# EM.Ferma

**Electrostatic and Magnetostatic Solvers For DC And Low Frequency Simulations**

## Contents

- 1 Product Overview
- 2 EM.Ferma Features at a Glance
- 3 Building the Physical Structure in EM.Ferma
- 4 EM.Ferma's Computational Domain
- 5 EM.Ferma's Simulation Data & Observables
- 6 Discretizing the Physical Structure in EM.Ferma
- 7 Running Static Simulations in EM.Ferma
- 8 The 2D Electrostatic Simulation Mode

## Product Overview

### EM.Ferma in a Nutshell

**EM.Ferma** is a 3D static solver. It features two distinct electrostatic and magnetostatic simulation engines that can be used to solve a variety of static and low-frequency electromagnetic problems. Both simulation engines are based on finite difference solutions of Poisson's equation for electric and magnetic potentials.

With **EM.Ferma**, you can explore the electric fields due to volume charge distributions or fixed-potential perfect conductors, and magnetic fields due to wire or volume current sources and permanent magnets. Your structure may include dielectric or magnetic (permeable) material blocks. You can also use **EM.Ferma**'s 2D quasi-static mode to compute the characteristic impedance (Z0) and effective permittivity of transmission line structures with complex cross section profiles.

Click here to learn more about the ** Theory of Electrostatic and Magnetostatic Methods**.

### EM.Ferma as the Static Module of EM.Cube

**EM.Ferma** is the low-frequency **Static Module** of **EM.Cube**, a comprehensive, integrated, modular electromagnetic modeling environment. **EM.Ferma** shares the visual interface, 3D parametric CAD modeler, data visualization tools, and many more utilities and features collectively known as CubeCAD with all of EM.Cube's other computational modules.

Click here to learn more about ** EM.Cube Modeling Environment**.

### Advantages & Limitations of EM.Ferma's Static Simulator

**EM.Ferma** computes the electric and magnetic fields independent of each other based on electrostatic and magnetostatic approximations, respectively. As a result, any "electromagnetic" coupling effects or wave retardation effects are ignored in the simulation process. In exchange, static or quasi-static solutions are computationally much more efficient than the full-wave solutions of Maxwell's equations. Therefore, for low-frequency electromagnetic modeling problems or for simulation of sub-wavelength devices, **EM.Ferma** offers a faster alternative to EM.Cube's full-wave modules like EM.Tempo, EM.Picasso or EM.Libera. **EM.Ferma** currently provides a fixed-cell brick volume mesh generator. To model highly irregular geometries or curved objects, you may have to use very small cell sizes, which may lead to a large computational problem.

## EM.Ferma Features at a Glance

### Physical Structure Definition

- Metal (PEC) solids and surfaces in free space (Electrostatics)
- Dielectric objects in free space (Electrostatics)
- Magnetic (permeable) objects in free space (Magnetostatics)

### Sources

- Fixed-potential PEC for maintaining equi-potential metal objects
- Volume charge sources
- Volume current sources
- Wire current sources
- Permanent magnets

### Mesh generation

- Fixed-size brick cells

### 3D Electrostatic & Magnetostatic Simulation

- Finite difference solution of Laplace and Poisson equations for the electric scalar potential with Dirichlet and Neumann domain boundary conditions
- Finite difference solution of Laplace and Poisson equations for the magnetic vector potential with Dirichlet domain boundary conditions
- Calculation of electric scalar potential and electric field
- Calculation of magnetic vector potential and magnetic field
- Calculation of electric flux over user defined flux boxes and capacitance
- Calculation of magnetic flux over user defined flux surfaces and inductance
- Calculation of electric and magnetic energies, Ohmic power loss and resistance
- Parametric sweep with variable object properties or source parameters

### 2D Quasi-Static Simulation

- 2D Finite difference solution of cross section of transmission line structures
- 3D domain solution as well as 2D solution of a longitudinally infinite version of the structure defined on a 2D plane
- Calculation of electric potential and electric field distribution
- Parametric sweep of transmission line's geometric and material parameters
- Optimization of transmission line's parameters for impedance design

### Data Generation & Visualization

- Electric and magnetic field intensity and vector plots on planes
- Electric potential intensity plots
- Animation of field and potential plots after parametric sweeps
- Graphs of characteristic impedance and effective permittivity of transmission line structures vs. sweep variables
- Custom output parameters defined as mathematical expressions of standard outputs

