Krasnikov G.E., Nagornov O., Starostin N.V. Simulation of physical processes using the Comsol Multiphysics package

Electrical cables are characterized by parameters such as impedance and attenuation coefficient. This topic will consider an example of modeling a coaxial cable, for which there is an analytical solution. We'll show you how to calculate cable parameters from electromagnetic field simulations in COMSOL Multiphysics. Having dealt with the principles of building a model of a coaxial cable, in the future we will be able to apply the knowledge gained to calculate the parameters of transmission lines or cables of an arbitrary type.

Electrical cable design issues

Electrical cables, also called transmission lines, are now widely used for the transmission of data and electricity. Even if you are reading this text from a screen on a cell phone or tablet computer using a “wireless” connection, there are still “wired” power lines inside your device connecting various electrical components into a single whole. And when you return home in the evening, you will most likely connect the power cable to the device for charging.

A wide variety of power lines are used, from small, made in the form of coplanar waveguides on printed circuit boards, to very large high-voltage power lines. They must also function in various and often extreme modes and operating conditions, from transatlantic telegraph cables to electrical wiring on spacecraft, the appearance of which is shown in the figure below. Transmission lines must be designed with all the necessary requirements in mind to ensure their reliable operation under given conditions. In addition, they can be the subject of research in order to further optimize the design, including meeting the requirements for mechanical strength and low weight.

Connecting wires in the cargo hold of the OV-095 shuttle mock-up at the Shuttle Avionics Integration Laboratory (SAIL).

When designing and using cables, engineers often work with distributed (or specific, i.e. per unit length) parameters for series resistance (R), series inductance (L), shunt capacitance (C), and shunt conductance (G, sometimes called insulation conductivity). These parameters can be used to calculate the quality of the cable, its characteristic impedance and losses in it during signal propagation. However, it is important to keep in mind that these parameters are found from the solution of Maxwell's equations for the electromagnetic field. To solve Maxwell's equations numerically to calculate electromagnetic fields, as well as to take into account the influence of multiphysics effects, you can use the COMSOL Multiphysics environment, which will allow you to determine how the parameters of the cable and its efficiency change under various operating modes and operating conditions. The developed model can then be converted into an intuitive application like this one, which calculates the parameters for standard and commonly used transmission lines.

In this topic, we will consider the case of coaxial cable - a fundamental problem that is usually contained in any standard curriculum on microwave technology or power lines. The coaxial cable is such a fundamental physical entity that Oliver Heaviside patented it in 1880, just a few years after Maxwell formulated his famous equations. For students of the history of science, this is the same Oliver Heaviside, who first formulated Maxwell's equations in the vector form that is now generally accepted; the one who first used the term "impedance"; and who made a significant contribution to the development of the theory of power lines.

Results of analytical solution for coaxial cable

Let's start our consideration with a coaxial cable, which has the characteristic dimensions indicated on the schematic representation of its cross section, presented below. The dielectric core between the inner and outer conductor has a relative permittivity ( \epsilon_r = \epsilon"-j\epsilon"") equal to 2.25 – j*0.01, relative magnetic permeability (\mu_r ) equal to 1 and zero conductivity, while the inner and outer conductors have a conductivity (\sigma ) equal to 5.98e7 S/m (Siemens/meter).

2D cross-section of a coaxial cable with characteristic dimensions: a = 0.405 mm, b = 1.45 mm, and t = 0.1 mm.

The standard solution for power lines is that the structure of electromagnetic fields in the cable is assumed to be known, namely, it is assumed that they will oscillate and attenuate in the direction of wave propagation, while in the transverse direction the field section profile remains unchanged. If then we find a solution that satisfies the original equations, then by virtue of the uniqueness theorem, the solution found will be correct.

In mathematical language, all of the above is equivalent to the fact that the solution of Maxwell's equations is sought in the form ansatz-forms

for an electromagnetic field , where (\gamma = \alpha + j\beta ) is the complex propagation constant, and \alpha and \beta are the damping and propagation coefficients, respectively. In cylindrical coordinates for coaxial cable, this leads to the well-known field solutions

\begin(align)
\mathbf(E)&= \frac(V_0\hat(r))(rln(b/a))e^(-\gamma z)\\
\mathbf(H)&= \frac(I_0\hat(\phi))(2\pi r)e^(-\gamma z)
\end(align)

from which the distributed parameters per unit length are then obtained

\begin(align)
L& = \frac(\mu_0\mu_r)(2\pi)ln\frac(b)(a) + \frac(\mu_0\mu_r\delta)(4\pi)(\frac(1)(a)+ \frac(1)(b))\\
C& = \frac(2\pi\epsilon_0\epsilon")(ln(b/a))\\
R& = \frac(R_s)(2\pi)(\frac(1)(a)+\frac(1)(b))\\
G& = \frac(2\pi\omega\epsilon_0\epsilon"")(ln(b/a))
\end(align)

where R_s = 1/\sigma\delta is the surface resistance, and \delta = \sqrt(2/\mu_0\mu_r\omega\sigma) is an .

It is extremely important to emphasize that the relationships for capacitance and shunt conductance hold for any frequency, while the expressions for resistance and inductance depend on the skin depth and, therefore, are applicable only at frequencies at which the skin depth is much less than the physical thickness. conductor. That is why the second term in the expression for inductance, also called internal inductance, may be unfamiliar to some readers, as it is usually neglected when metal is considered an ideal conductor. This term is the inductance caused by the penetration of a magnetic field into a metal of finite conductivity and is negligible at sufficiently high frequencies. (It can also be represented as L_(Internal) = R/\omega .)

