Understanding Grounding Concepts in EM Simulators

INTRODUCTION

Misunderstanding how ground is implemented in circuit simulation is one of the

most common misuses of electromagnetic (EM) simulators and their results.

This white paper discusses the defi nition of ground in EM simulators and how to

correctly choose among various grounding options, a topic of special importance

to designers using the results in a circuit simulator. Many modern simulators

now support the notion of local grounding, where different ports can use different

ground defi nitions. New features in AWR’s AXIEM™ 2009 3D planar EM simulator

offer extensive sources/ports and de-embedding options, including internal edge,

fi nite difference/gap and extraction ports, and per-port, coupled line and mutual

group de-embedding.

Before it can be understood how ground is used in EM simulators, it is fi rst

necessary to understand how it works in circuit simulation. The major portion

of the discussion involves examining where ground is defi ned in EM simulators

and how this is a function of the solver and port types. The paper demonstrates

how this fl exibility in selecting ground can help the engineer characterize board,

package, and interconnect performance. Several specifi c examples at the board,

package, and chip level are discussed. The paper concludes with circuit tricks that

can be used to aid in ground studies.

So, ground is a concept that, while perhaps a bit elusive, forms an important

basis for how transmission lines are modeled and simulated. Just as the notion

of “what is ground” and “where is it” changes as the underlying physics are

probed, so too when real circuits are being analyzed. AXIEM is a superb tool in

this instance as it has been engineered with this sort of fl exibility in mind. AXIEM

enables the designer to specify where the ground reference is in the design.

When adding a port, it’s just a single mouse-click on any port to reference that

port to the particular layer in the geometry that the designer wishes to treat

as ground. In more complex situations, for instance, with coplanar waveguide,

designers can even set up a system of ports to accurately represent how the

wave propagates and how the current “returns” in the structure through the use

of port groups. In addition, AXIEM has tremendous fl exibility in how ports are

de-embedded so that designers have the highest level of assurance that they are

measuring their structure with the ground where they intend it to be, and not

artifacts of the ports or the geometry “setup” required by the EM solver. AXIEM is

an excellent choice for EM wherever ground may be.

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Dr. John M. Dunn AWR Corporationjdunn@awrcorp.com

BIO:

John Dunn is a senior engineering consul-tant at AWR, where he is in charge of training and university program develop-ment. His areas of expertise include electromagnetic modeling and simulation for high speed circuit applications.

Dr. Dunn past experience includes both the worlds of industry and academia. Prior to joining AWR, he was head of the interconnect modeling group at Tektronix, Beaverton, Oregon, for four years. Before entering industry, Dr. Dunn was a professor of electrical engineering at the University of Colorado, Boulder, for fi fteen years, from 1986 to 2001, where he lead a research group in the areas of electromagnetic simulation and modeling.

Dr. Dunn received his Ph.D. and M.S. degrees in Applied Physics from Harvard University, Cambridge, MA, and his B.A. in Physics from Carleton College, Northfi eld, MN. He is a senior member of IEEE.

GROUND IN CIRCUIT SIMULATION

Ground is probably one of the fi rst concepts that the electrical engineering student learns

about. They learn that the two fundamental concepts they need are current and voltage.

They are then given Kirchoff’s two laws as facts: current (with sign convention) summed up

at any node is 0, and the sum of voltages around a loop is zero. Since the sum of voltages

around a loop is what matters, i.e., differences in voltage, there might as well be one

point in a circuit at 0 volts, which is ground. It also is commonly called “node 0” in circuit

simulators. Students then spend most of a semester solving Kirchoff’s laws with circuits

of varying degrees of diffi culty. It is interesting to note in the standard circuit class there is

very little discussion of what a physical ground is, and why it is good to be “well grounded.”

THE ELECTROMAGNETIC DEFINITION OF GROUND

The electromagnetic basis of electrical engineering says that Kirchoff’s laws derive from

Maxwell’s equations. Indeed, if one takes Ampere’s law and Gauss’s law for electric fi elds, one

obtains the equation for conservation of charge, which states:

where the summation is the net current coming into the node and Q is the charge buildup on

that node. In normal circuit theory, there is only one place where charge can occur, and that

is a capacitor. Therefore, there are two conclusions: at any node the sum of currents must

be zero, and at a capacitor:

Notice that the notion of voltage has been used in the capacitor equation. Voltage is not part

of Maxwell’s equations, and is what is called a derived quantity. Maxwell’s equations only have

a notion of electric and magnetic fi elds, current density, and charge density. Certain other

physical quantities can be consistently derived from Maxwell’s equations. For example, power

density, and electric and magnetic fi eld energy can be defi ned in a way that is consistent with

the defi nition of energy from mechanics.