## Building the Physical Structure in EM.Ferma

### Variety of Physical Objects in EM.Ferma

The simplest static problems involve a charge source in the free space that produces an electric field, or a current source in the free space that produces a magnetic field. In such cases, the only applicable boundary conditions are defined at the boundary of the computational domain. As soon as you introduce a dielectric object next to a charge source or a magnetic (permeable) material next to a current source, you have to deal with a complex boundary value problem. In other words, you need to solve the electric or magnetic Poisson equation subject to the domain boundary conditions as well as material interface boundary conditions. **EM.Ferma** uses the Finite Difference (FD) technique to find a numerical solution of your static boundary value problem.

**EM.Ferma** offers six types of physical objects:

Icon | Physical Object Type | Applications | Geometric Object Types Allowed | Notes & Restrictions |
---|---|---|---|---|

Fixed-Potential Perfect Electric Conductor (PEC) | Modeling perfect metals with a fixed voltage | Solid and surface objects | Can be considered an electric source if the fixed voltage is nonzero | |

Dielectric/Magnetic Material | Modeling any homogeneous material | Solid objects | non-source material | |

Volume Charge | Modeling volume charge sources with a fixed charge density or an expression in the global coordinates (x,y,z) | Solid objects | Acts as an electric source | |

Volume Current | Modeling volume current sources with a fixed volume current density vector or expressions in the global coordinates (x,y,z) | Solid objects | Acts as a magnetic source | |

Permanent Magnet | Modeling permanent magnet sources with a fixed magnetization vector or expressions in the global coordinates (x,y,z) | Solid objects | Acts as a magnetic source | |

Wire Current | Modeling wire current sources | line and polyline objects | Acts as a magnetic source |

Click on each category to learn more details about it in the Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types.

### Grouping Objects by Material or Source Type

Your physical structure in **EM.Ferma** is typically made up of some kind of source object either in the free space or in the presence of one or more material objects. **EM.Ferma**'s electrostatic and magnetostatic simulation engines then discretize the entire computational domain including these source and material objects and solve the Laplace or Poisson equations to find the electric or magnetic fields everywhere in the computational domain.

All the geometric objects in the project workspace are organized together into object groups which share the same properties including color and electric or magnetic parameters. It is recommended that you first create object groups, and then draw new objects under the active group. To create a new object group, right-click on its category name in the "Physical Structure" section of the navigation tree and select one of the "Insert New Group..." items from the contextual menu. However, if you start a new **EM.Ferma** project from scratch, and start drawing a new object without having previously defined any object groups, a new default "Fixed-Potential PEC" object group with a zero voltage is created and added to the navigation tree to hold your new geometric object.

Once a new object group node has been created in the navigation tree, it becomes and remains the "Active" object group, which is always listed in bold letters. When you draw a new geometric object such as a box or a sphere, its name is added under the currently active object group. There is only one object group that is active at any time. Any group can be made active by right-clicking on its name in the navigation tree and selecting the **Activate** item of the contextual menu.

Click here to learn more about ** Moving Objects among Different Groups**.

### A Note on Material and Source Types in EM.Ferma

In EM.Cube's other modules, material types are categorized under the "Physical Structure" section of the navigation tree, and sources are organized under a separate "Sources" section. In those modules, all the geometric objects you draw in your project workspace typically represent material bodies. All of EM.Cube modules except for **EM.Ferma** require at least one excitation source to be selected from the "Sources" section of the navigation tree before you can run a simulation.

In **EM.Ferma**, materials and sources are all lumped together and listed under the "Physical Structure" section of the navigation tree. In other words, there is no separate "Sources" section. For example, you can define default zero-potential perfect electric conductors (PEC) in your project to model metal objects. You can also define fixed-potential PEC objects with a nonzero voltage, which can effectively act as a voltage source for your boundary value problem. In this case, you will solve the Lapalce equation subject to the specified nonzero potential boundary values. Both types of PEC objects are defined from the same PEC node of the navigation tree by assigning different voltage values. Charge and current sources are also defined as geometric objects, and you have to draw them in the project workspace just like other material objects.

