Modeling cables and transmission lines in COMSOL Multiphysics. Krasnikov G.E., Nagornov O., Starostin N.V.

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. Results of calculation of the shielding coefficient.

V). The results of the calculation are the field pattern and the value of the spreading resistance for a two-layer soil. Results of calculation of the shielding coefficient.

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

Nonlinear surge suppressors (Fig. 2.1) are used to protect high-voltage equipment from overvoltages. A typical polymer insulated surge suppressor consists of a non-linear zinc oxide resistor (1) placed inside an insulating fiberglass cylinder (2), onto the outer surface of which a silicone insulating cover (3) is pressed. The insulating body of the limiter is closed at both ends by metal flanges (4), which have a threaded connection to the 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 design under consideration are well described by the equations of electrostatics

div gradU 0

EgradU,

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

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

Fig.2.1 Design of a nonlinear surge suppressor

Since the surge suppressor is a body of revolution, when calculating the electric field it is advisable to use a cylindrical coordinate system. As an example, we will consider a device with a voltage of 77 kW. The operating apparatus is mounted on a conductive cylindrical base. The computational domain indicating the dimensions and boundary conditions is presented in Fig. 2.2. The external dimensions of the computational area should be chosen equal to approximately 3-4 times the height of the apparatus together with the mounting base with a height of 2.5 m. The equation for the potential under conditions of cylindrical symmetry can be written in a cylindrical coordinate system with two independent variables in the form

Fig.2.2 Computational domain and boundary conditions

At the boundary of the calculation (shaded) 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 device, the surface of the lower flange and the base of the device are grounded, at the boundaries of the external

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

r=0 sets the axis symmetry condition.

From the physical properties of the materials used in the design of the surge suppressor, it is necessary to set the relative dielectric constant, the values ​​of which are given in Table 2.1

Relative permittivity of subregions of the computational domain

Rice. 2.3

Structural dimensions are shown in Fig. 2.3

surge suppressor and base

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

We select 1) type of geometry (space dimension) – 2D Axisymmetric, 2) Type of physical problem – 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 begin drawing the computational domain with a nonlinear resistor (1). To do this, in the Draw menu, select specify objects->rectangle and enter width 0.0425 and height 0.94, as well as the coordinates of the base point r=0 and z=0.08. Then we draw in the same way: the wall of a 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).

Next, rectangles of the flange sub-regions 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. 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, use the mouse to mark the vertex of the corner to be smoothed and enter the value of the rounding radius in the pop-up window. Using this method, we will round off the cross-sectional corners of flanges that have direct contact with air (Fig. 2.4), setting the initial rounding radius to 0.002 m. Then this radius should be selected based on the corona discharge limitation.

After completing the edge rounding operations, all that remains is to draw the base and the outer area. This can be done using the rectangle drawing commands described above. For the base (width=0.2, hight=2.4, r=0, z=-2.4) and for the outer area (width=10, hight=10, r=0, z=- 2.4).

	The next stage of preparation
	model is a task of physical
	properties of structural elements. IN
	our task			dielectric
	permeability.					facilities
	editing					let's create
	list constants using menu
	Options->constats. To table cells
	constants
	constants and their meaning, and
	names can be assigned arbitrarily.
Fig.2.4 Rounding areas (Fillet)	Numeric values			dielectric
	permeability				materials
	designs				limiter
	are given above. Let's give, for example,
	following				permanent

eps_var, eps_tube, eps_rubber, the numerical values ​​of which will determine the relative dielectric constant of the nonlinear resistor, fiberglass pipe, and external insulation, respectively.

Next, we switch Comsol Multiphysics c into the mode for setting the properties of subdomains using the Physics->Subdomain settings command. Using the zoom window command, you can enlarge parts of the drawing if necessary. To set the physical properties of a subarea, 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 ε r isotropic window of the subarea properties editor, enter the name of the corresponding constant. For the outer subregion, the default dielectric constant value of 1 should be maintained.