Voltage is defi ned to be the integral of the electric fi eld along a path (actually the tangential

part of the electric fi eld to the path). The immediate question is why is this important? The

reason is Faraday’s law, which states that the electric fi eld around any closed path is equal to

the change in time of the magnetic fl ux through that loop (with a minus sign):

where E is the electric fi eld, is the total fl ux of the magnetic fi eld density, B, of the loop. So,

the immediate conclusion is that the change in voltage around a loop is equal to the change

in time of the magnetic fl ux through the loop (with a minus sign). Circuit theory makes the

assumption that the changing magnetic fl ux through the loop is small enough that we can

assume it is zero. And if it’s zero, it means that the electric fi eld around any loop, i.e. the

voltage around any loop, is zero, which is Kirchoff’s voltage law.

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∑i

I = jωQ,

Q = CV.

∮�E · d� = −dΦ

dt,

Φ

There is of course one place where the magnetic fl ux change is large enough that it can’t be

neglected – at an inductor. This is the defi nition of inductance:

where the fl ux is produced by the current I. To fi nish up the derivation of circuit equations,

notice that Kirchoff’s laws have fallen out of Maxwell’s equations, as have the notion of a capacitor

and inductor. There also is an assumption that the capacitor and inductor are electrically small

enough that an idealized, perfect circuit capacitor and inductor can be talked about.

To conclude, since the voltage around any loop is zero, which in turn means the voltage is

path independent, one point in the circuit can be chosen and called 0 volts. The voltage at

any point in the circuit can just be given a number, and designers do not have to worry about

“with respect to what?” It is with respect to ground.

CIRCUIT SIMULATORS, S-PARAMETERS, AND THE NOTION OF GROUND

Circuit simulators work by solving Kirchoff’s laws. More specifi cally they solve a matrix equation

to determine the voltages and currents at all nodes in the circuit. Modern circuit simulators

set the matrix equation up by using modifi ed nodal analysis [1]. In this technique, equations are

written for every node based on Kirchoff’s laws, and the resulting matrix is then solved using

standard techniques. Circuit simulators use Y matrices to describe how sub-networks work. A

network is viewed as a collection of exterior nodes that are related to each other.

RF and microwave engineers often prefer to work with S-parameter matrices, as opposed

to Y matrices, which presents a problem. The circuit simulator has been given an S matrix,

for example from an EM simulator. Yet, it must work with Y matrices. Therefore, it needs

to convert the S matrix, which means it must work with currents and voltages. In order to

do this, it must know the characteristic impedances of the ports. S-parameters, in their

truest sense, are ratios of powers. If the characteristic impedances of the ports are a known

complex quantity, it can convert this notion of power into current and voltage and therefore a

Y matrix. The actual equation to do this is:

where S is the S matrix, I is the identity matrix, and Zg is a diagonal matrix whose elements

are characteristic impedances for each port—which can be complex numbers.

PORTS IN EM SIMULATORS

It is important to understand how S-parameters are generated in EM simulators. All EM

simulators need ports to derive the S-parameters. The ports inject energy into the system

(incident wave) and look at the refl ected power back into the port and the transmitted power

going into the other ports. The ratio of the refl ected or transmitted power to the incident

power is the defi nition of the S-parameter. It is assumed that all ports are perfectly matched

so that any wave going into the port is not re-refl ected back into the system. EM simulators

differ in how they carry out the details of this procedure. The various types of ports used in

EM simulators will now be examined, with an eye toward how ground is defi ned for these

ports. As previously argued, every S-parameter fi le imported from an EM simulator into

a circuit simulator makes the assumption that the ground is the same for all the ports.

Therefore, it is important to know what each port is assuming for the ground it uses.

