Response of Electrical Circuits

Application of an electrical perturbation (current, potential) to an electrical circuit causes a response. In this chapter, the system response to an arbitrary perturbation and later to an ac signal, is discussed. Knowledge of the Laplace transform technique is assumed, but the reader may consult numerous books on the subject if necessary. 

let us consider application of an arbitrary (but known) potential E(t) to a resistance R. The current i(t) is given as i(t) = E(t)IR. When the same potential is applied to the series connection of the resistance R and capacitance C, the total potential difference is the sum of potential drops on each element. Taking into account that for a capacitance E t) = Q t) C, where Q is the charge stored in a capacitor, the following equation is obtained  

This equation may be solved using either Laplace transform or differen- 

The Laplace transform is an integral transform in which a function of time /(t) is transformed into a new function of a parameter s called frequency, J[s) oi F s), according to 

The Laplace transform is often used in the solution of differential and integral equations. In general, the parameter y may be complex, y = v + j(f , where j = v=r. but in this chapter only the real transform will be considered, that is y = v. Direct application of the Laplace transform to Eq. (1), taking into account that 

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The above development could be considered to be a verification cf equation (1.112) and justifies the treatment of the current and potential response of electrical circuits expressed as equation (4.8). 

The impedance response of electrochemical systems is often normalized to the effective area of the electrode. Such a normalization applies only if the effective area can be well defined, and is not used in this chapter on the impedance response of electrical circuits. The capacitance used in this chapter, therefore, has units of F rather than F/cm, the resistance has units of O rather than fl cm, and the inductance has units of H rather than H cm. ... 

By his application of Laplace transforms to the transient response of electrical circuits, Oliver Heaviside created the foimdation for impedance spectroscopy. Heaviside coined the words inductance, capacitance, and impedance and introduced these concepts to the treatment of electrical circuits. His papers on the subject, published in The Electrician beginning in 1872, were compiled by Heaviside in book form in 1894.6/7 pj-om the perspective of the application to physical systems, however, the history of impedance spectroscopy begins in 1894 with the work of Nemst. ... 

In Chap. 2 we saw the responses of electrical circuits containing the elements R, C, and L. Because these are linear elements, their impedance is independent of the ac amplitude used. However, in electrochemical systems, we do not have such elements we have solution-electrode interfaces, redox species, adsorption, etc. In this and the following chapters, we will learn how to express the electrochemical interfaces and reactions in terms of equations that, in particular cases, can be represented by the electrical equivalent circuits. Of comse, such circuits are only the electrical representations of physicochemical phenomena, and electrical elements such as resistance, capacitance, or inductance do not exist physically in cells. However, such a presentation is useful and helps in our understanding of the physicochemical phenomena taking place in electrochemical cells. Before presenting the case of electrochemical reactions, the case of an ideally polarizable electrode will be presented. 

Before discussing the impedance of electrochemical systems it is useful to recall briefly the alternating current response of electrical circuit elements. Three passive elements are normally present in an electrical circuit ... 

Figure 2.2 Illustration of different exponential and other responses of electrical circuits.

Further information on this subject can be obtained by frequency response analysis and this technique has proved to be very valuable for studying the kinetics of polymer electrodes. Initially, it has been shown that the overall impedance response of polymer electrodes generally resembles that of intercalation electrodes, such as TiS2 and WO3 (Ho, Raistrick and Huggins, 1980 Naoi, Ueyama, Osaka and Smyrl, 1990). On the other hand this was to be expected since polymer and intercalation electrodes both undergo somewhat similar electrochemical redox reactions, which include the diffusion of ions in the bulk of the host structures. One aspect of this conclusion is that the impedance response of polymer electrodes may be interpreted on the basis of electrical circuits which are representative of the intercalation electrodes, such as the Randles circuit illustrated in Fig. 9.13. The figure also illustrates the idealised response of this circuit in the complex impedance jZ"-Z ) plane. 

As described in the subsequent chapters in Part m, models for the impedance response can be developed from proposed hypotheses involving reaction sequences (e.g., Chapters 10 and 12), mass transfer (e.g., Chapters 11 and 15), and physical phenomena (e.g.. Chapters 13 and 14). These models can often be expressed in the mathematical formalism of electrical circuits. Electrical circuits can also be used to construct a framework for accounting for the phenomena that influence the impedance response of electrochemical systems. A method for using electrical circuits is presented in this chapter. 

To calculate the response of complex circuits and stimuli in the s-plane, we will need to use the above impedance summing rules, along with the rather obvious s-plane versions of the electrical laws. For example, Ohm s law is now... 

