Institute for Eastern Europe
Electrical Engineering and InformationTechnology
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University "Lviv Stavropihin"
Institute for Eastern Europe
Electrical Engineering and InformationTechnology
Institute for Eastern Europe
Electrical Engineering and InformationTechnology
While there is no consensus yet as to a precise definition of this term, mathematical modeling is generally understood as the process of applying mathematics to a real world problem with a view of understanding the latter. One can argue that mathematical modeling is the same as applying mathematics where we also start with a real world problem, we apply the necessary mathematics, but after having found the solution we no longer think about the initial problem except perhaps to check if our answer makes sense. This is not the case with mathematical modeling where the use of mathematics is more for understanding the real world problem. The modeling process may or may not result to solving the problem entirely but it will shed light to the situation under investigation[1-10]. The figure below shows key steps in modeling process.
Mathematical modeling approaches can be categorized into four broad approaches: Empirical models, simulation models, deterministic models, and stochastic models. The first three models can very much be integrated in teaching high school mathematics. The last will need a little stretching.Empirical modeling involves examining data related to the problem with a view of formulating or constructing a mathematical relationship between the variables in the problem using the available data.Simulation modeling involve the use of a computer program or some technological tool to generate a scenario based on a set of rules. These rules arise from an interpretation of how a certain process is supposed to evolve or progress.Deterministic modeling in general involve the use of equation or set of equations to model or predict the outcome of an event or the value of a quantity.Stochastic modeling takes deterministic modeling one further step. In stochastic models, randomness and probabilities of events happening are taken into account when the equations are formulated. The reason behind this is the fact that events take place with some probability rather than with certainty. This kind of modeling is very popular in business and marketing.Mathematical modelling is a process of representing real world problems in mathematical terms in an attempt to find solutions to the problems. A mathematical model can be considered as a simplification or abstraction of a real world problem or situation into a mathematical form, thereby converting the real world problem into a mathematical problem. The mathematical problem can then be solved using whatever known techniques to obtain a mathematical solution. This solution is then interpreted and translated into real terms. Figure 2 shows a simplified view of the process of mathematical modelling[7-13].
Figure 2: A simple view of the mathematical modelling process
The above is, of course, a grossly simplified definition for the usually complex process of modelling. However, for the purpose of the present discussion, it is sufficient to note that in mathematical modelling, the starting point is a real world problem or situation. As we shall see, in mathematical modelling, the emphasis is in solving a problem rather than finding an answer that must exist. Sometimes, we may not even be able to solve the problem entirely, although we hope to move one step closer to obtaining a solution. At other times, we are happy with a good approximation to the solution of the problem when an “exact answer” either does not exist or is beyond reach.Hence, when we approach the teaching of mathematics through mathematical modelling, we are really teaching mathematical problem solving. We present mathematics in action, instead of as a confusing set of formulae scribbled on the chalkboard. We place mathematics in some context and focus on why mathematics exists in the first place. Moreover, many challenging and exciting skills are used in developing models and these have often been ignored in traditional school mathematics. Some of these will become apparent in the next section when we examine specific examples.Computer models and computer simulations have become an important part of the research repertoire, supplementing (and in some cases replacing) experimentation. Going from application area to computational results requires domain expertise, mathematical modeling, numerical analysis, algorithm development, software implementation, program execution, analysis, validation and visualization of results. CSE involves all of this[6-12].
One point we would like to emphasize in this document is that CSE is a legitimate and important academic enterprise, even if it has yet to be formally recognized as such at some institutions. Although it includes elements from computer science, applied mathematics, engineering and science, CSE focuses on the integration of knowledge and methodologies from all of these disciplines, and as such is a subject which is distinct from any of them.
Series RLC Example
You can often formulate the mathematical system you are modeling in several ways. Choosing the best-form mathematical model allows the simulation to execute faster and more accurately. For example, consider a simple series RLC circuit[1-12].
According to Kirchoff's voltage law, the voltage drop across this circuit is equal to the sum of the voltage drop across each element of the circuit.
Using Ohm's law to solve for the voltage across each element of the circuit, the equation for this circuit can be written as[2-7]
You can model this system in Simulink® by solving for either the resistor voltage or inductor voltage. Which you choose to solve for affects the structure of the model and its performance.
Solving Series RLC Using Resistor Voltage
Solving the RLC circuit for the resistor voltage yields
The following diagram shows this equation modeled in Simulink where R is 70, C is 0.00003, and L is 0.04. The resistor voltage is the sum of the voltage source, the capacitor voltage, and the inductor voltage. You need the current in the circuit to calculate the capacitor and inductor voltages. To calculate the current, multiply the resistor voltage by a gain of 1/R. Calculate the capacitor voltage by integrating the current and multiplying by a gain of 1/C. Calculate the inductor voltage by taking the derivative of the current and multiplying by a gain of L[2-13].
This formulation contains a Derivative block associated with the inductor. Whenever possible, you should avoid mathematical formulations that require Derivative blocks as they introduce discontinuities into your system. Numerical integration is used to solve the model dynamics though time. These integration solvers take small steps through time to satisfy an accuracy constraint on the solution. If the discontinuity introduced by the Derivative block is too large, it is not possible for the solver to step across it. In addition, in this model the Derivative, Sum, and two Gain blocks create an algebraic loop. Algebraic loops slow down the model's execution and can produce less accurate simulation results. See Algebraic Loops for more information.
Solving Series RLC Using Inductor Voltage
To avoid using a Derivative block, formulate the equation to solve for the inductor voltage[5-9].
