � �X1X2

1 � �X1X2

. (28)

The sign in b could be decided in two ways. Thefirst way is as follows. If we know approximately thevalue of Z, we can determine the sign of b by com-paring the value calculated from eq. (28) with theapproximate value. So the best way is to select thereference impedance Z0 to be far from the preevalu-ated characteristic impedance of the line standard. Incase that the reference impedance Z0 is selected to benear the real Z, then b is very small; however, wecannot distinguish which sign that corresponds to theright value. Another way to estimate the sign of b isto use the reflect (with high reflecting coefficient)standard. We suppose the reflect standard is an open(not necessarily an ideal open) standard.

From eq. (21a), we obtain

� � ��XaA21 � A11��1� A12 � �XaA22�. (29)

From k2b � X2k1 and k1b � X1k2 in eqs. (24) and(26), respectively, eq. (19) can be written as

A � �A0,11k1 � A0,12k2b A0,11k1b � A0,12k2

A0,21k1 � A0,22k2b A0,21k1b � A0,22k2�

� � (A0,11 � A0,12X2)k1 (A0,11 � A0,12/X1)k1b(A0,21 � A0,22X2)k1 (A0,21 � A0,22/X1)k1b

�� A11k1 A12k1b

A21k1 A22k1b� . (30)

When the error box A is connected with an open,we can obtain the following with reference to eqs.(29) and (30).

� �1

k1��XaA21 � A11�

�1� A12 � �XaA22�bk1 � �b,

(31)

where � � (�XaA21 � A11)�1( A12 � �XaA22).Because we know the reflection coefficients of the

open standard that is almost equal to 1, from eq. (31),we can decide the sign of b (� or �). From eq. (24),we obtain k2 � k1b/X1. So there is only one unknownk1 for now. It is enough to complete de-embedding ofthe error box without knowing the value of k1. If theerror box has the reciprocity property, we can obtainthe value of k1. The reciprocity property exists whenwe carry out the EM simulations with a deterministicMoM commercial simulator. According to the reci-procity property, we have

A11A22 � A12A21 � 1. (32)

From eqs. (30) and (32), we can obtain

A11k1A22k1b � A12k1bA21k1 � 1. (33)

Eq. (33) can be written as

k12 � G and k1 � ��G, (34)

where G � (1/( A11A22 � A12A21)b).So k1 is determined except for the sign. If we

directly make use of the TRL calibration with thenetwork analyzer in real measurement, there is noreciprocity property in the error boxes. But if we usea full two-port calibration first then use a TRL cali-bration, the error box has the reciprocity property.

In fact, we can use other terminators with knownreflection coefficient to get the characteristic imped-



ance of the line standard. If a reflector or a resistorstandard is used with a known value �, from eq. (21a)we obtain

� A0,11� � �XaA0,21��k1 � � A0,11 � �XaA0,21�k1b

� � A0,12 � �XaA0,22�k2

� � A0,12� � �XaA0,22��k2b � 0. (35)

From the previous analysis in the case of a matchconnection, we know that there should be a linearrelation between k1 and k2, as k2 � lk1. So we canwrite eq. (35) as

C1k1 � C2bk1 � C3lk1 � C4lbk1 � 0, (36)

where C1 � A0,11� � �XaA0,21�, C2 � A0,11 ��XaA0,21, C3 � A0,12 � �XaA0,22, and C4 �A0,12� � �XaA0,22�.

Eq. (36) can be written as

k1�C1 � C2b � C3l � C4lb� � 0. (37)

Because k1 cannot be zero, we have

C1 � C2b � C3l � C4lb � 0. (38)

In the same way, from eq. (21b) we can obtain

D1 � D2b � D3l � D4lb � 0. (39)

Solving eqs. (38) and (39), we can obtain twounknowns l and b. In this way, the calibration proce-dure is completed.

