Laplace current

Similarly, for an inductor Z(s) = s L. It can be seen that Eq. (3) has the same form as Ohm s law and the quantity Z(s) plays the same role as resistance in Ohm s law. That allows us to treat the voltage/current relationship for any network of capacitors, resistors and inductors in the Laplace domain as if they all were resistors, only substimting the quantity Z s) instead of resistance for each element. For example the Laplace current for a serially connected resistor R and a capacitor C will be I(s) = V(s)l Zis) + R) which can be rearranged as... 

Now, to find the current/voltage relationship in the time domain, we only need to substitute V(5) for the particular excitation of interest and carry out the inverse Laplace transform for the entire equation. However, first we need to know how the particular excitation we want to apply looks in the Laplace domain. Here is an example of how to do it for a voltage step. In the time domain it can be expressed as v(0 = Heaviside(t) Vi where the function Heaviside(/) takes a value 1 if i > 0 and 0 otherwise. Vi is the value of voltage applied during the step. The Laplace transform of this function, V s) =. [Heaviside(i) Vj] = VJs. To find the Laplace current of the above mentioned network to pulse excitation, we substitute this V(x) into Eq. (4) and perform inverse Laplace transformation of the resulting equation. We obtain, ... 

Equation (A3.3.73) is referred to as the Gibbs-Thomson boundary condition, equation (A3.3.74) detemiines p on the interfaces in temis of the curvature, and between the interfaces p satisfies Laplace s equation, equation (A3.3.71). Now, since ] = -Vp, an mterface moves due to the imbalance between the current flowing into and out of it. The interface velocity is therefore given by... 

Laplace s equation is appHcable to many electrochemical systems, and solutions are widely available (8). The current distribution is obtained from Ohm s law... 

The distribution of current (local rate of reaction) on an electrode surface is important in many appHcations. When surface overpotentials can also be neglected, the resulting current distribution is called primary. Primary current distributions depend on geometry only and are often highly nonuniform. If electrode kinetics is also considered, Laplace s equation stiU appHes but is subject to different boundary conditions. The resulting current distribution is called a secondary current distribution. Here, for linear kinetics the current distribution is characterized by the Wagner number, Wa, a dimensionless ratio of kinetic to ohmic resistance. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

Solving equation (1-8) (using Laplace transform techniques) yields the time evolution of the current of a spherical electrode ... 

Newman, J. Determination of Current Distributions Governed by Laplace s Equation 23... 

Determination of Current Distributions Governed by Laplace s Equation Direct Methanol Fuel Cells From a Twentieth Century Electrochemist s Dream to a Twenty-First Century Emerging Technology West, A. C. Newman, J. Lamy, C. Ueger, J.-M. Srinivasan, S. 23... 

The 2D Laplace inversion, such as Eq. (2.7.1), can in fact be cast into the ID form of Eq. (2.7.11). However, the size of the kernel matrix will be huge. For example, a Ti-T2 experiment may acquire 30 % points and 8192 echoes for each X assuming that 100 points for Ti and T2 are used, respectively. Thus, the kernel will be a matrix of (30 8192) 10 000 with 2.5 109 elements. SVD of such a matrix is not practical on current desktop computers. Thus the ID algorithm cannot be used directly. 

The spatial current distribution can be simulated by solving Laplace s equation with Neumann boundary conditions at the pore walls,... 

The Laplace equation also applies to the distribution of electrical potential and current flow in an electrically conducting medium as well as the temperature distribution and heat flow in a thermally conducting medium. For example, if => E, V => i, and fi/K => re, where re is the electrical resistivity (re = RA/Ax), Eq. (13-22) becomes Ohm s law ... 

Denoting the Laplace transforms of n(r, t) and I(r, t) respectively by n and I, where n is the electron density and I is the outward electron current as seen from the positive ion, the authors derive the following equations ... 

We introduce into consideration the Laplace transformations of the probability density and the probability current... 

Note that the probability current in Laplace transform terms is... 

Calculating from (9.4) the values of arbitrary constants and putting them into (9.3), one can obtain the following value for the probability current Laplace transform G(z, s) (9.2) at the point of symmetry z — 0. 

