Theory of Operation for Momentum

Momentum is based on a numerical discretization technique called the method of moments. This technique is used to solve Maxwell's electromagnetic equations for planar structures embedded in a multilayered dielectric substrate. The simulation modes available in Momentum (microwave and RF) are both based on this technique, but use different variations of the same technology to achieve their results.

Momentum has two modes of operation, Microwave or full-wave mode and the RF or quasi-static mode. The main difference between these two modes lies within the Green functions formulations that are used. The full-wave mode uses full-wave Green functions, these are general frequency dependent Green functions that fully characterize the substrate without making any simplification to the Maxwell equations. This results in L and C elements that are complex and frequency dependent. The quasi-static mode uses frequency independent Green functions resulting in L and C elements that are real and frequency independent. Because of the approximation made in the quasi-static mode, the RF simulations run a lot faster since the matrix L and C elements only have to be calculated for the first frequency simulation point. The approximation also implies that the quasi-static mode typically should be used for structures that are smaller than half the wavelength. Both engine modes are using the so-called star-loop basis function, ensuring a stable solution at all frequencies. Both engines also make use of a mesh reduction algorithm which reduces the number of unknowns in the simulation by generating a polygonal mesh. This mesh reduction algorithm can be turned on or off, with a toggle switch.

The sources applied at the ports of the circuit yield the excitations in the equivalent network model. The currents in the equivalent network are the unknown amplitudes of the rooftop expansion functions. Solving the equivalent network for a number of independent excitation states yields the unknown current amplitudes. A port calibration process is used to calculate the S-parameter data of the circuit from the current solution, when calibration is requested.

The following sections in this chapter contain more information about:

The Method of Moments Technology

The method of moments (MoM) technique is based upon the work of R.F. Harrington, an electrical engineer who worked extensively on the method and applied it to electromagnetic field problems, in the beginning of the 1960's. It is based on older theory which uses weighted residuals and variational calculus. More detailed information on the method of moments and Green's theorem can be found in Field Computation by Moment Methods (1).
In the method of moments, prior to the discretization, Maxwell's electromagnetic equations are transformed into integral equations. These follow from the definition of suitable electric and magnetic Green's functions in the multilayered substrate.

In Momentum, a mixed potential integral equation (MPIE) formulation is used. This formulation expresses the electric and magnetic field as a combination of a vector and a scalar potential. The unknowns are the electric and magnetic surface currents flowing in the planar circuit.

Using notations from linear algebra, we can write the mixed potential integral equation in very general form as a linear integral operator equation:

(1)

Here, J(r) represents the unknown surface currents and E(r) the known excitation of the problem. The Green's dyadic of the layered medium acts as the integral kernel. The unknown surface currents are discretized by meshing the planar metalization patterns and applying an expansion in a finite number of subsectional basis functions B1(r), ..., BN(r):

(2)

The standard basis functions used in planar EM simulators are the subsectional rooftop functions defined over the rectangular, triangular, and polygonal cells in the mesh. Each rooftop is associated with one edge of the mesh and represents a current with constant density flowing through that edge as shown in the following illustration. The unknown amplitudes Ij, j=1,..,N of the basis function expansion determine the currents flowing through all edges of the mesh.

Discretization of the surface currents using rooftop basis functions.

The integral equation (1) is discretized by inserting the rooftop expansion (2) of the currents. By applying the Galerkin testing procedure, that is, by testing the integral equation using test functions identical to the basis functions, the continuous integral equation (1) is transformed into a discrete matrix equation:

for i=1,...,N

(3)

with

(4)

(5)

The left hand side matrix [Z] is called the interaction matrix, as each element in this matrix describes the electromagnetic interaction between two rooftop basis functions. The dimension N of [Z] is equal to the number of basis functions. The right-hand side vector [V] represents the discretized contribution of the excitations applied at the ports of the circuit.

