project: icestars - cordis · the project. at the early stage of the project no transient ﬁeld...

Project: ICESTARSProject Number: FP7/2008/ICT/214911Work Package: WP3Task: T3.1Deliverable: D3.5

Title: Public Report on Selection and Characterization of the DevicesUsing State-of-the-Art Field Solvers

Authors: Wim Schoenmaker*, Peter MeurisRick JanssenMonica Selva Soto, Caren Tischendorf

Affiliations: MAGWEL, NXP, Koln University

Corresponding [email protected]:Date: September 30, 2010

1

Simulations of Selected Devices and their Couplings usingExisting Tools

Wim Schoenmaker, Peter Meuris(MAGWEL)

Rick Janssen, Wil Schilders (NXP)

Monica Selva Soto, Caren Tischendorf(University of Cologne)

October 5, 2010

Contents

1 Introduction 4

2 Review of the electromagnetic modeling 62.1 Use of potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 S-parameters, Y-parameters, Z-parameters . . . . . . . . . . . . . . . . 7

3 A simple conductive rod 83.1 Interesting aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Simulation set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Strip line above a conductive plate 124.1 Interesting aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Simulation set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Effects of finite strip thickness . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4.1 Finite tM results . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.5 Simulations with Opera - VectorFields . . . . . . . . . . . . . . . . . . . 184.6 Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2

5 Coax configuration 225.1 Interesting aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Simulation set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Inductor with grounded guard ring 256.1 Interesting aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Simulation set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.3 Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7 Inductor with narrow winding above a patterned semiconduc tor layer 267.1 Interesting aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Simulation set up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Simulation of single and coupled oscillators: phase macro model approach 318.1 Interesting aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.2.1 LC oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.2.2 Mutual inductive coupling . . . . . . . . . . . . . . . . . . . . . . 32

8.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3.1 LC oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3.2 Mutually coupled oscillators . . . . . . . . . . . . . . . . . . . . . 348.3.3 Inductively coupled oscillators under injection . . . . . . . . . . . 36

9 Conclusions 38

3

1 Introduction

This document is a public summary of the work performed in WP3 Task 3.1 in theICESTARS project. In order to support compact model building, we need ab-initiofield solving to verify / falsify approximations that are made, while building the compactmodels. In the frequency domain, a compact modeling procedure was developed inthe projects CODESTAR and CHAMELEON-RF. Such a procedure was not yet suc-cessfully implemented in the time domain. The main stumbling block is the so calledCourant limit1 which requires unrealistically small time steps for progressing in time.

Compact models are a necessary requirement for designing integrated circuits (ICs).IC consists of a large number of individual devices collaborating in coherent way toperform the tasks for which they are designed. Information processing is the mostwell-known application but cost-saving arguments have led to the paradigm of integrat-ing also passive devices into the ICs. Low-noise amplifiers (LNAs) voltage controlledoscillators (VCO) need such passive parts, being integrated inductors, capacitancesand resistances. Whereas, the characterization of such devices usually is done in thefrequency domain, it should be realized that the resulting compact models are usedin the transient regime. Such an approach has a serious risk of leading to incorrectresults because the characterization in the frequency domain via S-parameter calcu-lations implicitly relies on a small-signal analysis. In other words: We can achieve adetailed knowledge of the device provided that the test signals are taken small. Forpassive devices one may safely assume linearity and therefore the results obtainedusing small signals can be safely extrapolated to large signals. However, this approachworks fine if the device is characterized in isolation. When the device is located in acomplicated environment, as is the case when integrated into an IC, then the environ-ment will change the surrounding electro-magnetic fields. The ”back reaction” of thesefields on the device will modify the outcome of the characterization. Thus we are facingthe challenge of arriving at an accurate characterization of the passive devices in anenvironment whose response is not linear. The latter is the case for semiconductingdomains with doping variations. The impact of non-linear responses becomes moresevere as the frequencies increase or (what is equivalent) if the temporal variation ofthe signals increase. These consideration have led to the proposal to characterizethe passive devices located on-chip, i.e. having semiconducting materials in their sur-rounding, using a large-signal approach. Another important aspect of the environmentis the presence of near-by passive devices that tend to radiate at large frequenciesor short switching times. Such effect can also only be accurately addressed whenthe coupling (EM fields) are accurate computed. Large signals effects are most easilystudied in the transient regime using step-function like excitation.

Applications of ICs with integrated passives can be found in many applications such asdomotic, entertainment (games, video streamers,...), LAN in automotive.

In this work package, the transient regime is addressed by implicit methods. Thismeans that a time step is integrated using both the end-point as well as the begin pointof the time interval. Our method is inspired by the transient simulation technique whichis used in Technology Computer Aided Design (TCAD). In particular, a backward finitedifference is combined with the trapezoidal rule, leading to a rather enhanced time-step

1condition for convergence while solving certain PDEs numerically, arises when using explicit time-steppingmethods.

4

size.

However, the construction of the transient solver did not start from scratch but is buildon available software tools that are available from MAGWEL and the University of Kolnand NXP. Starting from these resources, it is possible to ’short-circuit’ huge amountsof development effort (reuse) that otherwise would be spent on activities not directlyrelated to the required work in this task but are indispensable for computing industrialapplications. The following set of library components that are available are :

• Structure loading and simulation set-up

• Mesh generation

• Linear system solving

• Field viewing

• Mesh analysis

• Time-step integration

Of course, in order not to turn the work into a futile exercise it is required to evaluatethe status of these tools with respect to what is available. In particular, the purpose oftask 3.1 is three-fold : The following questions need to be addressed.

