Application of EM- Simulators for Extraction of Line Parameters

Transcription

1 Chapter - 2 Application of EM- Simulators for Extraction of Line Parameters 2. 1 Introduction The EM-simulators-2D, 2.5D and 3D, are powerful tools for the analysis of the planar transmission lines structure. Fig.(2.1) shows structures to which these simulators are normally used. The EM-simulators are used to obtain equivalent circuits and equivalent transmission line parameters. Thus, this chapter presents the process of extraction of line parameters of different planar transmission lines using the EM-simulators. The results obtained using three EM-simulators are compared among themselves. The method of excitations and de-embedding techniques, in each one of the EM-simulators, are explained. (a) (b) (c) Fig.(2.1): EM-simulators classified by geometrical dimensionality:(a) 2D cross-section, (b) 2.5D mostly planar, and (c) 3D fully arbitrary. The main objective in this chapter is to realize different planar transmission lines in EMsimulators and compare their line parameters (de-embedded and non de-embedded) against results from experiments and analytical models. The study has been done in this 13

2 thesis for microstrip, CPW, CPS and slotline on six different substrates with permittivity ranges from in the frequency range 0.1 GHz 200 GHz and conductor thickness ranges from 0.25µm 9µm. This chapter also computes the RLCG parameters of planar lines using the EM-simulators. 2.2 EM Simulators for Extracting Data In this thesis, mainly three EM-simulators are used 1) Ansoft HFSS [15], 2) CST Microwave Studio [28] and 3) Sonnet Professional Suite [115]. In this section, we will briefly review issues that are common with these simulators. These special issues include method of excitation and de-embedding. Both aspects are essential for detailed study of the planar transmission lines. Table - 2.1: Characteristics of EM-simulators Ansoft HFSS [15] CST Microwave Studio [28] Sonnet Pro Suite [116] 3D Arbitrary geometry solver 3D Arbitrary geometry solver 2.5D Planar solver Numerical method: FEM Numerical method: FIT (FDTD) Numerical method: MoM Arbitrary geometry and resolution Conformal approx. in 3D Closed box formulation Tetrahedral edge elements; closed Transient solver, eigenmode - Suitable basis functions box formulation Eigenmode-solver solver, modal analysis simulator 2D eigenmode - solver for port modes Diagonal elements, dielectric bricks and calibrated internal ports Fast sweep Optimization capability Built in parametric sweep and optimization True surface object modeling Code is multithreaded Support modules for viewing currents Perfectly matched layers (PMLs) Eigenmode - solver supports two 3 rd party module for viewing processors on PC geometry in 3D Second order absorbing boundary Well integrated ACIS-based Optimization and parametric conditions (ABCs) interface analysis Dual processor support Bulk conductivity for semiconductor substrates 14

3 The main characteristics of each EM-simulator are summarized in Table-2.1. All of these simulators approximate the true fields or currents in the problem space by subdividing the problem into basic cells or elements that are roughly 1/10 to 1/20 of a guide wavelength in size Method of Excitation The boundary conditions are important to understand and are fundamental to solution of Maxwell s equations. These conditions enable to control the characteristics of planes, faces, or interfaces between objects. An excitation port is a type of boundary condition that permits energy to flow into and out of a structure. A port is a 2D surface on which the fields will be solved according to Maxwell s equations to determine appropriate RF modal excitations into the 3D model volume. It can be assigned to any 2D object or 3D object face. Before the full three-dimensional electromagnetic field inside a structure can be calculated, it is necessary to determine the excitation field pattern at each port. The method of excitations in each one of the EM-simulators, are discussed in this section. Ansoft HFSS There are two types of method of excitation available in Ansoft HFSS, as shown in Fig.(2.2): (i) Wave Ports are applied to the structure to indicate the area where the energy enters and exits the conductive shield. The HFSS port solver assumes that the wave port defined is connected to a semi-infinitely long waveguide that has the same crosssection and material properties as the port, then solves for the Eigen modes. This mode solution is a key point. Each wave port is excited individually and each mode incident on a port contains 1 Watt of time-averaged power. The wave ports calculate characteristic impedance, complex propagation constant and generalized S- parameters. 15