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Φ = LI,

Φ

Y = (I + S)−1Z−1g (I − S),

THE WORLD OF EM SIMULATORS

EM simulators solve Maxwell’s equations numerically. There are a wide

variety of simulators available, with widely differing features. It is necessary

to understand the different types of simulators available, so that it can be

better understood how they defi ne ports and their associated grounds. There

are many ways of classifying EM simulators; for example, one may look at

the mathematical techniques used, the generality of the problems they solve,

simulation speed, and of course features and user interface. For purposes of

this discussion, EM simulators will be classifi ed according to Figure 1.

The horizontal axis roughly equates to the generality of problems the EM simulator

covers. The various categories will be explained shortly. The vertical axis is the

computational complexity. The most general type of EM simulator is described

here as three-dimensional (3D), and is the most computationally intensive.

The class of solvers described as 2D solvers or cross sectional solvers are

used for getting transmission line parameters per unit length of a transmission

line system. For example, the solver will solve for the resistance, capacitance,

inductance, and conductance per unit length of a microstrip line. Once the

electrical properties of the transmission lines per unit length is known, it

is a simple matter to multiply by the length and get the transmission line

parameters. A distributed line model can then be used in the circuit simulator.

These types of simulators are commonly built into the circuit simulator, and run

the EM simulation behind the scenes. For example, in AWR’s Microwave Offi ce®

software, a variety of these models are available. They can be grouped into two

general classes: moment method techniques and fi nite element techniques. The

moment method models are shown in Figure 2.

The technique solves for the currents and charges on the surfaces of the

conductors using MoM, which is explained below. Once determined, the charge

gives the capacitance per unit length, and the current the inductance per unit

length. The resistance per unit length can also be determined. The model

only calculates the values at one frequency. The capacitance per unit length is

assumed not to change in frequency, and the resistance and inductance per

unit length are assumed to be changing with the square root of frequency,

as in the standard skin depth approximation. This approximation assumes

that the conductors are thick enough that they are several skin depths thick.

The skin depth of copper at 1 GHz is approximately 1 micron. Therefore,

the approximation is certainly valid for copper lines on boards, where the

thickness of the lines is 10’s of microns thick. As a matter of fact, the skin

depth of a ½ ounce copper line is not one skin depth thick until a frequency

of 10 KHz or lower. Where is the ground for this type of simulation? The

moment method simulator typically puts a ground plane at the bottom of the

structure for microstrip lines, and a ground plane on the top and bottom of

the cross section for a stripline simulation. Later on this paper will discuss

how the method-of-moments does this.

The other type of cross sectional solver used in AWR’s Microwave Offi ce is a

fi nite element method (FEM) solver. This is shown In Figure 3.

Figure 1. Classifi cations of EM simulators.

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Figure 2. 2D method-of-moments (MoM) EM simulators for determining line parameters.

Figure 3. FEM cross sectional solve for determining line parameters.

This type of simulation also gets the transmission line properties per unit length. The FEM

meshes the cross section in triangles and solves for the electric fi eld on each of the meshes.

The advantage of this technique is that can account for the loss and internal inductance in the

transmission lines without making the assumption of skin depth. This is important for silicon

chips, where the skin depth region is not clearly established. For example, with aluminum

lines, the skin depth at 1 GHz is about 1.5 micron, and with a 2 micron thick line, the skin

depth approximation is not satisfi ed. In addition, the silicon substrate is meshed up, as silicon

substrates cannot be assumed to be good conductors. Typically, the bulk conductivity is

between 0.01 and 10 S/m depending on the doping of the layer. The skin depth approximation

would lead to serious errors. The grounding for this type of simulation is the bottom conductor

of the silicon, and the side walls at the edge of the fi nite element space. Grounding straps can

be placed from the sides to close proximity of the line if desired. Of course, care must be taken

that these agree with the actual physical geometry, for example, in the case of a coplanar line,

where the side grounds would represent the grounds of the coplanar line.

The next class of EM simulators shown in Figure 1 is 3D planar simulators, which are

sometimes called 2 ½ D simulators. These simulators use MoM, which will be explained below.

In this method, the currents are solved on the conductors by meshing up the lines into a series

of triangles and rectangles. A matrix equation is solved to get the current on each mesh.

The currents are excited by attaching ports, and the S-parameters can then be calculated.