The surface currents contribute to the electromagnetic field in the circuit by means of the Green's dyadic of the layer stack. In the MPIE formulation, this Green's dyadic is decomposed into a contribution from the vector potential A(r) and a contribution from the scalar potential V(r):

(6)

The scalar potential originates from the dynamic surface charge distribution derived from the surface currents and is related to the vector potential through the Lorentz gauge.
By substituting the expression (6) for the Green's dyadic in the expression (4) for the interaction matrix elements, yields the following form:

(7)

with

(8)

(9)

This allows the interaction matrix equation to be given a physical interpretation by constructing an equivalent network model see Figure A-2. In this network, the nodes correspond to the cells in the mesh and hold the cell charges. Each cell corresponds to a capacitor to the ground. All nodes are connected with branches which carry the current flowing through the edges of the cells. Each branch has in inductor representing the magnetic self coupling of the associated current basis function. All capacitors and inductors in the network are complex, frequency dependent and mutually coupled, as all basis functions interact electrically and magnetically see Figure A-3. The ground in this equivalent network corresponds with the potential at the infinite metallization layers taken up in the layer stack. In the absence of infinite metallization layers, the ground corresponds with the sphere at infinity. The method of moments interaction matrix equation follows from applying the Kirchoff voltage laws in the equivalent network. The currents in the network follow from the solution of the matrix equation and represent the amplitudes of the basis functions.

Figure A-2. The equivalent circuit is built by replacing each cell in the mesh with a capacitor to the ground reference and inductors to the neighboring cells.

Figure A-3. Equivalent network representation of the discretized MoM problem.

The Momentum Solution Process

Different steps and technologies enable the Momentum solution process:

Calculation of the Substrate Green's Functions

The substrate Green's functions are the spatial impulse responses of the substrate to Dirac type excitations. They are calculated for each pair of signal (strip, slot and/or via) layers mapped to a substrate level. Although it is necessary to know which signal layers are mapped to a substrate level, since only impulse responses are being calculated, it is not necessary to know the patterns on these signal layers. This implies that the Green's functions can be pre-calculated and stored in a substrate database. This allows the substrate Green's functions to be reused for other circuits defined on the same substrate.

The high frequency electromagnetic Green's functions depend upon the radial distance and the frequency. The computations are performed up to very large radial distances over the entire frequency band specified by the user. The frequency points are selected adaptively to ensure an accurate interpolation with respect to frequency. Computations performed over very wide frequency ranges can consume more CPU time and disk space to store the results. To increase speed, the RF mode uses quasi-static electromagnetic Green's functions based on low-frequency approximation and scales the quasi-static Green functions at higher frequencies.

Meshing of the Planar Signal Layer Patterns

The planar metallization (strip, via) and aperture (slot) patterns defined on the signal layers are meshed with rectangular and triangular cells in the microwave simulation mode. As translational invariance can be used to speed up the interaction matrix load process, the meshing algorithm will maximize the number if uniform rectangular cells created in the mesh. The meshing algorithm is very flexible as different parameters can be set by the user (number of cells/wavelength, number of cells/width, edge meshing, and mesh seeding), resulting in a mesh with different density. It is clear that the mesh density has a high impact on both the efficiency and accuracy of the simulation results. Default mesh parameters are provided which give the best accuracy/efficiency trade-off. Both the RF and microwave modes use mesh reduction technology to combine rectangular and triangular cells to produce a mesh of polygonal cells, thus reducing demand for computer resources. Mesh reduction eliminates small rectangles and triangles which from an electrical modeling point of view only complicate the simulation process without adding accuracy. As mentioned previously, this feature may be turned on or off with a toggle switch.

Loading and Solving of the MoM Interaction Matrix Equation

The loading step of the solution process consists of the computation of all the electromagnetic interactions between the basis functions and the filling of the interaction matrix and the excitation vector. The interaction matrix as defined in the rooftop basis is a dense matrix, that is, each rooftop function interacts with every other rooftop function. This electromagnetic interaction between two basis functions can either be strong or weak, depending on their relative position and their length scale. The matrix filling process is essentially a process of order (N 2 ), (i.e. the computation time goes up with the square of the number of unknowns).