4 It also solves for the port impedance based on one of the three definitions using combinations of power, voltage or current, i.e. Z pi, Z pv and Z vi. Ansoft HFSS always calculates Z pi. The impedance calculation using power and current is well defined for a wave port. The wave ports permit de-embedding to remove excess uniform input transmission lengths. (ii) Lumped Ports are applied to model internal ports within a structure. They excite simplified field distributions to permit S-parameter outputs, where wave ports are not feasible and support only uniform field distributions. They must contact two surfaces that constitute a transmission line and definition includes a line which is parallel to the direction of current. They are normally defined on a 2D object (a rectangle). The proportions of this rectangle should not have a huge aspect ratio. These ports do not calculate characteristic impedance and propagation constants. The impedance and calibration line assignments are required for lumped port assignments. A terminal line may still be defined, but only one per port. As impedance is supplied by the user and not computed, no alternate definitions (Z pi, Z pv, Z vi ) are supplied. Since propagation constants are not computed, lumped port S-parameters may not be deembedded to remove or add uniform input transmission lengths. (a) (b) Fig.(2.2): Types of Method of Excitation in Ansoft HFSS:(a) Wave Port, and (b) Lumped Port. 16

5 CST Microwave Studio In this simulator, generally two different types of ports exist for the transient and frequency domain analyses, as shown in Fig.(2.3): (i) Waveguide Ports simulate an infinitely long wave guide connected to the structure and are definitely the most accurate way to terminate a waveguide. The waveguide modes travel out of the structure towards the boundary planes and thus leave the computation domain with very low levels of reflections. The transient analysis only uses so-called waveguide ports. (ii) Discrete Ports are sometimes more convenient to use than waveguide ports. They consist of a current source with an internal resistor and have two pins with which they can be connected to the structure. This kind of port is often used as feeding point source for antennas or as the termination of transmission lines at very low frequencies. At higher frequencies (e.g. the length of the discrete port is longer than a tenth of a wavelength) the S-parameters may differ from those when using waveguide ports because of the improper match between the port and the structure. For transient or frequency domain analyses, discrete ports can be used in the same way as waveguide ports. (a) Fig.(2.3): Types of Method of Excitation in CST Microwave Studio:(a) Waveguide Port and (b) Discrete Port. (b) 17

6 Sonnet Pro Suite - There are five types of ports used in Sonnet and all ports are twoterminal devices. The default normalizing impedance for a port is 50 Ohms. There is no limit on the number of ports and the number of ports has a minimal impact on the analysis time needed for de-embedding. By default, the ports are numbered by the order in which they are created. With this default method, all ports are positive and unique. However, there are some applications that require the ports to have duplicate or even negative, numbers. Each type of port is described below and is shown in Fig.(2.4). (i) Standard Box wall port is a grounded port, with the positive terminal attached to a polygon edge coincident with a box wall and the negative terminal attached to ground. It can be de-embedded and can also have reference planes. This type of port is the most commonly used. (ii) Co-calibrated ports are used in the interior of a circuit. Co-calibrated internal ports are identified as part of a calibration group with a common ground node connection. The ground node connection can be defined as floating or the Sonnet box. When the simulator performs the electromagnetic analysis, the co-calibrated ports within a group are simultaneously de-embedded using a high accuracy de-embedding technique; thus, coupling between all the ports within a calibration group is removed during de-embedding. This type of port is the most commonly used internal port. The reference planes may be used as well. (iii) Via port has the negative terminal connected to a polygon on a given circuit level and the positive terminal connected to a second polygon on another circuit level. This port can also have the negative terminal connected to the top or bottom of the box. Unlike co-calibrated ports, EM simulation cannot de-embed via ports. The most common case where a via port would be used is to attach a port between two adjacent levels in the circuit and to connect a port to the interior of a polygon, which 18

7 is not allowed for co-calibrated ports. The reference planes cannot be applied to via ports, since it is not possible for EM simulation to de-embed them. (a) (b) (c) (d) (e) Fig.(2.4): Types of Method of Excitation in Sonnet Pro Suite :(a) Standard Box-Wall Port, (b) Co- Calibrated Internal Port, (c) Via Port, (d) Auto-Grounded Port and (e) Ungrounded Internal Port. 19

8 (iv) Automatic-grounded port is a special type of port used in the interior of a circuit similar to a co-calibrated internal port. This port type has the positive terminal attached to the edge of a metal polygon located inside the box and the negative terminal attached to the ground plane through all intervening dielectric layers. Autogrounded ports can be de-embedded by the analysis engine. It is used in place of a co-calibrated port to reduce the de-embedding processing time at the cost of less accuracy. But the de-embedding for auto-grounded ports does not take into account the coupling between the auto-grounded ports. The reference planes may be used here as well. (v) Ungrounded internal port is located in the interior of a circuit and has its two terminals connected between abutted metal polygons. These ports can be deembedded by EM; however a reference plane or calibration length may not be set. These ports are not as accurately de-embedded as co-calibrated internal ports, but they do not require any space between the two polygons as is required for a cocalibrated port. Ungrounded internal ports are not allowed on the edge of a single polygon because this would leave one terminal of the port unattached. Also, care should be taken in interpreting the results for circuits which use these ports since all the ports do not access a common ground De-embedding At RF and microwave frequencies, it is often impractical to measure the impedance, admittance, or S-parameters of an active device directly at the device terminals. Instead the device is typically embedded in some form of test fixture and measurements are made at a reference plane some distance away from the actual device as shown in Fig.(2.5) [29]. Depending upon the nature of the analysis, this may or may not be desirable. De-embedding is the mathematical process of removing the embedding networks and determining the true parameters of the device under test (DUT) [68, 69]. 20

