Electric circuit example illustrated

The example presented above will now be developed, as it is a problem which arises frequently in many applications. The vibrations of mechanical systems and oscillations in electrical circuits are illustrated by the following simple examples. The analogous subject of molecular vibrations is treated with the use of matrix algebra in Chapter 9. 

Metals and alloys are malleable and ductile, and they conduct electricity. When a metal can be poimded or rolled into thin sheets, it is called malleable. Gold is an example of a malleable metal, as shown in Figure 9.11. A chimk of gold can be flattened and shaped by hammering imtil it is a thin sheet. Ductile metals can be drawn into wires. For example, copper can be pulled into thin strands of wire and used in electric circuits, as illustrated in Figure 9.11. Electrical conductivity is a measure of how easily electrons can flow through a material to produce an electric current. Metals such as silver are excellent conductors because there is low resistance to the movement of electrons in the metal. These properties— malleability, ductility, and electrical conductivity—are the result of the way that metal atoms bond with each other. 

Another specialized form of potentiometric endpoint detection is the use of dual-polarized electrodes, which consists of two metal pieces of electrode material, usually platinum, through which is imposed a small constant current, usually 2-10 /xA. The scheme of the electric circuit for this kind of titration is presented in Figure 4.1b. The differential potential created by the imposition of the ament is a function of the redox couples present in the titration solution. Examples of the resultant titration curve for three different systems are illustrated in Figure 4.3. In the case of two reversible couples, such as the titration of iron(II) with cerium(IV), curve a results in which there is little potential difference after initiation of the titration up to the equivalence point. Hie titration of arsenic(III) with iodine is representative of an irreversible couple that is titrated with a reversible system. Hence, prior to the equivalence point a large potential difference exists because the passage of current requires decomposition of the solvent for the cathode reaction (Figure 4.3b). Past the equivalence point the potential difference drops to zero because of the presence of both iodine and iodide ion. In contrast, when a reversible couple is titrated with an irreversible couple, the initial potential difference is equal to zero and the large potential difference appears after the equivalence point is reached. 

The physical imderstanding of the current paths and potential drops in the system serves to guide the structure of the corresponding electrical circuit. The mathematical expression for the interfaciai impedance can be obtained following the development presented in the subsequent chapters. Several examples are given in the following sections to illustrate the procedure. 

Corrosion can be represented in terms of a simple electrochemical cell as illustrated in Figure 1 for aqueous corrosion. In the electrochemical cell, the electrical circuit is completed by the transport of charge through the aqueous electrolyte. The source of potential may be, for example, a difference in electrode composition. For metals exposed to a corrosive environment, the anodic and cathodic reactions can occur at adjacent or widely separated sites on a single metal, or on different metals that are electrically connected. As in the electrochemical cell, the electrical circuit is typically completed by an aqueous electrolyte sources of potential include differences in electrolyte concentration and in electrode composition. 

The examples to be presented illustrate the diversity of fields of applications, but they are mentioned in outline form only. Many biological phenomena used to be modelled by real or formal kinetic models. A biochemical control theory that is partially based on non-mass-action-type enzyme kinetics seems to be under elaboration, and certain aspects will be illustrated. A few specific models of fluctuation and oscillation phenomena in neurochemical systems will be presented. The formal structure of population dynamics is quite similar to that of chemical kinetics, and models referring to different hierarchical levels from elementary genetics to ecology are well-known examples. Polymerisation, cluster formation and recombination kinetics from the physical literature will be mentioned briefly. Another question to be discussed is how electric-circuit-like elements can be constructed in terms of chemical kinetics. Finally, kinetic theories of selection will be mentioned. 

Examples 2, 4 and 5 illustrate a common feature implied by the principle of minimum entropy production (Fig. 17.4) in a series of coupled systems, entropy production is extremized when the flows are equal. In a chemical reaction it was the velocity Vk for heat conduction it was the heat flow Jq, for an electric circuit it is the electric current Ik-... 

The simplest and, despite its several drawbacks, the most widely used type of control is the on/off control system. An example is a contact thermometer, which closes or opens a heater circuit. The designation on/off means that the controller output, or the manipulated variable (electric current) is either fully on or completely off. To avoid oscillations around the setpoint, the real on/off controller has built into it, a small interval on either side of the setpoint, within which the controller does not respond, and which is called the differential gap or deadzone. When the controlled variable moves outside the deadzone, the manipulated variable is set either on or off. This is illustrated in Fig. 2.30. Such shifts from the set point are known as offset. 

Both Z and Z can be combined in a single plot A Nyquist plot is obtained by plotting Z on the horizontal axis and Z on the vertical axis. An example of a Nyqnist plot is illustrated in Figure 5. As compared to a Bode plot, a Nyquist plot does not indicate the frequency response of a material directly. A Nyquist plot represents the electrical characteristic of a material. This electrical characteristic can be represented by an equivalent circuit that may consist of a resistor and capacitor, resistor in series with capacitor, resistor in parallel with capacitor, and so oa... 

Another form of electrical energy is direct current, defined as the flow of electrons in one direction. A battery can generate direct current (DC). An example of this is a battery connected to a light connected to a switch connected to a battery. Figure 14-12 illustrates the key components of a DC circuit. 

The resistance between the protected structure and the environment includes the resistance of any electrically insulating paint or coating on the structure. This is illustrated in Fig. 13.31. detailing the various resistive components when a buried pipe is polarized by an ICCP system. In this example, the desirable place to measure potential would be across the interface between the pipe and the environment, as is represented by the terminals marked "polarization potential" on the equivalent circuit of Fig. 13.31. 

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