For subsequent comparison with the numerical results, the ratio for the DC resistance can be calculated from the expression for the conductivity and the cross-sectional area of the metal. The analytical expression for inductance (with respect to direct current) is a little more complicated, and therefore we include it here for reference.

L_(DC) = \frac(\mu)(2\pi)\left\(ln\left(\frac(b+t)(a)\right) + \frac(2\left(\frac(b) (a)\right)^2)(1- \left(\frac(b)(a)\right)^2)ln\left(\frac(b+t)(b)\right) – \frac( 3)(4) + \frac(\frac(\left(b+t\right)^4)(4) – \left(b+t\right)^2a^2+a^4\left(\frac (3)(4) + ln\frac(\left(b+t\right))(a)\right) )(\left(\left(b+t\right)^2-a^2\right) ^2)\right\)

Now that we have the C and G values over the entire frequency range, the DC values for R and L, and their asymptotic values in the high frequency region, we have excellent benchmarks to compare with the numerical results.

Modeling cables in an AC/DC module

When formulating a problem for numerical simulation, it is always important to consider the following point: is it possible to use the symmetry of the problem to reduce the size of the model and increase the speed of calculations. As we saw earlier, the exact solution will be \mathbf(E)\left(x,y,z\right) = \mathbf(\tilde(E))\left(x,y\right)e^(-\gamma z). Since the spatial change of fields of interest to us occurs primarily in xy-plane, then we only want to model the 2D cross section of the cable. However, this raises a problem, which is that for the 2D equations used in the AC/DC module, it is assumed that the fields remain invariant in the direction perpendicular to the simulation plane. This means that we will not be able to obtain information about the spatial variation of the ansatz solution from a single 2D AC/DC simulation. However, with the help of simulation in two different planes, this is possible. Series resistance and inductance depend on the current and energy stored in the magnetic field, while shunt conductance and capacitance depend on the energy in the electric field. Let's consider these aspects in more detail.

Distributed Parameters for Shunt Conductance and Capacitance

Since the shunt conductance and capacitance can be calculated from the distribution of the electric field, we start by applying the interface Electric currents.

Boundary Conditions and Material Properties for the Simulation Interface Electric currents.

Once the model geometry is defined and the material properties are assigned values, the assumption is made that the surface of the conductors is equipotential (which is absolutely justified, since the difference in conductivities between a conductor and a dielectric is typically almost 20 orders of magnitude). We then set the values of the physical parameters by assigning the electrical potential V 0 to the inner conductor and ground to the outer conductor to find the electrical potential in the dielectric. The above analytical expressions for the capacitance are obtained from the following most general relations

\begin(align)
W_e& = \frac(1)(4)\int_(S)()\mathbf(E)\cdot \mathbf(D^\ast)d\mathbf(S)\\
W_e& = \frac(C|V_0|^2)(4)\\
C& = \frac(1)(|V_0|^2)\int_(S)()\mathbf(E)\cdot \mathbf(D^\ast)d\mathbf(S)
\end(align)

where the first relation is the electromagnetic theory equation and the second is the circuit theory equation.

The third relation is a combination of the first and second equations. Substituting the above known expressions for the fields, we get the analytical result given earlier for C in a coaxial cable. As a result, these equations allow us to determine the capacitance through the field values for an arbitrary cable. Based on the simulation results, we can calculate the integral of the electrical energy density, which gives the capacitance a value of 98.142 pF/m, which is consistent with the theory. Since G and C and are related by the expression

G=\frac(\omega\epsilon"" C)(\epsilon")

we now have two of the four parameters.

It is worth repeating that we made the assumption that the conductivity of the dielectric region is zero. This is a standard assumption that is made in all textbooks, and we also follow this convention here, because it does not significantly affect the physics - in contrast to our inclusion of the internal inductance term, which was discussed earlier. Many materials for a dielectric core have non-zero conductivity, but this can easily be taken into account in modeling by simply substituting new values into the material properties. In this case, to ensure a proper comparison with the theory, it is also necessary to make appropriate corrections to the theoretical expressions.

Specific parameters for series resistance and inductance

Similarly, series resistance and inductance can be calculated by simulation using the interface Magnetic fields in the AC/DC module. Simulation settings are elementary, which is illustrated in the figure below.

Conductor regions are added to a node Single Turn Coil In chapter Coil group , and, the selected reverse current direction option ensures that the direction of the current in the inner conductor will be opposite to the current of the outer conductor, which is indicated in the figure by dots and crosses. When calculating the frequency dependence, the current distribution in the single-turn coil will be taken into account, and not the arbitrary current distribution shown in the figure.

To calculate the inductance, we turn to the following equations, which are the magnetic analogue of the previous equations.

\begin(align)
W_m& = \frac(1)(4)\int_(S)()\mathbf(B)\cdot \mathbf(H^\ast)d\mathbf(S)\\
W_m& = \frac(L|I_0|^2)(4)\\
L& = \frac(1)(|I_0|^2)\int_(S)()\mathbf(B)\cdot \mathbf(H^\ast)d\mathbf(S)
\end(align)

To calculate the resistance, a slightly different technique is used. First, we integrate the resistive losses to determine the power dissipation per unit length. And then we use the well-known relation P = I_0^2R/2 to calculate the resistance. Since R and L change with frequency, let's look at the calculated values and the analytical solution in the DC limit and in the high frequency region.