• Are the software tools sufficiently mature and competitive to justify its use in thepresent work?

• Are there critical issues that need to be resolved before the software can be suc-cessfully applied to transient simulations?

• Is the present available software capable of dealing with EM coupling in a suffi-ciently satisfactory way? (The frequency domain is under consideration, becausethe transient regime is the goal of the work)

• Are there specific alerts detected that need to be taken into account while per-forming the development work?

Above questions can be viewed as part of a SWOT (strengths, weakness, opportunityand threat) analysis. The last issue deals with identifying warnings in the develop-ment of the new tools by taking into account experiences collected in earlier softwaredevelopment cycles.

In order to give answer to above questions, we follow a very practical approach : bycomputing a collection of structures taken from various resources using different sim-ulation tools, we obtain a good insight into the positioning of the source codes that areavailable in the project in comparison to tools from other providers. Moreover, we learnabout the strengths and weaknesses of the tools and gain understanding on which as-pects need to be improved irrespective of the task related to the building of the transientsolver. The cases are selected on the following criteria : (1) availability, (2) ”intriguing”(the benchmark challenges insight) and (3) compact model availability. Using thesecriteria we have selected the following set a benchmarks :

5

• Simple conductive rod

• Two parallel transmission lines

• Strip line above a conductive plate

• Coax configuration

• Inductor with grounded guard ring

• Inductor with narrow winding above a patterned semiconductor layer

It is important to realize that some of these benchmarks have been acquired underNDA. The details of these benchmarks will be excluded from this report. However, theconclusions will be presented.

The test cases that are considered here were selected and elaborated at the start ofthe project. At the early stage of the project no transient field solver for semiconductingapplications was available. (This situation has changed thanks to the efforts in theICESTARS project.) Therefore, the simulations that are reported here were done in thefrequency regime.

2 Review of the electromagnetic modeling

2.1 Use of potentials

Our starting point will be the equations of Maxwell and D, E, B, H, J en ρ are the elec-tric induction, the electric field, magnetic induction and magnetic field, current densityand charge density.

The following constitutive laws are used:

B = µH, D = εE (1)

The charge density ρ and current density J = 0 in insulating materials and the chargedensity ρ consists of a fixed background charge. In conductive domains we rely on thecurrent continuity:

∇ · J +∂ρ

∂t= 0 (2)

If these conductive domains are metallic, then we apply Ohm’s law for the connectionbetween electric field intensity and current density. It should be noted that this is notthe most general expression and for Hall devices a magnetic field dependence mustalso be included. However, here we limit ourselves to situations where the magneticfields are sufficiently weak in order to ignore Hall currents:

J = σE (3)

It should also be noted that the charge density and current densities are determined bythe physical character of the materials under study. For example leakage currents can

6

flow in insulating layers and the current-field relation is highly non-linear since tunnelingmechanisms play an important role. Semiconductors also have more general current-field relations as given above and these will be discussed later.

We introduce the scalar potential V en the magnetic vector potential A that satisfy

B = ∇× A (4)

E = −∇V − ∂A

∂t(5)

then the Maxwell equations become in these variables;

• for insulators:

−∇ ·[

ε

(

∇V +∂A

∂t

)]

= 0 (6)

∇× 1

µ(∇× A) = −ε

∂

∂t

(

∇V +∂A

∂t

)

(7)

• for conductors:

−∇ · σ(

∇V +∂A

∂t

)

=∂

∂t

(

∇ · ε(

∇V +∂A

∂t

))

(8)

∇× 1

µ(∇× A) = −σ

(

∇V +∂A

∂t

)

− ε∂

∂t

(

∇V +∂A

∂t

)

(9)

2.2 S-parameters, Y-parameters, Z-parameters

In order to determine the S-matrix, a rather straightforward procedure is followed. Forthat purpose a collection of ports is needed and each port consists of two contacts.A contact is defined as a collection of nodes that are electrically identified. A ratherevident appearance of a contact is a surface segment on the boundary of the simulationdomain. A slightly less trivial contact consists of two or more of these surfaces on theboundary of the simulation domain. The nodes that are found on these surfaces are allat equal potential. Therefore, although there may be many nodes assigned to a singlecontact, all these nodes together generate only one potential variable to the systemof unknowns. Of course, when evaluating the current entering or leaving the contact,each node in the contact contributes to the total contact current. Assigning prescribedvalues for the contact potential can be seen as applying Dirichlet’s boundary conditionsto these contacts. This is a familiar technique in technology CAD. Outside the contactregions, Neumann boundary conditions are applied. Unfortunately, since we are nowdealing with the full system of Maxwell equations, providing boundary conditions forthe scalar potential will not suffice. We also need to provide boundary conditions forthe vector potential. Last but not least, since the set of variables V and A are notindependent, setting a boundary condition for one variable has an impact on the others.Moreover, the choice of the gauge condition also participates in the appearance of thevariables and their relations. A convenient set of boundary conditions is given by thefollowing set of rules :

7

• contact surface: V = V |ic. To each contact area is assigned a prescribed potentialvalue.

• outside the contact area on the simulation domain : Dn = 0. There is no electricfield component in the direction perpendicular to the surface of the simulationdomain.

• For the complete surface of the simulation domain, we set Bn = 0. There is nomagnetic induction perpendicular to the surface of the simulation domain.

We must next translate these boundary conditions to restrictions on A. Let us start withthe last one. Since there is no normal B, we may assume that the vector potential hasonly a normal component on the surface of the simulation domain. That means that thelinks at the surface of the simulation domain do not generate a degree of freedom. Itshould be noted that more general options exist. Nevertheless, above set of boundaryconditions provides the minimal extension of the TCAD boundary conditions if vectorpotentials are present.