The following diagram shows this equation modeled in Simulink. The inductor voltage is the sum of the voltage source, the resistor voltage, and the capacitor voltage. You need the current in the circuit to calculate the resistor and capacitor voltages. To calculate the current, integrate the inductor voltage and divide by L. Calculate the capacitor voltage by integrating the current and dividing by C. Calculate the resistor voltage by multiplying the current by a gain of R.
This model contains only integrator blocks and no algebraic loops. As a result, the model simulates faster and more accurately.
Kirchhoff's circuit laws
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of Maxwell. Widely used in electrical engineering, they are also called Kirchhoff's rules or simply Kirchhoff's laws.
Kirchhoff's laws of electric circuits
Both of Kirchhoff's laws can be understood as corollaries of the Maxwell equations in the low-frequency limit. They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits.
Kirchhoff's current law (KCL)
The current entering any junction is equal to the current leaving that junction. i2 + i3 = i1 + i4
This law is also called Kirchhoff's first law, Kirchhoff's point rule, or Kirchhoff's junction rule.
The principle of conservation of electric charge implies that:
At any node in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node
The algebraic sum of currents in a network of conductors meeting at a point is zero.
Recalling that current is a signed quantity reflecting direction towards or away from a node, this principle can be stated as:
n is the total number of branches with currents flowing towards or away from the node.
This formula is valid for complex currents:
The law is based on the conservation of charge whereby the charge is the product of the current and the time.
A matrix version of Kirchhoff's current law is the basis of most circuit simulation software, such as SPICE. Kirchhoff's current law combined with Ohm's Law is used in nodal analysis.KCL is applicable to any lumped network irrespective of the nature of the network; whether unilateral or bilateral, active or passive, linear or non-linear[2-9].
Kirchhoff's voltage law (KVL)
The sum of all the voltages around a loop is equal to zero.
v1 + v2 + v3 - v4 = 0
This law is also called Kirchhoff's second law, Kirchhoff's loop rule, and Kirchhoff's second rule.
The principle of conservation of energy implies that
The directed sum of the electrical potential differences (voltage) around any closed network is zero, or:
More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop, or:
The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop.
Similarly to KCL, it can be stated as:
Here, n is the total number of voltages measured. The voltages may also be complex:
This law is based on the conservation of energy whereby voltage is defined as the energy per unit charge. The total amount of energy gained per unit charge must be equal to the amount of energy lost per unit charge, as energy and charge are both conserved.
In the low-frequency limit, the voltage drop around any loop is zero. This includes imaginary loops arranged arbitrarily in space – not limited to the loops delineated by the circuit elements and conductors. In the low-frequency limit, this is a corollary of Faraday's law of induction.KCL and KVL both depend on the lumped element model being applicable to the circuit in question. When the model is not applicable, the laws do not apply.KCL, in its usual form, is dependent on the assumption that current flows only in conductors, and that whenever current flows into one end of a conductor it immediately flows out the other end. This is not a safe assumption for high-frequency AC circuits, where the lumped element model is no longer applicable. It is often possible to improve the applicability of KCL by considering "parasitic capacitances" distributed along the conductors. Significant violations of KCL can occur even at 60Hz, which is not a very high frequency.In other words, KCL is valid only if the total electric charge, , remains constant in the region being considered. In practical cases this is always so when KCL is applied at a geometric point. When investigating a finite region, however, it is possible that the charge density within the region may change. Since charge is conserved, this can only come about by a flow of charge across the region boundary. This flow represents a net current, and KCL is violated.KVL is based on the assumption that there is no fluctuating magnetic field linking the closed loop. This is not a safe assumption for high-frequency AC circuits. In the presence of a changing magnetic field the electric field is not a conservative vector field. Therefore the electric field cannot be the gradient of any potential. That is to say, the line integral of the electric field around the loop is not zero, directly contradicting KVL.It is often possible to improve the applicability of KVL by considering parasitic inductances distributed along the conductors. These are treated as imaginary circuit elements that produce a voltage drop equal to the rate-of-change of the flux.
Assume an electric network consisting of two voltage sources and three resistors.
According to the first law we have
The second law applied to the closed circuit s1 gives
The second law applied to the closed circuit s2 gives
Thus we get a linear system of equations in :
Which is equivalent to
the solution is
has a negative sign, which means that the direction of is opposite to the assumed direction.
In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid.
where and are the angular and linear velocity vectors at the point , respectively; is the moment of inertia tensor, is the body's mass; is a unit normal to the surface of the body at the point ; is a pressure at this point; and are the hydrodynamic torque and force acting on the body, respectively; and likewise denote all other torques and forces acting on the body. The integration is performed over the fluid-exposed portion of the body's surface.
If the body is completely submerged body in an infinitely large volume of irrotational, incompressible, inviscid fluid, that is at rest at infinity, then the vectors and can be found via explicit integration, and the dynamics of the body is described by the Kirchhoff – Clebsch equations:
Their first integrals read
Further integration produces explicit expressions for position and velocities.
In physics, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Some of its principles are even used in traffic engineering, where traffic is treated as a continuous fluid, and crowd dynamics.Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time.Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases[3-10].
Equations of fluid dynamics
The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy. These are based on classical mechanics and are modified in quantum mechanics and general relativity. They are expressed using the Reynolds Transport Theorem.In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption considers fluids as continuous, rather than discrete. Consequently, properties such as density, pressure, temperature, and flow velocity are assumed well-defined at infinitesimally small points, and are assumed to vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier–Stokes equations—which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have a general closed-form solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve. Some of them allow appropriate fluid dynamics problems be solved in closed form. In addition to the mass, momentum, and energy conservation equations, a thermodynamical equation of state giving the pressure as a function of other thermodynamic variables for the fluid is required to completely specify the problem. An example of this would be the perfect gas equation of state[2-9]:
where p is pressure, ρ is density, Ru is the gas constant, M is molar mass and T is temperature.
