Sinusoidal excitation

The behaviour is qualitatively similar when the exciting field is sinusoidal. Putting E, = E cos cot, the coupled equations for the charge and the curvature become  

We have seen in the previous section that in the conduction regime, q changes sign with the field but y/ does not. Let us make the simplifying assumption that y/ is sensibly constant over a period. Since oj Ijr and T X, vie may replace cos tot by its average value. Also expanding q t) as 

Thus the threshold field increases with to. Though the analysis is not strictly valid when toT 1, calculations show that (3.10.29) holds good quite well over the range 0 to cu,. where 

When ft)T , q may be taken as constant over a period. Expanding (/) as a Fourier series 

Equation (3.10.27) can now be integrated but solutions can only be obtained by numerical techniques. Calculations show that in the dielectric regime (a) the threshold field th is independent of the wavevector k, which results in athreshold (as distinct from a voltage threshold as in the conduction regime), (b) varies hnearly as to, (c) for a given H, k varies linearly as to, and ( /) for a given o,x H + 33 k is a constant. These predictions have been confirmed experimentally. 

2 General Solutions of Voltages and Currents 1.3.2.1 Sinusoidal Excitation 

Assuming v and i as sinusoidal steady-state solutions. Equations 1.36 and 1.37 can be differentiated with respect to time t. The derived partial differential equations are converted to ordinary differential equations, which makes it possible to obtain the solution of the earlier equations. By expressing v and i in polar coordinate, that is, in an exponential form, the derivation of the solution becomes straightforward. 

Either real parts or imaginary parts of the earlier equations represent v and i. 

Substituting Equation 1.38 into Equation 1.36 and differentiate partially with respect to time t, the following ordinary differential equations are obtained  

K+j(oL = Z series impedance of a conductor G + ycoC = Y shunt admittance of a conductor 

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A ring specimen is cut from the tube. Two coils are wrapped around this ring, one exciter coil and one receiver coil. The exciter coil N1 should cover the entire ring so that there are no field losses when ring is saturated. The sinusoidal exciter current which can be measured at... 

Phase-modulation fluorometry The sample is excited by a sinusoidally modulated light at high frequency. The fluorescence response, which is the convolution product (Eq. 6.9) of the pulse response by the sinusoidal excitation function, is sinusoidally... 

Relationship between harmonic response and rt-pulse response It is worth demonstrating that the harmonic response is the Fourier transform of the d-pulse response. The sinusoidal excitation function can be written as... 

Figure 9.9. Sinusoidal excitation can be regarded as a succession of sufficiently narrow rectangular pulses in lime Then, at any particular time r, the fluorescence response to the train of pulses sinusoidally modulated is given by the superposition of all single-pulse responses initiated at times l < t.

Let us regard the sinusoidal excitation of Eq. (9.46) as a succession of sufficiently narrow rectangular pulses in time t as illustrated in Figure 9.9. Let us also assume linear regime conditions such that only a small fraction of molecules is excited in the steady state with negligible stimulated emission and excited state reactions. 

Sinusoidal excitation provides only one harmonic at the modulation frequency. In contrast, pulsed light provides a large number of harmonics of the excitation repetition frequency. The harmonic content, the number of harmonics and their amplitude, is determined by the pulse width and shape.(25) For example, a train of infinitely short pulses provides an infinite number of harmonics all with equal amplitude. A square wave provides only three modulation frequencies with sufficient amplitude to be usable. Equation (9.74) gives the harmonic content of a train of rectangular pulses R(t) of D duty cycle (pulse width divided by period) and RP peak value ... 

Sinusoidal excitation of luminescent systems and the subsequent determination of either the degree of modulation or the phase shift of the emissions represent yet another class of lifetime-measuring techniques. The work of Lord and Rees (77) and of Tumerman (78) is notable in this respect. The former provides a good summary of the method. An adaptation of their analysis is as follows. 

Sometimes continuous sinusoidal excitation is used. In this case the emissions are not transient. 

We can understand how this is carried out by considering the waveforms of Figure 8.13a. At frequencies for which parallel capacitive components of the conductance cell impedance are negligible, sinusoidal excitation of the cell produces the waveforms of A, where es, eR, ec, and i have the same significance as previously discussed. In order to measure the real component of the impedance, the magnitude of the correlation integral cc must be determined. 

Furthermore, it is also not necessary to discuss different excitations in detail as long as we restrict ourselves to the linear response regime. There it holds that the response to any excitation allows the calculation of the response to other excitations via the convolution theorem of cybernetics.213 In the galvanostatic mode, e.g., we switch the current on from zero to /p (or switch it off from 7p to zero) and follow IKj) as a response to the current step. The response to a sinusoidal excitation then is determined through the complex impedance which is given by the Laplace transform of the response to the step function multiplied with jm (j = V-I,w = angular frequency). 

The linear model of Equation (4) gives a representation of damping of vibrations by internal friction 2J. When a steady sinusoidal excitation is involved, the internal friction causes a phase delay in the transmission of signals through the material which can be expressed as a loss-tangent, tan 5, which is related to i6, y and the frequency w, of the signal by... 

