Electric circuit example equations

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. 

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. 

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 ... 

An electrical circuit that yields the impedance response equivalent to equation (10.25) for a single Faradaic reaction is presented in Figure 10.2. Such a circuit may provide a building block for development of circuit models as shown in Chapter 9 for the impedance response of a more complicated system involving, for example, coupled reactions or more complicated 2- or 3-dimensional geometries. 

In the non-linear systems (5.2), a second type of attractor — a closed curve (limit cycle) is also possible. For example, the system of van der Pol equations (representing oscillations of current in electrical circuits and oscillations of concentrations, or more precisely the differences between the concentrations and their stationary values, in chemical systems)... 

The Fourier transform (FT) is ubiquitous in science and engineering. For example, it finds application in the solution of equations for the flow of heat, for the diffraction of electromagnetic radiation, and for the analysis of electrical circuits. The concept of the FT lies at the core of modern... 

As each conserved state determines a domain, additional connection constraints can be found for various port types. For instance, a bond connected to one side of a 0-junction may be connected to a C-type storage port or a source port, as these ports do not violate the balance equation. However, in principle, one should be more careful when connecting an I-type, R-type, TF-type, or GY-typeport, because these ports cannot absorb the conserved state related to the flow. However, all domains with relative equilibrium-determining variables have a non-displayed balance for the reference node (this balance equation is dependent on the balance equations for the rest of the network and corresponds to the row that is omitted in an incidence matrix to turn it into a reduced incidence matfix of an electrical circuit, for example). This additional balance compensates for this flow, such that it is still possible to connect these ports without violating the balance equation. Note that the I-type port in principle is a connection to a GY-type port that connects to the storage in another domain. Some domains have absolute equilibrium-determining variables, like temperature and pressure, but since in most cases it is not practical to choose the absolute zero point as a reference, usually another reference state is chosen, such that these variables are treated as differences with respect to an arbitrary reference and an additional balance too. 

The resistance of the electric circuit between anode and cathode is often negligible (/ ext 0) Furthermore, if the electrolyte, for example water or soil, has a low conductivity its resistance may far exceed the value of the polarization resistances. Rpj and i p n- Equation (7.9) thus simplifies to (12.33) where is the ohmic resistance of the electrolyte. 

This can be verified by substituting the expression for v t) into the differential equation model and performing the indicated operations. The fact that v t) can be shown to have this form indicates that it is possible for this circuit to sustain oscillatory voltage and current waveforms indefinitely. When the parametric expression for v(t) is substituted into the differential equation model the value of co that is compatible with the solution of the equation is revealed to be ct) = l/.-/(LC). This is an example of the important fact that the frequency at which an electrical circuit exhibits resonance is determined by the physical value of its components. The remaining parameters ofv(t), K, and (p are determined by the initial energy stored in the circuit (i.e., the boundary conditions for the solution to the differential equation model of the behavior of the circuit s voltage). 

Control circuits of various types demonstrate similar behavior whenever they can be characterized by equation systems of the same kind for their corresponding quantities. In such a case their real elements will be of no importance. This, e.g., affords the possibility of replacing a mechanical control circuit by an electrical circuit and studying the behavior of the former by means of the latter. Examples of this may be found in the literature [3-5]. Even for Wiener, one of the essential approaches in his collaboration with medical men was to develop mechanical, electrical, or even computational models for the simulation of control processes in living organisms. 

An example of a transfer function based on a physical model is the Nemst impedance of a transport controlled electrode reaction. The impedance spectra in Fig. 7-14, which were obtained on a rotating platinum disk electrode at the equilibrium potential of the iron hexacyanoferrate redox system, exhibit the typical shape of a transport-controlled process. The transfer function cannot be described by a limited number of electrical circuit elements but must be derived from the differential equations of Fick s 2nd law and the appropriate boundary conditions. For finite linear diffusion, the so-called Nemst impedance Z can be derived theoretically... 

As an example of such systems of equations, consider the electrical circuit shown in Figure 10.34 consisting of five resistors, two capacitors and one inductor plus a voltage source. It will be assumed that the circuit has been unexcited for a... 

Figure 10.34. Electrical circuit for example of differential-algebraic systems of equations.

Because the film growth rate depends so strongly on the electric field across it (equation 1.115), separation of the anodic and cathodic sites for metals in open circuit is of little consequence, provided film growth is the exclusive reaction. Thus if one site is anodic, and an adjacent site cathodic, film thickening on the anodic site itself causes the two sites to swap roles so that the film on the former cathodic site also thickens correspondingly. Thus the anodic and cathodic sites of the stably passive metal dance over the surface. If however, permanent separation of sites can occur, as for example, where the anodic site has restricted access to the cathodic component in the electrolyte (as in crevice), then breakdown of passivity and associated corrosion can follow. 

In equation (16-1), q is the heat absorbed from the surroundings and w is the work done on the system. A few examples of work are (1) if a chemical reaction occurs within a system, work may be done upon it if gases are consumed and its volume is decreased, or (2) the system may perform work if gases are produced, or (3) work may be done if the system delivers an electric current to an external circuit. 

As indicated previously, the electric charge passing through the cell and the external circuit is strictly proportional to the masses of reactants that have reacted at the electrodes. For example, if the Ag+/Ag reaction as represented in Equation (7) proceeds such that 1 g-equivalent (107.9 g Ag) is deposited, 1 mol of electrons is required, which is the charge of an electron (1.6021 10-19 C) multiplied with the Avogadro constant, that is,... 

These equations represent the basis of classical electrokinetics. For example, the magnitude of the electroosmotic volume flow per unit potential at zero pressure difference, = (= 0, and the streaming current per unit pressure difference at short circuit, (//AP), = 0, must be identical. Equations (10.89) and (10.90) indicate that the existence of a pressure difference will produce an electric flow if the coupling coefficient is nonvanishing when no pressure is applied, AP = 0, the action of the electric force will cause a volume flow of water. 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

III.l [see also Eq. (17) and Fig. 2], and that in the presence of a faradaic reaction [Section III. 2, Fig. 4(a)] are found experimentally on liquid electrodes (e.g., mercury, amalgams, and indium-gallium). On solid electrodes, deviations from the ideal behavior are often observed. On ideally polarizable solid electrodes, the electrically equivalent model usually cannot be represented (with the exception of monocrystalline electrodes in the absence of adsorption) as a smies connection of the solution resistance and double-layer capacitance. However, on solid electrodes a frequency dispersion is observed that is, the observed impedances cannot be represented by the connection of simple R-C-L elements. The impedance of such systems may be approximated by an infinite series of parallel R-C circuits, that is, a transmission line [see Section VI, Fig. 41(b), ladder circuit]. The impedances may often be represented by an equation without simple electrical representation, through distributed elements. The Warburg impedance is an example of a distributed element. 

The Daniell cell is an example of a galvanic cell, in this type of electrochemical cell, electrical work is done by the system. The potential difference, between the two half-cells can be measured (in volts, V) on a voltmeter in the circuit (Figure 7.1) and the value of is related to the change in Gibbs energy for the cell reaction. Equation 7.9 gives this relationship under standard conditions, where is°ceu is the standard cell potential. 

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