Three conservation laws are used to solve fluid dynamics problems, and may be written in integral or differential form. Mathematical formulations of these conservation laws may be interpreted by considering the concept of a control volume. A control volume is a specified volume in space through which air can flow in and out. Integral formulations of the conservation laws consider the change in mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply Stokes' theorem to yield an expression which may be interpreted as the integral form of the law applied to an infinitesimal volume at a point within the flow.
Mass continuity: The rate of change of fluid mass inside a control volume must be equal to the net rate of fluid flow into the volume. Physically, this statement requires that mass is neither created nor destroyed in the control volume, and can be translated into the integral form of the continuity equation[2-9]:
Above, is the fluid density, u is the flow velocity vector, and t is time. The left-hand side of the above expression contains a triple integral over the control volume, whereas the right-hand side contains a surface integral over the surface of the control volume. The differential form of the continuity equation is, by the divergence theorem:
Conservation of momentum: This equation applies Newton's second law of motion to the control volume, requiring that any change in momentum of the air within a control volume be due to the net flow of air into the volume and the action of external forces on the air within the volume. In the integral formulation of this equation, body forces here are represented by fbody, the body force per unit mass. Surface forces, such as viscous forces, are represented by , the net force due to stresses on the control volume surface[1-7].
The differential form of the momentum conservation equation is as follows. Here, both surface and body forces are accounted for in one total force, F. For example, F may be expanded into an expression for the frictional and gravitational forces acting on an internal flow.
In aerodynamics, air is assumed to be a Newtonian fluid, which posits a linear relationship between the shear stress and the rate of strain of the fluid. The equation above is a vector equation: in a three-dimensional flow, it can be expressed as three scalar equations. The conservation of momentum equations for the compressible, viscous flow case are called the Navier–Stokes equations. Conservation of energy: Although energy can be converted from one form to another, the total energy in a given closed system remains constant[1-9].
Above, h is enthalpy, k is the thermal conductivity of the fluid, T is temperature, and is the viscous dissipation function. The viscous dissipation function governs the rate at which mechanical energy of the flow is converted to heat. The second law of thermodynamics requires that the dissipation term is always positive: viscosity cannot create energy within the control volume. The expression on the left side is a material derivative.
Compressible vs incompressible flow
All fluids are compressible to some extent, that is, changes in pressure or temperature cause changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modelled as an incompressible flow. Otherwise the more general compressible flow equations must be used.Mathematically, incompressibility is expressed by saying that the density ρ of a fluid parcel does not change as it moves in the flow field, i.e.,
where D/Dt is the substantial derivative, which is the sum of local and convective derivatives. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density.For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible assumption is valid depends on the fluid properties and the flow conditions. Acoustic problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of the medium through which they propagate[2-9].
Inviscid vs Newtonian and non-Newtonian fluids
Potential flow around a wing
All fluids are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other. The velocity gradient is referred to as a strain rate; it has dimensions . Isaac Newton showed that for many familiar fluids such as water and air, the stress due to these viscous forces is linearly related to the strain rate. Such fluids are called Newtonian fluids. The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property independent of the strain rate.Non-Newtonian fluids have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology studies the stress-strain behaviours of these fluids, which include emulsions and slurries, some viscoelastic materials such as blood and some polymers, and sticky liquids such as latex, honey and lubricants. The dynamic of fluid parcels is described with the help of Newton's second law. An accelerating parcel of fluid is subject to inertial effects.The Reynolds number is a dimensionless quantity which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number (Re<<1) indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called Stokes or creeping flow.
On the contrary, high Reynolds numbers (Re>>1) indicate that the inertial effects have more effect on the velocity field than the viscous effects. In high Reynolds number flows, the flow is often modeled as an inviscid flow, an approximation in which viscosity is completely neglected. The Navier-Stokes equations then simplify into the Euler equations. Integrating these along a streamline in an inviscid flow yields Bernoulli's equation. When in addition to being inviscid, the flow is everywhere irrotational, Bernoulli's equation can be used throughout the flow field. Such flows are called potential flows, because the velocity field may be expressed as the gradient of a potential.This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the no-slip condition generates a thin region of large strain rate, the boundary layer, in which viscosity effects dominate and which thus generates vorticity. Therefore, to calculate net forces on bodies, viscous flow equations must be used: inviscid flow theory fails to predict drag forces, a limitation known as the d'Alembert's paradox.A commonly used model, especially in computational fluid dynamics, is to use two flow models: the Euler equations away from the body, and boundary layer equations in a region close to the body. The two solutions can then be matched with each other, using the method of matched asymptotic expansions.When all the time derivatives of a flow field vanish, the flow is considered steady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Otherwise, flow is called unsteady. Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over a sphere is steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady.Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. According to Pope: The random field U(x,t) is statistically stationary if all statistics are invariant under a shift in time.This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow.Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer than the governing equations of the same problem without taking advantage of the steadiness of the flow field[1-9].
Laminar vs turbulent flow
Turbulence is flow characterized by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar. It should be noted, however, that the presence of eddies or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via a Reynolds decomposition, in which the flow is broken down into the sum of an average component and a perturbation component.It is believed that turbulent flows can be described well through the use of the Navier–Stokes equations. Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows. Solving these real-life flow problems requires turbulence models for the foreseeable future. Reynolds-averaged Navier–Stokes equations (RANS) combined with turbulence modelling provides a model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by the Reynolds stresses, although the turbulence also enhances the heat and mass transfer. Another promising methodology is large eddy simulation (LES), especially in the guise of detached eddy simulation (DES)—which is a combination of RANS turbulence modelling and large eddy simulation[2-11].