From the above analysis, we can see that with theknown � of the reflect standard or the resistor stan-dard, we can derive the needed parameters of the errorbox and the characteristic impedance of the line stan-dard. Sometimes the fourth standard can facilitate thedecision of the sign or the calculation. As indicated in[20], we can also obtain the propagation constant ofthe line standard.

In various calibration procedures, the generallyused standards are open, short, or resistor. We choosethe resistor standard because the error in the resistorbrought by the parasitic effect could be smaller thanthat in a short or an open. The error in the standardcan be caused by serial inductance or parallel capac-itance. So it will give error to the reflection coeffi-cient. The reflect coefficient of a terminator is givenby

� �Z � Z0

Z � Z0�

Y0 � Y

Y0 � Y. (40)

From eq. (40), we obtain

�

Z�

2Z0

�Z � Z0�2 �

�2Y0

�Y0 � Y�2 . (41)

For the short connection,

�

Z�

2

Z0. (42a)

For the open connection,

�

Y� �

2

Y0. (42b)

For match connection,

�

Z�

1

2Z0�

�1

2Y0. (42c)

From eqs. (42a)–(42c), we can see that the para-sitic inductance or capacitance will bring less error inthe case of the match connection than the other con-nections.

After we obtain S-parameters of the core circuit byremoving the error box, we can build up a circuitmodel of the core circuit. Because the TRL calibrationprocedure works with the T- and S-parameters, wecannot directly generate the Z- or Y-parameters of thecircuit. In this case, a circuit model has to be gener-ated on the basis of a conversion from the S-param-eters to the Z- or Y-parameters.

For simplicity, we consider a lossless and recipro-cal two-port network. The S-parameters can simply beexpressed by three variables: a, 1, and 2, as for-mulated in eqs. (43a)–(43c). The normalized Z-pa-rameters can then be written as in eqs. (44a)–(44c),which have the same denominator. Similarly, the nor-malized Y-parameters can also be formulated in eqs.(44e)–(44g), which also have the same denominator.But the common denominator in the Z-parameters isdifferent from that in the Y-parameters. If the denom-inator approaches zero, errors in the calculated S-parameters of the circuit will be amplified and pro-nounced, resulting in much large errors in theconverted Z-parameters. Therefore, we should becareful when the S-parameters happens to be close tothe region of a zero denominator. In the following, anerror analysis can be made regarding this aspect.



First of all, we can formulate dZ/da and dZ/d asin eqs. (45a)–(45i). Some typical examples of corre-lation between dZ/da, dZ/d and a, (Z1 � Z� 11 �Z� 12, Z2 � Z� 22 � Z� 12, Z3 � Z� 12) are shown inFigures 2(a)–2(g). In Figure 2(a), 1 � 20°, 2 �170°, and d(1/Z3)/da versus a are shown in the rangeof a from 0 to 1. In Figures 2(b)–2(d), 2 � 0° anda � 0.1, 0.5, 0.9, respectively, dZ/d1 versus 1

are shown in the range of 1 from �180° to 180°. InFigures 2(e)–2(g), 2 � 90° and a � 0.1, 0.5, 0.9,respectively, dZ/d1 versus 1 are also shown in therange of 1 from �180° to 180°. From such numer-ical examples in Figures 2(a)–2(g), we can see that in

some regions, the errors on the calculated a, of thecore circuit will cause much larger error on Z-param-eters of the core circuit. These regions move aroundas a and change and such regions become largerwhen a approaches 1. Therefore, if we obtain a moreaccurate characteristic impedance of the line standard,we can obtain more accurate extracted circuit param-eters so that we can reduce the error in the circuitmodel of the core circuit. In Figure 2(a), we can seethat when a approaches 0, dZ3/da becomes muchmore significant than d(1/Z3)/da, which means thatthe error in a becomes less pronounced in 1/Z3. Thatis, because they have different denominators, the re-

Figure 2. Examples of correlations between dZ/da, dZ/d and a, : (a) dZ/da when 1 � 20°and 2 � 170°; (b), (c), (d) dZ/d1 when 2 � 0 and a � 0.1, 0.5, 0.9, respectively; (e), (f), (g)dZ/d1 when 2 � 90° and a � 0.1, 0.5, 0.9, respectively.