The method solves for horizontal currents (the planar or “2” part) and vertical currents for

vias or thick metal lines (the full or ½ part). There are geometry restrictions. The structure

consists of planar, homogenous, dielectric layers, with an optional ground plane at the bottom

of the geometry, and an optional top conductor. Fortunately, this type of restriction is not a

serious problem for many geometries of interest in the world of boards, packages, and chips.

The structure can either be in a rectangular conducting box or not in a box, depending on

the implementation of the code. As is shown in Figure 1, theses simulators have been used

successfully for interconnect simulation, planar antennas, spiral inductors, and other structures

of interest to the signal integrity engineer.

The third class of EM simulators is described in Figure 1 as the 3D simulators. These

simulators can simulate the most general structures, as they mesh up all the space of interest

using small elements. Typically the elements are either 3D triangular pyramids, usually called

tetrahedral, or rectangular bricks. The two most popular methods for solving for the fi elds

are the FEM, which solves in frequency domain, and the fi nite difference time-domain (FDTD),

which solves in time domain. It should be mentioned that the nomenclature for these methods

varies. For example, an FDTD is really an FEM in the sense that fi nite elements are used.

There are FEM methods in the time domain. Since this discussion focuses on how grounds are

defi ned for the ports in these methods, it is not necessary to delve into the subtle and technical

distinctions between the methods in this article. Finally, it should be mentioned that there are

other methods for solving for 3D problems. For example, the boundary element method (BEM)

is popular in Europe. In this method, the boundary on each object is meshed, and fi ctitious

electric and magnetic currents are inserted. This method will be discussed later in this article.

The 3D solvers are more general than the 3D planar solvers. They typically are computationally

more intensive, and there is therefore a tradeoff between simulators. There is no one best

simulator for all applications.

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3D SIMULATORS AND PORTS

3D simulators solve for the electric fi elds in a region of space bounded by the edges of the

simulation space. The space is meshed up in simple, small shapes, typically either tetrahedral or

parallelepipeds. The electric fi eld has a simple, approximate form for each region, for example

linear variation in all three spatial directions. The electric fi elds are excited by means of a port,

which is the focus of this chapter. The fi elds are solved so that they satisfy Maxwell’s equations

with the port excitations and the correct boundary conditions on the edges of the structure. The

details of how this is carried out are not relevant to this discussion. The interested reader can

look at fi nite element methods in the literature [4].

The discussion of ports in 3D simulators begins by considering the analogy to

the network analyzer in a laboratory. The network analyzer works by launching a

wave out of a port, and down a transmission medium. Typically, this is a coaxial

cable, or possibly a waveguide for millimeter wave applications. The incident

wave hits the device under test (DUT). The power in the wave eventually either

refl ects, giving S11, or transmits to the other port, S21, or is dissipated.

Dissipation of energy can occur by absorption or radiation in the DUT, or by

being absorbed at other ports that have been terminated in matched loads

(loads equal to the wave’s characteristic impedance). The equivalent of a

numerical network analyzer will be made for this EM simulator. Therefore, a

wave port will be created. A simple example is shown in Figure 4.

Think of the port as the end face of a waveguide, the other end of which is the

network analyzer. The network analyzer is perfectly matched to the waveguide.

Waves coming down a waveguide are described in modes. For example, in the

case of a transmission line, for example a coaxial line, the dominant mode of transmission is

the transverse –electric-magnetic (TEM) mode. Higher order modes are possible, although

at normal operating frequencies they are cutoff. See standard electromagnetic texts for an

explanation of waveguide modal theory [5]. Now, modes can be characterized by three features:

their fi eld pattern, their characteristic impedance, and their propagation constant. Details on

each of these three features follows.

The fi eld pattern of the mode is determined by Maxwell’s equations. The port will be used

to launch the wave on the signal line. Notice the wave port has an outer boundary, and it is

certainly a reasonable question if this matters. The answer is that yes, in principle it changes

the answer. For example, the microstrip line in fi gure 4 works in a quasi-TEM mode.

(The “quasi” is because it is not a true TEM mode. There are small components of electric

and magnetic fi elds in the direction of propagation because of the dielectric/air interface.)

The current fl ows down the microstrip line, and returns on the ground return beneath it.