Subregions located inside the potential electrodes (flanges and base) should be excluded from the analysis. To do this, in the subdomain property editor window, remove the active in this domain checkbox. This command should be executed for example for the subareas shown in

		The next stage of model preparation is
		setting boundary conditions. For
		transition to	editing		boundary
		conditions the command Physucs- is used
		the desired line is highlighted with the mouse and
			given
		The boundary conditions editor starts.
		Type and value		borderline	conditions for
		each segment of the border is assigned to
		compliance		rice. 2.2. When setting

potential of the upper flange, it is also advisable to include it in 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 constructing a finite element mesh. To ensure high accuracy in calculating the field near the edges, you should use manual adjustment of the size of the finite elements in the area of ​​fillets. To do this, in the boundary conditions editing mode, select the fillet directly using 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

you should 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 fillet. The mesh is created using the Mesh->Initialize mesh command. The mesh can be made more dense using the Mesh->refine mesh command. Mesh->Refine selection command

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

The problem is solved using the Solve->solve problem command. After completing the calculation, Comsol Multiphysics switches to postprocessor mode. In this case, a graphical representation of the calculation results is displayed on the screen. (By default, this is the color picture of the electrical potential distribution.)

To obtain a more convenient representation of the field picture when printing on a printer, you can change the presentation method, for example, as follows. The Postprocesing->Plot parameters command opens the postprocessor editor. On the General tab, activate two items: Contour and Streamline. As a result, a 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 radii of rounding of the edges of the electrodes bordering the air, according to the conditions for the occurrence of a corona discharge and calculation of the electrical capacitance of the surge suppressor.

a) Selection of edge rounding radii

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

Typical field picture of a surge suppressor

Selecting a section of the flange boundary to plot the distribution of electric field strength

Next, using the command Postprocessing -> Domain plot parameters-> Line extrusion, follow the value editor for drawing linear distributions 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 section of the border will be plotted. If the field strength exceeds the value indicated above, then you should return to building the geometric model (Draw->Draw mode) and increase the radii of rounding the edges. After selecting suitable fillet radii, compare the stress distribution along the flange surface with the initial option.

2) Calculation of electrical capacitance

IN As part of this work, we will use the energy method for estimating capacity. To do this, the volume integral is calculated over the entire

calculation 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 subregions, you should select all subregions containing a dielectric, including air, and select the field energy density -We_emes as the integrated quantity. It is important that the mode for calculating the integral taking into account axial symmetry is activated. IN

result of calculating the integral (after clicking OK) at the bottom

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

If we replace the dielectric constant in the region of the nonlinear resistor with a value corresponding to fiberglass, then the properties of the structure under study will fully correspond to a rod-type polymer support insulator. The capacity of the support insulator should be calculated and compared with the capacity of the surge suppressor.

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

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

3. Calculation results for the capacity of surge arrester and support insulator

4. Explanation of results, conclusions

3. Optimization of the electrostatic shield for a nonlinear surge suppressor.

Within the framework of this work, based on calculations of the electrostatic field, it is necessary to select the geometric parameters of the toroidal screen of a nonlinear surge suppressor for a voltage of 220 kV. This device consists of two identical modules connected in series by installing on top of each other. The entire 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 nonlinear resistor, which is a column of circular cross-section. The upper and lower parts of the module end with metal flanges used as a contact connection (Fig. 3.1).

Fig. 3.1 Design of two-module surge 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 noticeable unevenness. This causes an uneven distribution of currents in the arrester resistor when exposed to 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, installed on the upper flange of the device, the dimensions and location of which are selected based on achieving the most uniform distribution of the electric field along the height of the device.