In order to evaluate the scattering matrix, say of an N-port system, we iterate over allports and put a voltage difference over one port and put an impedance load over allother ports. Thus the potential variables of the contacts belonging to all but one port,become degrees of freedom that need to be evaluated. The following variables arerequired to understand the scattering matrices, where Z0 is a real impedance that isusual taken to be 50 Ohms.

ai =Vi + Z0Ii

2√

Z0

(10)

bi =Vi − Z0Ii

2√

Z0

(11)

The variables ai represent the voltage waves incident on the ports labeled with index iand the variables bi represent the reflected voltages at ports i. The scattering parame-ters sij describe the relationship between the incident and reflected waves.

bi =N∑

j=1

sijaj (12)

The scattering matrix element sij can be found by putting a voltage signal on port i andplace an impedance of Z0 over all other ports. Then aj is zero by construction, sincefor those ports we have that Vj = −Z0Ij. Note that Ij is defined positive if the currentis ingoing. In this configuration sij = bi/aj. In a simulation setup, we may put the inputsignal directly over the contacts that correspond to the input port. This would implythat the input load is equal to zero. The S-parameter evaluation set up is illustrated inFig. 1.

3 A simple conductive rod

3.1 Interesting aspects

The simple conductive rod is a circular bar with constant conductance. In the staticregime, a magnetic field is created, that behaves as 1/r outside the rod. As a conse-

8

Figure 1: Set up of the S-parameter evaluation: 1 port is excited and all others are floating.

quence, the total magnetic energy diverges, i.e.

Emagn =

∫

dr B · H ∝∫

∞

0

drr2(1/r)2 (13)

Therefore, we would like to learn if the simulation tool can deal with such configurations.Furthermore, the MAGWEL solver has a unique discretization algorithm based on atwo-fold application of Stokes’ law in order to deal with the term ∇ (1/µ ×∇×A).Recently this term was programmed also for unstructured grids. This benchmark willtest if these implementations can accurately produce analytic known results.

3.2 Simulation set up

The simulation set up is done with the following parameters :

• Wire cylindrical

• Radius 2 micron

• Length 100 micron

• Sigma=108 S/m

• µ0 = 4π ∗ 10−7 H/m

An illustration of the wire is shown in Fig. 2.

3.3 Results

Check if the B field matches the analytic expectation.Outside the wire we have :

B(r) =4π × 10−7 ∗ I

2πr(14)

9

Figure 2: Illustration of the wire in the MAGWEL editor; the red square is a contact area.

We put a voltage of V=10 volt over the wire. This gives the following values for thecurrent and resistance.

parameter valueAnalytic current 1.257861 102 A

Analytic resistance 7.95 10−2 Ohm

The result of the simulation is :

parameter valueAnalytic current 1.23323376 102 A

Analytic resistance 8.10876 10−2 Ohm

We plot B-field magnitudes along a line through the center of the wire. The resultis shown in Fig. 3. The MAGWEL results (red) are the outcome of a full numericalsimulation on this mesh. Then there are two analytic curves. The green curve (analyticnum I) refers to an analytic computation using the current as computed numerically onthis mesh, whereas the blue curves also refers to the current computed analytically.

The 1/r drop is very well reproduced. This is only possible if the curl-curl operator iscorrectly implemented (ignoring finite-size effects), no loss of B field propagation ofthe inverse operator is observed. The numerical B field (red) is in agreement with theanalytic current (blue). The numerical current value induces an analytic B field (green)in agreement with the numerical B field (red).

10

Figure 3: View of the magnetic field strength

11

0

2

4

6

8

10

12

14

0 20 40 60 80 100

B

x

B field around cylindrical wire

MGWMGW 10KHz

MGW 100KHzMGW 1MHz

MGW 10MHzMGW 100MHz

MGW 1GHz


We expect that at 1 GHz the conclusion is still more or less valid, the skin depth = 1.59micron < R= 2 micron.

The next plot is a scan over frequencies from 1KHz to 1GHz. It is very boring up to 100MHz and then there is a collapse.

3.4 Outcome

From this benchmark we learned that the numerical treatment of the curl-curl operatoris leading to correct analytic results in the static regime. No specific techniques forcorrecting or calibrating the outcome is needed.

4 Strip line above a conductive plate


When computing capacitances using the MAGWEL field solver there is a tendency foroverestimation. This problem is not seen for capacitors which extend in two dimensionsall the way over the simulation domain, i.e. for infinitely large parallel plates, but whenfinite segments are taken, then the measured capacitance is below the computed ca-pacitance. Therefore, this benchmark is of interest to get a grip on this systematicalcomputational error which is related to fringe capacitance. The benchmark is of interestfor support and selection of adaptive meshing strategies.

The strip line theory is explained in full detail in an on line book: http://www.ece.rutgers.edu/ or-fanidi/ewa/ch10.pdf :pp. 401-403. This modeling is due to Hammerstad and Jensen

12

Figure 5: Layout of the parallel strip above a conductive plate

[13, 11].

The model has the following ingredients :

• Effective relative permeability: εeff

• Characteristic impedance: Z

• Relations between Z, L and C

The capacitance, C, per unit length is C = ε η

Z, where ε = ε0 ∗ εr and ε0 = 8.85418e−12

F/m.

A cross sectional view of the strip line above a plate is shown in Fig. 5. The parameterw is the width and h is the height of the dielectric layer. Furthermore the parameters ηand η0 are defined as:

η =

√

µ

ε, η0 =

√

µ0

ε0

, (15)

(16)

where µ = µ0 = 4π ∗ 10−7 H/m.