First, the applied time-dependent stimulus (one-shot or repetitive — voltage or current) is mapped into the complex-frequency domain, that is, the s-plane. Then, by using the s-plane versions of the impedances, we can transform the entire circuit into the s-plane. To this transformed circuit we apply the s-plane versions of the basic electrical laws and thereby analyze the circuit. We will then need to solve the resultant (transformed) differential equation (now in terms of, v rather than t). But as mentioned, we will be happy to discover that the manipulation and solution of such differential equations is much easier to do in the s-plane than in the time domain. In addition, there are also several lookup tables for the Laplace transforms of common functions available, to help along the way. We will thus get the response of the circuit in the frequency domain. Thereafter, if so desired, we can use the... 

The potential difference (p.d.) is the change in energy levels measured across the load terminals. This is also called the volt drop or terminal voltage, since e.m.f. and p.d. are both measured in volts. Resistance in every circuit offers some opposition to current flow, which we call the circuit resistance, measured in ohms (symbol O), to commemorate the famous German physicist Georg Simon Ohm, who was responsible for the analysis of electrical circuits. 

Impedance is an essential characterization of the current intensity response of the corrosion system to the sinusoidal perturbation of the potential applied to the metal. The results of impedance measurements made in a suitably wide range of frequencies provide valuable information about the system and electrochemical corrosion occurring therein. The majority of electrochemical as well as physical processes can be interpreted within the impedance spectroscopy method as elements of electrical circuits with appropriate time constants. Thus, to interpret the results of electrochemical impedance measurements surrogate models of electrical circuits, known as Randles models, can be used. 

The mechanical endurance of the current-carrying parts of all the equipment, bus system, deviees and components, used in a particular circuit as well as the load-bearing members and supports on which they are mounted. The electrical parts of a device (breakers and switches, etc.) are the responsibility of the component manufacturers. The manufacturer of the switchgear assembly is responsible for the busbar systems, metallic links and wires. 

Combustible gas detection systems are frequently used in areas of poor ventilation. By the early detection of combustible gas releases before ignitible concentration levels occur, corrective procedures such as shutting down equipment, deactivating electrical circuits and activating ventilation fans can be implemented prior to fire or explosion. Combustible gas detectors are also used to substantiate adequate ventilation. Most combustible gas detection systems, although responsive to a wide range of combustible gases and vapors, are normally calibrated specifically to indicate concentrations of methane since most natural gas is comprised primarily of methane. 

Inductance (L) is the property of an electric circuit that produces an emf in the circuit in response to a change in the rate of current, i.e.. 

Eigenstates of a crystal, 725 Eigenvalues of quantum mechanical angular momentum, 396 Electrical filter response, 180 Electrical oscillatory circuit, 380 Electric charge operator, total, 542 Electrodynamics, quantum (see Quantum electrodynamics) Electromagnetic field, quantization of, 486, 560... 

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

A model for the ac response of real electrodes is the simple electric equivalent circuit consisting of a resistance R and capacitance Q conneeted in series (Fig. 12.12a). It follows from the rules for ac circuits that for this combination... 

IZI=J(Z )2+(Z ), and phase angle shift,, vs. f). The electrochemical system is then simulated with an electrical circuit that gives the same impedance response. Ideally this electrical circuit is composed of linear passive elements, e.g. resistors and capacitors, each of which represents individual physicochemical steps in the electrochemical reaction. ... 

This impedance response, in general, is similar to that elicited from an Armstrong electrical circuit, shown in Figure 3, which we represent by Rfl+Cd/(Rt+Ca/Ra). Rfl is identified with the ohmic resistance of the solution, leads, etc. Cj with the double-layer capacitance of the solution/metal interface Rfc with its resistance to charge transfer and Ca and Ra with the capacitance and resistance... 

Equivalent Electrical Circuit, In spite of the complex nature of the inhibition process, the inhibited systems actually display simple impedance responses. 

With analogy to electric circuits, a transfer function of the antenna can be calculated and the response of the antenna to an incoming wave obtained. The output signal is usually expressed as antenna cross-section. It is defined as the ratio between the total energy absorbed by the antenna and the incident spectral density function of the incident wave. In the case of Nautilus antenna (2300 kg, 3 x 0.6 m) the cross-section is of the order of 10 25m2 Hz. 

Fabrication of the prototype is an important step in product development. It demonstrates that the various components can indeed be physically integrated to form the final product with the desired functionalities. Consider a UV sensor. While its functionality depends on the physical response of a certain nanomaterial in the presence of UV light, an electric circuit and a display system are required for a functional consumer product. The availability of a prototype is essential in test marketing, safety tests, reliability tests and so on. However, the development of consumer-oriented products often involves a considerable amount of trial-and-error, which can lead to costly delays in product launching [10]. 

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