Since the design of an 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 calculations

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

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

Physical properties of subregions determined by the relative dielectric constant of the materials used, the values ​​of which are given in the table

Table 3.1

Relative dielectric constant of arrester construction materials

	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 Potential leveling screen – 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 earth - ground; 3) Surface of intermediate flanges (bottom flange of the upper and upper flange of the lower module) Floating Potential; 4) Line of axial symmetry (r=0) – Axial Symmetry; 5)

Remote boundaries of the Zero Charge/Symmetry calculation area The floating potential boundary condition applied on the intermediate flange is physically based on the equality of zero total electrical

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

Burlington, Massachusetts June 17, 2016. COMSOL, Inc., a leading provider of multiphysics simulation software, today announces the release of a new version of its COMSOL Multiphysics® and COMSOL Server™ simulation software. Hundreds of new user-expected features and enhancements have been added to COMSOL Multiphysics®, COMSOL Server™, and extensions to improve the accuracy, usability, and performance of the product. From new solvers and methods to application development and deployment tools, the new COMSOL® 5.2a software release enhances electrical, mechanical, fluid dynamics, and chemical modeling and optimization capabilities.

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Overview of new features and tools in version 5.2a

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Availability

To view an overview video and download COMSOL Multiphysics® and COMSOL Server™ 5.2a software, visit https://www.comsol.ru/release/5.2a.

About COMSOL

COMSOL is a global provider of computer simulation software used by technology companies, scientific laboratories and universities to design products and conduct research. COMSOL Multiphysics® software is an integrated software environment for creating physics models and simulation applications. The special value of the program is the ability to take into account interdisciplinary or multiphysics phenomena. Additional modules extend the simulation platform's capabilities to electrical, mechanical, fluid dynamics, and chemical application areas. Rich import/export tools allow COMSOL Multiphysics® to integrate with all major CAD tools available in the engineering software market. Computer simulation professionals use COMSOL Server™ to enable application development teams, manufacturing departments, test labs, and customers anywhere in the world. COMSOL was founded in 1986. Today we have more than 400 employees in 22 offices in various countries, and we collaborate with a network of distributors to promote our solutions.

COMSOL, COMSOL Multiphysics, Capture the Concept, and COMSOL Desktop are registered trademarks of COMSOL AB. COMSOL Server, LiveLink, and Simulation for Everyone are trademarks of COMSOL AB. Other product and brand names are trademarks or registered trademarks of their respective owners.

Electrical cables are characterized by parameters such as impedance and attenuation coefficient. In this topic we 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 based on electromagnetic field simulations in COMSOL Multiphysics. Having understood the principles of constructing a coaxial cable model, in the future we will be able to apply the acquired knowledge to calculate the parameters of transmission lines or cables of any type.

Electrical cable design considerations

Electrical cables, also called power lines, are now widely used to transmit data and electricity. Even if you are reading this text from the screen on a cell phone or tablet computer using a “wireless” connection, there are still “wired” power lines inside your device that connect 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 to charge.

Power lines are used in a wide variety of applications, from small coplanar waveguides on printed circuit boards to very large high-voltage power lines. They must also function under a variety of and often extreme operating conditions, from transatlantic telegraph cables to electrical wiring on spacecraft, as shown in the figure below. Transmission lines must be designed to meet all necessary requirements to ensure they operate reliably under specified conditions. In addition, they can be the subject of research with a view to further optimizing the design, including meeting the requirements for mechanical strength and low weight.

Connecting wires in the cargo bay of the shuttle mockup OV-095 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 admittance (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 solving Maxwell's equations for the electromagnetic field. To numerically solve Maxwell's equations 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 cable parameters and its efficiency change under different operating modes and operating conditions. The developed model can then be converted into an intuitive application like this one, which calculates parameters for standard and commonly used transmission lines.

In this topic we will analyze the case of coaxial cable - a fundamental problem that is usually contained in any standard training course on microwave technology or power lines. The coaxial cable is such a fundamental physical object that Oliver Heaviside patented it in 1880, just a few years after Maxwell formulated his famous equations. For students studying 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.

Analytical Solution Results for Coaxial Cable

Let's begin our consideration with a coaxial cable, which has characteristic dimensions indicated in the schematic representation of its cross section presented below. The dielectric core between the inner and outer conductor has a relative dielectric constant ( \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 dimension values: a = 0.405 mm, b = 1.45 mm, and t = 0.1 mm.