9 Fig.(2.5): Typical measurement situation. It is a useful laboratory technique and an equally powerful tool when applied to EMsimulators. Each port in a circuit analyzed by EM simulator, introduces a discontinuity into the analysis results. In addition, any transmission lines that might be present introduce phase shift, and possibly, impedance mismatch and loss. Thus, it is required to remove the discontinuities at numerical ports and to remove lengths of line from numerical fixture. The de-embedding in active device and EM-simulators is shown in Fig. (2.6). (a) (b) Fig.(2.6): Typical de-embedding problems:(a) Measurement-based Active Device De-embedding, and (b) EM- Simulator-based De-embedding; to remove port discontinuity. The de-embedding is achieved through a calibration process in which the S-parameters of two boxes, shown in Fig.(2.7), are quantified. The box represents s in the S-parameters due to cables and connectors connecting the device to the external circuit ports [76]. The S or [ABCD] parameters representation of the device at device ports (1-21

10 2 ) along with the boxes is shown in Fig. (2.7). The circuit network is the device under test (DUT). Fig.(2.7): Calibration process in measurement of S-parameters of a device. The measured A m B m C m D m parameters are related to the device A B C D parameters as follows, ' A ' C ' B A = ' D C B D 1 A C m m B D m m A C B D (2.1) The de-embedded device A B C D parameters are converted to the de-embedded S- parameters of the device. The de-embedded S-parameters are further converted to Z and Y- parameters. Thus any two port device can be characterized though measurements using the suitable parameters - S, Z, or Y. The above mentioned concept of de-embedding of the device S- parameters at the devices port is applicable to the EM simulators - both 2.5D and 3D simulators [29-30, 76]. Every simulator vendors have developed their own de-embedding algorithms. These are usually not known. In next section, we examine deembedding of S-parameters of several planar transmission lines using EM-simulators. 22

11 2.3 Comparison of De-embedded and Non De-embedded Results The investigator should examine the EM-simulator satisfactorily before adopting it for the data generation of the structure. The de-embedding process of an EM-simulator determines the accuracy of results on the S-parameter and the extracted line parameters. Thus, comparison of the de-embedded and non de-embedded results obtained from EMsimulators - HFSS, Sonnet and CST are to be examined in this section. Let us first examine, using all the three EM-simulators, the S-parameters of the conventional planar transmission lines without and with de-embedding i.e. the S- parameters at the external circuit ports and at the internal device ports. The S 11 and S 22 are identical and also S 21 and S12 are identical. Fig. (2.8a) Fig. (2.8d) show S11 and S21 parameters for the microstrip line ( ε r =12.9, w/h=5), CPW ( ε r =9.8, w=20 µm, s=15 µm), CPS ( ε r =3.78, w=4.5 µm, s=7.5 µm) and slotline ( ε r =2.5, w/h=0.5) in frequency range 1 GHz 100 GHz for substrate thickness h = 635 µm, conductor thickness t = 6 µm and line length of 500 µm is carried out at the de-embedding distance, L=15 µm. As frequency increases from 2 GHz, the de-embedded and non de-embedded curves experience a gradually depart from each other for both S11 and S 21 in all the four cases. This is due to the increase in discontinuity at the ports with frequency, resulting in higher insertion loss. The de-embedded S 21 results show less frequency dependence as compared to original S21 -parameters. The HFSS shows some in results for CPW and slotline above 60 GHz. The slotline is treated as the limiting case of the CPW with its central conductor width 1µm. Some in results are seen for the CPS around 10 GHz. The CST de-embedded results are much smoother. Overall, the de-embedded S21 parameters obtained from HFSS show more frequency dependency w.r.t. Sonnet and CST. There is small increase in the de-embedded S 11 with frequency for all the four lines in all the three EM-simulators. 23

12 (a) (b) (c) (d) Fig.(2.8): S 11 and S 21 from EM-simulators of conventional transmission lines: (a) microstrip line, (b) CPW, c) CPS and d) slotline before and after de-embedding. The de-embedded results should not be dependent on the de-embedding line length. It is examined in Fig.(2.9). The figure shows the comparison of effect of de-embedding length on effective relative permittivity and total loss of microstrip line, on GaAs substrate, using EM-simulators. A both negative and positive value of de-embedded distance has been considered. Unlike HFSS and CST, Sonnet doesn t have provision to incorporate negative de-embedding distance. In Fig.(2.9a), at f = 10 GHz and 20 GHz, εeff of the line 24