The contacting is shown in Fig. 6.

The modeling is based on the geometrical parameter u = w/h and the effectivematerial parameter εeff :

εeff =ǫr + 1

2+

εr − 1

2× (1 +

10

u)−ab (17)

Z =η0

2π√

εeff

ln

[

f(u)

u+

√

1 + 4

u2

]

=Z01(u)√

εeff

(18)

The variables a and b are given below :

a = 1 +1

49ln[

u4 + ( u52

)2

u4 + 0.432] +

1

18.7ln[

1 + (u

18.1)3]

(19)

13

Figure 6: Illustration of the contacts

b = 0.564

(

εr − 0.9

εr + 3

)0.053

(20)

These expression were programmed and the text book examples are confirmed (page403 of the on-line book S.J. Orfanidis [22])


In MAGWEL we take 1 micron length strip line

variable valueh 1 micronw 10 micronεr 3.9

Tmetal 0.5Tair 1 micron

The input variables for the numerical experiment

4.3 Results

The results of this set up are shown below :

14


variable valueMAGWEL 3.951870e-16 F

Hammerstad and Jensen 3.958997e-016 FParallel plate model C = εA/h 3.4531302e-16 F

Static capacitance results

Note that the result from the simple parallel-plate model has a substantial error. A fewplots of the electric field are shown in Fig. 7 and Fig. 8.

At first sight, the result for the capacitance looks very acceptable. However, we havenot accounted for the finite thickness of the strip (here we used 0.5 micron) nor did wetake into account a substantial layer of air (1 micron). So far, the height of the strip is notaccounted for by Hammerstad and Jensen. By taking very thin strips, we can deducefrom the equations of Hammerstad and Jensen (HJ) that εHJ

eff < εr. When w/h → ∞then εHJ

eff → εr. We have studied this limit and found serious deviations between theanalytic and numerical results. The reason is that there also needs to be included asufficient amount of air in the field solver set up. We re-computed the capacitance withtmetal =0.01 micron, tair = 10 micron. In Fig. 9, the spread out of the electric field isillustrated.

4.4 Effects of finite strip thickness

A follow-up paper of Hammerstad and Jensen [11] deals width the impact of the finitestrip thickness. The modified theory is given below. The input parameters are :

• w : strip width ,

15


Figure 9: Separation in two domains of the electric field. red: E > 0.01Emax, blue: E <

0.01Emax

16

• h : thickness of the dielectric layer ,

• tM : thickness of the metal strip .

The normalized thickness is t = tM/h. Furthermore we define :

u1 = u + δu1

ur = u + δur

in which :

δu1 =t

πln

(

1 +4e(1)

t · coth2(√

6.517u

)

δur =1

2

(

1 +1

cosh√

εr − 1

)

δu1

The parameter Z0(u, t, r) is defined as

Z0(u, t, r) =Z01(ur)

eff (ur, r)(21)

where the function Z01 was defined as (see eq.(18)

Z01(u) =η0

2πln

[

f(u)

u+

√

1 +4

u2

]

(22)

A thickness-dependent permittivity is given by

εtMeff = εeff(ur, εr)

(

Z01(u1)

Z01(ur)

)2

(23)

The capacitance is obtained from

C =

√µ0ε0 × εtM

eff

Z(24)

4.4.1 Finite tM results

Hammerstad and Jenssen, with a correction for the strip thickness, give for the choiceof parameters : w=10 , εr = 3.9 and h=1, the following results :

tM C in F/m0.1 3.9790333515E-0160.3 4.0054405631E-0160.4 4.0162279229E-0160.5 4.0260254404E-016

17

tM capacitances

We refined the capacitance computation. Indeed, the results that where obtained inour first experiment are accidentally in agreement. The HJ model is an underestimation

because the thickness of the metal was not included and the MAGWEL simulation wasan underestimation because the air layer was only 1 micron (instead of ’infinity’).

By taking a layer of air of 10 micron we obtain the following numerical results :

tM CMGW in F/m0.01 3.964446958408e-16 F/m0.5 4.015557092879e-16 F/m

Numerical results

This last number is in close agreement with the corrected HJ result:

tM = 0.5, CHJcorrected = 4.02602544041718e− 016 F/m

Capacitances

Capacitance refinement level1.038975073324e-15 no refinement6.966594046402e-16 1 sweep5.296263512904e-16 2 sweeps4.525636287160e-16 3 sweeps4.211632499841e-16 4 sweeps

Result moves towards analytic result when the mesh refines.

We now know where we want our refinement to occur. The choice of metric does notyet generate the refinement at this desired location.Notes: How does the aspect ratio tag interfere with the refinement request? Adaptationwas based on HF testing. The C plots were for static capacitances, but they are verysimilar to 1 GHz plots for V and E.

4.5 Simulations with Opera - VectorFields

Strip line simulations with Opera-2D

The commercial software of Vector Fields [1] provides design solutions for a wide va-riety of electromagnetic applications. Opera is a state-of-the-art software package forthe modeling of static and time varying electromagnetic fields, using 2D and 3D fi-nite elements. For benchmarking purposes for the strip line model, the 2D electrostaticadaptive solver was used, where capacitance calculations can be performed in 2 ways:

• Energy calculation: C1 = 2Eelec/V2

• Gauss’ law: C2 =∮

D · dA

18

For the adaptive meshing stopping criterion, we use the Internal Error < 0.5% criterion,which means that the relative difference between the electric field solution in eachmesh element adjacent to a node and the averaged field in a node is smaller than0.5%. The meshing is base on a triangular Delauney algorithm. Furthermore, we putthe electrostatic potential V=0 on the bottom of the substrate (ground plane) and V=1on the strip line, while the rest of the boundary has a Neumann condition.