Subsonic vs transonic, supersonic and hypersonic flows
While many terrestrial flows occur at low mach numbers, many flows of practical interest occur at high fractions of the Mach Number M=1 or in excess of it. New phenomena occur at these Mach number regimes and it is necessary to treat each of these flow regimes separately.
Magnetohydrodynamics is the multi-disciplinary study of the flow of electrically conducting fluids in electromagnetic fields. Examples of such fluids include plasmas, liquid metals, and salt water. The fluid flow equations are solved simultaneously with Maxwell's equations of electromagnetism.
There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below[2-10].
Terminology in fluid dynamics
The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods.Some of the terminology that is necessary in the study of fluid dynamics is not found in other similar areas of study. In particular, some of the terminology used in fluid dynamics is not used in fluid statics.
Terminology in incompressible fluid dynamics
The concepts of total pressure and dynamic pressure arise from Bernoulli's equation and are significant in the study of all fluid flows. To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure to distinguish it from total pressure and dynamic pressure. Static pressure is identical to pressure and can be identified for every point in a fluid flow field.A point in a fluid flow where the flow has come to rest is of special significance. It is of such importance that it is given a special name—a stagnation point. The static pressure at the stagnation point is of special significance and is given its own name—stagnation pressure. In incompressible flows, the stagnation pressure at a stagnation point is equal to the total pressure throughout the flow field[3-14].
Terminology in compressible fluid dynamics
In a compressible fluid, such as air, the temperature and density are essential when determining the state of the fluid. In addition to the concept of total pressure, the concepts of total temperature and total density are also essential in any study of compressible fluid flows. To avoid potential ambiguity when referring to temperature and density, many authors use the terms static temperature and static density. Static temperature is identical to temperature; and static density is identical to density; and both can be identified for every point in a fluid flow field.The temperature and density at a stagnation point are called stagnation temperature and stagnation density.
A similar approach is also taken with the thermodynamic properties of compressible fluids. Many authors use the terms total enthalpy and total entropy. The terms static enthalpy and static entropy appear less common, but where they are used they mean enthalpy and entropy respectively, using the prefix "static" to avoid ambiguity with their 'total' or 'stagnation' counterparts. Because the 'total' flow conditions are defined by isentropically bringing the fluid to rest, the total entropy is by definition always equal to the "static" entropy.
Both Kirchhoff’s current law and Kirchhoff’s voltage law can be directly derived from Maxwell’s equations. However, Kirchhoff’s circuit laws are simplified form of Maxwell’s equations designated for electric circuit analysis.
Kirchhoff’s current law known also as KCL, Kirchhoff’s first law or Kirchhoff’s junction rule is based upon conservation of charge law and says that is, the algebraic sum of all branch currents flowing into any node must be zero. Kirchhoff’s current law is valid for DC and AC stationary electric networks alike. For given electric circuit, number of independent equations according to Kirchhoff’s current law is N – 1, where N is the number of nodes.
Kirchhoff’s voltage law known also as KVL, Kirchhoff’s second law or Kirchhoff’s mesh rule is based upon energy conservation law and says that the algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop. Kirchhoff’s voltage law is valid for DC and AC stationary electric networks alike. For given electric circuit, number of independent equations according to Kirchhoff’s voltage law is M, where M is the number of meshes.
Even though Kirchhoff’s circuit laws are simplified form of Maxwell’s equations, there are still very complex to be utilized directly for solving electric circuit. For example, if we need to solve circuit given in figure above we would need to write six equations with six variables. It is more appropriate to solve electric circuit by writing smaller number of equations and smaller number of variables. This can be achieved by using one of the following solving methods[3-11]:
All above mentioned methods and theorems are in accordance with Kirchhoff circuit laws yet less complex because lots of the magnitudes are already included into equations written.
Nodal analysis is practical method for electric circuit analysis with small number of nodes and possibly large number of mesh. All equations are written according KCL, so it is important to have small number of nodes i.e. small number of equations to solve.
Steps of the analysis[3-13]:
Find one node and make it reference node.
Assign a variable for each node whose voltage is unknown. If the voltage is already known, it is not necessary to assign a variable.
If there is a branch with only ideal voltage source in it, set of equations has to be modified, since it is not possible to calculate current to/from ideal voltage source, apart from other currents in node given.
Write set of equations based on KCL, where each of individual currents is calculated simply by applying Ohm’s law – as difference of voltage potential between two nodes over resistance.
Solve formed system of equations.
In the following example, given scheme is analyzed with node voltage analysis.
In given scheme, node V0, is chosen to be reference voltage. Remaining nod is marked as VA, and only one equating should be written:
By separating values:
Final solution is:
Now, currents I3 and I2 are easy to calculate as:
System of equations is significantly reduced comparing to direct application of Kirchhoff circuit laws, where should be one equation written according to Kirchhoff’s current law and two according to Kirchhoff’s voltage law. For comparison, the same circuit is analyzed with mesh analysis.Mesh analysis is a method that is used to solve planar circuits for the currents. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. The mesh method is a well organized technique which relies on KVL. All equations are written according KVL, so it is important to have small number of loops.
Steps of the analysis[2-9]:
Find NB-(NN-1) independent loops in electric circuit. NB is number of branches, and NN is the number of nodes.
In every chosen loop, draw loop current to be independent variable.