The standard solution method for power lines is that the structure of the 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 cross-section profile remains unchanged. If we then 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 to Maxwell’s equations is sought in the form ansatz-forms

for the electromagnetic field, where (\gamma = \alpha + j\beta ) is the complex propagation constant, and \alpha and \beta are the attenuation 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 .

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, since it is usually neglected when metal is considered as an ideal conductor. This term represents the inductance caused by the penetration of a magnetic field into a metal with finite conductivity, and is negligible at sufficiently high frequencies. (It can also be represented as L_(Internal) = R/\omega .)

For subsequent comparison with numerical results, the relationship for DC resistance can be calculated from the expression for conductivity and cross-sectional area of ​​the metal. The analytical expression for inductance (DC) is a little more complicated, so we present 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 values ​​for C and G over the entire frequency range, the values ​​for DC R and L, and their asymptotic values ​​at high frequencies, we have an excellent reference point for comparison with the numerical results.

Modeling cables in an AC/DC module

When setting up a problem for numerical modeling, 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 of the form \mathbf(E)\left(x,y,z\right) = \mathbf(\tilde(E))\left(x,y\right)e^(-\gamma z). Since the spatial change in fields that interests us occurs primarily in xy-plane, then we want to model only the 2D cross-section of the cable. However, this raises the problem that the 2D equations used in the AC/DC module assume that the fields remain invariant in the direction perpendicular to the modeling 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, by modeling 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 look at these aspects in more detail.

Distributed parameters for shunt conductance and capacitance

Since shunt conductance and capacitance can be calculated from the electric field distribution, let's start by using the interface Electric currents.

Boundary conditions and material properties for the modeling interface Electric currents.

Once the geometry of the model is determined and values ​​are assigned to the material properties, the assumption is made that the surface of the conductors is equipotential (which is absolutely justified, since the difference in conductivity between the conductor and the dielectric is usually almost 20 orders of magnitude). We then set the values ​​of the physical parameters by assigning an electrical potential V 0 to the inner conductor and a ground to the outer conductor to find the electrical potential in the dielectric. The above analytical expressions for capacity 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 an equation of electromagnetic theory, and the second is an equation of circuit theory.

The third relationship is a combination of the first and second equations. Substituting the above known expressions for the fields, we obtain the previously given analytical result for C in the 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 and coincides with the theory. Since G and C 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 have made the assumption that the conductivity of the dielectric region is zero. This is the standard assumption that is made in all tutorials, and we follow this convention here as well because it does not have a significant impact on the physics - unlike our inclusion of the internal inductance term discussed earlier. Many dielectric core materials have non-zero conductivity, but this can easily be taken into account in the simulation by simply plugging in new values ​​into the material properties. In this case, to ensure proper comparison with theory, it is also necessary to make appropriate adjustments to the theoretical expressions.

Specific parameters for series resistance and inductance

Similarly, series resistance and inductance can be calculated using simulation when using the interface Magnetic fields in the AC/DC module. The simulation settings are simple, as illustrated in the figure below.

Wire areas are added to the 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 in the outer conductor, which is indicated in the figure by dots and crosses. When calculating the frequency dependence, the current distribution in a 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 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 vary with frequency, let's look at the calculated values ​​and analytical solution in the DC limit and in the high frequency region.

“Analytical solution for direct current” and “Analytical solution for high frequencies” graphical dependencies correspond to 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 shown on a logarithmic scale along the frequency axis.

It is clearly seen that the calculated values ​​smoothly transition 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 thickness of the conductor. It is reasonable to assume that the transition region is located approximately at the point 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 is exactly the expected result.

Characteristic impedance and propagation constant

Now that we have completed the time-consuming work of calculating R, L, C, and G, there are still two other parameters essential to power line analysis 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 attenuation coefficient and \beta is the propagation coefficient.

\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 the DC and RF modes, compared with the values ​​​​determined from the simulation results. In addition, the fourth relationship in the graph is impedance, calculated in COMSOL Multiphysics using the Radio Frequency module, which we will briefly look at a little later. As can be seen, the results of numerical simulations are in good agreement with the analytical solutions for the corresponding limiting regimes, and also give correct values ​​in the transition region.