Fig. (10) shows the model (using different scales for the X- and Y-axis).

The next figures, Fig. (11,12), where X- and Y-axis now have the same scale, showsthe potential distribution and electric fields (arrows).

In the table we plot the different capacitance values C1 and C2 where we vary the Errorand air height. The Hammerstad and Jensen value is 4.0260e-016 F/m for the stripline case with thickness T= 0.5 µm. On average a number of 15 iterations in the meshadaptation process were needed to get an internal accuracy of less than 0.5%, with afinal average (of C1 and C2) difference of 0.2 % compared to the Hammerstad & Jensenvalue, which of course may also contain some error itself.

T= 0.5 µm, air height Internal Error C1 (in e-16 F/m) C2 (in e-16 F/m) # elem. # nodes2.5 µm 2 % 3.9089 3.9152 11151 57262.5 µm 1 % 3.9093 3.9121 28745 145582.5 µm 0.5 % 3.9096 3.9108 96719 486112.5 µm 0.2 % 3.9106 3.9111 285335 1430436.5 µm 0.5 % 3.9821 3.9831 116517 5846811.5 µm 0.5 % 4.0054 4.0061 189073 9473616.5 µm 0.5 % 4.0121 4.0127 149111 7471650 µm 0.5 % 4.0181 4.0189 165182 82744

Numerical results of the OPERA simulations

In the next table we plot C1 and C2 and also the Hammerstad & Jensen values forvarious values of the strip line thickness, with average (of C1 and C2) differences of0.02-0.2 % with the H-J values.

air height = 50 µm H&J C1 C2 Aver. Diff.Error = 0.5 % %

T= 0.4 µm 4.01623 4.01016 4.0115 0.13T= 0.3 µm 4.00544 4.0010 4.0024 0.09T= 0.2 µm 3.99330 3.99046 3.9916 0.06T= 0.1 µm 3.97033 3.97802 3.9787 0.20

T= 0.01 µm - 3.96148 3.9638 -T= 0 µm 3.95900 3.95926 3.9600 0.02

Numerical results of the OPERA simulations for various strip line thicknesses

4.6 Outcome

Many effects influence the final result of the fringe capacitance, including air layer,mesh size, strip thickness, domain size, etc. Refined analytic and refined numericalresults converge towards each other.

19

Difference: (CMGW − CHJ) /CHJ = 0.011/4.02 = 0.003 = 0.3 %This is very important for deciding on stopping criteria for adaptive meshing meth-ods. Furthermore, we have demonstrated that the MAGWEL solver can compete withOPERA (and therefore with any other finite-element based field solver) as far as accu-racy vs. node consumption is concerned.

20

Figure 10: Illustration of the model with X and Y at different scales.

Figure 11: Potential of the model with X and Y at equal scales.

Figure 12: Electric field of the model with X and Y at equal scales.

21

Figure 13: Illustration of the coax wire in the MAGWEL editor

5 Coax configuration


The simple coax example is used for testing a circular geometry. In the MAGWELsolver these circles are converted to polygons, and either a staircase Manhattan ap-proximation is used or a general unstructured Delaunay mesh. This example enablesus to compare with analytical expressions.


The simulation set up is done with the following parameters :

• Cylindrical coax

• Inner radius 1 micron (R0)

• Inner radius of outer conductor 5 micron (R1)

• Outer radius of outer conductor 5.5 micron (R2)

• Length 10 micron

• µ0 =4 π*10-7 H/m

• µr=1

An illustration of the wire is shown in Fig. 13

22

5.3 Results

We compare the static simulation results with the analytical magnetic field.

0 ≤ r ≤ R0 : B(r) =Bmaxr

R0(25)

R0 ≤ r ≤ R1 : B(r) =BmaxR0

r(26)

R1 ≤ r ≤ R2 : B(r) =BmaxR0

r

(

R22 − r2

R22 − R2

1

)

(27)

Also the resulting inductance per unit length is analytically known as:

L =µo

2πln

R1

R0+

µo

2π

(

R22

R22 − R2

1

)2

lnR2

R1− µo

4π

(

R22

R22 − R2

1

)

(28)

and for the given configuration this results in a value of Lanalytical=3.7837 10−7 H/m.This result is reproduced by MAGWEL simulations using different meshes (Manhattanand Delaunay), to give the following results:

nodes Lsim error mesh type8448 3.7709 10−7 H/m 0.38% Delaunay

31000 3.7768 10−7 H/m 0.18% Delaunay13920 3.7372 10−7 H/m 1.40% Manhattan209760 3.7950 10−7 H/m 0.15% Manhattan

The RF simulations for the same structure get exactly the same resulting inductance,when we stay below a frequency of 1 GHz. For higher frequencies, extra phenomenaplay a role, like skin effect, current re-distributions, etc.

5.4 Outcome

From this benchmark we learned that the numerical treatment of the magnetic operatoris leading to correct analytic results in the static regime. Typical circular geometries areadequately simulated with the MAGWEL software.

23


24

6 Inductor with grounded guard ring


This design considers an inductor that is shielded by a closed-loop grounded guardring. When an alternating current is injected into the inductor, an induced currentin opposite phase is induced in the closed loop. Although the loop is grounded atboth ends, the vector potential is still present and therefore the induced current existsdespite the fact that the loop is grounded. This configuration is also instructive to obtainan in-depth understanding of the measurement set-up for S-parameters.