“Analytical solution for direct current” and “Analytical solution for high frequencies” graphical dependences correspond to the solutions of analytical equations for direct current and high frequencies, which were discussed earlier in the text of the article. Note that both dependences are given on a logarithmic scale along the frequency axis.

It is clearly seen that the calculated values smoothly pass from the solution for direct current in the low-frequency region to the high-frequency solution, which will be valid at a skin depth much smaller than the conductor thickness. It is reasonable to assume that the transition region is located approximately at the place along the frequency axis where the skin depth and conductor thickness differ by no more than an order of magnitude. This region lies in the range from 4.2e3 Hz to 4.2e7 Hz, which exactly corresponds to the expected result.

Characteristic impedance and propagation constant

Now that we have completed the laborious work of calculating R, L, C, and G, there are two other parameters that are essential for the analysis of power lines that need to be determined. These are the characteristic impedance (Z c) and the complex propagation constant (\gamma = \alpha + j\beta ), where \alpha is the damping factor and \beta is the propagation factor.

\begin(align)
Z_c& = \sqrt(\frac((R+j\omega L))((G+j\omega C)))\\
\gamma& = \sqrt((R+j\omega L)(G+j\omega C))
\end(align)

The figure below shows these values calculated using analytical formulas in DC and RF modes, compared with the values determined from the simulation results. In addition, the fourth relationship in the graph is the impedance calculated in the COMSOL Multiphysics environment using the RF module, which we will briefly discuss a little later. As can be seen, the results of numerical simulation are in good agreement with the analytical solutions for the corresponding limit modes, and also give the correct values in the transition region.

Comparison of characteristic impedance calculated using analytic expressions and determined from simulation results in COMSOL Multiphysics. Analytic curves were generated using the appropriate DC and RF limit expressions discussed earlier, while the AC/DC and RF modules were used for simulations in COMSOL Multiphysics. For clarity, the thickness of the “RF module” line has been specially increased.

Modeling a cable in the high frequency region

The energy of the electromagnetic field propagates in the form of waves, which means that the operating frequency and wavelength are inversely proportional to each other. As we move into higher and higher frequencies, we have to take into account the relative size of the wavelength and the electrical size of the cable. As discussed in the previous entry, we should change AC/DC to an RF module at an electrical size of approximately λ/100 (see ibid for the concept of "electrical size"). If we choose the diameter of the cable as the electrical dimension, and instead of the speed of light in a vacuum - the speed of light in the dielectric core of the cable, then we get the frequency for the transition in the region of 690 MHz.

At such high frequencies, the cable itself is more appropriately considered as a waveguide, and the excitation of the cable can be considered as waveguide modes. Using waveguide terminology, so far we have considered a special type of mode called TEM a mode that can propagate at any frequency. When the cable cross-section and wavelength become comparable, we must also take into account the possibility of the existence of higher order modes. Unlike the TEM mode, most guiding modes can only propagate at an excitation frequency above a certain characteristic cutoff frequency. Due to the cylindrical symmetry in our example, there is an expression for the cutoff frequency of the first higher order mode - TE11. This cutoff frequency is f c = 35.3 GHz, but even with this relatively simple geometry, the cutoff frequency is a solution to a transcendental equation that we will not consider in this article.

So what does this cutoff frequency mean for our results? Above this frequency, the wave energy transported in the TEM mode we are interested in has the potential to interact with the TE11 mode. In an idealized geometry like the one modeled here, there will be no interaction. In a real situation, however, any defects in the cable design can lead to mode interaction at frequencies above the cutoff frequency. This can be the result of a range of uncontrollable factors, from manufacturing errors to gradients in material properties. This situation is most easily avoided at the cable design stage by designing to operate at frequencies known to be lower than the cutoff frequency of the higher order modes, so that only one mode can propagate. If it's of interest, you can also use the COMSOL Multiphysics environment to simulate the interaction between higher order modes, as done in this one (although this is beyond the scope of this article).

Modal Analysis in the Radio Frequency Module and the Wave Optics Module

Modeling higher order modes is ideally implemented using modal analysis in the Radio Frequency module and the Wave Optics module. The ansatz form of the solution in this case is the expression \mathbf(E)\left(x,y,z\right) = \mathbf(\tilde(E))\left(x,y\right)e^(-\gamma z), which exactly matches the mode structure, which is our goal. As a result, the modal analysis immediately gives a solution for the spatial distribution of the field and the complex propagation constant for each of a given number of modes. In this case, we can use the same model geometry as before, except that it is enough for us to use only the dielectric core as the modeling area and .

The results of calculating the damping constant and the effective refractive index of the wave mode from the Mode Analysis. The analytical curve on the left graph, damping factor versus frequency, is calculated using the same expressions as for the RF curves used to compare with simulation results in the AC/DC module. The analytic curve in the right plot, the effective refractive index versus frequency, is simply n = \sqrt(\epsilon_r\mu_r) . For clarity, the size of the "COMSOL - TEM" line has been deliberately increased on both graphs.

It is clearly seen that the results of the TEM Mode Mode Analysis agree with the analytical theory and that the calculated higher order mode appears at the predetermined cutoff frequency. It is convenient that the complex propagation constant is directly calculated during the simulation and does not require intermediate calculations of R, L, C, and G. This becomes possible due to the fact that \gamma is explicitly included in the desired form of the ansatz solution and is found when solving by substituting it into the main equation. If desired, other parameters can also be calculated for the TEM mode, and more information about this can be found in the Application Gallery. It is also noteworthy that the same method of modal analysis can be used to calculate dielectric waveguides, as implemented in .

