Sinusoidal transfer function

Flence, for a sinusoidal input, the steady-state system response may be calculated by substituting. v = )lu into the transfer function and using the laws of complex algebra to calculate the modulus and phase angle. 

Our analysis is based on the mathematical properly that given a stable process (or system) and a sinusoidal input, the response will eventually become a purely sinusoidal function. This output will have the same frequency as the input, but with different amplitude and phase angle. The two latter quantities can be derived from the transfer function. 

We now generalize our simple illustration. Consider a general transfer function of a stable model G(s), which we also denote as the ratio of two polynomials, G(s) = Q(s)/P(s). We impose a sinusoidal input f(t) = A sin cot such that the output is... 

Transfer functions involving polynomials of higher degree than two and decaying exponentials (distance-velocity lags) may be dealt with in the same manner as above, i.e. by the use of partial fractions and inverse transforms if the step response or the transient part of the sinusoidal response is required, or by the substitution method if the frequency response is desired. For example, a typical fourth-order transfer function ... 

FIG. 13 Illustration of the buffer capacity measurement by means of an applied sinusoidal perturbation of the titrant. Basically, the method can be described by an electrochemo-electrical transfer function, parametrically dependent on the buffer capacity in the chemical domain. 

Therefore, the elementary transfer functions that give the response of current and potential to the perturbation of the m input quantity are related by the electrochemical impedance. For a given experiment, only one input quantity is sinusoidally modulated aroimd a mean value, and all others are maintained constant by different regulations. 

The potentials and currents were measured and controlled by a Solatron 1286 potentiostat, and a Solatron 1250 frequency response emalyzer was used to apply the sinusoidal perturbation and to calculate the transfer function. The impedance data analyzed in this section were taken after 12 hours of immersion and were found by the methods described in Chapter 22 to be consistent with the Kramers-Kronig relations. 

Another means of expressing the fidelity of an optical system is in terms of its modulation transfer function. The modulation transfer function describes the ability of the optical system to accurately reproduce an object whose pattern of luminance varies in a sinusoidal manner. An optical system which can precisely duplicate the modulation pattern of the object in the modulation pattern of the image has a modulation transfer function equal to 1.0 this represents the performance of a perfect system. The greater the difference between the modulation pattern in the image compared with that in the object, the lower the modulation transfer function. In practice, the modulation transfer pattern is usually evaluated with a square-wave pattern produced by a periodic array of lines. The modulation transfer function approaches zero as the spatial frequency of the lines increases. The limiting resolution in terms of the modulation transfer function is... 

Active query methods measure cell impedance, which is then correlated to SoC. The technique often superimposes an active signal (a low amplitude, characteristic high-frequency square or sinusoidal current pulse) onto the battery and then uses a transfer function on the response waveform to determine the ohmic polarization or a direct correlation to SoC. One permutation of this technique uses the voltage response to indigenous current spikes to map impedance in a similar way. This method provides reasonable results, but if hardware is involved it is often complex and expensive even sensors will require a relatively high-speed data acquisition bus to minimize the slew between voltage and current. 

To use the Bode criterion, we need the Bode plots for the open-loop transfer function of the controlled system. These can be constructed in two ways (a) numerically, if the transfer functions of the process, measuring device, controller, and final control element are known and (b) experimentally, if all or some of the transfer functions are unknown. In the second case the system is disturbed with a sinusoidal input at various frequencies, and the amplitude and phase lag of the open-loop response are recorded. From these data we can construct the Bode plots. 

III-61 For each of the systems with transfer functions given below, (a) draw the corresponding block diagram, (b) identify the poles and zeros of the transfer function, (c) plot the response to a unit step input change, and determine the ultimate response to a sinusoidal input sin 21. 

What are the characteristics of the ultimate response of a linear system with a transfer function G(s) to a sustained sinusoidal input ... 

Methods for measuring the impedance can be divided into controlled current and controlled potential [2, 4, 81]. Under controlled potential conditions, the potential of the electrode is sinusoidal at a given frequency with the amplitude being chosen to be sufficiently small to assure that the response of the system can be considered linear. The ratio of the response to the perturbation is the transfer function, or impedance, Z, when considering the response of an AC current to an AC voltage imposition and is defined asE = IZ, where E and I are the waveform amplitudes for the potential and the current respectively. Impedance may also be envisaged as the resistance to the flow of an alternating current. 

To give a physical interpretation of the transfer function, let consider a sinusoidal input to a linear and stable SISO system ... 

Applying the superposition principle, the response to an initially sinusoidal modulation of the photoacid concentration, also called the modulation transfer function (MTF), can be calculated with Eq. (17.3) ... 

Spatial resolution can be characterized quantitatively and more usefully through the modulation transfer function (MTF). The MTF describes how well the imaging system or one of its components such as the detector transfers the contrast of sinusoidal patterns from the incident X-ray pattern to the output. A sinusoid is a repetitive function, characterized as having a frequency (in this case a spatial frequency specified in cydes/mm) and an amplitude. The concept of spatial frequency can be visualized by considering ripples in a pond. Low spatial frequencies (long distance between wave peaks) represent coarse structures and high spatial frequencies (short wavelengths) describe fine detail. 

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... 

Many time-resolved methods do not record the transient response as outlined in the earlier example. In the case of linear systems, all information on the dynamics may be obtained by using sinusoidally varying perturbations x(t) (harmonic modulation techniques) [27], a method far less sensitive to noise. In this section, the complex representation of sinusoidally varying signals is used, that is, A (r) = Re[X( ) exp(I r)]> where i = The quantity X ( ) contains the amplitude and the phase information of the sinusoidal signal, whereas the complex exponential exp(I )f) expresses the time dependence. A harmonically perturbed linear system has a response that is - after a certain transition time - also harmonic, differing from the perturbation only by its amplitude and phase (i.e. y t) = Re[T( ) exp(i > )]). In this case, all the information on the dynamics of the system is contained in its transfer function which is a complex function of the angular frequency, defined as [27, 28]... 

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