The structure is simulated using the Y-parameter extraction method. This design has ahigh-resistive substrate. Therefore, the decay of the field strength is rather slow and afull stack of 625 micron substrate is included in the simulation.

6.3 Outcome

From this simulation work we learned that the implementation of the Hodge operatorsis very important for getting results that are within 1% accuracy for the inductance. Anaive implementation gives errors of 10%. These errors can be counteracted by usingan enhancement factor for the permeability µr. This factor is tuned once at zero fre-quency. Furthermore, in order to get results that are very accurate for inductors withwide windings, the integrating factor method improves the results at higher frequency.This is in agreement with the expectations since the integrating factor becomes impor-tant for high frequency and large mesh elements.

25

Figure 15: View of the inductor (no details)

7 Inductor with narrow winding above a patterned semiconduc-

tor layer


This inductor is interesting because the narrow windings require a rather advancedmeshing strategy. There are numerous calibration issues involved for the simulation ofthese type of inductors. The layout of the inductor is shown in Fig. 15.

However, the details of this design are found in the Nwell pattern that is supposed tolimit the eddy currents in the substrate. The underlying idea is that conductive patternswill induce an equipotential in the top layer and thereby induce a collapse of the eddycurrents.


These inductor configurations were simulated in finalized IST projects. However, thereare a number of issues that were not fully appreciated while performing the measure-ments. For instance, detailed knowledge on the guarding and contact engineering ismissing.

7.3 Results

The following figures show how good we can reach the experimental data. Further-more, a comparison is given using HFSS results [2]. Figures (16), (17), (18), (19),(20) and (21) show the S-parameters. We have several techniques at out disposal forobtaining these parameters. A direct method exploits the termination of all but one portand extract the S-parameters. This simulation method mimics the experimental situa-tion. Another technique that is more robust for simulation, grounds all but one port andextracts the Y-parameters. The latter method is applied here. Both methods give equalresults.

26

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2e+09 4e+09 6e+09 8e+09 1e+10

S11

re

Frequency GHz

S11 re

AMS HFSSMGW

Figure 16: Re(S11)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2e+09 4e+09 6e+09 8e+09 1e+10

S11

im

Frequency GHz

S11 im

AMSHFSSMGW

Figure 17: Im(S11)

27

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2e+09 4e+09 6e+09 8e+09 1e+10

S12

re

Frequency GHz

S12 re

AMS HFSSMGW

Figure 18: Re(S12)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0 2e+09 4e+09 6e+09 8e+09 1e+10

S12

im

Frequency GHz

S12 im

AMSHFSSMGW

Figure 19: Im(S12)

28

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2e+09 4e+09 6e+09 8e+09 1e+10

S22

re

Frequency GHz

S22 re

AMSHFSSMGW

Figure 20: Re(S22)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 2e+09 4e+09 6e+09 8e+09 1e+10

S22

im

Frequency GHz

S22 im

AMSHFSSMGW

Figure 21: Im(S22)

29

7.4 Outcome

This simulation work has learned us numerous facts. First of all, in order to positionthe simulation tools with respect to each other it is important to eliminate all disputableuncertainties. In this example, there were two important aspects. The first deals withguard ring design and the second one deals with the contact design. Both uncertain-ties led to some guess work and in particular, this guess work manifests it self in thedefinition of the ports. As a consequence, both the MAGWEL solver as well as HFSSproduce outcomes that do not match the measurements over the full frequency range.Nevertheless, by pursuing this simulation case, we have obtained a much better un-derstanding of which aspects play an important role in obtaining successful Q-factorresults. Moreover, we were able to define a calibration strategy for good inductor sim-ulation. The strategy is based on first getting the static inductance and resistance inplace. Next the determination of the Q-factor is set by an accurate modeling of the sub-strate. In practice it means that the substrate resistance and permittivity must be accu-rately determined. Finally, the high-frequency behavior must be settled by obtaining agood value for the inter-winding capacitive coupling. Based on these observations weconclude that accurate simulation at high frequency requires adaptive meshing. With-out adaptive meshing the capacitive couplings are over-estimated. We suspect that thisis the reason why the 2-port inductance of shows a resonance at a too low resonancefrequency. Since HFSS is equipped with adaptive measurement facilities, this reso-nance is shifted to a higher frequency since the over-estimation of the inter-windingcapacitance is avoided. In order to verify/falsify this interpretation of the results we re-peated the simulation and replaced the permittivity of the oxide with 1.0, This leads toa smaller inter-winding capacitance and that should give a higher self-resonance fre-quency. Besides, the specific knowledge that was generated for simulating inductorsabove substrates, a few more interesting facts were gathered:

• At low frequencies, the computation of inductance requires that Hodge operatoreffects are taken into account.

• At high frequencies a modified discretization is needed.

• S-parameter, Y-parameter based simulations provide equivalent compact modelresults.

Note: We also have performed inductor simulations using Momentum [3]. The conclu-sions remain unaltered.

30

8 Simulation of single and coupled oscillators: phase macro-

model approach


Design of integrated RF circuits requires detailed insight in the behavior of the usedcomponents. Unintended coupling and perturbation effects need to be accounted forbefore production, but full time-domain simulation of these effects is expensive or in-feasible. Oscillators are one of the main building blocks which are running at highfrequencies (in GHz range), hence full time-domain simulations require a very smalltime step and are time consuming. Here we present a method to build non-linearphase macromodels for voltage controlled oscillators. These models can be used toaccurately predict the behavior of individual and mutually coupled oscillators under per-turbation, at much lower costs than full circuit simulations. In this section we considerLC oscillators, see Section 8.2.1, where the L and C parameters of a given oscillatorare computed with the MAGWEL software. The approach is illustrated by numericalexperiments with realistic designs.