Final Notes on Cable Modeling

By now, we have thoroughly analyzed the coaxial cable model. We calculated the distributed parameters from the constant current mode to the high frequency region and considered the first higher order mode. It is important that the results of modal analysis depend only on the geometric dimensions and properties of the cable material. The results for simulation in the AC/DC module require more information about how the cable is driven, but hopefully you are aware of what is connected to your cable! We have used analytical theory solely to compare the results of numerical simulations with well-known results for the reference model. This means that the analysis can be generalized to other cables, as well as adding relationships for multiphysics simulations that include temperature changes and structural deformations.

A few interesting nuances for building a model (in the form of answers to possible questions):

“Why didn’t you mention and/or give graphs of the characteristic impedance and all distributed parameters for the TE11 mode?”
- Because only TEM modes have uniquely defined voltage, current, and characteristic impedance. In principle, it is possible to assign some of these values to higher-order modes, and this issue will be considered in more detail in future articles, as well as in various works on the theory of transmission lines and microwave technology.
“When I solve a mod problem using Modal Analysis, they are labeled with their working indexes. Where do the designations TEM and TE11 modes come from?”
- These designations appear in the theoretical analysis and are used for convenience in discussing the results. Such a name is not always possible with an arbitrary waveguide geometry (or cable in waveguide mode), but it should be borne in mind that this designation is just a “name”. Whatever the name of fashion, does it still carry electromagnetic energy (excluding, of course, non-tunneling evanescent waves)?
“Why do some of your formulas have an extra factor of ½?”
- This happens when solving problems of electrodynamics in the frequency domain, namely, when multiplying two complex quantities. When performing time averaging, there is an additional ½ multiplier, as opposed to time domain (or DC) expressions. For more information, you can refer to the works on classical electrodynamics.

Literature

The following monographs were used in writing this note and will serve as excellent references when looking for additional information:

Microwave Engineering (microwave technology), by David M. Pozar
Foundations for Microwave Engineering (Fundamentals of Microwave Engineering), by Robert E. Collin
Inductance Calculations by Frederick W. Grover
Classical Electrodynamics (Classical electrodynamics) by John D. Jackson

The latest release of COMSOL Multiphysics® and COMSOL Server™ provides a state-of-the-art integrated engineering analysis environment that enables numerical simulation professionals to create multiphysics models and develop simulation applications that can be easily deployed to employees and customers around the world.

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a). Drawing of the computational domain indicating the boundary conditions and the equation to be solved b). Calculation results - field pattern and spreading resistance value

for homogeneous soil. Screening factor calculation results.

in). The results of the calculation are the field pattern and the value of the spreading resistance for a two-layer soil. Screening factor calculation results.

2. Study of the electric field in a nonlinear surge arrester

Nonlinear surge arresters (Fig. 2.1) are used to protect high-voltage equipment from surges. A typical polymer insulated surge arrester consists of a non-linear zinc oxide resistor (1) placed inside an insulating glass-reinforced plastic cylinder (2), on the outer surface of which a silicone insulating cover (3) is pressed. The insulating body of the limiter is closed at both ends with metal flanges (4) having a threaded connection to a fiberglass pipe.

If the limiter is under the operating voltage of the network, then the active current flowing through the resistor is negligible and the electric fields in the considered design are well described by the equations of electrostatics

divgradU 0

EgradU,

where is the electric potential, is the electric field strength vector.

As part of this work, it is necessary to investigate the distribution of the electric field in the limiter and calculate its capacitance.

Fig.2.1 Construction of a non-linear surge arrester

Since the surge arrester is a body of revolution, it is advisable to use a cylindrical coordinate system when calculating the electric field. As an example, a device for a voltage of 77 kV will be considered. The operating apparatus is mounted on a conductive cylindrical base. The calculation area with indication of dimensions and boundary conditions is shown in Fig. 2.2. The external dimensions of the computational domain should be chosen to be approximately 3–4 times the height of the device together with the installation base 2.5 m high. The equation for the potential under cylindrical symmetry conditions can be written in a cylindrical coordinate system with two independent variables in the form

Fig.2.2 Computational domain and boundary conditions

On the border of the calculated (hatched) area (Fig. 2.2), the following boundary conditions are established: on the surface of the upper flange, the potential corresponding to the operating voltage U = U 0 of the apparatus, the surface of the lower flange and the base of the apparatus are grounded, on the boundaries of the external

the region is given the conditions for the disappearance of the field U 0; on the border with

r=0 the condition of axial symmetry (axis symmetry) is set.

From the physical properties of the construction materials of the surge arrester, it is necessary to set the relative permittivity, the values of which are given in Table 2.1

Relative permittivity of subdomains of the computational domain

Rice. 2.3

Structural dimensions are shown in Fig.2.3

surge arrester and base

The construction of the calculation model begins with the launch of Comsol Multiphysics and on the start tab

Choose 1) geometry type (space dimension) – 2D Axisymmetric, 2) Physical task type – AC/DC module->static->electrostatics.

It is important to note that all geometric dimensions and other parameters of the problem should be specified using the SI system of units.

We start drawing the computational domain with a non-linear resistor (1). To do this, in the Draw menu, select specify objects->rectangle and enter width 0.0425 and hight 0.94, as well as the coordinates of the base point r=0 and z=0.08. Then similarly draw: the wall of the fiberglass pipe: (Width= 0.0205, hight=1.05, r=0.0425, z=0.025); rubber insulation wall

(width=0.055, hight=0.94, r=0.063, z=0.08).

