Electrical-circuit equation

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

Another important point, closely connected with electron kinetics, concerns the self-consistent treatment of electron kinetics, of the particle and/or power balance equations for heavy particles (such as different ions and excited atoms or molecules), and of the Maxwell equations (or a reduced version such as the Poisson equation or appropriate electric circuit equations) to obtain a more complete description of all important plasma components as well as of the internal electric field. This self-consistent treatment is usually tricky and is based on an iterative approach to the solution of the various types of equations involved (Loffhagen and Winkler, 1994 Uhrlandt and Winkler, 1996 Yang and Wu, 1996). To integrate the electron kinetic equation in such an approach adequately, a very effective solution procedure for this equation is of particular importance, although remarkable progress with respect to the speed of computation has been achieved in recent years. 

The thermal equation is analogous to the electrical-circuit equation. Electrical power consumption P corresponds to a current source. Static heat-transfer properties are usually specified using a thermal resistance Rq that defines a relation between heat flow per unit time... 

Programs to solve electric circuit equations using odeivse() requ i re"odeiv"... 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

It is clear that if the pendulum oscillates, the inductance L of the circuit is a function of the angle d, since the magnetic reluctance of the coil C varies because of the presence of the soft iron P. The first equation is the equation of an electric circuit to which is applied an e.m.f., E sin ad, subject to the condition that the inductance of this circuit is a function of d and has the frequency of the pendulum. 

Edwards equation, 230-232 Electrical circuits, analogs to mechanisms, 138-139 Electron exchange, 243 Electrostatic effects, 203 Elementary reaction, 2, 4, 12, 55 rate of, 5... 

An important early paper on fluctuation processes is that of Harry Nyquist (1928), who suggested an equation linking the mean-square amplitude of thermal noise in an electrical circuit to the resistance R of the noise EMF (or current) generator ... 

Other integral transforms are obtained with the use of the kernels e" or xk among the infinite number of possibilities. The former yields the Laplace transform, which is of particular importance in the analysis of electrical circuits and the solution of certain differential equations. The latter was already introduced in the discussion of the gamma function (Section 5.5.4). 

To have a better appreciation of the utility of these representations let us first consider the laws that govern flow rates and pressure drops in a pipeline network. These are the counterparts to KirchofTs laws for electrical circuits, namely, (i) the algebraic sum of flows at each vertex must be zero (ii) the algebraic sum of pressure drops around any cyclic path must be zero. For a connected network with N vertices and P edges there will be (N — 1) independent equations corresponding to the first law (KirchofTs current... 

An ammeter placed in the electrical circuit between the two electrodes would register the electrons from equation (3.1) as current. 

As mentioned in Section 2.4, in the ionic model the chemical bond is an electrical capacitor. It is therefore possible to replace the bond network by an equivalent electric circuit consisting of links which contain capacitors as shown in Fig. 2.6. The appropriate Kirchhoff equations for this electrical network are eqns (2.7) and (2.11). It is thus possible in principle to determine the bond fluxes for a bond network in exactly the same way as one solves for the charges on the capacitors of an electrical network. While solving these equations is simple in principle providing the capacitances are known, the calculation itself can be... 

In Chapter 2 it was shown that the Madelung field of a crystal is equivalent to a capacitive electric circuit which can be solved using a set of Kirchhoff equations. In Sections 3.1 and 3.2 it was shown that for unstrained structures the capacitances are all equal and that there is a simple relationship between the bond flux (or experimental bond valence) and the bond length. These ideas are brought together here in a summary of the three basic rules of the bond valence model, Rules 3.3, 3.4, and 3.5. 

Of course, the equations are simpler, since they arbitrarily discard that very large class of Maxwellian systems in local thermodynamic c/Asequilibrium with the active vacuum They arbitrarily discard all those Maxwellian systems that are capable of producing electrical circuits and power systems with a coefficient of performance (COP) > 1.0. 

Consider the simple electric circuit in Figure 1, where R is an ohmic resistance. The charge Q on the condenser obeys the linear equation... 

External noise denotes fluctuations created in an otherwise deterministic system by the application of a random force, whose stochastic properties are supposed to be known. Examples are a noise generator inserted into an electric circuit, a random signal fed into a transmission line, the growth of a species under influence of the weather, random loading of a bridge, and most other stochastic problems that occur in engineering. In all these cases clearly (4.5) holds if one inserts for A(y) the deterministic equation of motion for the isolated system, while L(t) is approximately but never completely white. Thus for external noise the Stratonovich result (4.8) and (4.9) applies, in which A(y) represents the dynamics of the system with the noise turned off. 