8.2 Mathematical model

A general free-running oscillator can be expressed as an autonomous system of differ-ential equations:

dq(x)

dt+ j(x) = 0, (29a)

x(0) = x(T ), (29b)

where x(t) ∈ Rn are the state variables, T is the period of the free running oscillator,

which is in general unknown, and q, j : Rn → R

n are (non-linear) functions describingthe oscillator’s behavior. The solution of (29) is called periodic steady state (PSS) andis denoted by xpss. A general oscillator under perturbation can be expressed as asystem of differential equations

dq(x)

dt+ j(x) = b(t), (30)

where b(t) are perturbations to the free running oscillator. For small perturbations b(t)it can be shown [9] that the solution of (30) can be computed as

xp(t) = xpss(t + α(t)), (31)

where α(t) is called the phase shift. The phase shift α(t) satisfies the following scalarnon-linear differential equation:

α(t) = V T (t + α(t)) · b(t), (32a)

α(0) = 0, (32b)

with V (t) ∈ Rn being the perturbation projection vector (PPV) [9] of (30) and n is the

system size. Using this simple and numerically cheap method one can do many kindsof analysis for oscillators, e.g. , injection locking, pulling, a priori estimate of the lockingrange [9, 19].

31

v

CR L

i=f(

) v

Figure 22: Voltage controlled oscillator.

8.2.1 LC oscillator

For many applications oscillators can be modeled as an LC tank with a non-linear resis-tor as shown in Fig. 22. This circuit is governed by the following differential equations:

Cdv(t)

dt+

v(t)

R+ i(t) + S tanh(

Gn

Sv(t)) = b(t), (33a)

Ldi(t)

dt− v(t) = 0, (33b)

where C, L and R are the capacitance, inductance and resistance, respectively, whichare calculated by using the MAGWEL software. The nodal voltage is denoted by vand the branch current of the inductor is denoted by i. The voltage controlled non-linear resistor is defined by S and Gn parameters, where S determines the oscillationamplitude and Gn is the gain.

8.2.2 Mutual inductive coupling

Let us consider the two mutually coupled LC oscillators shown in Fig. 23. The inductivecoupling between these two oscillators can be modeled as

L1di1(t)

dt+ M

di2(t)

dt= v1(t), (34a)

L2di2(t)

dt+ M

di1(t)

dt= v2(t), (34b)

where M = k√

L1L2 is the mutual inductance and |k| < 1 is the coupling factor. Inthis section all the parameters with a subindex refer to the parameters of the oscillatorwith the same subindex. If we combine the mathematical model (33) of each oscillatorwith (34), then the two inductively coupled oscillators can be described by the following

32

L C R

v v

CR L

vi=

f(

) v

M

i=f( )

2 2 2

1 2

1 1 1

1 2

Figure 23: Two inductively coupled LC oscillators.

differential equations

C1dv1(t)

dt+ v1(t)/R + i1(t) + S tanh(

Gn

Sv1(t)) = 0, (35a)

L1di1(t)

dt− v1(t) = −M

di2(t)

dt, (35b)

C2dv2(t)

dt+ v2(t)/R + i2(t) + S tanh(

Gn

Sv2(t)) = 0, (35c)

L2di2(t)

dt− v2(t) = −M

di1(t)

dt. (35d)

For small values of the coupling factor k, the right hand side of (35b) and (35d) can beconsidered as a small perturbation to the corresponding oscillator and we can apply thephase shift macromodel [14]. Then we obtain the following simple non-linear equationsfor the phase shift of each oscillator:

α1(t) = V T1 (t + α1(t)) ·

(

0

−Mdi2(t)

dt

)

, (36a)

α2(t) = V T2 (t + α2(t)) ·

(

0

−Mdi1(t)

dt

)

, (36b)

where the currents and voltages are evaluated by using (31):

[v1(t), i1(t)]T = x1

pss(t + α1(t)), (36c)

[v2(t), i2(t)]T = x2

pss(t + α2(t)). (36d)

8.3 Results

8.3.1 LC oscillator

Consider an oscillator with the following parameters L = 0.93 nH,C = 1.145 pF andR = 1000 Ω. It has a free running frequency f0 = 4.8773 ∗ 109 and the amplitude of the

33

0 200 400 600 800 1000−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

−9

time/T

α−ph

ase

shift

4.5 5 5.5 6 6.5

x 109

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

frequency

PS

D(d

B)

simulationfin

Figure 24: Unlocked oscillator with b(t) = 0.05A0 sin(0.97f0 ∗ 2πt). Left: phase shift, right:output spectrum.

0 200 400 600 800 1000−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

−9

time/T

α−ph

ase

shift

2 3 4 5 6 7 8

x 109

−300

−250

−200

−150

−100

−50

0

frequency

PS

D(d

B)

simulationfin

Figure 25: Locked oscillator with b(t) = 0.05A0 sin(0.98f0 ∗2πt). Left: phase shift, right: outputspectrum.

capacitance response voltage v is about 0.5844 and the inductor current has amplitudeA0 = 0.0205.

Consider the injected signal to be sinusoidal of the form

b(t) = Ainj sin(ω1t), (37)

where Ainj is the amplitude of the injected signal and ω = 2πf is the frequency inradians.

For the injection amplitude 0.05 ·A0 and the injection frequency 0.97 · f0, then the oscil-lator under this perturbation will not lock, see Fig.24.

A slight change in the injection frequency will lock the oscillator. If the injection fre-quency is 2% less than the oscillator’s frequency, i.e. f1 = 0.98 · f0 then the oscillatorwill lock to the injected signal, see Fig. 25.