13 obtained from EM-simulators is compared against Finlay et. al. [53]. It has been observed that Sonnet provides nearly length independent values with 0.8 deviation, whereas deembedding does improve the results obtained from HFSS and CST with less than 0.1 deviation. However, improvement is distance dependent in both HFSS and CST. (a) Fig.(2.9): Effect of de-embedding length on: (a) Effective relative permittivity and (b) Loss of microstrip line. (b) Similarly, in Fig.(2.9b), at f = 15 GHz and 40 GHz, loss α T obtained from EMsimulators is compared against Goldfarb et. al. [86]. The de-embedded result obtained from HFSS, Sonnet and CST is within 0.9 average deviation against experimental results. In this case also, Sonnet de-embedded αt is the most independent of deembedding distance. However, it is not constant at 40 GHz. The results of HFSS are fluctuating with distance. Its de-embedded results are not constant. Thus it is difficult to fix a proper de-embedding distance to remove effect of port discontinuity. It appears 2 mm to 3 mm distance is appropriate. Next we compare results of three simulators; CST shows variation of 0.3 in ε eff at 20 GHz and 0.2 variation at 10 GHz. Whereas HFSS shows 0.2 and 0.6 variation at 10 GHz and 20 GHz respectively due to deembedding distance. 25

14 2.4 Comparison of Different EM Simulators In this section, we compare accuracy of computation of ε eff and αt using HFSS, Sonnet and CST. In case of microstrip line, CPW and CPS results are compared against the experimental results. In case of the slotline results are compared against the results of mode matching method (MMM).amongst each other. Table- 2.2a 2.2d presents deviation in line parameters obtained from EM-simulators for the microstrip line, CPW, CPS and slotline respectively on GaAs dielectric substrate. Table-2.2a shows comparison of dispersive ε eff of microstrip line against experimental results of Finlay et. al. [53] over frequency range 2 GHz - 22 GHz. The loss α T of the microstrip line is also compared in Table-2.2a against experimental results of Goldfarb et. al.[86] over frequency range 1 GHz 40 GHz. The average deviation in results obtained for ε eff and α T from HFSS, Sonnet and CST are (2.3, 4.3), (2.2, 4.2) and (2.6, 5.1) respectively. Table-2.2b shows comparison of ε eff and α T of CPW against experimental results of Papapolymerou et. al. [64] over frequency range 2 GHz GHz. The average deviation in results obtained for ε eff and α T from HFSS, Sonnet and CST are (4.9, 8.8), (8.5, 6.2) and (8.2, 11) respectively. Table-2.2c shows comparison of ε eff and α T of CPS against experimental results of Kiziloglu et. al. [74] over frequency range 2 GHz - 18 GHz. The average deviation in results obtained for ε eff and α T from HFSS, Sonnet and CST are (2.3, 3.7), (2.2, 2) and (2.6, 1.2) respectively. 26

15 Table-2.2a: Comparison of ε eff and α T of microstrip line from EM-simulators against experimental results [53] and [86] on GaAs substrate with σ=3.33x10 7 S/m respectively: [53] w= 30 µm, h=200µm, t= 6 µm & [86] w=20 µm, h=100µm, t=3µm. Freq. f (GHz) Exp. [53] ε eff (f) HFSS Sonnet CST Freq. f (GHz) Exp. [86] α T (db/cm) HFSS Sonnet CST av. deviation av. deviation max. deviation max. deviation Table-2.2b: Comparison of ε eff and α T of CPW from EM-simulators against experimental results [64] on GaAs substrate: w= 45 µm, s=50µm, h=525µm, t =1µm, σ=3.33x10 7 S/m. Freq. f (GHz) Exp. [64] ε eff (f) HFSS Sonnet CST Freq. f (GHz) Exp. [64] α T (db/cm) HFSS Sonnet CST av. deviation av. deviation max. deviation max. deviation

16 Table-2.2c: Comparison of ε eff and α T of CPS from EM-simulators against experimental results [74] on GaAs substrate: w= 4 µm, s= 8µm, h=670µm, t=0.5µm, σ=4.1x10 7 S/m. Freq. f (GHz) Exp. [74] ε eff (f) HFSS Sonnet CST Freq. f (GHz) Exp. [74] α T (Np/cm) HFSS Sonnet CST av. deviation av. deviation max. deviation max. deviation Table-2.2d: Comparison of ε eff and α T of slotline from EM-simulators against MMM results [132] on GaAs substrate: s= 40µm, h=600µm, t=6µm, σ=3x10 7 S/m. Freq. f (GHz) MMM [132] ε eff (f) HFSS Sonnet CST Freq. f (GHz) MMM [132] α T (db/mm) HFSS Sonnet CST av. deviation av. deviation max. deviation max. deviation