## EM.Ferma's Computational Domain

### The Domain Box

In **EM.Ferma**, the Poisson or Laplace equations are solved subject to boundary conditions using the Finite Difference technique. As a result, you need to specify a finite computational domain and then specify the domain boundary conditions. **EM.Ferma**'s computational domain defines where the domain boundary condition will be specified. A default domain box is always placed in the project workspace as soon as you draw your first object. The domain can be seen as a blue cubic wireframe that surrounds all of the CAD objects in the project workspace.

To modify the domain settings, click the Domain button of the Simulate Toolbar or right-click on "3D Static Domain" entry in the Navigation Tree and select "Domain Settings..." from the contextual menu. In the Domain Settings Dialog, the computational domain can be defined in two different ways: Default and Custom. The default type places an enclosing box with a specified offset from the largest bounding box of your project's CAD objects. The default offset value is 20 project units, but you can change this value arbitrarily. The custom type defines a fixed domain box by specifying the coordinates of its two opposite corners labeled Min and Max in the world coordinate system.

### Domain Boundary Conditions

**EM.Ferma** allows you to specify the electric potential boundary conditions on the domain box. Two options are available. The Dirichlet boundary condition is the default option and is specified as a fixed potential value on the surface of the domain walls. By default, this value is 0 Volts. The Neumann boundary condition specifies the normal derivative of the electric scalar potential on the surface of the domain walls. This is equivalent to the normal electric field component on the domain walls and its value is specified in V/m. The magnetostatic simulation engine always assumes Dirichlet domain boundary conditions and sets the values of the magnetic vector potential to zero on all the domain walls. To modify the boundary conditions, right-click on "Boundary Conditions" in the Navigation Tree, and select "Boundary Conditions..." from the contextual menu to open the Boundary Conditions Dialog.

## EM.Ferma's Simulation Data & Observables

At the end of an electrostatic simulation, the electric field vector and electric scalar potential values are computed at all the mesh grid points of the entire computational domain. At the end of an magnetostatic simulation, the magnetic field vector and magnetic vector potential values are computed at all the grid nodes. Besides the electric and magnetic fields, **EM.Ferma** can compute a number of field integral quantities such as voltage, current, flux, energy, etc. The field components, potential values and field integrals are written into output data files and can be visualized on the screen or graphed in EM.Grid only if you define a field sensor or a field integral observable. In the absence of any observable defined in the navigation tree, the static simulation will be carried out and completed, but no output simulation data will be generated.

**EM.Ferma** offers the following types of output simulation data:

Icon | Simulation Data Type | Observable Type | Applications |
---|---|---|---|

Near-Field Distribution Maps | Near-Field Sensor | Computing electric and magnetic field components, electri scalar potential and magnitude of magnetic vector potential on a planar cross section of the computational domain | |

Field Integral Quantities | Field Integral | Computing line, surface and volume integrals of the electric and magnetic fields |

Click on each category to learn more details about it in the Glossary of EM.Cube's Simulation Observables & Graph Types.

The table below list the different types of field integrals and their definitions:

Field Integral | Definition | Output Data File |
---|---|---|

Voltage | [math] V = - \int_C \mathbf{E(r)} . \mathbf{dl} [/math] | voltage.DAT |

Current | [math] I = \oint_{C_o} \mathbf{H(r)} . \mathbf{dl} [/math] | current.DAT |

Conduction Current | [math] I_{cond} = \int\int_S \mathbf{J(r)} . \mathbf{ds} = \int\int_S \sigma \mathbf{E(r)} . \mathbf{ds} [/math] | conduction_current.DAT |

Electric Flux | [math] \Phi_E = \int\int_{S_o} \mathbf{D(r)} . \mathbf{ds} = \int\int_{S_o} \epsilon \mathbf{E(r)} . \mathbf{ds} [/math] | flux_E.DAT |

Magnetic Flux | [math] \Phi_H = \int\int_S \mathbf{B(r)} . \mathbf{ds} = \int\int_S \mu \mathbf{H(r)} . \mathbf{ds} [/math] | flux_H.DAT |

Electric Energy | [math] W_E = \frac{1}{2} \int \int \int_V \epsilon \vert \mathbf{E(r)} \vert ^2 dv [/math] | energy_E.DAT |

Magnetic Energy | [math] W_H = \frac{1}{2} \int\int\int_V \mu \vert \mathbf{H(r)} \vert ^2 dv [/math] | energy_H.DAT |