In the solving step, the interaction matrix equation is solved for the unknown current expansion coefficients. The solution yields the amplitudes of the rooftop basis functions which span the surface current in the planar circuit. Once the currents are known, the field problem is solved because all physical quantities can be expressed in terms of the currents. With the release 2005A, an iterative matrix solve scheme was introduced in Momentum. For large problem sizes, the iterative matrix solve performs as an order (N2) process. Momentum still uses a direct matrix solve process if the structure is small or when convergence problems are detected in the iterative matrix solver process.

Calibration and De-embedding of the S-parameters

Momentum performs a calibration process on the single type port, the same as any accurate measurement system, to eliminate the effect of the sources connected to the transmission line ports in the S-parameter results. Feedlines of finite length (typically half a wavelength at high frequencies, short lines are used at low frequencies) are added to the transmission line ports of the circuit. Lumped sources are connected to the far end of the feedlines. These sources excite the eigenmodi of the transmission lines without interfering with the circuit. The effect of the feedlines is computed by the simulation of a calibration standard and subsequently removed from the S-parameter data. A built-in cross section solver calculates the characteristic impedance and propagation constant of the transmission lines. This allows to shift the phase reference planes of the S-parameters, a process called de-embedding. Results of the calibration process includes the elimination of low-order mode mismatches at the port boundary, elimination of high-order modes, and removal of all port excitation parasitics.

Besides transmission line ports, Momentum offers the user the ability to define direct excitation or internal ports. These ports can be specified at any location on the planar metallization patterns as either a point or a line feed. They allow to connect both passive and active lumped components to the distributed model of the planar circuits. The S-parameters associated with these ports are calculated from the excitation consisting of a lumped source connected to the equivalent network model at the locations of the internal ports. The parasitic effects of these lumped sources are not calibrated out of the S-parameters results.

Reduced Order Modeling by Adaptive Frequency Sampling

A key element to providing fast, highly accurate solutions using a minimum of computer resources is the Adaptive Frequency Sampling (AFS) technology. When simulating over a large frequency range, oversampling and straight line interpolation can be used to obtain smooth curves for the S-parameters. Oversampling however implies a huge amount of wasted resources. Momentum allows the user to benefit from a smart interpolation scheme based on reduced order modeling techniques to generate a rational pole/zero model for the S-parameter data. The Adaptive Frequency Sampling algorithm selects the frequency samples automatically and interpolates the data using the adaptively constructed rational pole/zero model. This feature allows important details to be modeled by sampling the response of the structure more densely where the S-parameters are changing significantly. It minimizes the total number of samples needed and maximizes the information provided by each new sample. In fact, all kinds of structure can take advantage of the AFS module. The Adaptive Frequency Sampling technology reduces the computation time needed for simulating large frequency ranges with Momentum significantly.

Special Simulation Topics

Some special simulation topics are discussed in this section:

Simulating Slots in Ground Planes

Slots in ground planes are treated in a special manner by Momentum. An electromagnetic theorem called the equivalence principle is applied. Instead of attempting to simulate the flow of electric current in the wide extent of the ground plane, only the electric field in the slot is considered. This electric field is modeled as an equivalent magnetic current that flows in the slot.
Momentum does not model finite ground plane metallization thickness. Ground planes and their losses are part of the substrate definition.

By using slot metallization definitions, entire structures, such as slot lines and coplanar waveguide circuits, can be built. Slots in ground planes can also be used to simulate aperture coupling through ground planes for multi-level circuits. Structure components on opposite sides of a ground plane are isolated from each other, except for intermediate coupling that occurs through the slots. This treatment of slots allows Momentum to simulate slot-based circuits and aperture coupling very efficiently.

Simulating Metallization Loss

When using Momentum, losses in the metallization patterns can be included in the simulation. Momentum can either treat the conductors as having zero thickness or include the effects of finite thickness in the simulation. In the substrate definition, the expansion of conductors to a finite thickness can be turned on/off for every layer. For more information, refer to Via Structures Limitation.