17 Table-2.2d shows comparison of ε eff and α T of slotline against MMM based results of Heinrich [132] over frequency range 10 GHz - 50 GHz. The average deviation in results obtained for ε eff and α T from HFSS, Sonnet and CST are (3.4, 3.5), (5.4, 4.9) and (4.2, 5.4) respectively. Next we compare the performance of microstrip, CPW, CPS and slotline over frequency 1 GHz - 60 GHz using three simulators. For this purpose, the characteristic impedance Z 0 is kept constant at 50 Ω,75 Ω and 90 Ω for ε r = 9.8, t = 6µm, tan δ = , σ = 4.1 x 10 7 S/m. The results on εeff and α T are shown in Fig.(2.10a)-(2.10f). It has been observed that the microstrip is the most dispersive line, followed by slotline, CPS and CPW. The dispersion is technically present at all frequencies. However, it becomes significant above 4 GHz. The total loss is observed to be more in CPW, followed by CPS, slotline and microstrip line. The losses are frequency dependent in the same order. In all the four lines, with increasing Z 0 from 50 Ω to 90 Ω, there is subsequent increase in ε eff and decrease inα T. Finally, Table-2.3 compares average and maximum deviation in computation for all the three cases of Z 0, in Sonnet and CST w.r.t. HFSS for microstrip line, CPW, CPS and slotline. The results of CST and HFSS are in closer agreement for computation of ε eff. However results of Sonnet and HFSS show better agreement for computation of α T. In general, we can say results of simulations for εeff can deviate between 1.1 to 4.3 i.e. about 2.5 on average. The results for computation of α T can deviate between i.e. about 4.5 on average. 29

18 (a) (b) (c) (d) (e) (f) Fig.(2.10): Comparison of characteristics of different planar transmission lines keeping Z 0 constant at 50 Ω, 75 Ω and 90 Ω. 30

19 Table-2.3: Percentage deviations of EM-simulators w.r.t. HFSS for all the four lines ε r = 9.8, t = 6µm, tan δ = , σ = 4.1 x 10 7 S/m. 2.5 Comparison of Characteristics of Transmission Lines In this section, line parameters of microstrip line obtained from EM-simulators, are compared against results from experiments and analytical models available in open literature. The same methodology for study of characteristics of CPW, CPS and slotline has been carried out in Chapter-5, 6 and 7 respectively. The line parameters considered for study are effective relative permittivity ( ε eff ), characteristic impedance (Z 0 ), total loss ( α T ) and quality factor (Q) factor. The study has been done for different line geometries of all the four lines on six different substrates PTFE/glass ( ε r = 2.5); Quartz ( ε r = 3.78); Alumina ( ε r = 9.8); Gallium Arsenide, GaAs ( ε r = 12.9); Zirkonate ( ε r = 20), and Barium Tetratitanate, BaTi 4 O 9 ( ε r = 37) in the frequency range 0.1 GHz 200 GHz and conductor thickness (t) of gold (σ = 4.1 x 10 7 S/m) ranges from 0.25µm 9 µm. The gold is not a practical conductor for all cases; however for comparison we have used it in all cases of simulation. 31

20 Microstrip line is the most commonly used transmission medium in RF and microwave circuits, due to its quasi-tem nature and excellent layout flexibility. A cross-sectional view of a microstrip line is shown in Fig. (1.1b). The important parameters for designing these transmission lines are the characteristic impedance, effective relative permittivity, attenuation constant, discontinuity reactances, frequency dispersion, surface-wave excitation and radiation. Several methods to determine these parameters are summarized in the open literature [3, 36,51, 56-58, 73,87]. Our study of effective relative permittivity, characteristic impedance and losses of microstrip line w.r.t. frequency, conductor thickness and w/h ratio are based on the following models: Effective relative permittivity (LDM) [9] ε r ε eff ( f,t ) = E ε eff ( f K( f / f ) 1 + Me 1 ) (a) Characteristic impedance [103] R R Z0 ( f,t) = Z0 ( 0) R (b) 14 Wheeler s incremental inductance rule [6] Z( ε r = 1,w,h, f,t, δ s ) α c = ε eff ( ε r,w,h, f,t ) db/m (c) λ0 Z 0( ε r = 1,w,h, f,t ) Holloway and Kuester model [24,157] Strip conductor R ln w t t 1 sm : α = Np/m (d) sc Z ( f,t) 2 0 2π w 32