Write set of equations based on KVL, where each of individual currents is taken into account by multiplying it with impedance on it’s path.
Solve formed system of equations.
Find branch currents as sum or difference of loop currents depending on current direction.
In the following example, the same scheme analyzed with nodal analysis, will be analyzed with mesh analysis.
In given scheme, there are two loops, so two equations should be written. However loop equation can’t be written if current source is on the mesh. In this example, current source is contained between two essential meshes. In order to overcome problem, for writing first equation according to KVL, one loop that bypasses current source should be chosen.
For chosen mesh, equation is:
Another equation must be written according to KCL.
Now, we have system of two linear equations with two variables (I2, I3) to be solved.
Signal-flow graphs are another method for visually representing a system. Signal Flow Diagrams are especially useful, because they allow for particular methods of analysis, such as Mason's Gain Formula.Signal flow diagrams typically use curved lines to represent wires and systems, instead of using lines at right-angles, and boxes, respectively. Every curved line is considered to have a multiplier value, which can be a constant gain value, or an entire transfer function. Signals travel from one end of a line to the other, and lines that are placed in series with one another have their total multiplier values multiplied together.Signal flow diagrams help us to identify structures called "loops" in a system, which can be analyzed individually to determine the complete response of the system.
An example of a signal flow diagram.
A forward path is a path in the signal flow diagram that connects the input to the output without touching any single node or path more than once. A single system can have multiple forward paths.
A loop is a structure in a signal flow diagram that leads back to itself. A loop does not contain the beginning and ending points, and the end of the loop is the same node as the beginning of a loop.Loops are said to touch if they share a node or a line in common.The Loop gain is the total gain of the loop, as you travel from one point, around the loop, back to the starting point.
The Delta value of a system, denoted with a Greek Δ is computed as follows:
A is the sum of all individual loop gains
B is the sum of the products of all the pairs of non-touching loops
C is the sum of the products of all the sets of 3 non-touching loops
D is the sum of the products of all the sets of 4 non-touching loops
If the given system has no pairs of loops that do not touch, for instance, B and all additional letters after B will be zero.
Mason's rule is a rule for determining the gain of a system. Mason's rule can be used with block diagrams, but it is most commonly used with signal flow diagrams.If we have computed our delta values, we can then use Mason's Gain Rule to find the complete gain of the system:
Where M is the total gain of the system, represented as the ratio of the output gain (yout) to the input gain (yin) of the system. Mk is the gain of the kth forward path, and Δk is the loop gain of the kth loop[3-9].
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Hello, the original circuit above is where I am trying to find I1, and I simplify it the figure below and assuming the following current directions.
Using Kirchoff current and voltage laws I end up with
However, plugging in the equations into my calculator it says there is no solution. Is it with my kirchoff current equations (first 4 equations) because if I add up all current law equations I end up with 0 = 0. Thanks for any help.
Solving a signal-flow graph by systematic reduction : Two interlocking loops
This example shows how a system of five equations in five unknowns is solved using systematic reduction rules. The independent variable is . The dependent variables are , , , , . The coefficients are labeled .
Here is the starting flowgraph[4-14]:
The steps for solving follow.
Removing edge c from x2 to x3
Removing node x2 and its inflows
has no outflows, and is not a node of interest.
Removing edge e from x3 to x1
Remove edge d from x3 to x4
Node has no outflows and is not a node of interest. It is deleted along with its inflows.
Removing self-loop at x1
Removing self-loop at x4
Remove edge from x4 to x1
Remove outflow from x4 to xout
's outflow is then eliminated: is connected directly to using the product of the gains from the two edges replaced.is not a variable of interest; thus, its node and its inflows are eliminated[3-12].
Eliminating self-loop at x1
Eliminating outflow from x1, then eliminating x1 and its inflows
is not a variable of interest; and its inflows are eliminated
Simplifying the gain expression
Solving a signal-flow graph by systematic reduction: Three equations in three unknowns
This example shows how a system of three equations in three unknowns is solved using systematic reduction rules.The independent variables are , , . The dependent variables are , , . The coefficients are labeled . The steps for solving follow[2-13]:
Electrical engineering: Construction of a flow graph for a RC circuit
This illustration shows the physical connections of the circuit. Independent voltage source S is connected in series with a resistor R and capacitor C. The example is developed from the physical circuit equations and solved using signal-flow graph techniques. Polarity is important[4-9]:
S is a source with the positive terminal at N1 and the negative terminal at N3
R is a resistor with the positive terminal at N1 and the negative terminal at N2
C is a capacitor with the positive terminal at N2 and the negative terminal at N3.
The unknown variable of interest is the voltage across capacitor C.
Approach to the solution[3-12]:
Find the set of equations from the physical network. These equations are acausal in nature.
Build a signal-flow graph from the equations.
Solve the signal-flow graph.
The branch equations are shown for R and C.
Resistor R (Branch equation )
The resistor's branch equation in the time domain is:
In the Laplace-transformed signal space:
Capacitor C (Branch equation )
The capacitor's branch equation in the time domain is:
Assuming the capacitor is initially discharged, the equation becomes:
Taking the derivative of this and multiplying by C yields the derivative form:
In the Laplace-transformed signal space:
Kirchhoff's laws equations
Kirchhoff's Voltage Law equation
This circuit has only one independent loop. Its equation in the time domain is:
In the Laplace-transformed signal space:
Kirchhoff's Current Law equations
The circuit has three nodes, thus three Kirchhoff's current equations:
In the Laplace-transformed signal space:
A set of independent equations must be chosen. For the current laws, it is necessary to drop one of these equations. In this example, let us choose .