If the side walls and top of the wave port are far enough way, the current will predominantly

still return underneath the microstrip line and a microstrip mode will be the result. However,

is the wave port walls are too close, serious discrepancies between the actual mode and

the microstrip mode will occur. There will therefore be a refl ection when the wave leaves the

wave port due to the mismatch. On the other hand, if the wave port walls are too far away,

computational resources are wasted, and numerical errors can occur. Reasonable rules of

thumb are to make the side walls three substrate heights away, and the top of the port three

heights substrates above the line.

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Figure 4. Meshed wave port at the end of a microstrip line.

How are the S-parameters at these ports determined? The simulator fi rst of all calculates all

the necessary modes. Normally, the user must tell the software how many modes to calculate.

The modes are determined by solving Maxwell’s equations across the wave port numerically,

by meshing it up and performing a fi nite element solution. Technically speaking, the solver

determines the eigenmodes and eigenvalues of the system. An eigenmode is a natural state of

solution for a system. For example, when you hit the surface of a drum, it vibrates in the natural

eigenmodes of the drum head. When you send power into a waveguide, the wave travels in the

natural eigenmodes of the system. The eigenvalues are the wavenumbers, of the modes:

The is the phase constant = . The is the decay constant in Np/m. A non-zero

means that the wave is decaying as it goes down the guide. Normally, the mode of most

interest is the one with the largest and the smallest .. This is the mode that is closest to

TEM, and has the least decay, and is called mode 1. The software determines the various

modes requested, getting for each the fi eld pattern, and propagation constant. At this point,

the S-parameters can be determined. Power is injected into the mode(s), Maxwell’s equations

are solved, and the power distribution at all the ports in all the modes is determined. An

S-parameter is the ratio of two powers. For example, S11 is the ratio of the power coming back

into the fi rst port, divided by the power incident from the port. Notice that the power in which

mode must be specifi ed when this calculation is carried out.

Now the question, “where is the ground for this type of port?” can be answered. The answer is

that one really isn’t used! The modal distributions and powers are being addressed, and these

are electromagnetic concepts, so there is no need to discuss ground. The need for ground

occurs when trying to get the characteristic impedance for the mode. Recall that impedance

is necessary if the S-parameter fi le is going to be used in a circuit simulator. For a general

waveguide mode, there is no unique defi nition of impedance. Now this discussion will look at a

number of these defi nitions and see how ground is used. It has already been stated that the

power going down the waveguide is known. This can be uniquely determined by the electric and

magnetic fi elds across the waveguide. The current on the conductors is also known. The current

on any of the conductors can be found by realizing it is related to the tangential magnetic fi eld

next to the conductor (see Figure 5).

One reasonable defi nition of impedance uses the ratio of total current going down the waveguide

to power:

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Figure 5. The current in the conductor is found by using the tangential magnetic fi eld on the conductors.

γ = α + jβ.

β

β

α α

α

ZPV =V 2

2P.

Notice that the net power going down the waveguide must be zero; i.e., there

has to be power fl owing back down the guide equal and opposite to the current, I,

given in the equation for modal impedance. Why? The reason is shown in Figure

6. The magnetic and electric fi elds in the outer conductors are zero (as they are

perfect conductors). Therefore, the integral of the magnetic fi eld around the outer

conducting loop is zero. But Ampere’s law then says that the net current enclosed

by the loop (which is the waveguide) is zero. In the case of the microstrip line, for

example, the return current is on the outside conductor of the waveguide. And

this is what is assumed is ground, because the S-parameter port has the return

current come back on its ground. Furthermore, the current I in the equation for

impedance is the net current going down the guide on the interior conductors.

Unfortunately, there are two problems with this defi nition. First, it is not unique

as will be seen in a moment. Second, it is not always useful. Take the case of a

differential pair working in the differential mode, as shown in Figure 7.

The current in the two lines are equal and opposite. Therefore, the current I is

0, and the impedance of the mode is infi nite! This is certainly not a very useful

defi nition of impedance. Normally the differential pair example would be thought of

in terms of the voltage across the lines. Indeed, if the voltage between the signal

line and “ground” impedance could be defi ned as:

How is the voltage obtained? Voltage is the integral of the electric fi eld along

a path. Make the two end points of the path the ground conductor and the

signal conductor. Of course, it is also necessary to specify what path is wanted.

Remember that there will be a different answer for different paths, unless there

is a TEM mode. This defi nition of impedance solves the problem of the differential

pair. The voltage calibration line could be drawn from the side of the waveguide

to the positive signal line, as shown in Figure 7. This defi nition would give the odd

impedance. Alternatively, the line could be drawn between the two signal lines

giving the differential mode impedance, also shown in Figure 7.