Sample Computation Results. The Lagrangian scheme of Equation (9) to (15) was used to compute the steady state response to sinusoidal excitation of an anechoic coating glued on a steel plate shown in Figure 1. Symmetry considerations permit the calculations to be limited to the regions between the dotted lines, the unit cell. 

In phase-modulation fluorometry, the sample is excited by a sinusoidally modulated light at high frequency. The fluorescence response, which is the convolution product (Eq. (7.6)) of the d-pulse response by the sinusoidal excitation function, is sinusoidally modulated at the same frequency but delayed in phase and partially demodulated with respect to the excitation. The phase shift and the modulation ratio M (equal to m/mo), that is the ratio of the modulation depth m (AC/DC ratio) of the fluorescence and the modulation depth of the excitation mg, characterize the harmonic response of the system. These parameters are measured as a function of the modulation frequency. No deconvolution is necessary because the data are directly analyzed in the frequency domain. 

By the start of World War II, a new approach to control system synthesis was being developed from Nyquist s theoretical treatment (Nl) of feedback amplifiers in 1932. This approach utilized the response of components and systems to steady-state sinusoidal excitation or frequency response as it is more usually called. The frequency response approach provides an important basis for present-day methods of handling control problems by affording a simply manipulable characterization which avoids the need for obtaining the complete solutions of system equations. 

ACIS measurements were performed at frequencies between 1 mHz and 1 kHz using a Solartron Model 1250 Frequency Response Analyzer. Output from the comb specimens was amplified with a Keithley Model 427 Current-to-Voltage Converter before waveform analysis. Reference electrodes were not used owing to the geometry of the encapsulated test specimens. The data reported herein were obtained with a 0.1 V rms amplitude sinusoidal excitation waveform. In one experiment, DC bias was superimposed on this waveform. 

Finally, the global error of the enhanced PMLs is explored in the frequency domain for a 3-DFDTD2.4ur x 1.8ur x 2.Our lattice in the presence of a cubic 0.5ur x 0.5ur x 0.5ur lossy scatterer (sr = 2.5). The structure is excited by a ramped sinusoidal excitation Ez n = [1 — e fr/30)2] sin(27r/ z2 /) with the exponential term reducing any form of transients, while the reference domain is 4.0ur x 2.5ur x 3.0ur. Figure 4.3(b) proves that the errors induced by the higher order absorbers are very low for a broad frequency range. 

Given that nonlinearities are ubiquitous, testing with oscillatory excitation is of less practical importance in contact mechanics than in other fields of material science. For instance, stick-slip motion is most easily studied by steadily pulling the object of interest across the supporting substrate. Oscillatory testing will result in complicated trajectories [17]. Sinusoidal excitation mostly... 

Y = Vosiniln ft), of fixed frequency, /, and amplitude, Vq, is typically applied through an electrode to the chemically-sensitive film and the AC current is monitored. The cunent can be written as 7 = /q sin(2 r/r + 4>), where 7o is the current amplitude and 0 is a phase shift induced by the sensor film. For a purely resistive sensor, there is no phase shift (p = 0) and for a purely capacitive sensor, the phase shift is 90°. In general, the device admittance is given by Y = I/V and, for sinusoidal excitations, can be written using complex number notation as... 

In the past, clever methods were devised to deier-mine values for / fj and For example, suppose that we apply a small-amplitude sinusoidal excitation signal to a cell containing the working electrode represented by Figure 25-7h. We further assume that the value of the applied voltage is iiisufticieni to initiate faradaic... 

Fig. 3.10.5. Threshold voltage versus frequency for MBBA. Sample thickness 50/ m. Open circles sinusoidal excitation triangles square wave excitation. (After the Orsay Liquid Crystals Group. )...

One can envision three types of perturbation an infinitesimally narrow light pulse (a Dirac or S-functional), a rectangular pulse (characteristic of chopped or interrupted irradiation), or periodic (usually sinusoidal) excitation. All three types of excitation and the corresponding responses have been treated on a common platform using the Laplace transform approach and transfer functions [170]. These perturbations refer to the temporal behavior adopted for the excitation light. However, classical AC impedance spectroscopy methods employing periodic potential excitation can be combined with steady state irradiation (the so-called PEIS experiment). In the extreme case, both the light intensity and potential can be modulated (at different frequencies) and the (nonlinear) response can be measured at sum and difference frequencies. The response parameters measured in all these cases are many but include... 

The d3niamic instrument uses the method of sinusoidal excitation and response. In this case, the applied force and the resulting deformation both vary sinusoidally with time, the rate usually being specified by the frequency f in cycles/sec or w = 2 tt f in radians/sec. For linear viscoelastic behavior, the strain will alternate sinusoidally but will be out of phase with the stress. 

Immittance theory is based upon sinusoidal excitation and sinusoidal response. In relaxation theory (and cell excitation studies), a step waveform excitation is used, and the time constant is then an important concept. If the response of a step excitation is an exponential curve, the time constant is the time to reach 63% of the final, total response. Let us for instance consider a series resistor-capacitor (RC)-connection, excited with a controlled voltage step, and record file current response. The current as a function of time I(t) after the step is I(t) = (V/R)e , file time constant x = RC, and I( oo) = 0. 

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