Ohmic Power Loss | [math] P_{ohmic} = \int\int\int_V \sigma \vert \mathbf{E(r)} \vert ^2 dv [/math] | ohmic.DAT |

Resistance | [math] R = V/I_{cond} = - \int_C \mathbf{E(r)} . \mathbf{dl} / \int\int_S \sigma \mathbf{E(r)} . \mathbf{ds} [/math] | resistance.DAT |

Capacitance | [math] C = \Phi_E/V = \int\int_{S_o} \epsilon \mathbf{E(r)} . \mathbf{ds} / \int_C \mathbf{E(r)} . \mathbf{dl} [/math] | capacitance.DAT |

Self-Inductance | [math] L = \Phi_H/I = \int\int_S \mu \mathbf{H(r)} . \mathbf{ds} / \oint_{C_o} \mathbf{H(r)} . \mathbf{dl} [/math] | inductance.DAT |

Mutual Inductance | [math] M = \Phi_H^{\prime}/I = \int\int_{S^{\prime}} \mu \mathbf{H(r)} . \mathbf{ds} / \oint_{C_o} \mathbf{H(r)} . \mathbf{dl} [/math] | inductance.DAT |

## Discretizing the Physical Structure in EM.Ferma

### The Static Mesh

The Finite Difference technique discretizes the computational domain using a 3D rectangular grid. **EM.Ferma** generates a fixed-cell mesh. This means that the extents of the mesh cells along the principal axes are fixed: Δx, Δy, Δz. By default, the mesh cell size is set to one unit project along all the three directions (with Δx = Δy = Δz). To modify the cell size, click the Mesh Settings button of the Simulate Toolbar or right-click on "Static Mesh" in the Navigation Tree, and select "Mesh Settings..." from the contextual menu to open the Mesh Settings Dialog.

To obtain accurate results, it is highly recommended to use a square mesh as much as possible. |

Click here to learn more about ** Working with Mesh Generator**.

Click here to learn more about the properties of ** EM.Ferma's Fixed-Cell Brick Mesh Generator**.

## Running Static Simulations in EM.Ferma

### EM.Ferma's Simulation Modes

**EM.Ferma** currently offers three different simulation modes as follows:

Simulation Mode | Usage | Number of Engine Runs | Frequency | Restrictions |
---|---|---|---|---|

Analysis | Simulates the physical structure "As Is" | Single run | N/A | None |

Parametric Sweep | Varies the value(s) of one or more project variables | Multiple runs | N/A | None |

Optimization | Optimizes the value(s) of one or more project variables to achieve a design goal | Multiple runs | N/A | None |

### Running an Electrostatic or Magnetostatic Analysis

**EM.Ferma** has two independent but functionally similar static simulation engines: Electrostatic and Magnetostatic. The electrostatic engine solves the electric form of Poisson's equation for electric scalar potential subject to electric field boundary conditions, in the presence of electric sources (volume charges and fixed-potential PEC blocks) and dielectric material media. The magnetostatic engine solves the magnetic form of Poisson's equation for magnetic vector potential subject to magnetic field boundary conditions, in the presence of magnetic sources (wire and volume currents and permanent magnetic blocks) and magnetic material media.

In **EM.Ferma** you don't have to select any specific simulation engine. The program looks at the types sources and material objects present in your project workspace and then it determines whether an electrostatic analysis or a magnetostatic analysis or possibly both should be performed. When there are only electric sources present, you will get nonzero electric fields and zero magnetic fields. When there are only magnetic sources present, you will get nonzero magnetic fields and zero electric fields.

An "Analysis" is the simplest simulation mode of **EM.Ferma**. It is a single-shot finite difference solution of your static problem. The physical structure of your project workspace is first discretized using a fixed-cell mesh and the Poisson equation is solved numerically everywhere in the computational domain. The field and potential values at each mesh node are computed, and the specified observables are written into data files.

To run a static simulation, first you have to open the Run Dialog. This is done by clicking the "Run" button of the Simulate Toolbar, or by selecting the "Run" item of the Simulate Menu, or simply using the keyboard shortcut "Ctrl+R". The only available simulation engine is "Static". Clicking the Run button of this dialog starts a static analysis. A separate window pops up which reports the progress of the current simulation.