There are other impedance defi nitions possible. For example the impedance

as some combination of the voltage and current defi nitions could be defi ned.

The geometric mean of the current and voltage impedance defi nitions could be

taken as the new defi nition of ground. The philosophy used here is that some

combination of our previous defi nitions gives a better average of the other

grounds. Which defi nition to use? There is no one right answer. For example,

the voltage defi nition for microstrip lines, with the voltage line drawn from the

signal line to the bottom of the waveguide might fi t better into our intuition of

what impedance should be. However it is chosen, the impedance of the mode

must be determined if the S-parameters are to be put into a circuit simulator.

There are other ports that can be used in a 3D simulator. These types of ports are

more closely related to the types of ports used in moment method simulators.

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Figure 6. The net current fl owing down the wave port is zero.

Figure 7. Odd and differential voltage, calibration lines for impedance.

ZPI =2P

I2.

MOMENT METHOD SIMULATORS AND PORTS

As has already been mentioned, moment methods work by solving for currents

on the conducting lines. The port is used to excite the currents, and to determine

the resulting S-parameters. The discussion begins by understanding the basic

way in which moment methods work. Figure 8 shows a microstrip bend. The

conductor is meshed into a series of rectangles. Triangles could also be used.

Triangles have the advantage of being able to go around curved lines better. On

each mesh, the current is approximated by a simple basis function. Typically, the

current is assumed to vary linearly from one end of the rectangle to the other.

Two rectangles together make a so called rooftop basis function.

The currents are changing spatially. Conservation of charge (as required by

Maxwell’s equations) says that current varying with distance requires there to

be a charge on the line. As the user goes down the line, there are alternating

positive and negative charges so that the total charge on the line is zero, as

is true for the real line. The charges and currents interact with each other.

For example, two charges have a capacitance between them. Two currents

have a mutual inductance between them. Furthermore, each current has a

self inductance between itself and ground, and each capacitance has a self

capacitance to ground. In this case the ground is the bottom plane beneath

the microstrip line.

Now the discussion will turn to how ports are used in a moment method

simulator. The port injects current into the EM simulator. This can be carried

out in a variety of ways. For example, Figure 9 pictures what is known as an

edge port, because it is at the end of the line. These ports can be viewed as

a voltage source in series with an impedance that injects current into the line.

The meshing for this example is also shown.

Note the narrow mesh along the edge of the line, called edge meshing. This

type of mesh has proven over the years to be accurate and effi cient when

modeling lines, as the current tends to concentrate at the edges of the line.

The ground for the port can be seen in the equivalent circuit in Figure 9 for

the edge port. The question is, where is this in the EM simulator? It depends

on the details of the port. Consider a few different cases.

The fi rst case is for MoM simulators in a box, for example EMSight™ or

Sonnet [6].

Figure 10 shows an edge port, which has been calibrated. (Calibration is

discussed below.) The port has been deembedded into the box, as shown

by the arrow from the port. The reference plane for the port (phase of the

incident wave is 0 degrees here) is at the end of arrow.

The voltage source is applied across a small gap with the sidewall of the box, which

is a perfect conductor. Current is injected into the line. The current comes from

the sidewall, which is the ground reference of the port.

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Figure 8. Moment methods mesh up the current as rooftop basis functions.

Figure 9. Edge port and the equivalend circuit in a MoM simulator. edge meshing is shown.

Figure 10. An edge port in a boxed simulator.

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Figure 11. The port is specifi cally attached by a current sheet to the bottom conductor (ground).

Figure 12. Edge port and the equivalend circuit in a MoM simulator. edge meshing is shown.

Figure 13. A coplanar line. The six ports are combined in the schematic to get the coplanar modes.

The second situation is a simulator with no box, for example AXIEM, Momentum

[7], or IE3D [8]. It is not nearly as obvious where the ground is for this port. The

answer depends on the how the port is implemented in the vendor’s software.

In AXIEM, the ground can be defi ned in a variety of ways. If implicit grounding is

used, the ground is actually at infi nity. At fi rst this does not sound like a useful

defi nition. However, recall that if the ground plane is a perfect conductor extending

to infi nity, it also will be at ground, as there is no voltage difference between it and

the point at infi nity. It is also possible to have the edge port connected by means of

a vertical current sheet to the metal above or below it.