Further, rectangles of blanks of flange subregions are drawn: upper (width=0.125, hight=0.04, r=0, z=1.06), (width=0.073, hight=0.04, r=0, z=1.02) and lower (width=0.073, hight=0.04, r=0, z=0.04), (width=0.125, hight=0.04, r=0, z=0). At this stage of constructing the geometry of the model, the sharp edges of the electrodes should be rounded off. To do this, use the Fillet command of the Draw menu. In order to use this command, select with the mouse a rectangle one of the corners of which will be smoothed and execute Draw-> Fillet. Next, mark the vertex of the corner to be smoothed with the mouse and enter the value of the rounding radius in the pop-up window. Using this method, we will perform rounding of the corners of the section of flanges that have direct contact with air (Fig. 2.4), setting the initial rounding radius equal to 0.002 m. Further, this radius should be selected based on the limitation of the corona discharge.

After performing the rounding operations, it remains to draw the base (base) and the outer area. This can be done with the rectangle drawing commands described above. For the base (width=0.2, hight=2.4, r=0, z=-2.4) and for the outer region (width=10, hight=10, r=0, z=-2.4).


	The next stage of preparation

	model is a task of physical
	properties of structural elements. AT
	our task	dielectric
	permeability.			facilities
	editing			create
	list of constants using menu
	Options->constants. To table cells
	constants
	constants and its meaning, moreover
	names can be assigned arbitrarily.
Fig.2.4 Fillet areas	Numeric values	dielectric
	permeability		materials
	designs		limiter

	given above. Let's give, for example,
	the following		permanent

eps_var, eps_tube, eps_rubber, the numerical values of which will determine the relative permittivity of the non-linear resistor, fiberglass pipe, external insulation, respectively.

Next, switch Сomsol Мultiphysis c to the mode of setting subdomain properties using the Physics->Subdomain settings command. Using the zoom window command, you can enlarge drawing fragments if necessary. To set the physical properties of a subregion, select it with the mouse in the drawing or select it from the list that appears on the screen after executing the above command. The selected area is colored in the drawing. In the window ε r isotropic of the subdomain properties editor, enter the name of the corresponding constant. Keep the default dielectric constant of 1 for the outer sub-region.

The subregions inside the potential electrodes (flange and base) should be excluded from the analysis. To do this, remove the active in this domain pointer in the subdomain properties editor window. This command must be executed, for example, for the sub areas shown in

The next stage of model preparation is


setting of boundary conditions. For
transition to	editing		boundary
conditions, use the Physucs-

the desired line is highlighted with the mouse and
	given
the boundary conditions editor starts.
Type and meaning		boundary	conditions for
each segment of the boundary is assigned in
accordance		rice. 2.2. When asked

potential of the upper flange, it is also advisable to add it to the list of constants, for example, under the name U0 and with a numerical value of 77000.

The preparation of the model for calculation is completed by building a mesh of finite elements. To ensure high accuracy of the calculation of the field near the edges, you should use the manual setting of the size of the finite elements in the fillet area. To do this, in the boundary conditions editing mode, select the rounding directly with the mouse cursor. To select all fillets, hold down the Ctrl key. Next, select the menu item Mesh-Free mesh parameters->Boundary. To window maximum element size

enter a numerical value obtained by multiplying the rounding radius by 0.1. This will provide a mesh that is adapted to the curvature of the flange chamfer. Mesh construction is performed by the Mesh->Initialize mesh command. The mesh can be made denser with the Mesh->refine mesh command. Mesh->Refine selection command

makes it possible to obtain a local mesh refinement, for example, near lines with a small radius of curvature. When this command is executed with the mouse, a rectangular area is selected in the drawing, within which the mesh will be refined. In order to view the already built mesh, you can use the Mesh-> mesh mode command.

The solution of the problem is performed by the Solve->solve problem command. After the calculation is completed, Comsol Multiphysis enters the postprocessor mode. In this case, a graphical representation of the calculation results is displayed on the screen. (By default, this is a color picture of the electric potential distribution).

To obtain a more convenient presentation of the field picture when printing on a printer, you can change the presentation method, for example, as follows. The Postprocessing->Plot parameters command opens the postprocessor editor. On the General tab, activate two items: Contour and Streamline. As a result, the picture of the role will be displayed, consisting of lines of equal potential and lines of force (electric field strength) - Fig. 2.6.

Within the framework of this work, two tasks are solved:

selection of the rounding radii of the edges of the electrodes adjoining the air, according to the condition of the occurrence of a corona discharge and the calculation of the electric capacitance of the surge arrester.

a) Choice of chamfer radii

When solving this problem, one should proceed from the intensity of the beginning of the corona discharge equal to approximately 2.5*106 V/m. After the formation and solution of the problem to assess the distribution of the electric field strength along the surface of the upper flange, switch Сomsol Мultiphysis to the mode of editing the boundary conditions and select the necessary section of the boundary of the upper flange (Fig. 9)

Typical field picture of a surge arrester

Selection of a section of the flange boundary for constructing the electric field strength distribution

Next, using the Postprocessing -> Domain plot parameters-> Line extrusion command, the value editor for drawing linear distributions follows and enter the name of the electric field strength module - normE_emes into the displayed value window. After clicking OK, a graph of the field strength distribution along the selected boundary section will be plotted. If the field strength exceeds the above value, then you should return to the construction of a geometric model (Draw->Draw mode) and increase the radius of the edges. After choosing suitable rounding radii, compare the stress distribution along the flange surface with the initial version.