The great advantage of this method of describing the state of affairs is that the mutual action of two atoms, which in the earlier forms of the quantum theory was very indeterminate, can be calculated by combining their wave equations, in a manner more or less analogous to that which would be used to calculate the resonance of two oscillating electric circuits. 

In order to solve the transport problem we have to complete the set of necessary equations and, therefore, boundary conditions must be formulated. Depending on the boundary conditions we impose, quite different transport situations will arise. Let us analyze the one-dimensional transport in a binary electrolyte as an illustration. Two different boundary conditions will be introduced. 1) AX is brought between different chemical potentials relative to one of its component (open electrical circuit). 2) AX is brought between two inert electrodes to which a voltage A U is applied. Figures 4-3a and 4-3b show the experimental schemes. Let us examine them separately. 

Many other types of groups have been studied. They are of interest in geometry, differential equations, topology, and other branches of mathematics. In physics and chemistry, groups are used in the study of quantum mechanics molecular, crystal, and nuclear structure electrical circuits, etc. 

P, and T, are the pressure and temperature at one end and P2 and T2 refer to the other end, M is the molecular weight of the gas, R is the gas constant, and N is Avogardo s number. This equation applies in the above-mentioned (molecular flow) region, which commences at about 10 J torr. Of particular importance is the direct proportionality of the flow to the cube of the tube diameter. Thus, large-diameter tubing and large-bore stopcocks improve the pumping speed at low pressures. As with a series electrical circuit, the total impedance (proportional to 1 /q) is equal to the sum of the individual impedances, Eq. (2). 

Commercial impedance analyzers offer equivalent circuit interpretation software that greatly simplifies the interpretation of results. In this Appendix we show two simple steps that were encountered in Chapters 3 and 4 and that illustrate the approach to the solution of equivalent electrical circuits. First is the conversion of parallel to series resistor/capacitor combination (Fig. D.l). This is a very useful procedure that can be used to simplify complex RC networks. Second is the step for separation of real and imaginary parts of the complex equations. 

The equivalent electrical circuit in the case of a three-electrode setup is given in Fig. 2.9. Working and counter electrode are identical as for a two-electrode setup, while the reference electrode, as a non-current conducting electrode, only has the role of potential reference and therefore does not contribute to the impedance. However, the position of the Haber-Luggin capillary determines the contribution of Re and Rcomp to Ra given by the following equation ... 

Although the shortest way to the tunneling gap 8 is the solution of Landau and Lifshits [27], here we consider the problem from a different perspective. Like in the theory of electric circuits, instead of a detailed consideration of each particle, one can apply some simple rules that provide enough equations to solve the problem. One is the junction rule. It is based upon the probability conservation law for a stationary state, PiQ, t). At any point Q in the domain of 77(2, t), the probability density, I PiQ, t) 2 remains constant, dl P(Q. f)P/df = 0. Consider the part of a vibronic state that is located in a potential well. In this region, the probability density, P(Q, t) 2, looks like an octopus with its tentacles extended into the restricted areas under the barriers.2 If we construct a closed surface S around the body of the octopus , then, due to conservation of probability density, the total flux of probability through the surface S must be equal to zero,... 

Numerical calculations using MATHEMATICA software were made based on a theoretical model which assumes flow distribution in circular pipes under laminar conditions as described by the Bernoulli equation and applies an electrical circuit model based on Ohm s law [164],... 

Both 2m and s may be measured quite accurately by a variety of techniques such as precisely spaced pins that close electrical circuits and high-speed cameras. Then, from Eqs. (16) to (19) and the initial conditions, one can And P, E, and V for the compressed material behind the shock front and the equation of state E(P, V) of the material near the Hugoniot curve. Various other reasonable assumptions ultimately permit fairly accurate determinations of E(P, V) for pressures and densities further removed from the Hugoniot curve. For each value of P and V, a separate experiment producing particular values of x and m is needed. 

For example [146], a system of interconnected electrical circuits and a mechanical system of masses connected by springs satisfy the same linear equations if system parameters are related by the following definitions ... 

Equation (4) is analogous to electrical current flowing through electrical resistances in series. Figure 3 illustrates such an electrical circuit, where an electrical current, i, flows through a series of three resistances, Ri, i 2 and R3. The overall voltage difference of F0 — V3, is given by... 

Using Ohm s law the unit of voltage, called the volt (V), can be defined by the two precedent units as the potential difference required for a flow of current of one ampere through a conductor having a resistance of one ohm. In a closed electric circuit wo can write the equation E — IR, E meaning the electromotive force of the source of current connected to the circuit. Ohm s law can also be applied, however, to a part of an electric circuit then E = IR stands for the voltage or potential difference across the givon part of the conductor. 

---