Momentum uses a complex surface impedance model for all metals that is a function of conductor thickness, conductivity, and frequency. At low frequencies, current flow will be approximately uniformly distributed across the thickness of the metal. Momentum uses this minimum resistance and an appropriate internal inductance to form the complex surface impedance. At high frequencies, the current flow is dominantly on the outside of the conductor and Momentum uses a complex surface impedance that closely models this skin effect. In the ADS 2005A release, with the introduction of horizontal currents on the side metallization of finite thickness conductors, conductors with arbitrary height/width ratio can be accurately modeled with Momentum. The surface impedance model for thick conductors includes mutual internal coupling for currents at the top and bottom plane of the thick conductor.

The meshing density can affect the simulated behavior of a structure. A more dense mesh allows current flow to be better represented and can slightly increase the loss. This is because a more uniform distribution of current for a low density mesh corresponds to a lower resistance.

Losses can be defined for ground planes defined in the substrate definition. This uses the same formulation as for loss in microstrips (i.e., through a surface impedance approximation). It should be noted however that since the ground planes in the substrate description are defined as infinite in size, only HF losses are incorporated effectively. DC losses are zero by definition in any infinite ground plane. DC metallization losses in ground planes can only be taken into account by simulating a finite size ground plane as a strip metallization level.

Note
For infinite ground planes with a loss conductivity specification, the MW mode of Momentum incorporates the HF losses in ground planes, however, the RF mode of Momentum will make an abstraction of these HF losses.

Simulating with Internal Ports and Ground References

Momentum offers the ability to use internal ports within a structure. Internal ports can be specified at any location on the planar metallization patterns, and they make possible a connection for both passive and active lumped components to the distributed model of the planar circuits. Refer to Figure A-4.

The S-parameters associated with these ports are calculated from the excitation consisting of a lumped voltage source connected to the equivalent network model, as shown in Figure A-3. The ground reference for these ports in the resulting S-parameter model is the ground of the equivalent network, and this ground corresponds physically to the infinite metallization layers taken up in the layer stack. In the absence of infinite metallization layers, the ground no longer has a physical meaning and corresponds mathematically with the sphere at infinity.

It is important to mention that in this case, the associated S-parameters also lose their physical meaning, as the applied voltage source is assumed to be lumped, that is electrically small, since it sustains a current flow from the ground to the circuit without phase delay. To overcome this problem, a ground reference must be specified at a distant electrically small from the internal port. Failure to do so may yield erroneous simulation results.

Figure A-4. Internal port and equivalent network model.

The following sections illustrate the use of internal ports with ground planes and with ground references, and the results.

Internal Ports and Ground Planes in a PCB Structure

Figure A-5 shows the layout for a PCB island structure with two internal ports. The infinite ground plane is taken up in the substrate layer stack and provides the ground reference for the internal ports. The magnitude and phase of the S21-parameter calculated with Momentum are shown in Figure A-6. The simulation results are validated by comparing them with the measured data for the magnitude and the phase of S21.

Figure A-5. PCB Substrate layer stack and metallization layer.

Figure A-6. Magnitude and phase of S21.

The thicker line is Momentum results, the thinner line is the measurement.

Finite Ground Plane, no Ground Ports

The same structure was resimulated with a finite groundplane (Figure A-7). The substrate layer stack contains no infinite metallization layers. No ground reference was specified for the ports, so the sphere at infinity acts as the ground reference for the internal ports. The S-parameters obtained from a simulation for these ports no longer have a physical meaning. Using such ports in the simulation yields incorrect simulation results as shown in Figure A-8.

Figure A-7. PCB substrate layer stack and metallization layers.

Figure A-8. Magnitude and phase of S21.

The thicker line is Momentum results, the thinner line is the measurement.

Finite Ground Plane, Internal Ports in the Ground Plane

One way to define the proper grounding for the internal ports is to add two extra internal ports in the ground plane (Figure A-9). The resulting four-port structure is simulated with Momentum. Note that because no ground reference is specified for the internal ports, the resulting S-parameters for the four port structure have no physical meaning. However, by properly recombining the four ports, a two-port structure is obtained with the correct ground references for each of the ports. Care should be taken with this recombination, as the ground reference for the four individual ports does not act as the ground reference for the two recombined ports. Therefore, the top recombination scheme shown in Figure A-10 is incorrect. The correct recombination scheme is the bottom one shown in Figure A-10.

