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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 definition 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 definitions. New features in AWR’s AXIEM™ 2009 3D planar EM simulator
offer extensive sources/ports and de-embedding options, including internal edge,
finite 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 first
necessary to understand how it works in circuit simulation. The major portion
of the discussion involves examining where ground is defined in EM simulators
Understanding
Grounding
Concepts in EM
Simulators
and how this is a function of the solver and port types. The paper demonstrates
how this flexibility in selecting ground can help the engineer characterize board,
package, and interconnect performance. Several specific 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 flexibility 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
Dr. John M. Dunn
AWR Corporation
as ground. In more complex situations, for instance, with coplanar waveguide,
[email protected]
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 flexibility 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.
BIO:
John Dunn is a senior engineering consultant at AWR, where he is in charge of
training and university program development. 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 fifteen 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, Northfield,
MN. He is a senior member of IEEE.
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GROUND IN CIRCUIT SIMULATION
Ground is probably one of the first 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 difficulty. 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 fields, one
obtains the equation for conservation of charge, which states:
I = jωQ,
i
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:
Q = CV.
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 fields, 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 field energy can be defined in a way that is consistent with
the definition of energy from mechanics.
Voltage is defined to be the integral of the electric field along a path (actually the tangential
part of the electric field to the path). The immediate question is why is this important? The
reason is Faraday’s law, which states that the electric field around any closed path is equal to
the change in time of the magnetic flux through that loop (with a minus sign):
· d = − dΦ ,
E
dt
where E is the electric field, Φ is the total flux of the magnetic field 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 flux through the loop (with a minus sign). Circuit theory makes the
assumption that the changing magnetic flux through the loop is small enough that we can
assume it is zero. And if it’s zero, it means that the electric field around any loop, i.e. the
voltage around any loop, is zero, which is Kirchoff’s voltage law.
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There is of course one place where the magnetic flux change is large enough that it can’t be
neglected – at an inductor. This is the definition of inductance:
Φ = LI,
where the flux Φ is produced by the current I. To finish 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 specifically 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 modified 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:
Y = (I + S)−1 Zg−1 (I − S),
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 reflected power back into the port and the transmitted power
going into the other ports. The ratio of the reflected or transmitted power to the incident
power is the definition of the S-parameter. It is assumed that all ports are perfectly matched
so that any wave going into the port is not re-reflected 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 defined for these
ports. As previously argued, every S-parameter file 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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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 define 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 classified 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
Figure 1. Classifications of EM simulators.
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 Office®
software, a variety of these models are available. They can be grouped into two
general classes: moment method techniques and finite element techniques. The
moment method models are shown in Figure 2.
The technique solves for the currents and charges on the surfaces of the
Figure 2. 2D method-of-moments (MoM) EM
simulators for determining line parameters.
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 Office is a
finite element method (FEM) solver. This is shown In Figure 3.
Figure 3. FEM cross sectional solve for determining line
parameters.
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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 field 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 satisfied. 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 finite 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 fields
are the FEM, which solves in frequency domain, and the finite 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 finite elements are used.
There are FEM methods in the time domain. Since this discussion focuses on how grounds are
defined 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 fictitious
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 fields 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 field has a simple, approximate form for each region, for example
linear variation in all three spatial directions. The electric fields are excited by means of a port,
which is the focus of this chapter. The fields 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 finite 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
reflects, 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
Figure 4. Meshed wave port at the end of a
microstrip line.
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 field pattern, their characteristic impedance, and their propagation constant. Details on
each of these three features follows.
The field 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 figure 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 fields in the direction of propagation because of the dielectric/air interface.)
The current flows 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 reflection 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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How are the S-parameters at these ports determined? The simulator first 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 finite 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:
γ = α + jβ.
The β is the phase constant =
. The α is the decay constant in Np/m. A non-zero α
means that the wave is decaying as it goes down the guide. Normally, the mode of most
interest is the one with the largest β and the smallest α.. 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 field 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 first port, divided by the power incident from the port. Notice that the power in which
mode must be specified 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 file is going to be used in a circuit simulator. For a general
waveguide mode, there is no unique definition of impedance. Now this discussion will look at a
number of these definitions 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 fields 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 field
next to the conductor (see Figure 5).
Figure 5. The current in the conductor is found by using the tangential magnetic
field on the conductors.
One reasonable definition of impedance uses the ratio of total current going down the waveguide
to power:
ZP V =
V2
.
2P
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Notice that the net power going down the waveguide must be zero; i.e., there
has to be power flowing 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 fields in the outer conductors are zero (as they are
perfect conductors). Therefore, the integral of the magnetic field 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.
Figure 6. The net current flowing down the wave
port is zero.
Unfortunately, there are two problems with this definition. 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 infinite! This is certainly not a very useful
definition 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 defined as:
ZP I =
2P
.
I2
How is the voltage obtained? Voltage is the integral of the electric field 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 definition 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 definition 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 definitions possible. For example the impedance
as some combination of the voltage and current definitions could be defined.
The geometric mean of the current and voltage impedance definitions could be
taken as the new definition of ground. The philosophy used here is that some
combination of our previous definitions gives a better average of the other
grounds. Which definition to use? There is no one right answer. For example,
the voltage definition for microstrip lines, with the voltage line drawn from the
signal line to the bottom of the waveguide might fit 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.
Figure 7. Odd and differential voltage, calibration
lines for impedance.
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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
Figure 8. Moment methods mesh up the current as
rooftop basis functions.
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
Figure 9. Edge port and the equivalend circuit in a
MoM simulator. edge meshing is shown.
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 efficient 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 first 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.
Figure 10. An edge port in a boxed simulator.
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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 defined in a variety of ways. If implicit grounding is
used, the ground is actually at infinity. At first this does not sound like a useful
definition. However, recall that if the ground plane is a perfect conductor extending
to infinity, it also will be at ground, as there is no voltage difference between it and
the point at infinity. 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
Figure 11. The port is specifically attached by a
current sheet to the bottom conductor
(ground).
connection to the bottom ground. In this situation, the bottom ground plane is the
definition 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 definition. 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
Figure 12. Edge port and the equivalend circuit in a
MoM simulator. edge meshing is shown.
Signal integrity engineers are often interested in differential signals. Recall that
the ground for the two ports is at infinity (with implicit grounding). In differential
operation, positive current goes into the first port, and comes out the second
port. The first edge port has current coming in from infinity. The second edge
port has current going out to infinity. The two currents going to infinity 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 infinity 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 infinity. This particular geometry does not have a
ground plane underneath. The current at infinity appears to have no way to
return to the port, unlike the case of the microstrip line with a well defined
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
infinity 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.
Figure 13. A coplanar line. The six ports are
combined in the schematic to get the
coplanar modes.
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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.
Figure 14. A board to package transition with bond
wires. The local ground is carried to the
package.
AWR’s unique AXIEM tool is an exceptional choice for these types of applications
because it provides true flexibility 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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