21 Ground plane : 2 2 R 1 w 2x w + 2x sm 1 1 α gc = tan tan dx Z ( f,t) + Np/m (e) wπ 2h h α = α + α Np/m (f) c sc gc Dielectric loss [73] α d ε ε ( f,t ) 1 tanδ r eff = db/m (g) (2.2) ε eff ( f,t ) ε r 1 λ0 The detailed definitions of the above mentioned closed-form models can be found in the open literature [6,9,24,73,103,157]. Fig. (2.11), Fig.(2.12) and Fig.(2.13) present variation in ε eff, Z 0, α T and Q-factor of microstrip line with respect to the frequency range 0.1 GHz 200 GHz, strip conductor thickness 0.25 µm 9 µm and w/h ratio respectively. We have considered substrate relative permittivity in the range The closed-form model with LDM [9] for dispersion has average and maximum deviation of (1.5, 4.8), (2.2, 9.2) and (3.4, 7.9) against HFSS, Sonnet and CST respectively, as shown in Fig.(2.11). With increasing permittivity, the deviation in closed-form model is evidently increasing. The deviation is also more at higher frequencies. Fig.(2.12) shows the effect of dispersion in Z 0 of a microstrip line. The closed-form model for the frequency-dependent Z 0 a microstrip line [103] is based on power-current definition. For ε r < 20 and 0 h. f 30 GHz-mm, the model is within 6.2 average deviation against the results obtained from EM-simulators. The closed-form model includes the correct asymptotic dependences on the various parameters, however outside the specified ranges, the increases slowly but significantly as shown in Fig.(2.12a). When compared against each EM simulator for 2. 5 ε r 20, 0.25 µm t 9 µm and 0. 1 w / h 10, the closed-form model has average and maximum deviation of 33

22 (6.9, 13.8), (8.2, 15.2) and (7.6, 12.9) against HFSS, Sonnet and CST respectively, as shown in Fig.(2.12b) and Fig.(2.12c). (a) (b) (c) Fig.(2.11): Comparisons of ε eff ( f, t) computed by the closed-form model against the EM-simulators as a function of: (a) Frequency, (b) Conductor thickness, and (c) w/h ratio for microstrip lines on various substrates. The losses depend on the nature of conducting material used in the strip conductor and ground plane of a microstrip, material of the substrate and physical structure of a microstrip. The total loss in a microstrip structure can be obtained by adding mainly three loss components: radiation loss ( α ), conductor loss ( α ) and dielectric loss ( α ).The r c d 34

23 α d and αc are dissipative effects whereas αr is an essentially parasitic phenomenon [116]. Moreover, the attenuation is usually dominated by αr for frequencies above 200 GHz. As THz frequencies have not been accounted in this study, so radiation losses have been neglected in this thesis. The conductor loss is caused by the finite conductivity of strip conductor and ground plane conductor, whereas the dielectric loss is caused by an imperfect lossy substrate material. (a) (b) Fig.(2.12): Comparisons of Z 0 ( f, t) computed by the closed-form model against the EM-simulators as a function of: (a) Frequency, (b) Conductor thickness, and (c) w/h ratio for microstrip lines on various substrates. (c) 35

24 The Q or Quality factor of a microstrip [57] can be related to α T in the line by: β π ε eff ( f ) Q u = = (2.3) 2α T αt λ 0 where Q u is the total or unloaded Q of the line, β is the phase coefficient and λ0 is the free-space wavelength. Unloaded Q factors are of great importance in resonant circuit applications (e.g. matching networks and filters) which indicate the selectivity, the performance of the resonator and the losses of the resonator. It is expressed as 1 Qu Qr Qc Qd = (2.4) where Q r is the radiation Q due to α r, Q c is the conductor Q due to α c and Q d is the dielectric Q due to α d. In this study, Q u is computed from EM- simulators and closedform model (both ε eff and α T ), using equation-(2.3). Fig.(2.13) shows the characteristics of the total loss α T and unloaded Q factors in a microstrip line on different substrates with ε r = 3.78, 9.8 and 12.9; w/h =1, h = 635 µm and t = 3 µm as a function of frequency. Two closed-form models are used for the conductor loss computation of the lines in our study: Wheeler s incremental inductance rule [13] and perturbation method with the concept of the stopping distance by Holloway [24]. Holloway s model along with the closed-form model for dielectric loss [73] are used for the computation of α T and compared against the EM-simulators in Fig.(2.13a) with 6.7 average deviation till 0 h. f 30 GHz-mm condition is getting satisfied and beyond that, increases steadily for high permittivity substrate at higher frequencies with average deviation reaching upto The deviation at frequencies above 50 GHz and for high permittivity substrate is due to the surface-wave generation that is not accounted for in the closed-form models. The EM-simulator accounts surface-wave loss. 36