Figure A-9. PCB substrate layer stack and metallization layers.

Figure A-10. Simulation results.

The bottom illustration shows the correct recombination of ports.

Finite Ground Plane, Ground References

The process of adding two extra internal ports in the ground plane and recombining the ports in the correct way can be automated in Momentum by defining two ground reference ports (Figure A-11). The S-parameters obtained with Momentum are identical to the S-parameters of the two-port after the correct recombination in the section above, also shown in Figure A-12.

Figure A-11. Substrate layer stack and metallization layers.

Figure A-12. Magnitude and phase of S21.

The thicker line is Momentum results, the thinner line is the measurement.

Internal Ports with a CPW Structure

Figure A-13 and Figure A-14 show the layout for a CPW step in width structure with an integrated sheet resistor (45 m x 50 m, 50!mom-15-1-24.gif!). Six internal ports are added to the metallization layers. The six-port S-parameter model is recombined into a two-port model to describe the CPW-mode S-parameters. Different recombination schemes are possible, yielding different simulation results. Scheme (c) is the correct one, as the ground reference for the internal ports is the sphere at infinity and this cannot act as the ground reference for the CPW ports.

Figure A-13. CPW step-in-width with integrated sheet resistor.

Figure A-14. CPW Step-in-width structure with integrated sheet resistor, port configurations and results.

The process of adding extra internal ports to the ground metallization patterns and recombining the ports in the correct way can be automated in Momentum by defining ground reference ports (Figure A-15). The S-parameters obtained with Momentum are identical to the S-parameters of the two-port for the correct recombination scheme (Figure A-14 (c)).

Figure A-15. CPW step-in-width structure with ground references for the internal ports.

Limitations and Considerations

This section describes some software limitations and physical considerations which need to be taken into account when using Momentum:

Comparing the Microwave and RF Simulation Modes

Momentum has two simulation modes: the Microwave and RF mode. The Microwave mode uses full-wave formulation, the RF mode uses a quasi-static formulation. In the quasi-static formulation, the Green functions are low-frequency approximations to the full-wave and more general Green functions. Because of the approximations made in the RF mode, the simulations are more efficient. The approximation is valid for structures that are small compared to the wavelength (size of the circuit smaller than half a wavelength).

Note
For infinite ground planes with a loss conductivity specification, the MW mode of Momentum incorporates the HF losses in ground planes, however, the RF mode of Momentum will make an abstraction of these HF losses.

Matching the Simulation Mode to Circuit Characteristics

The Momentum RF mode can be used to simulate RF and microwave circuits, depending on your requirements. However, Momentum RF is usually the more efficient mode when a circuit is electrically small, geometrically complex, and does not radiate. This section describes these characteristics.

Radiation

Momentum RF provides accurate electromagnetic simulation performance at RF frequencies. However, this upper limit depends on the size of your physical design. At higher frequencies, as radiation effects increase, the accuracy of the Momentum RF models declines smoothly with increased frequency.

Similarly, if the substrate allows the propagation of surface waves (these are guided waves that propagate in the substrate layers) the accuracy of the RF mode will gradually decline because surface waves are not included in the RF calculation.

Electrically Small Circuits

Momentum RF works best for electrically small circuits as its accuracy smoothly decreases with increasing electrical size relative to a given frequency. A circuit is considered electrically small relative to a given frequency if its physical dimension is smaller than half the wavelength of the frequency. Depending on which value you know, maximum circuit dimension or maximum simulation frequency, you can determine a qualitative approximation of the circuit's electrical size. Suppose you have a circuit with dimension D as shown in the following figure:

For space wave radiation, you can use one of the following two expressions strictly as a guideline to have an awareness about the circuit's electrical size relative to the maximum frequency you plan to run the simulation. When you know the value of D , use the first expression to approximate the maximum frequency up to which the circuit is electrically small. When you know the maximum simulation frequency, use the second expression to approximate the maximum allowable dimension:

or

where:
D = the maximum length in mm diagonally across the circuit.
F = the maximum frequency in GHz .
The following expression provides a guideline up to which frequency the substrate is electrically small for surface wave radiation.