2) Calculation of electrical capacitance

AT In the framework of this work, we will use the energy method for estimating the capacitance. For this, the volume integral is calculated over the entire

computational domain on the energy density of the electrostatic field using the Postprocessing->Subdomain integration command. In this case, in the window that appears with a list of subdomains, all subdomains containing a dielectric, including air, should be selected, and the field energy density -We_emes should be selected as the integrable quantity. It is important that the integral calculation mode taking into account axial symmetry is activated. AT

the result of the integral calculation (after pressing OK) at the bottom

C 2We _emes /U 2 calculates the capacity of the object.

If we replace the permittivity in the region of the non-linear resistor with a value corresponding to glass-reinforced plastic, then the properties of the structure under study will fully correspond to a rod-type polymer support insulator. Calculate the capacitance of the post insulator and compare it with the capacitance of the surge arrester.

1. Model (equation, geometry, physical properties, boundary conditions)

2. Table of the results of calculating the maximum electric field strengths on the surface of the upper flange for various rounding radii. The distribution of the electric field strength on the flange surface should be given at the minimum and maximum of the investigated values of the radius of curvature

3. The results of the calculation of the capacitance of the surge arrester and the support insulator

4. Explanation of results, conclusions

3. Optimization of the electrostatic screen for a non-linear surge arrester.

As part of this work, based on the calculations of the electrostatic field, it is necessary to select the geometric parameters of the toroidal screen of a non-linear surge arrester for a voltage of 220 kV. This device consists of two identical modules connected in series by installing on top of each other. The whole apparatus is installed on a vertical base 2.5 m high (Fig. 3.1).

The modules of the device are a hollow cylindrical insulating structure, inside of which there is a non-linear resistor, which is a column of circular cross section. The top and bottom parts of the module end with metal flanges used as a contact connection (Fig. 3.1).

Fig.3.1 Design of two-module arrester -220 with leveling screen

The height of the assembled apparatus is about 2 m. Therefore, the electric field is distributed along its height with a noticeable unevenness. This causes uneven distribution of currents in the resistor of the arrester when exposed to the operating voltage. As a result, part of the resistor receives increased heating, while other parts of the column are not loaded. In order to avoid this phenomenon during long-term operation, toroidal screens are used that are installed on the upper flange of the apparatus, the dimensions and location of which are chosen based on the achievement of the most uniform distribution of the electric field along the height of the apparatus.

Since the design of the surge arrester with a toroidal screen has axial symmetry, it is advisable to use a two-dimensional equation for the potential in a cylindrical coordinate system for the calculation

Comsol MultiPhysics uses the 2-D Axial Symmetry AC/DC module->Static->Electrostatics model to solve the problem. The computational area is drawn in accordance with Fig. 3.1, taking into account axial symmetry.

The preparation of the calculation area is performed by analogy with work 2. It is advisable to exclude the internal areas of metal flanges from the calculation area (Fig. 3.2) using the Create composite object commands of the Draw menu. The external dimensions of the computational domain are 3-4 of the total height of the structure. The sharp edges of the flanges should be rounded with a radius of 5-8 mm.

Physical properties of subregions determined by the value of the relative permittivity of the materials used, the values of which are given in the table

Table 3.1

Relative permittivity of construction materials of arrester

	Relative Permittivity

Tube (Glass plastic)
External insulation (rubber)

Border conditions: 1) The surface of the upper flange of the upper module and the surface of the leveling screen Potential - the phase voltage of the network is 154000 * √2 V; 2) The surface of the lower flange of the lower module, the surface of the base, the surface of the ground - ground; 3) Surface of intermediate flanges (bottom flange of upper and upper flange of lower module) Floating Potential; 4) Line of axial symmetry (r=0) - Axial Symmetry; 5)

Remote boundaries of the computational domain Zero Charge/Symmetry

2. COMSOL Quick Start Guide

The purpose of this section is to introduce the reader to the COMSOL environment, focusing primarily on how to use its graphical user interface. To facilitate this quick start, this subsection provides an overview of the workflow for creating simple models and obtaining simulation results.

Two-dimensional model of heat transfer from a copper cable in a simple heatsink

This model explores some of the effects of thermoelectric heating. It is strongly recommended that you follow the simulation steps described in this example, even if you are not a heat transfer expert; the discussion focuses primarily on how to use the COMSOL GUI application, rather than the physical basis of the phenomenon being modeled.

Consider an aluminum heat sink that removes heat from an insulated high voltage copper cable. The current in the cable generates heat due to the fact that the cable has electrical resistance. This heat passes through the heatsink and is dissipated into the surrounding air. Let the temperature of the outer surface of the radiator be constant and equal to 273 K.

Rice. 2.1. The geometry of the cross section of a copper core with a radiator: 1 - radiator; 2 - electrically insulated copper core.

In this example, the geometry of a radiator is modeled, the cross section of which is a regular eight-pointed star (Fig. 2.1). Let the geometry of the radiator be plane-parallel. Let the length of the radiator in the direction of the z axis be many

greater than the diameter of the circumscribed circle of the star. In this case, temperature variations in the direction of the z axis can be ignored, i.e. the temperature field can also be considered plane-parallel. The temperature distribution can be calculated in a two-dimensional geometric model in Cartesian coordinates x ,y .

This technique of neglecting variations in physical quantities in one direction is often convenient when setting up real physical models. You can often use symmetry to create high fidelity 2D or 1D models, saving significant computational time and memory.

Modeling Technology in the COMSOL GUI Application

To start modeling, you need to launch the COMSOL GUI application. If MATLAB and COMSOL are installed on your computer, you can start COMSOL from the Windows desktop or by clicking the Start button ("Programs", "COMSOL with MATLAB").