gions of denominator approaching zero are different.Sometimes if one of the inductances in the circuitmodel has a large error, we may replace it by acapacitance in order to reduce the error; so, how wechose the Z-parameters (T-type circuit) or Y-parame-ters (�-type circuit) to build the circuit model is veryimportant for reducing the error:

S11 � a � ej1, (43a)

S22 � a � ej2, (43b)

S12 � �1 � a2 � ej�1�2��/ 2, (43c)

Z� 11 �1 � a � �ej1 � ej2� � ej�1�2�

1 � a � �ej1 � ej2� � ej�1�2� , (44a)

Z� 22 �1 � a � �ej1 � ej2� � ej�1�2�

1 � a � �ej1 � ej2� � ej�1�2� , (44b)

Z� 12 �2�1 � a2 � ej�1�2��/ 2

1 � a � �ej1 � ej2� � ej�1�2� , (44c)

Y� 11 �1 � a � �ej1 � ej2� � ej�1�2�

1 � a � �ej1 � ej2� � ej�1�2� , (44d)

Y� 22 �1 � a � �ej1 � ej2� � ej�1�2�

1 � a � �ej1 � ej2� � ej�1�2� , (44e)

Y� 12 � �2�1 � a2 � ej�1�2��/ 2

1 � a � �ej1 � ej2� � ej�1�2� , (44f)

dZ12

da

��2�ej1 � ej2 � a � a � ej�1�2� ej�1�2��/ 2

�1 � a2 � �1 � a � �ej1 � ej2� � ej�1�2� 2 ,

(45a)

dZ22

da�

2ej2�1 � ej21

�1 � a � �ej1 � ej2� � ej�1�2� 2 , (45b)

dZ12

da

��2�ej1 � ej2 � a � a � ej�1�2� ej�1�2��/ 2

�1 � a2 � �1 � a � �ej1 � ej2� � ej�1�2� 2 ,

(45c)

dZ11

d1�

j2�a � ej1 � ej�1�2� � �1 � a � ej2

�1 � a � �ej1 � ej2� � ej�1�2� 2 ,

(45d)

dZ22

d2�

j2�aej2 � ej�1�2�� 1 � aej1�

�1 � a � �ej1 � ej2� � ej�1�2� 2 , (45e)

dZ11

d2�

j2�a2 � 1�ej�1�2�

�1 � a � �ej1 � ej2� � ej�1�2� 2 , (45f)

dZ22

d1�

j2�a2 � 1�ej�1�2�

�1 � a � �ej1 � ej2� � ej�1�2� 2 , (45g)

dZ12

d1�

�j�1 � a2ej�1�2��/ 2�1 � a � �ej1 � ej2� � ej�1�2�

�1 � a � �ej1 � ej2� � ej�1�2� 2 , (45h)

dZ12

d2�

�j�1 � a2ej�1�2��/ 2�1 � a � �ej1 � ej2� � ej�1�2�

�1 � a � �ej1 � ej2� � ej�1�2� 2 . (45i)

III. RESULTS AND DISCUSSION

To prove the validity and effectiveness of the pro-posed numerical TRL method, we now examine thecharacteristic impedance of a microstrip line. In ourcalculation example, the permittivity of substrate is9.9, and the substrate thickness and line width are all

equal to 0.635 mm. It is the same microstrip line as in[16]. An electromagnetic simulator is used (Agilent’sMomentum in our case). The mesh size is 20 cells perwavelength at the highest frequency.