Figure 11 shows the edge port with the current sheet inserted to make

connection to the bottom ground. In this situation, the bottom ground plane is the

defi nition of ground; it is where the current is coming from. Note that the box is

for graphing purposes only. AXIEM is not in a box.

In conclusion, MoM simulators support a variety of ports. Each of them has its

own ground defi nition. The easiest way to think of the ground is to appreciate that

the current going out of the port into the circuit is coming from the port’s ground.

DIFFERENTIAL AND COPLANAR PORTS

Signal integrity engineers are often interested in differential signals. Recall that

the ground for the two ports is at infi nity (with implicit grounding). In differential

operation, positive current goes into the fi rst port, and comes out the second

port. The fi rst edge port has current coming in from infi nity. The second edge

port has current going out to infi nity. The two currents going to infi nity are spatially

distributed on the ground plane almost identically once they are away from the

lines. They therefore will almost cancel perfectly, and the grounding at infi nity will

not lead to much error. Differential lines are used in systems to reduce noise due

to imperfections in the interconnect. The idea is that is whatever happens in one

line, also happens in the other. Essentially, the line carries it own ground return

with it; i.e. the other line. The differential mode is excited in the circuit simulators

by exciting the two ports with opposite polarities. This is shown in Figure 12, using

an MMCONV element, which is an easy way of generating a differential signal.

Coplanar lines have a signal line with two grounds, one on each side of the

line. Figure 13 shows the EM structure with six edge ports. Again, the implicit

ground for all ports is at infi nity. This particular geometry does not have a

ground plane underneath. The current at infi nity appears to have no way to

return to the port, unlike the case of the microstrip line with a well defi ned

ground plane. However, when the ports are excited properly in the schematic,

it is possible to excite the coplanar modes, as well as the unwanted odd

modes and radiation modes. The coplanar mode current contributions at

infi nity again cancel, leading to reasonable results, without the need of an

explicit ground plane underneath. This would not be possible if the implicit edge

port were forced to have its ground on a ground plane beneath it.

AWR AXIEM White Paper

Figure 14. A board to package transition with bond wires. The local ground is carried to the package.

CONCLUSION

It is important that the designer using S-parameters from an EM simulator

understand where the ground is in the EM simulator, because ground is an

important framework for how transmission lines are modeled and simulated.

S-parameters assume there is a common ground between the ports, at least if

they are to be used in a circuit simulator. This paper has discussed how circuit

simulators work with S-parameters, including the tricks of balanced ports and

exposed ground nodes. In order to understand grounding assumptions of the

EM simulator, one must start by looking at the ports. All ports have some kind of

grounding assumption. A good way to investigate this problem is to understand

where the current going into the port is coming from. 3D simulators use this

notion when they calculate the impedance of the port. Moment method simulators

excite ports by a delta voltage source, which must be connected to ground.

AWR’s unique AXIEM tool is an exceptional choice for these types of applications

because it provides true fl exibility that enbles the designer to specify where the

ground reference is in the design.

REFERENCES

[1] Computer Methods for Circuit Analysis and Design, J. Vlach and K. Singhai,

Kluwer Academic Publishers, Norwell, MA, 2003.

[2] “A general waveguide circuit theory”, R.B. Marks and D.F. Williams,

NIST Journal of Research, 97(5), pp. 533 – 562, 1992.

[3] Microwave Engineering, Third Edition, David Pozar,

John Wiley and Sons, NY, NY, 2005.

[4] The Finite Element Method in Electromagnetics, Second Edition, Jianming Jin,

John Wiley and Sons, NY, NY, 2002.

[5] Field Theory of Guided Waves, Second Edition, Robert E. Collin,

IEEE Press, NY, NY, 1999.

[6] Sonnet Software, Syracuse, NY, www.sonnetsoftware.com

[7] Momentum EM Simulator, Agilent Technologies, Santa Rosa, CA,

eesof.tm.agilent.com.

[8] IE3D EM Simulator, Zeland Software, Inc., www.zeland.com

[9] “De-embedding the effect of a local ground plane in electromagnetic analysis”,

James Rautio, Microwave Theory and Techniques, 53(2), pp. 770-776.

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