As a result of executing this command, the COMSOL figure and the figure of the Model Navigator will be expanded on the screen (Fig. 2.2).

Rice. 2.2. General view of the Model Navigator figure

Since we are now interested in a two-dimensional heat transfer model, on the New tab of the Navigator, in the Space dimension field, select 2D , select the model Application Modes/ COMSOL Multiphysics/ Heat transfer/conduction/steady-state analysis and click OK.

As a result of these actions, the figure of the Model Navigator and the COMSOL axes field will take the form shown in fig. 2.3, 2.4. By default, modeling is performed in the SI system of units (the system of units is selected on the Settings tab of the Model Navigator).

Rice. 2.3, 2.4. Model Navigator Shape and COMSOL Axes Field in Application Mode

Drawing geometry

The COMSOL GUI application is now ready to draw the geometry (Draw Mode is in effect). Geometry can be drawn using the commands in the Draw group of the main menu or by using the vertical toolbar located on the left side of the COMSOL shape.

Let the origin of coordinates be in the center of the copper core. Let the core radius be 2 mm. Since the radiator is a regular star, half of its vertices lie on the inscribed circle, and the other half lie on the circumscribed circle. Let the radius of the inscribed circle be 3 mm, the angles at the inner vertices be straight.

There are several ways to draw geometry. The simplest of them are direct drawing with the mouse in the axes field and inserting geometric objects from the MATLAB workspace.

For example, you can draw a copper core as follows. We press the button of the vertical toolbar, set the mouse pointer to the origin, press and hold the Ctrl key and the left mouse button, move the mouse pointer from the origin until the radius of the drawn circle becomes equal to 2, release the mouse button and the Ctrl key. Drawing the correct star of the radiator is much more

more difficult. You can use the button to draw a polygon, then double-click on it with the mouse and correct the coordinate values of all star vertices in the expanded dialog box. Such an operation is too complicated and time consuming. You can draw a star

represent a combination of squares, which are convenient to create with the , buttons (when drawing with the mouse, you must also hold down the Ctrl key to get squares, not rectangles). For precise positioning of the squares, you need to double-click on them and adjust their parameters in the expanded dialog boxes (coordinates, lengths, and rotation angles can be set using MATLAB expressions). After the exact positioning of the squares, you need to create a composite geometric object from them by performing the following sequence of actions. Select the squares by making a single mouse click on them and holding down the Ctrl key (the selected objects will be

highlighted in brown), press the button , correct the compound object formula in the expanded dialog box, and press the OK button. Composite object formula

is an expression containing operations on sets (in this case, you need the union of sets (+) and the subtraction of sets (-)). Now the circle and the star are ready. As you can see, both ways of drawing a star are quite laborious.

It is much easier and faster to create geometry objects in the MATLAB workspace and then insert them into the axes field with the COMSOL GUI application command. To do this, use the m-file editor to create and execute the following computational script:

C1=circ2(0,0,2e-3); % Circle object r_radiator=3e-3; % Heatsink inner radius

R_radiator=r_radiator*sqrt(0.5)/sin(pi/8); % Radius Outer Radius r_vertex=repmat(,1,8); % Radial coordinates of star vertices al_vertex=0:pi/8:2*pi-pi/8; % Angular coordinates of star vertices x_vertex=r_vertex.*cos(al_vertex);

y_vertex=r_vertex.*sin(al_vertex); % Cartesian coordinates of star vertices

P1=poly2(x_vertex,y_vertex); % polygon object

To insert geometric objects into the axes field, you need to run the command File/ Import/ Geometry Objects. Execution of this command will lead to the deployment of a dialog box, the view of which is shown in Fig. 2.5.

Rice. 2.5. General view of the dialog box for inserting geometric objects from the workspace

Pressing the OK button will insert geometric objects (Fig. 2.6). The objects will be selected and highlighted in brown. As a result of this import, the grid settings in the COMSOL GUI application are automatically adjusted when you click

on the button. On this, the drawing of geometry can be considered complete. The next stage of modeling is setting the PDE coefficients and setting the boundary conditions.

Rice. 2.6. General view of the traced geometry of a current-carrying copper core with a radiator: C1, P1 - names (labels) of geometric objects (C1 - circle, P1 - polygon).

Specifying PDE Factors

Switching to the PDE coefficients setting mode is carried out by the Physics/ Subdomain Settings command. In this mode, in the axes field, the geometry of the computational domain is displayed as a union of non-overlapping subdomains, which are called zones. To see the zone numbers, you need to run the command Options/ Labels/ Show Subdomain Labels. The general view of the axes field with the computational domain in the PDE Mode with zone numbers is shown in fig. 2.7. As you can see, in this problem, the calculation area consists of two zones: zone No. 1 is a radiator, zone No. 2 is a copper current-carrying core.

Rice. 2.7. Image of the computational domain in the PDE Mode

To enter parameters of material properties (PDE coefficients), use the PDE/PDE Specification command. This command will open the dialog box for entering PDE coefficients, shown in fig. 2.8 (in general, the appearance of this window depends on the current application mode of the COMSOL GUI application).

Rice. 2.8. Dialog box for entering PDE coefficients in heat transfer application mode Zones 1 and 2 consist of materials with different thermophysical properties, the heat source is only a copper core. Let the current density in the core d = 5e7A/m2; electrical conductivity of copper g = 5.998e7 S/m; coefficient of thermal conductivity of copperk = 400; let the radiator be made of aluminum, having a thermal conductivity coefficient k = 160. It is known that the volumetric power density of heat losses during the flow of electric current through the substance is equal to Q=d2 /g. Select zone No. 2 in the Subdomain Selection panel and load the appropriate parameters for copper from the Library material / Load (Fig. 2.9).