Building the signal-flow graph
The next step consists in assigning to each equation a signal that will be represented as a node. Each independent source signal is represented in the signal-flow graph as a source node, therefore no equation is assigned to the independent source . There are many possible valid signal flow graphs from this set of equations. An equation must only be used once, and the variables of interest must be represented[12-19].
The resulting flow graph is then drawn
The next step consists in solving the signal-flow graph.
Using either Mason or systematic reduction, the resulting signal flow graph is:
Angular position servo and signal flow graph. θC = desired angle command, θL = actual load angle, KP = position loop gain, VωC = velocity command, VωM = motor velocity sense voltage, KV = velocity loop gain, VIC = current command, VIM = current sense voltage, KC = current loop gain, VA = power amplifier output voltage, LM = motor inductance, VM = voltage across motor inductance, IM = motor current, RM = motor resistance, RS = current sense resistance, KM = motor torque constant (Nm/amp) , T = torque, M = momment of inertia of all rotating components α = angular acceleration, ω = angular velocity, β = mechanical damping, GM = motor back EMF constant, GT = tachometer conversion gain constant. There is one forward path (shown in a different color) and six feedback loops. The drive shaft assumed to be stiff enough to not treat as a spring. Constants are shown in black and variables in purple.Mathematical Modeling is information of a program using mathematical ideas and language. Mathematical models are used not only in the natural sciences and technological innovation professions but also in the social sciences, physicists, technicians, statisticians, research experts and economic experts use mathematical designs most substantially[2-12].
A model may help to describe a program and to study the effects of different elements, and to make forecasts about behavior. Mathematical model can take several types, but not restricted to dynamical systems, differential equations, or game theoretic designs. These and additional types of models can overlap, with a specified model along with a variety of subjective elements. In general, mathematical models may include sensible designs, as far as reasoning is taken as a part of mathematics.
Principles of Mathematical Modeling:
Mathematical modeling is a principled action that has both principles behind it and techniques that can be efficiently used. The concepts are over-arching or meta-principles phrased as concerns about the objectives and reasons of mathematical modeling. In characteristics meta-principles are almost philosophical .We will now summarize the concepts, and in the next area we will temporarily evaluation some of the techniques.Initially recognize the need for the model and next find Record the information we are looking for. Then recognize the available appropriate information. Then assume and recognize the conditions that implement. Then recognize the governing physical concepts. We have to predict the equations that will be used, the computations that will be created, and the solutions that will result. The model predicts assessments that can be made to confirm the model, i.e., is it reliable with its concepts and assumptions. Finally verify the assessments that can be made to confirm the design, i.e., is it useful with regards to the preliminary purposes it were done.
Process of building a computer model, and the interplay between experiment, simulation, and theory.
The Electro Mechanical Behavioral Library of Engineering Models EMBLEM-MATH is the first one of the lineage and it is comprised of mathematical equations for functions which allow building complex behavioral models with a netlist approach for composing them at the ViC and the SoC level.The first release of EMBLEM-MATH 1.0 enables calculations with real numbers such as sources, addition, log, floor, round, derivation, gain, etc.With this release analog designers are able to create, by mere graphical means, their own library at high abstraction level based on mathematical functions, without the need to implement these functions in any specific HDL. These models can be used in any application or domain like analog, logic and mixed-signal electronic domains plus different physical domains like:thermal, electro-mechanics, electro-magnetic electro-optic and electro-fluidic.In the above schematic, the stepper motor and the PWM models were build with the help of the mathematical functions in EMBLEM-Math. They are seen as hierarchical blocks in the design.With SLASH bundling the schematic editor SLED and the mixed signal simulator SMASH, it is easy to drag and drop the symbols and to instantiate models composed out of EMBLEM MATH equations in SPICE, VHDL-AMS and Verilog-A circuits.
For more information on EMBLEM MA
An Eagle Schematic Example:
Building the model with Simulink
This system will be modeled by summing the torques acting on the rotor inertia and integrating the acceleration to give velocity. Also, Kirchoff's laws will be applied to the armature circuit. First, we will model the integrals of the rotational acceleration and of the rate of change of the armature current.
To build the simulation model, open Simulink and open a new model window. Then follow the steps listed below.
Insert an Integrator block from the Simulink/Continuous library and draw lines to and from its input and output terminals.
Label the input line "d2/dt2(theta)" and the output line "d/dt(theta)" as shown below. To add such a label, double-click in the empty space just below the line.
Insert another Integrator block above the previous one and draw lines to and from its input and output terminals.
Label the input line "d/dt(i)" and the output line "i".
Next, we will apply Newton's law and Kirchoff's law to the motor system to generate the following equations[14-23]:
The angular acceleration is equal to 1 / J multiplied by the sum of two terms. Similarly, the derivative of current is equal to 1 / L multiplied by the sum of three terms. Continuing to model these equations in Simulink, follow the steps given below.
Insert two Gain blocks from the Simulink/Math Operations library, one attached to each of the integrators.
Edit the Gain block corresponding to angular acceleration by double-clicking it and changing its value to "1/J".
Change the label of this Gain block to "Inertia" by clicking on the word "Gain" underneath the block.
Similarly, edit the other Gain's value to "1/L" and its label to "Inductance".
Insert two Add blocks from the Simulink/Math Operations library, one attached by a line to each of the Gain blocks.
Edit the signs of the Add block corresponding to rotation to "+-" since one term is positive and one is negative.
Edit the signs of the other Add block to "-+-" to represent the signs of the terms in the electrical equation.
Now, we will add in the torques which are represented in the rotational equation. First, we will add in the damping torque.