Before finding the Laplace-transformed probability density wj(s, zo) of FPT for the potential, depicted in Fig. A 1(b), let us obtain the Laplace-transformed probability density wx s, zo) of transition time for the system whose potential is depicted in Fig. Al(c). This potential is transformed from the original profile [Fig. Al(a)] by the vertical shift of the right-hand part of the profile by step p which is arbitrary in value and sign. So far as in this case the derivative dpoints except z = 0, we can use again linear-independent solutions U(z) and V(z), and the potential jump that equals p at the point z = 0 may be taken into account by the new joint condition at z = 0. The probability current at this point is continuous as before, but the probability density W(z, t) has now the step, so the second condition of (9.4) is the same, but instead of the first one we should write Y (0) + v1 (0) = YiiOje f1. It gives new values of arbitrary constants C and C2 and a new value of the probability current at the point z = 0. Now the Laplace transformation of the probability current is... 

From the Laplace transforms of Eqs. (19) and (21) to (23), a generalized Cottrell equation which describes the current response on the applied potential step is obtained as following equation... 

This being stated, applying Laplace s transform one obtains from Fick s second law that the maximum current (i.e. the current at the potential corresponding to the maximum of the peak) for a planar electrode is expressed by ... 

Laplace transformation, 1215 Nemst s equation and. 1217 non-steady, 1254 as rate determining step, 1261 Schlieren method, 1235 semi-infinite linear, 1216, 1234, 1255 in solution and electrodeposition, 1335 spherical. 1216. 1239 time dependence of current under, 1224 Diffusion control, 1248... 

The Laplace transfonnation method of solving nonsteady-state diffusion problems was briefly treated in Chapter 4. Thus, one can study all sorts of problems by using various types of current or potential stimuli (as in researches using transients see Section 7.7) and analyzing how transport in solution influences the response of the system. For example, a sinusoidally varying current, density can be used with... 

This equation can be deduced from the general Laplace formula that expresses the force exerted on a conductor of length dl, through which a current / passes, in a magnetic field of intensity B. The orientation of the Lorentz force (F = I dl AB) can be found by different approaches such as the right-handed three-finger rule or using the orientations of a direct trihedron. 

First of all, the mathematical background will be developed for the case of a simple electrode reaction O + n e = R. In this treatment, contrasts like potential versus current perturbation, large amplitude versus small amplitude, and single step versus periodical perturbation are emphasized. While discussing these principles, the most common methods derived from them will be briefly mentioned. On the other hand, it will be shown that, by virtue of the method of Laplace transformation, these methods have much in common and contain, in principle, the same information if the detected cell response is of the same order. 

As the Laplace transformed current—potential relationship is a much simpler function than its counterpart in the time domain, it has been suggested [76, 77] that it may be advantageous to analyze experimental data in the Laplace domain. In the present case, this would require some procedure to perform Laplace transformation of current datajF(0> i.e. to calculate the integral, according to eqn. (88)... 

If the electrode reaction proceeds via a non-linear mechanism, a rate equation of the type of eqn. (123) or (124) serves as a boundary condition in the mathematics of the diffusion problem. Then, a rigorous analytical derivation of the eventual current—potential characteristic is not feasible because the Laplace transfrom method fails if terms like Co and c are present. The most rigorous numerical approach will be... 

In this section, a description of the state of the art is attempted by (i) a review of the most fundamental types of reaction schemes, illustrated by some examples (ii) formulation of corresponding sets of differential equations and boundary conditions and derivation of their solutions in Laplace form (iii) description of rigorous and approximate expressions for the response in the current and/or potential step methods and (iv) discussion of the faradaic impedance or admittance. Not all the underlying conditions and fundamentals will be treated in depth. The... 

Using the solution of the Laplace equation for diffusion in cylindrical coordinates given by Eq. 5.10, fitting it to the boundary conditions given by Eq. 16.80, and employing Eq. 13.3 for the flux, the total diffusion current of atoms (per unit pipe length) passing radially from R,n to f out is... 

For the purposes of considering diffusion at microelectrodes, it is convenient to introduce two categories of electrodes those to which diffusion occurs in a linear fashion and those to which diffusion occurs in a nonlinear fashion. The former category consists of cylindrical and spherical electrodes. As shown schematically in Figure 12.2A, the lines of flux (i.e., the pathway followed by material diffusing to the electrode) are straight, and the current density is the same at all points on the electrode. Thus, the diffusion problem is one-dimensional (i.e., distance from the electrode surface) and involves solution of the appropriate form of Fick s second law, Equation 12.7 or 12.8, either by Laplace transform methods or by digital simulation (Chap. 20). 

The expression of the transverse current autocorrelation function can also be derived from the linearized hydrodynamic equations. Because it is decoupled from all the longitudinal modes, the derivation is simple and the final expression in wavenumber and Laplace frequency plane can be written as... 

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