In the scheme of simulation, local ports are usedinstead of other extension port modes, which is im-portant for the validity of the improved numerical



TRL method. The value of the resistor standard Zt isset to be equal to the reference impedance of 50�.The resistor is realized by thin film resistor which isprovided in the commercial tools. In the calibrationprocedure, an open standard is used to decide the signof b. The calculated characteristic impedance of themicrostrip line is shown in Figures 3(a) and 3(b). Thecalculated results from [16] are also shown in Figure3. We can see that the real part of the characteristicimpedance of the line standard is equal to that re-ported in [16]. The imaginary part of the characteristicimpedance is caused by the loss of the line standardand it is much smaller than the real part. We canobserve that without a TRL calibration, the calculatedcharacteristic impedance is greatly influenced by the

error terms and the results are completely out of order.It can be seen that the numerical calibration makes ahuge difference in this example.

To further our analysis, we use the proposed nu-merical TRL calibration to extract the parameters of astep discontinuity of microstrip line. The physicallayout of the step discontinuity simulated with theMomentum is shown in Figure 4(a). The networkmodel of this layout consists of three parts, namely,error box A, error box B, and the step discontinuity orcore circuit for which the characteristic parameters areto be extracted. The error boxes include the portdiscontinuity and the connect line from the port to thestep discontinuity. Using the proposed numerical TRLcalibration, we can effectively remove the error boxesand obtain the parameters for the step discontinuity.As the characteristic impedances of the lines at bothsides are different, we need to use two TRL proce-dures based on different reference impedances to cal-culate the two error boxes, respectively.

Figure 4. Geometry and equivalent circuit model of a stepdiscontinuity of microstrip line: (a) physical layout in Mo-mentum; (b) equivalent network; (c) circuit model of thestep discontinuity.

Figure 3. (a) Characteristic impedance of a microstrip lineobtained from the improved numeric TRL method togetherwith the data obtained in [16] (�r � 9.9, w � h � 0.635mm); (b) calculated data in [16] which shows the resultfrom our previous calibration method together with the dataobtained without calibration and the data obtained from the2D method.



The equivalent full-wave circuit model of the stepdiscontinuity can be expressed in terms of one capac-itive shunt admittance and two series inductive im-pedances [7], as shown in Figure 4(c). The extractedparameters of the circuit model are shown in Figures5(a) and 5(b). The results of the resistor and conduc-tance in the circuit model of Figure 4(c) are very smalland not shown here. We can see that the resultsobtained from the proposed TRL calibration schemeare comparable to those of the SOC calibration [7].Without the calibration, the port discontinuity willmake the accurate parameter extraction impossible.At 10 GHz, the S-parameters of the step discontinuityare a � 0.051, 1 � �105.5�, and 2 � �86.64°.From the equations (45a)–(45c), the correlations be-tween L, C, and a, are shown in Figure 6. We can

Figure 6. Typical numerical correlations between L, C,and a, : (a) dCg/da and dLp/da; (b) dCg/d1 and dLp/d1; (c) dCg/d2 and dLp/d2.

Figure 5. TRL-extracted parameters of a step discontinu-ity compared with those generated from other three meth-ods: static, SOC technique [7], and Z0 scheme (withoutcalibration) for �r � 10.2, h � 0.635 mm, W1 � 2.0 mm,W2 � 0.4 mm (Xp � Xp1 � Xp2): (a) Bg/ ; (b) Xp/ .



obtain dL/da � 8.2 � 103 (pH/unit), dC/da ��100 (pF/unit). From this value, we can see that asmall port discontinuity can bring up a large error inthe calculated inductance and capacitance as evi-denced in Figure 5. This is why the calibration pro-cedure is absolutely necessary for accurate circuit-model extraction.

IV. CONCLUSION

In this article, we have introduced an additional stan-dard called the “resistor standard” into the numericalTRL calibration on the basis of a deterministic full-wave MoM simulator in order to solve the ambiguityproblem of the characteristic impedance of the linestandard in such a calibration procedure. From a de-tailed comparative investigation, we have validatedthe proposed scheme in the determination of the char-acteristic impedance of the line standard. In addition,a new error analysis in connection with the parameter-extraction procedure that requires the conversion be-tween the S-parameters and the Y- or Z-parametershas been proposed and demonstrated. It has beenshown that the improved TRL calibration is com-pletely compatible with commercial full-wave simu-lator packages. The new additional resistor standardin the numerical TRL calibration can also be made forthe practical TRL measurements.