Comparison of characteristic impedance calculated using analytical expressions and determined from simulation results in COMSOL Multiphysics. Analytical curves were generated using the corresponding 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 was specially increased.

High frequency cable modeling

Electromagnetic field energy travels in the form of waves, which means the operating frequency and wavelength are inversely proportional to each other. As we move into higher and higher frequencies, we are forced to take into account the relative size of the wavelength and the electrical size of the cable. As discussed in the previous post, we must change AC/DC to RF module at an electrical size of approximately λ/100 (for the concept of "electrical size" see ibid.). If we choose the diameter of the cable as the electrical size, and instead of the speed of light in a vacuum, the speed of light in the dielectric core of the cable, we will obtain a 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 modes of the waveguide. Using waveguide terminology, so far we have considered a special type of mode called TEM-mode, which can propagate at any frequency. When cable cross-section and wavelength become comparable, we must also consider the possibility of higher order modes. Unlike the TEM mode, most waveguide 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 fc = 35.3 GHz, but even with this relatively simple geometry, the cutoff frequency is a solution to the transcendental equation, which we will not consider in this article.

So what does this cutoff frequency mean for our results? Above this frequency, the wave energy carried in the TEM mode in which we are interested has the potential to interact with the TE11 mode. In an idealized geometry like the one modeled here, there would be no interaction. In a real situation, however, any defects in the cable design can lead to mode coupling at frequencies above the cutoff frequency. This may be the result of a variety of uncontrollable factors, from manufacturing errors to gradients in material properties. The easiest way to avoid this situation is at the cable design stage by designing operation at obviously lower frequencies than the cutoff frequency of higher-order modes, so that only one mode can propagate. If this is of interest, you can also use the COMSOL Multiphysics environment to model the interactions between higher order modes, as done in this article (although that is beyond the scope of this article).

Modal analysis in the Radio Frequency module and Wave Optics module

Modeling of 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, modal analysis immediately provides a solution for the spatial distribution of the field and the complex propagation constant for each of a given number of modes. With this, we can use the same model geometry as before, except that we only need to use the dielectric core as the modeling region and .

Calculation results of the attenuation constant and effective refractive index of the wave mode from Mode Analysis. The analytical curve in the left graph - attenuation coefficient versus frequency - is calculated using the same expressions as for the RF curves used for comparison with the simulation results in the AC/DC module. The analytical curve in the right graph - the effective refractive index versus frequency - is simply n = \sqrt(\epsilon_r\mu_r) . For clarity, the size of the “COMSOL - TEM” line was intentionally increased in both graphs.

It can be clearly seen that the TEM Mode Mode Analysis results are consistent 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 modeling process 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 modal analysis method 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 DC 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 the AC/DC module simulation require more information about how the cable is driven, but hopefully you're aware of what's connecting to your cable! We used analytical theory solely to compare the numerical simulation results with well-known results for a reference model. This means the analysis can be generalized to other cables, as well as multiphysics modeling relationships 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 provide graphs of the characteristic impedance and all distributed parameters for the TE11 mode?”
  • Because only TEM modes have a uniquely defined voltage, current and characteristic impedance. In principle, it is possible to attribute some of these values ​​to higher order modes, and this issue will be discussed 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 mode problem using Modal Analysis, they are labeled with their work indices. Where do the TEM and TE11 mode designations come from?”
  • These notations appear in theoretical analysis and are used for convenience when discussing results. Such a name is not always possible with an arbitrary waveguide geometry (or cable in waveguide mode), but it is worth considering that this designation is just a “name”. Whatever the name of a fashion, does it still carry electromagnetic energy (excluding, of course, non-tunneling evanescent waves)?
• “Why is there an additional factor of ½ in some of your formulas?”
  • 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 factor of ½, unlike time domain (or DC) expressions. For more information, you can refer to works on classical electrodynamics.

Literature

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

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

2. COMSOL Quick Start Guide

The purpose of this section is to familiarize the reader with 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 steps involved in creating simple models and obtaining simulation results.