25 (a) (b) (c) (d) (e) (f) Fig.(2.13): Comparisons of the closed-form models against the EM-simulators for computation of: α T ( f,t ) as a function of (a) Frequency, (b) w/h ratio, (c) Conductor thickness; Q as a function of (e) Frequency, and (f) w/h ratio for microstrip lines on various substrates. u 37

26 Fig.(2.13b) shows the comparison of the total loss, computed using Wheeler s model and Holloway s model for conductor loss, against EM-simulators as a function of w/h ratio with average and maximum deviation of (7.2,16.6) and (3.9,11.4) respectively. Both the closed-form models have 7 average deviation between each other and deviation seems to be increasing with increase in w/h ratio. In Fig.(2.13c), both the closed-form models for conductor loss and all the three EM-simulators are compared against the MMM based results of Heinrich [133] for 0.25 µm t 9 µm. The average and maximum deviation of Holloway s model, Wheeler s model, HFSS, Sonnet and CST w.r.t. Heinrich s results are (3.4,5.2), (5.4,28), (1.9,2.7), (4.4,7.6) and (4.6,7.9) respectively. It has been observed that, for a wide range of parameters, t both the closed-form models work well. However, if the ratio 1. 1 (where δ s δ s is the skin depth and t is the thickness of the conductor), then Wheeler rule breaks down and gives poor results. Fig.(2.13d) compares the closed-form model for dielectric loss against the EM-simulators w.r.t. conductor thickness and the model is within 1.54 average deviation. Fig.(2.13e) shows the variation of Q u in a microstrip line on different substrates with ε r = 3.78, 9.8 and 12.9; w/h =1, h = 635 µm and t = 3 µm in the frequency range 0.1 GHz 200 GHz. Fig.(2.13f) compares computation of Q u for ε r = 2.5 and 12.9; f = 60 GHz as a function of w/h ratio. Overall, average and maximum deviation in the closed-form model against EM-simulators are 8.9 and 19.5 respectively. The in closed-form model is observed to be increasing with increase in frequency and w/h ratio. 2.6 Comparison of R, L, C and G against EM Simulators The planar transmission lines, which are like the TEM mode supporting two conductor transmission lines, are described by the secondary line parameters- the complex propagation constant γ and the characteristic impedance Z 0. These secondary line 38

27 parameters are described by the primary line constants - R, L, C, G i.e. the resistance, inductance, capacitance and conductance per unit length (p.u.l.) of a line. The secondary parameters are directly measurable quantities and are useful in the design. The measurement results are analyzed by converting the measured S- parameters to ABCD parameters of the transmission line system, from which γ and Z 0 of the lines can be extracted. As postulated by Kiziloglu et. al. [74] based on theoretical estimations, the quasi- TEM model can be used to describe digital circuit interconnects [149]. The measured line parameters can be determined from γ and Z 0. It has been observed that the propagation characteristics of lines, obtained from EMsimulators, deviate from linearity at the lower frequency range which is due to the finite conductivity of the strip conductor. At the lower frequencies the EM-fields penetrate to the strip conductors that cause dispersion in εeff and Z 0 in the lower frequency range. It really indicates increase in ε eff at the lower frequency range. Due to losses, Z 0 also has an imaginary part and shows increase at the lower frequency range. Fig.(2.14): Circuit model applicable to planar transmission lines The closed-form models that compute εeff and Z 0 of the planar transmission lines, do not account for the effect of conductor loss on these parameters. This can be accounted for with the help of the circuit model of the lines shown in Fig. (2.14). Thus, the circuit model provides an opportunity for the interaction of the line parameters, i.e. the 39

28 propagation constant and the characteristic impedances are influenced by the losses in the line. The EM-simulators - HFSS, Sonnet and CST, exhibit these low frequency features as these parameters are extracted from the frequency dependent S-parameters and also Y parameters and these parameters are influenced by the line structure and its losses. Either of these forms of data can be used to extract the line parameters of structures. The characteristic impedance Z 0 and complex propagation constant γ are related to S- parameters by the following expressions [138], s11 s12 = = s 22 s 21 = = 2 Z 0 Z r 2 Z 0 Z r 2 2 ( Z 0 Z r ) sinh ( γl ) 2 2 ( γl ) + ( Z + Z ) sinh ( γl ) cosh cosh 0 2 Z 0 Z r 2 2 ( γl ) + ( Z + Z ) sinh ( γl ) 0 r r ( a ) ( b ) (2.5) where Z r = 50 Ω is the reference impedance and l is the line length of transmission line. The expression for complex propagation constant γ giving the attenuation constant p.u.l is obtained from above equations, α T s + 11 s22 s = + = 11 s s s γ α T jβ cosh (2.6) l 2s21 The S-parameters obtained from the software are first converted to the ABCD-parameters. Next we get expression for γ from the ABCD-parameters. We can also convert the S- parameters to the Y-parameters and then obtain expression for the complex propagation constant γ in terms of it [81], γ = where 2 l tanh 1 y c 4 y d, 4 y d = y 11 y 12 y c = y 11 + y 12 y 21 + y 22 + y 21 + y 22 ( a ) ( b ) ( c ) (2.7) 40