ACKNOWLEDGMENTS

The authors wish to express their gratitude to Natural Sci-ences and Engineering Research Council (NSERC) of Can-ada for its financial support of this project, and also theDeutsche Forschungsgemeinschaft for sponsoring the re-search stay of Prof. Ke Wu at the Munich University ofTechnology.

REFERENCES

1. J.C. Rautio and R.F. Harrington, An electromagnetictime-harmonic analysis of shielded microstrip circuits,IEEE Trans Microwave Theory Tech MTT-35 (1987),726–730.

2. A. Skrivervik and J.R. Mosig, Equivalent circuits ofmicrostrip discontinuities including radiation effects,IEEE MTT-S Int Microwave Symp Dig 3 (1989),1147–1150.

3. J.C. Rautio, A deembedding algorithm for electromag-netics, Int J Microwave Millimeter-Wave CAE 1(1991), 282–287.

4. A. Hill, J. Burke, and K. Kottapalli, Three-dimensionalelectromagnetic analysis of shielded microstrip circuits,Int J Microwave Millimeter-Wave CAE 2 (1992), 286–296.

5. B. Linot, M.F. Wong, V.F. Hanna, and O. Picon, Anumerical TRL deembedding technique for the extrac-tion of S-parameters in a 2 1/2D planar electromagneticsimulator, IEEE MTT-S Int Microwave Symp Dig 2(1995), 809–812.

6. L. Zhu and K. Wu, Short-open calibration technique forfield theory-based parameter extraction of lumped ele-ments of planar integrated circuits, IEEE Trans Micro-wave Theory Tech 50 (2002), 1861–1869.

7. L. Zhu and K. Wu, Unified equivalent-circuit model ofplanar discontinuities suitable for field theory-basedCAD and optimization of M(H)MICs, IEEE Trans Mi-crowave Theory Tech 47 (1999), 1589–1602.

8. L. Zhu and K. Wu, Field-extracted lumped-element mod-els of coplanar stripline circuits and discontinuities foraccurate radio-frequency design and optimization, IEEETrans Microwave Theory Tech 50 (2002), 1207–1215.

9. L. Zhu and K. Wu, Characterization of finite-groundCPW reactive series-connected elements for innovativedesign of uniplanar M(H)MICs, IEEE Trans Micro-wave Theory Tech 50 (2002), 549–557.

10. L. Zhu and K. Wu, Revisiting characteristic impedanceand its definition of microstrip line with a self-cali-brated 3D MoM scheme, IEEE Microwave GuidedWave Lett 8 (1998), 87–89.

11. V.I. Okhmatovski, J. Morsey, and A.C. Cangellaris, Ondeembedding of port discontinuities in full-wave CADmodels of multiport circuits, IEEE Trans MicrowaveTheory Tech 51 (2003), 2355–2365.

12. J.C. Rautio, Comments on ‘On deembedding of portdiscontinuities in full-wave CAD models of multiportcircuits’, IEEE Trans Microwave Theory Tech 52(2004), 2448–2449.

13. Commercial MoM simulators: Agilent Momentum,Agilent Technologies, Santa Clara, CA; Sonnet LiteEM simulators, Sonnet Software, Liverpool, NY; Ze-land IE3D, Zeland Software, Inc, Fremont, CA.

14. L. Li, K. Wu, and L. Zhu, Numerical TRL calibrationtechnique for parameter extraction of planar inte-grated discontinuities in a deterministic MoM algo-rithm, IEEE Microwave Guided Wave Lett 12(2002), 485– 487.

15. L. Li and K. Wu, Numerical through-resistor (TR)calibration technique for modeling of microwave inte-grated circuits, IEEE Microwave Guided Wave Lett 14(2004), 139–141.