Insert a Gain block below the "Inertia" block. Next right-click on the block and select Format > Flip Block from the resulting menu to flip the block from left to right. You can also flip a selected block by holding down Ctrl-I.
Set the Gain value to "b" and rename this block to "Damping".
Tap a line (hold Ctrl while drawing or right-click on the line) off the rotational Integrator's output and connect it to the input of the "Damping" block.
Draw a line from the "Damping" block output to the negative input of the rotational Add block.
Next, we will add in the torque from the armature.
Insert a Gain block attached to the positive input of the rotational Add block with a line.
Edit its value to "K" to represent the motor constant and Label it "Kt".
Continue drawing the line leading from the current Integrator and connect it to the "Kt" block.
Now, we will add in the voltage terms which are represented in the electrical equation. First, we will add in the voltage drop across the armature resistance.
Insert a Gain block above the "Inductance" block and flip it from left to right.
Set the Gain value to "R" and rename this block to "Resistance".
Tap a line off the current Integrator's output and connect it to the input of the "Resistance" block.
Draw a line from the "Resistance" block's output to the upper negative input of the current equation Add block.
Next, we will add in the back emf from the motor.
Insert a Gain block attached to the other negative input of the current Add block with a line.
Edit it's value to "K" to represent the motor back emf constant and Label it "Ke".
Tap a line off the rotational Integrator's output and connect it to the "Ke" block.
Add In1 and Out1 blocks from the Simulink/Ports & Subsystems library and respectively label them "Voltage" and "Speed".
The final design should look like the example shown in the figure below.
In order to save all of these components as a single subsystem block, first select all of the blocks, then select Create Subsystem from the Edit menu. Name the subsystem "DC Motor" and then save the model. Your model should appear as follows. You can also download the file for this system here, Motor_Model.mdl. We use this model in the DC Motor Speed: Simulink Controller Design section.
Building the model with Simscape
In this section, we alternatively show how to build the DC Motor model using the physical modeling blocks of the Simscape extension to Simulink. The blocks in the Simscape library represent actual physical components; therefore, complex multi-domain models can be built without the need to build mathematical equations from physical principles as was done above by applying Newton's laws and Kirchoff's laws. Open a new Simulink model and insert the following blocks to represent the electrical and mechanical elements of the DC motor[11-23].
Resistor, Inductor and Rotational Electromechanical Converter blocks from the Simscape/Foundation Library/Electrical/Electrical Elements library
Rotational Damper and Inertia blocks from the Simscape/Foundation Library/Mechanical/Rotational Elements library
Four Connection Port blocks from the Simscape/Utilities library
Double-click on the Connection Port blocks to make the location of ports 1 and 2 from the Left and the location of ports 3 and 4 from the Right. Connect and label the components as shown in the following figure. You can rotate a block in a similar manner to the way you flipped blocks, that is, by right-clicking on the block then selecting Rotate Block from the Format menu.
Complete the design of the DC motor Simscape model by assigning values to the physical parameters of each of the blocks to match our assumed values.
The Rotational Damper block serves to model the viscous friction of the motor. This type of friction model was chosen because it is linear. In most cases real friction is more complicated than this. If you wish to employ a more complicated friction model, for instance to add Coulomb friction to the model, then you may use the Rotational Friction block from the Simscape/Foundation Library/Mechanical/Rotational Elements library. Also note that in the above you generated a DC Motor model from the individual mechanical and electrical aspects of the motor. The Simscape library also includes a DC Motor block under the Simscape/SimElectronics/Actuators library. This block is used in the DC Motor Position: Simulink Modeling section.
The physical parameters must now be set. Enter the following commands at the MATLAB prompt.These values are the same ones listed in the physical setup section.
You can then save these components in a single subsystem. Select all of the blocks and then choose Create Subsystem from the Edit menu. You can also change the subsystem block color by right-clicking on the block and choosing Format > Background Color from the resulting menu. This subsystem block can then be used to simulate the DC motor.
In order to simulate the response of this system it is further necessary to add sensor blocks to the model to simulate the measurement of various physical parameters and a voltage source to provide excitation to the motor. Furthermore, blocks are needed to interface Simscape blocks with tradtional Simulink blocks since the Simscape signals represent physical quantities with units, while the Simulink signals are dimensionless numbers. Add the following blocks to the model you just built to address these functions[20—29].
Current Sensor block from the Simscape/Foundation Library/Electrical/Electrical Sensors library
Controlled Voltage Source block from the Simscape/Foundation Library/Electrical/Electrical Sources library
Two PS-Simulink Converter blocks and a Solver Configuration block from the Simscape/Utilities library
Electrical Reference block from the Simscape/Foundation Library/Electrical/Electrical Elements library
Ideal Rotational Motion Sensor block from the Simscape/Foundation Library/Mechanical/Mechanical Sensors library
Mechanical Rotational Reference block from the Simscape/Foundation Library/Mechanical/Rotational Elements library
Three Out1 blocks and one In1 block from the Simulink/Ports & Subsystems library
The Ideal Rotational Motion Sensor block represents a device that measures the difference in angular position and angular velocity between two nodes. In this case, we employ the block to measure the position and velocity of the motor shaft as compared to a fixed reference represented by the Mechanical Rotational Reference block. You can leave the Initial angle of the Rotational Motion Sensor block as the default 0 radians. The Current Sensor block represents another sensor, specifically it measures the current drawn by the motor. The ground for the electrical portion of our system is defined by the Electrical Reference block. The Controlled Voltage Source block serves as the power source for the motor where you can externally define the voltage signal by connecting an input to the block. The PS-Simulink blocks convert physical signals to Simulink output signals, while the Simulink-PS block conversely converts a Simulink input signal to a physical signal. These blocks can be employed to convert the Simscape signals, which represent physical quantities with units, to Simulink signals, which don't explicitly have units attached to them. These blocks, in essence, can perform a units conversion between the physical signals and the Simulink signals. In our case, we can leave the units undefined since the input and output of each of the conversion blocks have the same units. In general, the Simscape blockset is employed to model the physical plant, while the Simulink blockset is employed to model the controller. The Solver Configuration block is employed for defining the details of the numerical solver employed in running the Simscape simulation. We will use the default settings for this block. Next, connect and label the components so that they appear as in the figure below. Double-click on the lines which are connected to the Out1 blocks and label them "Current", "Position", and "Speed". Also double-click on the In1 block and label it "Voltage".