29 In this section we present the comparison of low frequency features in the line parameters of planar transmission lines using different EM-simulators. In circuit model, the resistance and inductor of the equivalent circuits are in series; whereas the conductance and capacitors are connected in shunt. Their expressions in per unit length (p.u.l.) are summarized below, Resistance p.u.l. : Conductanc e p.u.l. : Capacitanc e p.u.l. : R(f, t) = 2 Z 0 ( f,t ) α c ( 2α d ( f,t ) G(f, t) = Z 0 ( f,t ) C (f,t) = ε eff ( f,t ) cz 0 ( f,t ) f,t ) ( a ) ( b ) ( c ) (2.8) Inductance p.u.l. : L (f,t) = Z 0 ( f,t ) c ε eff ( f,t ) ( d ) where, c is the free-space velocity of the EM wave. The ε eff ( f,t ), Z0( f,t ), αd ( f,t ) and αc (f, t) of the lines are computed using the closed-form models. The complex characteristic impedance * Z 0 (f, t) and complex propagation constant γ * (f, t) are computed from above primary line constants as follows: * Z0( γ * ( f,t ) f,t ) = = αt ( R( f, t ) + jω L( f, t ) G( f ) + jω C( f, t) f,t ) + jβ( f,t ) = ( a ) ( R( f, t ) + jω L( f, t ))( G( f, t ) + jω C( f, t )) ( b ) (2.9) The extracted line parameters of the microstrip line, using HFSS, Sonnet, CST and LINPAR, are compared against each other on substrates with 2. 5 ε r 37, 0.25µm t 9µm, 0. 1 w / h 5 and σ = 4.1 x 10 7 S/m. The frequency range is 0.01 GHz 10 GHz. This frequency range is selected to emphasize the dispersion in the lower frequency and also to examine changes in other line parameters. 41

30 (a) (b) Fig.(2.15): Extraction of RLCG parameters of lossy microstrip line using circuit model and EM-simulators. Fig.(2.15) and Fig.(2.16) show the characteristics of microstrip line on the substrate with ε r = 3.78, w/h = 2, h = 635µm and t = 3 µm. In Fig.(2.15a), resistance R increases as a function of frequency and the value of conductance G also increases with increasing frequency in EM simulators. The capacitance C remains relatively constant till 1 GHz, then onwards there is gradual decrease in the value and the inductance L of the microstrip line saturates gradually 1 GHz onwards as shown in Fig.(2.15b). The results of LINPAR are higher than those of the EM-simulators. The results of the circuit model follow results of EM-simulators closely, especially to CST. Then we obtain frequency dependent effective relative permittivity ε eff, characteristic impedance- real and imaginary parts, and total loss from the RLCG parameters. These results obtained from the circuit models are presented in Fig.(2.16) and compared against the results of EM-simulators. The results of circuit model follow results of EMsimulators; whereas results of our previous closed-form individual models do not follow EM-simulators, especially at lower end of frequency. It is true for all parameterseffective relative permittivity, losses and characteristic impedance. 42

31 Fig. (2.16a) shows that ε eff of the microstrip line has marginal increase in value with decrease in frequency, which is supported by the results obtained from the EMsimulators. There is continuous rise in attenuation with increase in frequency, as shown by results from all the three softwares in Fig.(2.16a). All the EM- simulators show slower decrease in loss. In Fig. (2.16b) the real part of Z 0 of the microstrip line also increases marginally with decrease in the frequency below 1 GHz. The imaginary part of Z 0 remains nearly constant throughout the frequency range. The circuit model accurately predicts and improves the dispersive nature of the microstrip line at the lower frequency range for computation of ε eff ( f, t), Z 0( f, t) and α T ( f,t ) with average and maximum deviation of (3.3, 7.2), (3.9, 6.5) and (4.6, 7.2) respectively. (a) (b) Fig.(2.16): Comparison of circuit model, closed-form model against EM-simulators for line parameters of lossy microstrip line in respect of eff ( f,t ) ε and ( f,t) α T, Re( Z * 0 ( f,t )) and Im(Z* 0 ( f,t )). The detailed study of the circuit models of CPW, CPS and slotline on planar and curved surfaces and comparison against experimental results will be discussed in Chapter-5, 6 and 7 respectively. 43