16. L. Li, K. Wu, and P. Russer, Improved numerical TRLcalibration technique in a deterministic MOM algo-rithm, 34th Euro Microwave Conf, Amsterdam, 2004,pp. 447–450.

17. J.C. Rautio, A new definition of characteristic imped-ance, IEEE MTT-S Int Microwave Symp Dig 2 (1991),761–764.

18. L. Zhu and K. Wu, Network equivalence of port dis-



continuity related to source plane in a deterministic 3Dmethod of moments, IEEE Microwave Guided WaveLett 8 (1998), 130–132.

19. D.M. Pozar, Microwave engineering 2nd ed., Wiley,New York, 1998, pp. 217–222.

20. H.-J. Eul and B. Schiek, A generalized theory and newcalibration procedures for network analyzer self-cali-

bration, IEEE Trans Microwave Theory Tech 39(1991), 724–731.

21. C. Seguinot, P. Kennis, J.-F. Legier, F. Huret, E. Pa-leczny, and L. Hayden, Multimode TRL. A new con-cept in microwave measurements: Theory and experi-mental verification, Trans Microwave Theory Tech 46(1998), 536–542.

BIOGRAPHIES

Lin Li received a B.S. degree in electricalengineering and an M.S. degree in micro-wave engineering from Nanjing Universityof Science & Technology, China, in 1994and 1997, respectively, and is currentlyworking toward a Ph.D. degree at the EcolePolytechnique de Montreal.

Ke Wu is Professor of Electrical Engineer-ing and Tier-I Canada Research Chair inRadio-Frequency and Millimeter-Wave En-gineering at the Ecole Polytechnique (Uni-versity of Montreal). He has been a Visitingor Guest Professor with many universitiesaround the world. He also holds an honoraryvisiting professorship and a Cheung Kongendowed chair professorship at the Southeast

University, and an honorary professorship at the Nanjing Universityof Science and Technology, China. He has been the Director of thePoly-Grames Research Center, as well as the Founding Director ofthe Canadian Facility for Advanced Millimeter-wave Engineering(FAME). He has authored or co-authored more than 460 referredpapers, and also several books/book chapters. His current researchinterests involve substrate integrated circuits, antenna arrays, ad-vanced CAD and modeling techniques, and development of low-cost RF and millimeter-wave transceivers. He is also interested inthe modeling and design of microwave photonic circuits and sys-tems. He serves on the Editorial Board of Microwave Journal,Microwave and Optical Technology Letters, and Wiley’s Encyclo-pedia of RF and Microwave Engineering. He is an Associate Editorof International Journal of RF and Microwave Computer-AidedEngineering (RFMiCAE). He is a member of ElectromagneticsAcademy, the Sigma Xi Honorary Society, and URSI. He has heldmany positions in and has served on various international commit-tees, including the vice-chairperson of the Technical Program Com-mittee (TPC) for the 1997 Asia-Pacific Microwave Conference, theGeneral Co-Chair of the 1999 and 2000 SPIE’s International Sym-posium on Terahertz and Gigahertz Electronics and Photonics, theGeneral Chair of 8th International Microwave and Optical Technol-ogy (ISMOT’2001), the TPC Chair of the 2003 IEEE Radio andWireless Conference (RAWCON’2003), the General Co-Chair ofthe RAWCON’2004. He will be the General Chair of the 2012IEEE MTT-S International Microwave Symposium to be held inMontreal. He has served on the Editorial or Review Boards ofvarious technical journals, including the IEEE Transactions onMicrowave Theory and Techniques, the IEEE Transactions onAntennas and Propagation, and the IEEE Microwave and WirelessComponents Letters. He served on the 1996 IEEE Admission and