You can save these components in a single subsystem with one input and three outputs. Select all of the blocks and then choose Create Subsystem from the Edit menu. Also label the subsystem and signals as shown in the following figure.
You can download the complete model file here, Motor Model Simscape.mdl, but note that you will need the Simscape addition to Simulink in order to run the file. Note that the two models generated above will behave equivalently as long as they are built using the same parameter values. The difference between them is then only the ease with which they are built and interfaced with, and how transparent they are in presenting information to the user. If you would like to actually run the models developed above and use them to simulate and develop control algorithms, you may continue on to the DC Motor Speed: Simulink Control page.
In fluid dynamics, the Kirchhoff equations, named after Gustav Kirchhoff, describe the motion of a rigid body in an ideal fluid.
where and are the angular and linear velocity vectors at the point , respectively; is the moment of inertia tensor, is the body's mass; is a unit normal to the surface of the body at the point ; is a pressure at this point; and are the hydrodynamic torque and force acting on the body, respectively; and likewise denote all other torques and forces acting on the body. The integration is performed over the fluid-exposed portion of the body's surface.If the body is completely submerged body in an infinitely large volume of irrotational, incompressible, inviscid fluid, that is at rest at infinity, then the vectors and can be found via explicit integration, and the dynamics of the body is described by the Kirchhoff – Clebsch equations:
Their first integrals read
Further integration produces explicit expressions for position and velocities.
In physics, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation. Some of its principles are even used in traffic engineering, where traffic is treated as a continuous fluid, and crowd dynamics.Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time.Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.
Equations of fluid dynamics
The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum (also known as Newton's Second Law of Motion), and conservation of energy (also known as First Law of Thermodynamics). These are based on classical mechanics and are modified in quantum mechanics and general relativity. They are expressed using the Reynolds Transport Theorem.In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption considers fluids as continuous, rather than discrete. Consequently, properties such as density, pressure, temperature, and flow velocity are assumed well-defined at infinitesimally small points, and are assumed to vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.
Compressible vs incompressible flow
All fluids are compressible to some extent, that is, changes in pressure or temperature cause changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modelled as an incompressible flow. Otherwise the more general compressible flow equations must be used.
Mathematically, incompressibility is expressed by saying that the density ρ of a fluid parcel does not change as it moves in the flow field, i.e.,
where D/Dt is the substantial derivative, which is the sum of local and convective derivatives. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density.
For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the Mach number of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible assumption is valid depends on the fluid properties and the flow conditions. Acoustic problems always require allowing compressibility, since sound waves are compression waves involving changes in pressure and density of the medium through which they propagate.
All fluids are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other. The velocity gradient is referred to as a strain rate; it has dimensions . Isaac Newton showed that for many familiar fluids such as water and air, the stress due to these viscous forces is linearly related to the strain rate. Such fluids are called Newtonian fluids. The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property independent of the strain rate.
Non-Newtonian fluids have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology studies the stress-strain behaviours of these fluids, which include emulsions and slurries, some viscoelastic materials such as blood and some polymers, and sticky liquids such as latex, honey and lubricants. The dynamic of fluid parcels is described with the help of Newton's second law. An accelerating parcel of fluid is subject to inertial effects.The commutation control device (CCD) for the furnace transformers switching designed at the expense of preceding excitation of their magnet system. The optimum scheme of CCU provides usage of the additional winding of the control autotransformer and presence of the exciting transformer for the coordination of a voltage level and phase with additional power source. To research the ASMF power supply system complex the mathematical models of structural elements in view of nonlinear characteristics of electrical equipment of the calculated circuits, and common mathematical models of PSS of ASMF with designed devices were created also. Thus the time of the transient process damping can be reduced by introduction of resistance in to circuits of additional windings of filter reactor.Research of the commutation control unit has shown that its usage ensures a complete reduction of amplitude of the switching current in the case of rated magnetic flux of the furnace transformer. For this purpose the supplemental power source of the reduced voltage is used. In this case the additional winding of the furnace transformer unit joins to the source with the help of the thyristor switch.
Keywords: power supply systems, filter of higher harmonics, switching, mathematical models, electromagnet processes.
List of sources and literature
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27.Marushchak Y.Y., Kopczak B.L. Strefowe sterowanie autonomicznym generatorem asynchronicznym z samowzbudzeniem. Zeszyty Naukowe Charkowskiego Narodowego Uniwersytetu Technicznego. Problemy Automatyzowanego Napędu Elektrycznego. Teoria i praktyka. - Charków: CNUT. - 2003. Tem.wyd.10. T.2 s.469-471.
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University "Lviv Stavropigion"
Institute for Eastern Europe
Institute for Eastern Europe
79016 Львів, вул. Замкнена 9, 3. Університет "Львівський Ставропігіон"
Інститут Східної Європи