Sinusoidal input

Figure Bl.5.2 Nonlinear dependence of tire polarization P on the electric field E. (a) For small sinusoidal input fields, P depends linearly on hence its hannonic content is mainly tiiat of E. (b) For a stronger driving electric field E, the polarization wavefomi becomes distorted, giving rise to new hannonic components. The second-hamionic and DC components are shown.

Conditions that give rise to unsteadiness are changes in feed rate, composition, or temperature. In the case of Fig. 7-6, a sinusoidal input of feed rate is introduced. The output concentration also appears to vary sinusoidally. The amphtude oi the response is lower as the specific rate is increased. 

Positioner Application Positioners are widelv used on pneumatic valve actuators, VIore often than not, thev provide improved process-loop control because thev reduce valve-related nonlinearitv, Dvnarnicallv, positioners maintain their abilitv to improve control-valve performance for sinusoidal input frequencies up to about one half of the positioner bandwidth. At input frequencies greater than this, the attenuation in the positioner amplifier netvv ork gets large, and valve nonlinearitv begins to affect final control-element performance more significantlv. Because of this, the most successful use of the positioner occurs when the positioner-response bandwidth is greater than twice that of the most dominant time lag in the process loop. 

Frequeney domain analysis is eoneerned with the ealeulation or measurement of the steady-state system output when responding to a eonstant amplitude, variable frequeney sinusoidal input. Steady-state errors, in terms of amplitude and phase relate direetly to the dynamie eharaeteristies, i.e. the transfer funetion, of the system. 

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. 

Theoretically, we are making the presumption that we can study and understand the dynamic behavior of a process or system by imposing a sinusoidal input and measuring the frequency response. With chemical systems that cannot be subject to frequency response experiments easily, it is very difficult for a beginner to appreciate what we will go through. So until then, take frequency response as a math problem. 

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. 

Figure 8.1. Schematic response (solid curve) of a first order function to a sinusoidal input (dashed). The response has a smaller amplitude, a phase lag, and its exponential term decays away quickly to become a pure sinusoidal response. 

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... 

We want to plot y(t) if we have a sinusoidal input, x(t) = sin(t). Here we need the function lsimo, a general simulation function which takes any given input vector. 

A sinusoidal input. The frequency of the sinusoidal variation is changed and the steady-state response of the effluent at different input frequencies is determined, thus generating a frequency-response diagram for the system. 

The plot of this function is shown in problem P5.01.ll. (7) Sinusoidal input, transform solution. 

The frequency response of most processes is defined as the steadystate behavior of the system when forced by a sinusoidal input Suppose the input to the process is a sine wave of amplitude m and frequency o) as shown in Fig. 12.1. 

A process is forced by sinusoidal input m,). The output is a sine wave If these two signals arc connected to an x — y recorder, we get a Lissajous plot. Time is the parameter along the curve, which repeats itself with each cycle. The shape of the curve will change if the frequency is changed and will be different for different kinds of processes. 

Same assumptions as in model 1 except Two sinusoidal inputs with a phase difference 6 between them ... 

For a periodic sinusoidal input of tracer. Turner (T14) showed how to find the values of Ir and /rom the experimental data taken at different frequencies, . Aris (A9) generalized the model by taking y3(Z) to be a continuous distribution of pocket volume rather than a set of discrete values, ySr. The problem of finding /3(l) then becomes one of solving the integral equation. 

First, we analyze the effect of the stray capacitance in parallel with the feedback resistance. Consider a sinusoidal input current with frequency /. From Fig. 11.1, the magnitude of the output voltage is... 

The center frequency is 1 kHz, This frequency was chosen to match the frequency of the sinusoidal input voltage. The harmonics that will be calculated are the first nine 1 kHz, 2 kHz, 3 kHz, 4 kHz, 5 kHz, 6 kHz, 7 kHz, 8 kHz, and 9 kHz. There may be others, but we want numerical values for only the first nine. The output variable for the Fourier analysis is the voltage at node Out, VfOUt). This is the output of the amplifier. We could look at the frequency components of any voltage or current, but for this example we are interested only in the output. Click the OK button twice to accept the settings and then run PSpice. 

Figure 25.7 Nondimensional attenuation coefficient, a, and 10-1 specific phase shift, P, for sinusoidal input into a saturated groundwater system as a function 10 2 of the Peclet Number Pe. The numbers attributed to the different 1 q 3 lines give the nondimensional angular frequency 2. See text for definitions. 10-4...

Sinusoidal input All properties of the sinusoidal input scenario can be expressed by the two nondimensional numbers, Pe and Q. Whereas Pe is not affected by sorption, Q is modified (Eq. 25-35) ... 

With sinusoid input signals, the output signal, and therefore Er, will be larger than anticipated where the frequency is in the range where the phase shift... 

Fio. 2.2. Tracer measurements types of input signals and output responses (a) Step input—F-curve (A) Pulse input—C-curve (c) Sinusoidal input... 

Many non-linearities are such that, for a sinusoidal input (e.g. = Afsinto/ = Ms n(2nft)), the output will exhibit the same period (where the period =1//= 2/r/o)) as the input signal. Moreover, these outputs satisfy the condition pertaining to odd periodic functions, viz. that the signal over the second half of the period is identical to that which would be obtained if the signal over the first half of the period were rotated n radians about the mid-point of the whole period. An example of such a function of period 2 is shown in Fig. 7.79. Clearly, if this function over the period 0 to it is rotated it radians about the point at = it, it will then coincide with itself over the period it to 2it. Odd functions of period 2/rcan be represented by a Fourier series of the form(l7) ... 

The output of a dead-zone or dead-band element in response to a sinusoidal input is shown in Fig. 7.82 and is written ... 

It is important to remember that this equation depends on the assumption that the quantizer is a fixed point, mid-tread converter with sufficient resolution so that the resulting quantization noise (enoise) is white. Furthermore, the input is assumed to be a full scale sinusoidal input. Clearly, few real world signals fit this description, however, it suffices for an upper bound. In reality, the RMS energy of the input is quite different due to the wide amplitude probability distribution function of real signals. One must also remember that the auditory system is not flat (see the chapter by Kates) and therefore SNR is at best an upper bound. 

Inputs Although some arbitrary variation of input concentration with time may be employed, five mathematically simple tracer input signals meet most needs. These are impulse, step, square pulse (started at time a, kept constant for an interval, then reduced to the original value), ramp (increased at a constant rate for a period of interest), and sinusoidal. Sinusoidal inputs are difficult to generate experimentally. 

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