During a simulation, Momentum RF calculates the maximum frequency up to which the circuit is considered electrically small, and displays that value in the status display. This is similar to using the expressions above since the dimension and thickness of a layout is typically fixed, and it is the simulation frequency that is swept.

Geometrically Complex Circuits

The mesh generated for a simulation establishes a geometric complexity for a circuit. A circuit is considered geometrically complex if its shape does not fit into a uniform, rectangular mesh, and the mesh generation produces a lot of triangles. Layouts containing shapes such as circles, arcs, and non-rectangular polygons usually result in meshes with many triangles. A measure of increasing geometric complexity is when the ratio of triangular cells to rectangular cells grows larger:

The mesh-reduction technology offered by Momentum RF and Microwave eliminates electromagnetically redundant rectangular and triangular cells. This reduces the time required to complete the simulation thus increasing efficiency. This mesh reduction algorithm can be turned on or off, using a toggle switch.

Higher-order Modes and High Frequency Limitation

Since Momentum does not account for higher-order modes in the calibration and de-embedding process, the highest frequency for which the calibrated and de-embedded S-parameters are valid is determined by the cutoff frequency of the port transmission line higher-order modes. As a rule of thumb for microstrip transmission lines, the cutoff frequency (in GHz) for the first higher-order mode is approximately calculated by:

Cutoff frequency fc = 0.4 Z0 / height where Z0 is the characteristic impedance of the transmission line. For a 10 mil alumina substrate with 50 ohm microstrip transmission line, we obtain a high frequency limit of approximately fc = 80 GHz.

Parallel Plate Modes

In the region between two infinite parallel plates or ground planes, parallel plate modes exist. Any current flowing in the circuit will excite all of these modes. How strong a mode will be excited depends on how well the field generated by the current source matches with the field distribution of the parallel plate mode.

A distinction can be made between the fundamental mode and the higher order modes. The fundamental mode, which has no cut-off frequency, propagates at any frequency. The higher order modes do have a cut-off frequency. Below this frequency, they decay exponentially and only influence the local (reactive) field around the source. Above this frequency, they propagate as well and may take real energy away from the source.

Momentum takes the effect of the fundamental mode into account. The higher order modes are taken into account as long as they are well below cut-off. If this is not the case, a warning is issued saying that higher-order parallel plate modes were detected close to their cut-off frequency. Simulation results will start to degrade from then on.

The Effect of Parallel Plate Modes

Let's concentrate on the fundamental mode since this one is always present. The fundamental mode has its electric field predominantly aligned along the Z-axis from the top plate to the bottom plate. This means that this mode creates a difference in the potential between both plates. This mode can be excited by any current in your circuit: electric currents on a strip or via, magnetic currents on a slot. It behaves as a cylindrical wave that propagates to infinity where it feels a short circuit (since both plates are connected there). However, since infinity is very far away, no reflected wave ever comes back.

In a symmetric strip line structure, the current on the strip won't excite the fundamental mode due to this symmetry. However, the presence of a feature such as a slot in one of the plates creates an asymmetry. The slot will excite the fundamental mode!

The accuracy of the Momentum results for calibrated ports degrades when the parallel plate modes become important. For calibrated ports is assumed that they excite the circuit via the fundamental transmission line mode. However, this excitation is not pure. The source will excite the circuit via the parallel plate modes, too. This effect cannot be calibrated out, since those contributions are not orthogonal. The consequence is that the excitation doesn't correspond with a pure fundamental transmission line mode excitation.

Avoiding Parallel Plate Modes

Since we know the field orientation of the parallel plate modes, it is easy to understand that you can short circuit them by adding vias to your structure at those places where parallel plate modes will be excited.