Fig.2.9. Entering Copper Properties Parameters

Now let's select zone No. 1 and enter the parameters of aluminum (Fig. 2.10).

Fig.2.10. Entering Aluminum Properties Parameters

Clicking the Apply button will cause the PDE coefficients to be accepted. You can close the dialog box with the OK button. This completes the entry of the PDE coefficients.

Specifying boundary conditions

To set the boundary conditions, you must put the COMSOL GUI application into boundary condition input mode. This transition is carried out by the command Physics/ Boundary Settings . In this mode, the axes field displays the inner and outer boundary segments (by default, in the form of arrows indicating the positive directions of the segments). The general view of the model in this mode is shown in Fig. 2.11.

Fig.2.11. Showing Boundary Segments in Boundary Settings Mode

According to the condition of the problem, the temperature on the outer surface of the radiator is 273 K. To set such a boundary condition, you must first select all the outer boundary segments. To do this, hold down the Ctrl key and click on all external segments with the mouse. Selected segments will be highlighted in red (see Fig. 2.12).

Rice. 2.12. Highlighted Outer Boundary Segments

The Physics/ Boundary Settings command will also open a dialog box, the view of which is shown in Fig. 2.13. In general, its appearance depends on the current application simulation mode.

Fig.2.13. Dialog box for entering boundary conditions

On fig. 2.13 shows the entered temperature value on the selected segments. There is also a segment selection panel in this dialog box. So, it is not necessary to select them directly in the axes field. If you press the OK or Apply, OK button, the entered boundary conditions will be accepted. At this point, in this problem, the introduction of boundary conditions can be considered complete. The next stage of modeling is the generation of a finite element mesh.

Finite Element Mesh Generation

To generate a mesh, it is enough to execute the command Mesh/ Initialise Mesh . The mesh will be automatically generated according to the current mesh generator settings. The automatically generated mesh is shown in fig. 2.13.

M.: NRNU MEPhI, 2012. - 184 p. Description:
Designed to study the mathematical modeling environment Comsol Multiphysics. The manual discusses in detail the key methods of working with this system and understands specific typical tasks. The book also contains a guide to mathematical programming in Comsol Script and features of the interaction of the Comsol Multiphysics package with the Matlab system.
This manual is the first Comsol Multiphysics manual in Russian.
Useful for 3rd and 4th year students studying mathematical modeling. Contents:
Finite element method.
Theoretical introduction.
Types of finite elements. Getting started with FEMLAB.
Installation.
General principles of work.
Application modes.
The process of setting and solving a problem.
Comsol Multiphysics 3.5a environment.
Model navigator.
Working environment of the program.
Setting areas.
Drawing basic geometric objects.
Object transformations.
Logical operations with objects.
Analytical assignment of objects.
Formulation of the problem.
Specifying equation coefficients.
Setting of boundary conditions.
Mesh generation.
triangular grid.
Quadrangular elements.
Choice of basic functions.
The solution of the problem.
Stationary solvers.
Visualization of results.
Construction of the main graph.
Export graph to file.
Construction of graphs on sections and points.
Construction of graphs on the borders and at the key points of the area.
Expressions and functions in FEMLAB.
Introduction.
Setting constants and regular expressions.
Using constants and regular expressions.
Functions.
Axes and grid properties. Practical simulation on FEMLAB.
Solution of non-stationary problems.
Formulation of the problem.
The solution of the problem.
Solution visualization.
Accounting for the initial conditions of the problem.
Solution of differential-algebraic systems of equations.
Solving problems for eigenvalues.
Solving problems with a parameter.
Solution of acoustics equations.
General information.
Mathematical statement of the problem.
Applied mode of acoustics equations.
Border conditions.
An example of a sound propagation problem. Reactive silencer acoustics.
Solving problems of structural mechanics.
Theoretical introduction.
Applied mode of structural mechanics equations.
Fixings.
Loads.
An example of the problem of stress distribution in a trapezoidal membrane.
Solution of the problem of finding ice flow velocities by the FEMLAB system.
Theoretical information.
Statement and solution of the problem.
Implementation of the multiphysics mode.
Solving problems with changing geometry.
Solution of the problem of heating a liquid drop.
Forms of equations.
General information.
Application modes.
The coefficient form of the equation.
General form.
Weak form.
Solution of one-dimensional problems.
Solution of three-dimensional problems.
Specifying 3D geometry.
Defining Equations and Mesh Generation.
Visualization of results.
Transition from two-dimensional geometry to three-dimensional. Communication with matlab. Comsol Script.
Introduction.
Launching joint work with Matlab and Comsol Script.
Getting started with Comsol Script.
Basic information.
Working with memory Comsol Script.
Vectors, matrices and arrays in Comsol Script.
Elements of programming in Comsol Script.
The if branch operator.
Conditional loop.
Cycle with a counter.
Choice operator.
Task modeling in Maltab and Comsol Script.
FEMLAB object model.
Solution of the Poisson equation.
Import and export of the model.
Creation of geometric objects.
Creation of basic geometric objects.
Creation of complex objects.
Object transformations and logical operations.
Interpolation of geometric objects.
Model assignment.
Basic provisions.
Formulation of the problem.
Setting equations.
Mesh generation.
test functions.
Constants and expressions.
Solver choice.
Visualization and data processing.