### Static Simulation Engine Settings

**EM.Ferma** currently uses a single iterative linear system solver based on the stabilized Bi-Conjugate Gradient (BiCG) method to solve the matrix equations which result from the discretization of Poisson's equation. You can specify some numerical parameters related to the Bi-CG solver. To do that, you need to open the Simulation Engine Settings Dialog by clicking the "Settings" button located next to the "Select Engine" drop-down list. From this dialog you can set the maximum number of BiCG iterations, which has a default value of 10,000. You can also set a value for "Convergence Error". The default value for electrostatic analysis is 0.001. For magnetostatic analysis, the specified value of convergence error is reduced by a factor 1000 automatically. Therefore, the default convergence error in this case is 1e-6.

## The 2D Electrostatic Simulation Mode

### Using EM.Ferma to Simulate 2D Transmission Lines

You can use **EM.Ferma** to perform a quasi-static analysis of multi-conductor transmission line structures, which usually provides good results at lower microwave frequencies (f < 10GHz). **EM.Ferma** computes the characteristics impedance Z_{0} and effective permittivity ε_{eff} of your TEM or quasi-TEM transmission line. The "quasi-static approach" involves two steps as described above, which **EM.Ferma**'s 2D Quasi-Static mode automatically performs to calculate ε_{eff} and Z_{0}.

To perform a transmission line simulation, first draw your structure in the project workspace just like a typical 3D structure. Define a "Field Sensor" observable in the navigation tree so as to capture the cross section of your structure as your desired transmission line profile.

Next, define a "2D Solution Plane" in the navigation tree based on your existing field sensor. When defining the 2D plane, check the box labeled "Perform 2D Quasi-Static Simulation". If an analysis is run with this option checked, the characteristic impedance Z_{0} and effective permittivity ε_{eff} will be computed for the corresponding 2D Solution Plane. The results are written to two output data files named "solution_plane_Z0.DAT" and "solution_plane_EpsEff.DAT", respectively, where "solution_plane" is the default name of your 2D plane.

Many 2D quasi-static solutions can be obtained in the same analysis, for example, when your design contains many types of transmission lines. At the end of a quasi-static analysis, the electric field components and scalar potential at the selected 2D planes will still be computed and can be visualized. In the case of a parametric sweep, the data files will contain multiple data entries listed against the corresponding variable samples. Such data files can be plotted in EM.Grid.

**EM.Ferma**'s electrostatic simulation engine features a 2D solution mode where your physical model is treated as a longitudinally infinite structure in the direction normal to specified "2D Solution Plane". More than one 2D solution plane may be defined. In that case, multiple 2D solutions are obtained. A 2D solution plane is defined based on a "Field Sensor" definition that already exists in your project.

To explore **EM.Ferma**'s 2D mode, right-click on **2D Solution Planes** in the "Computational Domain" section of the navigation tree and select **2D Domain Settings...** from the contextual menu. In the 2D Static Domain dialog, check the checkbox labeled "Treat Structure as Longitudinally Infinite across Each 2D Plane Specified Below". This would enable you to add new 2D Solution Plane definitions to the list or edit the existing ones. In the Add/Edit 2D Solution Plane dialog, you can choose a name other than the default name and select one of the available field sensor definitions in your project. At the end of a 2D electrostatic analysis, you can view the electric field and potential results on the respective field sensor planes. It is assumed that your structure is invariant along the direction normal to the 2D solution plane. Therefore, your computed field and potential profiles must be valid at all the planes perpendicular to the specified longitudinal direction.

You can also use **EM.Ferma** to perform a quasi-static analysis of multi-conductor transmission line structures, which usually provides good results at lower microwave frequencies (f < 10GHz). For that purpose, check the box labeled "Perform 2D Quasi-Static Simulation" when defining the 2D solution plane. **EM.Ferma** computes the characteristics impedance Z_{0} and effective permittivity ε_{eff} of your TEM or quasi-TEM transmission line. The results are written to two output data files named "solution_plane_Z0.DAT" and "solution_plane_EpsEff.DAT", respectively, where "solution_plane" is the default name of your 2D plane. At the end of a quasi-static analysis, the electric field components and scalar potential at the selected 2D planes will still be computed and can be visualized. In the case of a parametric sweep, the data files will contain multiple data entries listed against the corresponding variable samples. Such data files can be plotted in EM.Grid.

Click here to learn more about the theory of ** 2D Quasi-Static Analysis of Transmission Lines**.