2D model of heat transfer from copper cable in a simple heatsink

This model explores some of the effects of thermoelectric heating. It is strongly recommended that you follow the modeling sequence 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 on the physics of the phenomenon being modeled.

Consider an aluminum heatsink 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 radiator 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. 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 radiator geometry 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 of 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 computation time and memory.

Modeling Technology in a COMSOL GUI Application

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

As a result of executing this command, the COMSOL figure and the Model Navigator figure 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, we need to select 2D on the New tab of the Navigator in the Space dimension field, select the model Application Modes/ COMSOL Multiphysics/ Heat transfer/Conduction/Steady-state analysis and click OK.

As a result of these actions, the Model Navigator figure and the COMSOL axes field will take on the appearance 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 geometry (Draw Mode is in effect). You can draw geometry using the Draw commands in the main menu or using the vertical toolbar located on the left side of the COMSOL shape.

Let the origin of coordinates be at 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 on the circumscribed circle. Let the radius of the inscribed circle be 3 mm, the angles at the internal vertices be right.

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 wire as follows. Press the button on the vertical toolbar, position the mouse pointer at the origin, press the Ctrl key and the left mouse button and hold them down, move the mouse pointer from the origin until the radius of the circle being drawn becomes equal to 2, release the mouse button and the Ctrl key. Drawing the correct radiator star is much easier to do

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 vertices of the star in the expanded dialog box. Such an operation is too complex and time-consuming. You can draw a star

represent it as a combination of squares, which can be conveniently created using the , buttons (when drawing with a mouse, you also need to hold down the Ctrl key so that you get squares, not rectangles). To accurately position the squares, you need to double-click on them and adjust their parameters in the expanding dialog boxes (coordinates, lengths and rotation angles can be specified using MATLAB expressions). After accurately positioning the squares, you need to create a composite geometric object from them by performing the following sequence of actions. Select the squares by single-clicking on them and holding down the Ctrl key (the selected objects will be

highlighted in brown), press the button, in the expanded dialog box, correct the formula of the composite object, press the OK button. Compound Object Formula

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

It is much easier and faster to create geometric objects in the MATLAB workspace and then insert them into the axes field using a 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; % Inner radius of radiator

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

y_vertex=r_vertex.*sin(al_vertex); % Cartesian coordinates of the 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. Executing this command will lead to the expansion of a dialog box, the appearance 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 lead to the insertion of geometric objects (Fig. 2.6). Objects will be selected and highlighted in brown. As a result of this import, the grid settings in the COMSOL GUI application are automatically configured when you click

to the button. At this point, the drawing of the 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 drawn geometry of a current-carrying copper core with a radiator: C1, P1 – names (labels) of geometric objects (C1 – circle, P1 – polygon).

Setting PDE coefficients

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

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

To enter parameters of material properties (PDE coefficients), you need to 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 applied heat transfer 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; specific electrical conductivity of copper g = 5.998e7 S/m; thermal conductivity coefficient medik = 400; Let the radiator be made of aluminum, which has a thermal conductivity coefficient k = 160. It is known that the volumetric power density of heat losses when electric current flows through a substance is equal to Q = d2 /g. Let's select zone No. 2 in the Subdomain Selection panel and load the corresponding 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 PDE coefficients.

Setting boundary conditions

To set boundary conditions, you need to 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, as 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 problem conditions, 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, you can hold down the Ctrl key and click on all outer segments with the mouse. The selected segments will be highlighted in red (see Fig. 2.12).

Rice. 2.12. Selected outer boundary segments

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

Fig.2.13. Dialog box for entering boundary conditions

In Fig. Figure 2.13 shows the entered temperature value on the selected segments. This dialog box also contains a segment selection panel. So, it is not necessary to select them directly in the axes field. If you click OK or Apply, OK, the entered boundary conditions will be accepted. At this point, the introduction of boundary conditions in this problem 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, just run the Mesh/Initialize Mesh command. The mesh will be automatically generated according to the current mesh generator settings. The automatically generated mesh is shown in Fig. 2.13.