8.3.2 Mutually coupled oscillators

Let us consider two oscillators with inductance and resistance L1 = L2 = 0.64 nH andR1 = R2 = 50 Ω, respectively. The first oscillator is designed to have a free running

34

4.2 4.4 4.6 4.8 5 5.2 5.4

x 109

−150

−100

−50

0

frequency

PS

D(d

B)

macromodelfull simulation

(a) k=0.0005

4.2 4.4 4.6 4.8 5 5.2 5.4

x 109

−180

−160

−140

−120

−100

−80

−60

−40

−20

frequency

PS

D(d

B)


(b) k=0.001

4 4.5 5 5.5

x 109

−150

−100

−50

0

frequency

PS

D(d

B)


(c) k=0.005

3.5 4 4.5 5 5.5 6

x 109

−180

−160

−140

−120

−100

−80

−60

−40

−20

frequency

PS

D(d

B)


5.2 5.25 5.3 5.35 5.4

x 109

−95

−90

−85

−80

−75

−70

−65

−60

−55

frequency

PS

D(d

B)


@@

(d) k=0.01

Figure 26: Inductive coupling. Comparison of the output spectrum obtained by the phasemacromodel and by the full simulation for different coupling factors.

frequency f1 = 4.8 GHz with capacitance C1 = 1/(4L1π2f 2

1 ). Then the inductor currentin the first oscillator is A1 = 0.0303 A and capacitor voltage is V1 = 0.5844 V. In a similarway the second oscillator is designed to have a free running frequency f2 = 4.6 GHzwith the inductor current A2 = 0.0316 A and capacitor voltage V2 = 0.5844 V.

For both oscillators we choose Si = 1/Ri, Gn = −1.1/Ri with i = 1, 2. Simulationresults with the phase shift macromodel are compared with Pstar (in-house NXP circuitsimulator ) simulations of the full circuit, hereafter called full simulation.

In all the numerical experiments the simulations are run until Tfinal = 6 · 10−7 s with thetime step τ = 5 · 10−13. Numerical simulation results of two inductively coupled os-cillators for different coupling factors k are shown in Fig. 26. For small values of thecoupling factor we observe a very good approximation with the full simulation results.As the coupling factor grows, small deviations in the frequency occur, see Fig. 26(d).Because of the mutual pulling effects between the two oscillators, a double sided spec-trum is formed around each oscillator carrier frequency. The additional sidebands areequally spaced by the frequency difference of the two oscillators.

The phase shift α1(t) of the first oscillator for a certain time interval is given in Fig. 27.We note that it has a sinusoidal behavior. Recall that for a single oscillator underperturbation a completely different behavior is observed: in locked condition the phaseshift changes linearly, whereas in the unlocked case the phase shift has a non-linearbehavior different from a sinusoidal, see for example [20].

35

5.4 5.6 5.8 6

x 10−7

−4

−3

−2

−1

0

1

2

3

x 10−13

phas

e sh

ift

time

Figure 27: Inductive coupling. Phase shift of the first oscillator with k=0.001.

8.3.3 Inductively coupled oscillators under injection

As a final example let us consider two inductively coupled oscillators where in one ofthe oscillators an injected current is applied. Let us consider a case when a sinusoidalcurrent of the form

I(t) = A sin(2π(f1 − foff)t) (38)

is injected in the first oscillator. Then (36) is modified to

α1(t) = V T1 (t + α1(t)) ·

( −I(t)

−Mdi2(t)

dt

)

, (39)

For a small current injection with A = 10 µA and an offset frequency foff = 20 MHz,the spectrum of both oscillators with the coupling factor k = 0.001 is given in Fig.28.We observe that the phase macromodel is a good approximation of the full simulationresults.

36

0 1 2 3 4 5 6

x 10−7

−5

−4

−3

−2

−1

0

1x 10

−12

phas

e sh

ift

time

(a) oscillator 1

0 1 2 3 4 5 6

x 10−7

−5

0

5x 10

−13

phas

e sh

ift

time

(b) oscillator 2

4.6 4.7 4.8 4.9 5 5.1

x 109

−150

−100

−50

0

frequency

PS

D(d

B)


(c) oscillator 1

4.4 4.5 4.6 4.7 4.8

x 109

−160

−140

−120

−100

−80

−60

−40

−20

0

frequency

PS

D(d

B)


(d) oscillator 2

Figure 28: Inductive coupling with injection and k = 0.001. Top: phase shift. Bottom: com-parison of the output spectrum obtained by the phase macromodel and by the full simulationwith a small current injection .

37

9 Conclusions

This document presents the non-confidential results of the task 3.1 in the ICESTARSproject. A key concern has been to decide if the MAGWEL solver can serve as anappropriate base for constructing a transient solver in the ICESTARS project. We haveshown that the test cases designed for testing elementary qualities give excellent re-sults. However, for industrial examples, the situation is more involved. We comparedresults from MAGWEL with results from Momentum and HFSS. The details of the re-sults differ, but the overall conclusion is that the MAGWEL solver is able to generateoutcomes of comparable quality as Momentum and HFSS. Whereas the MAGWELsolver is not as complete in terms of ease-of-use, adaptive frequency stepping andother conveniences that are collected over many years, the tool is indeed a good start-ing point for further development and for constructing a transient version. In particular,the need for a faithful computation of the semiconductor properties was decisive inselecting this solver. Furthermore, the meshing facilities of the MAGWEL solver arevery unique and useful for dealing with combined small and large scales into a singlesimulation set up. In particular, the simulation of high-resistive substrate inductors hasbeen a serious challenge for many tools.

38

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