Similar problems will exist with the measurements. Both plates must be connected explicitly with each other! This won't take place at infinity but somewhere at the side of the board. If no vias are used in the real structure, the fundamental mode may be excited. This mode will propagate to the borders and reflect. Thus, the real structure differs in that from the simulations where no reflection will be seen since a reflection only happens at infinity.

One way to counter the difference between measurements in reality and simulation is to take into account the package (box) around the structure. Another possibility is adding vias to your structure, both in real life and in the simulations. Try to make the physical representation of your circuit as close as possible to reality. However, the actual shape of the via should not be that important. Small sheets can be used to represent the small circular vias that are realized in the real structure.

Surface Wave Modes

If the substrate is not closed (open or half open), and not homogeneous, surface wave modes can exist. The parallel plate mode can be seen as a special case of a surface wave mode. Their behavior is identical. Both are cylindrical waves that propagate radially away from the source. They are guided by the substrate. Both fundamental and higher order surface wave modes exist. Similar conclusions can be drawn with respect to limitations such as the effects of the modes.

Slotline Structures and High Frequency Limitation

The surface wave accuracy deterioration of the calibration process is somewhat more prominent for slotline transmission lines than for microstrip transmission lines. Therefore, slotline structures simulated with Momentum will exhibit a somewhat higher noise floor (10-20 dB higher) than microstrip structures. For best results, the highest simulation frequency for slotline structures should satisfy:

Substrate thickness < 0.15 * effective wavelength

Simulation of narrow slot lines are accurate at frequencies somewhat higher than the high frequency limit determined by this inequality, but wide slotlines will show deteriorated accuracy at this frequency limit.

Via Structures Limitation

In Momentum, the current flow on via structures is only allowed in the vertical direction; horizontal or circulating currents are not modeled. As via structures are metallization patterns, the modeling of metallization losses in via's is identical to the modeling of metallization loss in microstrip and slotline structures.

Via Structures and Substrate Thickness Limitation

The vertical electrical currents on via structures are modeled with rooftop basis functions. In this modeling, the vertical via structure is treated as one cell. This places an upper limit to the substrate layer thickness as the cell dimensions should not exceed 1/20 of a wavelength for accurate simulation results. Momentum simulations with via structures passing through electrically thick substrate layers will become less accurate at higher frequencies. By splitting the thick substrate layer into more than one layer, more via-cells are created and a more accurate solution is obtained.

CPU Time and Memory Requirements

Both CPU time and memory needed for a Momentum simulation increase with the complexity of the circuit and the density of the mesh. The size N of the interaction matrix equation is equal to the number of edges in the mesh. For calibrated ports, the number of unknowns is increased with the edges in the feedlines added to the transmission line ports.

CPU Time

The CPU time requirements for a Momentum simulation can be expressed as:

CPU time = A + B N + C N 2 + D N 3

where:

N = number of unknowns

A, B, C, D = constants independent of N

The constant term A accounts for the simulation set up time. The meshing of the structure is responsible for the linear term, BN. The loading of the interaction matrix is responsible for the quadratic term and the solving of the matrix equation accounts for:

It is difficult to predict the value of the constants A, B, C and D, because they depend on the problem at hand.

Memory Usage

The memory requirement for a Momentum simulation can be expressed as:

Memory = X + Y N + Z N 2

where

N = number of unknowns

X, Y, Z constants independent of N

Like with the CPU time expression, the constants X, Y and Z are difficult to predict for any given structure.

For medium to large size problems, the quadratic term, which accounts for storing of the interaction matrix, always dominates the overall memory requirement. For small structures, memory usage can also be dependent upon the substrate. The substrate database must be read and interpolated, which requires a certain amount of memory. Algorithms to make trade-off between time and memory resources are implemented in the simulator. These algorithms result in additional usage above that required to solve the matrix. Total memory consumption is typically less than 1.5 times what is required to store the matrix, for large matrices.

References

  1. R. F. Harrington, Field Computation by Moment Methods . Maxillan, New York (1968).
 

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