Advancement Committee, the Steering Committee for the 1997joint IEEE Antennas and Propagation Society (AP-S)/URSI Inter-national Symposium. He has also served as a TPC member for theIEEE Microwave Theory and Techniques Society (IEEE MTT-S)International Microwave Symposium. He was elected into theBoard of Directors of Canadian Institute for TelecommunicationResearch (CITR). He is currently the chair of the joint IEEEchapters of MTTS/APS/LEOS in Montreal, Canada, the chaptercoordinator for MTT-S Region 7. He is an elected IEEE MTT-SAdministrative Committee (AdCom) member for 2006–2009 andserves as the chair of the IEEE MTT-S Transnational Committee.He was the recipient of a URSI Young Scientist Award, the OliverLodge Premium Award of the Institute of Electrical Engineers(IEE) of the U.K., the Asia-Pacific Microwave Prize, the 2006CCECE Best Paper Award, the University Research Award “PrixPoly 1873 pour l’Excellence en Recherche” presented by the EcolePolytechnique on the occasion of its 125th anniversary, the Urgel-Archambault Prize (the highest honor) in the field of physicalsciences, mathematics, and engineering from the French-CanadianAssociation for the Advancement of Science (ACFAS), and the2004 Fessenden Medal of the IEEE Canada. In 2002, he was thefirst recipient of the IEEE MTT-S Outstanding Young EngineerAward. He is a Fellow of the IEEE, a Fellow of the CanadianAcademy of Engineering (CAE) and a Fellow of the Royal Societyof Canada (The Canadian Academy of the Sciences and Humani-ties).

Peter Russer received his Dipl.-Ing. degreein 1967 and his Dr. techn. degree in 1971,both in electrical engineering, from the Vi-enna University of Technology, Austria,where he was Assistant Professor from 1968to 1971. In 1971 he joined the ResearchInstitute of AEG-Telefunken in Ulm, Ger-many, where he worked on fiber-optic com-munication, broadband solid-state electronic

circuits, statistical noise analysis of microwave circuits, laser mod-ulation, and fiber optic gyroscopes. In 1979 he was co-recipient ofthe NTG Award for the publication “Electronic Circuits for High-Bit-Rate Digital Fiber Optic Communication Systems.” Since 1981he has been professor and head of the Institute of High FrequencyEngineering at the Technische Universitat Munchen, Germany. In1990 he was Visiting Professor at the University of Ottawa, and in1993 he was Visiting Professor at the University of Victoria. FromOctober 1992 through March 1995 he was Director of the Ferdi-nand-Braun-Institut fur Hochstfrequenztechnik, Berlin, Germany.In 1994 he was elected to the grade of Fellow of the IEEE forfundamental contributions to noise analysis and low-noise optimi-zation of linear electronic circuits with general topology. From1997 to 1999 he was Dean of the Department of Electrical Engi-



neering and Information Technology of the Technische UniversitatMunchen. From 1997 to 2004 he was been member of the Board ofDirectors of the European Microwave Association and in 1999 hewas the General Chair of the European Microwave Week held inMunich. He has published more than 500 scientific papers inrefereed journals and conference proceedings. He has developed avariety of courses in RF techniques, microwaves, quantum elec-tronics, and optical communications. He is the program director ofthe international graduate program “Master of Science in Micro-wave Engineering” at the TUM. Over the years he has graduatedmore than 400 students of which more than 50 received theirPh.D. degree. He has served as a member of the technicalprogramme committees and steering committees of various in-

ternational conferences (IEEE MTT-S, European MicrowaveConference) and as the member of the editorial board of severalinternational journals (Electromagnetics, International Journalof Numerical Modeling). From 1999 to 2002 he was Co-Chair ofURSI Commission D and he was elected URSI Commission Dfor the period from 2002 to 2005. His current research interestsare electromagnetic fields, integrated microwave and millimeter-wave circuits, statistical noise analysis of microwave circuits,and methods for computer-aided design of microwave circuits.He is the author of more than 300 scientific papers in these areas.He is Fellow of the IEEE, and a member of the German Infor-mationstechnische Gesellschaft (ITG) and the German as well asthe Austrian Physical Societies.



on the thru-reflect-line (trl) numerical calibration and error analysis for parameter extraction of...

Documents