Sinusoidal potential

Example 8.1 Guideline for Linearity We wish to establish a guideline for the perturbation amplitude needed to maintain linearity under potentiostatic regulation. An electrochemical system that follows a Tafel law is polarized at a potential V. If a large potential sinusoidal modulation is superimposed, write the current response in the form of a Taylor series and calculate the complete expression of the dc current. By considering only the first three terms of the Taylor series, write the expression cf the current under the form of the first three harmonics. 

In the presence of only resistive components, the ac waveform is defined only by its magnitudes. The current response to a potential sinusoidal wave, as shown in Fig. 5.19, indicates that the potential and the current traces differ not only in magnitude but also in phase. The time-dependent current response I t) of an electrode interface is a function of sinusoidal ac. Potential signal V t) is given in Eq. (5.41a) by frequency (a ) dependent impedance Z(a ) [68] ... 

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

Let us imagine a solenoid traversed by an alternating sinusoidal current near a conducting piece. The tension U on the coil is the sum of the tension Rsl due to the ohmic drop of potential in the coil of resistance Rs in the absence of eddy current and of the tension e opposing to the tension e given by the LENZS law ... 

Kinds oi Inputs Since a tracer material balance is represented by a linear differential equation, the response to anv one kind of input is derivable from some other known input, either analytically or numerically. Although in practice some arbitrary variation of input concentration with time may be employed, five mathematically simple input signals supply most needs. Impulse and step are defined in the Glossaiy (Table 23-3). Square pulse is changed at time a, kept constant for an interval, then reduced to the original value. Ramp is changed at a constant rate for a period of interest. A sinusoid is a signal that varies sinusoidally with time. Sinusoidal concentrations are not easy to achieve, but such variations of flow rate and temperature are treated in the vast literature of automatic control and may have potential in tracer studies. 

Since the potential and current are sinusoidal, the impedance has a magnitude and a phase, which can be represented as a vector. A sinusoidal potential or current can be pictured as a rotating vec tor. For standard AC current, the rotation is at a constant angular velocity of 60 Hz. 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

For thin-film samples, abrupt changes in refractive indices at interfrees give rise to several complicated multiple reflection effects. Baselines become distorted into complex, sinusoidal, fringing patterns, and the intensities of absorption bands can be distorted by multiple reflections of the probe beam. These artifacts are difficult to model realistically and at present are probably the greatest limiters for quantitative work in thin films. Note, however, that these interferences are functions of the complex refractive index, thickness, and morphology of the layers. Thus, properly analyzed, useful information beyond that of chemical bonding potentially may be extracted from the FTIR speara. 

Circuits that carry AC current employing two, three, or more sinusoidal potentials are C2 ed polyphase circuits. Polyphase circuits provide for more efficient generation and transmission of power than single-phase circuits. Power in a three- (or more) phase circuit is constant rather than pulsating like the single-phase circuit. As a result, three-phase motors operate more efficiently than single-phase motors. 

Another important feature of polymer adsorption is the influence exerted on it by the surface roughness. Ball et al. [22] proposed that if the surface potential is not attractive enough to bind the polymer when flat, then corrugation can aid binding as follows. For a sinusoidal corrugation, one might anticipate that some... 

The detection of the AC component allows one to separate the contributions of the faradaic and charging currents. The former is phase shifted 45° relative to the applied sinusoidal potential, while the background component is 90° out of phase. The charging current is thus rejected using a phase-sensitive lock-in amplifier (able to separate the in-phase and out-of-phase current components). As a result, reversible electrode reactions yield a detection limit around 5 x 10 7m. 

Here A is the amplitude, cp the initial phase, and coo the frequency of free vibrations. Thus, in the absence of attenuation free vibrations are sinusoidal functions and this result can be easily predicted since mass is subjected to the action of the elastic force only. In other words, the sum of the kinetic and potential energy of the system remains the same at all times and the mass performs a periodic motion with respect to the origin that is accompanied by periodic expansion and compression of the spring. As follows from Equation (3.105) the period of free vibrations is... 

Rather high charging currents cross the electrode when a variable potential component is applied. Therefore, to reduce the influence of these currents in the case of rectangular pulses, the measurements are made at a specific time after the potential change, when the charging current has decreased drastically. In the case of sinusoidal superimposed currents, one uses another device based on the fact that the... 

The basic experimental arrangements for photocurrent measurements under periodic square and sinusoidal light perturbation are schematically depicted in Fig. 19. In the previous section, we have already discussed experimental results based on chopped light and lock-in detection. This approach is particularly useful for measurement at a single frequency, generally above 5 Hz. At lower frequencies the performance of lock-in amplifier and mechanical choppers diminishes considerably. For rather slow dynamics, DC photocurrent transients employing optical shutters are more advisable. On the other hand, for kinetic studies of the various reaction steps under illumination, intensity modulated photocurrent spectroscopy (IMPS) has proved to be a very powerful approach [132,133,148-156]. For IMPS, the applied potential is kept constant and the light intensity is sinusoid-... 

In our opinion, the interesting photoresponses described by Dvorak et al. were incorrectly interpreted by the spurious definition of the photoinduced charge transfer impedance [157]. Formally, the impedance under illumination is determined by the AC admittance under constant illumination associated with a sinusoidal potential perturbation, i.e., under short-circuit conditions. From a simple phenomenological model, the dynamics of photoinduced charge transfer affect the charge distribution across the interface, thus according to the frequency of potential perturbation, the time constants associated with the various rate constants can be obtained [156,159-163]. It can be concluded from the magnitude of the photoeffects observed in the systems studied by Dvorak et al., that the impedance of the system is mostly determined by the time constant. 

In connection with the square-wave technique, mention can be made of high-frequency polarography, also called radiofrequency polarography and developed by Barker53, in which a sinusoidal radio-frequency cu, (100kHz to 6.4 MHz) square-wave modulated at co2 (225 Hz) is superimposed on to the dc potential ramp as the wave form includes (apart from additional higher... 

In an EIS experiment, a low amplitude (5 to 10 mV peak-to-peak) sine wave potential signal is superimposed on a fixed DC potential applied to an electrochemical system. Based on Ohm s law, the impedance can be computed from the applied sinusoidal potential and the measured sinusoidal current. As the sinusoidal potential and current will... 

In impedance spectroscopy a sinusoidally varying potential with a small amplitude is applied to the interface, and the resulting response of the current measured. It is convenient to use a complex notation, and write the applied signal in the form ... 

Figure 8.8 In impedance analysis, a sinusoidally varying potential V is applied across a sample, and the time-dependent current / is measured as a function of the frequency co. The current induced in response to the varying potential will be out of phase, by a time lag 9, and of different magnitude.

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