Exponential Responses

The response of this model to creep, relaxation and recovery situations is the sum of the effects described for the previous two models and is illustrated in Fig. 2.39. It can be seen that although the exponential responses predicted in these models are not a true representation of the complex viscoelastic response of polymeric materials, the overall picture is, for many purposes, an acceptable approximation to the actual behaviour. As more and more elements are added to the model then the simulation becomes better but the mathematics become complex. 

Figure 2.2. First-order exponential response to an imposed step-change disturbance.

The hrsl-order system considered in the previous section yields well-behaved exponential responses. Second-order systems can be much more exciting since they can give an oscillatory or underdamped response. 

An exponential response is also important in regulation of enzyme activity. When a small percentage change in a regulator concentration is required to increase a flux by several- or many-fold, the mechanisms by which this is achieved may produce an exponential response. This is discussed below. 

The complex permittivity is obtained as follows For nondisperse materials (frequency-independent permittivity), the reflected signal follows the exponential response of the RC line-cell arrangement for disperse materials, the signal follows a convolution of the line-cell response with the frequency response of the sample. The actual sample response is found by writing the total voltage across the sample as follows ... 

As the current increases due to the presence of organic vapor, the voltage drop across the linearizing resistance also increases and reduces the voltage across the electrode. For example if 1300 volts is applied to the detector and when a solute is eluted, the current increases to 10" amp, this will cause a 300 volt drop across the linearizing resistance of 3 x 10 and consequently reduce the voltage across the electrodes to 1000 volts. In this way the natural exponential response of the detector can be made sensibly linear. 

Transient measurements can be of two types small-amplitude transients, which give rise to a linear response and large-amplitude transients, which result in a nonlinear, often exponential, response. We have already seen (cf. Sections 12.4and 14.7) that a system at equilibrium responds linearly to a small perturbation in potential or in current, according to the equation... 

FIGURE 20.11 Surface salinity at Gedser Rev, low-pass filtered with an exponential response function with a memory of 10 days. The line at 17 psu indicates a threshold for significant barotropic inflow situations. 

Figure 3.6 c and d illustrate amplitude and phase responses of oscillators having different damping coefficients. The step response of a sensor is usually determined by the time constant as well as by the typical rise and response times of the system. Figure 3.6 b shows the response of a critical damped system to a steplike change in the input signal 0 The time constant r (as defined for an exponential response), the 10% to 90% rise time t(o.i/o.9) and the 95% response time t(0 95) are marked. 

Inside a mixing chamber (Fig. 3.7), the inlet solutions are thoroughly mixed by the action of centripetal forces, usually assisted by a stirring device, e.g., a small magnetic stirring bar [71]. The mathematical function describing the analyte concentration as a function of time is that of the tanks-in-series model with N = 1, i.e., the concentration at the chamber outlet exhibits an exponential response to a stepwise change in the... 

The step-function and exponential residence-time distributions of Figure 4.5 can be modeled by two different types of flow systems. For the step-function response we have already alluded to the model of plug flow through a tube, whieh is, indeed, a standard model for this response. The exponential response, deseribed previously as the result of the equality of the internal- and exit-age distributions, requires a bit more thought. In the following we will derive the equations for the mixing models and then the corresponding reactor models for these two limits. 

Thus, the equation is satisfied by the trial solution. Situations where the response x = are called exponential responses, and are very common in biology at aU levels. 

FIGURE 4.3.1 Exponential curves. The upper left curve is an nnbonnded exponential cnrve where tlx is positive. This cnrve can be used to represent the unrestricted growth or death of cells. The middle curve is exponentially decreasing, and represents some kind of biological decay. The lowest curve is an exponential response to a step input, and is very commonly seen in biology when conditions change suddenly. 

Discuss the advantages to a biological systan of the exponential response to a sudden change (step input) as given in Figure 4.3.1. 

Camara, C.G. Harding, J.W. 1984. Thymidine incorporation in the olfactory epithelium of mice early exponential response induced by olfactory neurectomy. Brain Res., 308, 63—68. 

However, if we excite the same series RC-circuit with a controlled current step and record the voltage across the RC circuit, the voltage will increase linearly with time ad infinitum. The time constant is infinite. Clearly, the time constant is dependent not only on the network itself, but on how it is excited. The time constant of a network is not a parameter uniquely defined by the network itself. Just as immittance must be divided between impedance and admittance dependent on voltage or current driven excitation, there are two time constants dependent on how the circuit is driven. The network may also be a three-or four-terminal network. The time constant is then defined with a step excitation signal at the first port, and the possibly exponential response is recorded at the second port. 

We will now discuss the simplest equivalent circuits mimicking the immittance found in tissue measurements. In this section, the R-C components are considered ideal that is, frequency independent and linear. Immittance values are examined with sine waves, relaxation times with step functions. A sine wave excitation results in a sine wave response. A square wave excitation results in a single exponential response with a simple R-C combination. 

The characteristic time constant Tc in the form of tz and xyof Eqs (9.26) and (9.29) deserves some explanation. A two-component RC circuit with ideal components has a time constant t = RC. A step excitation results in an exponential response. With a CPE, the response will not be exponential. In Section 9.2.5, the time constant was introduced simply as a frequency scale factor. However, the characteristic time constant Tc may be regarded as a mean time constant because of a DRT. When transforming the Colez impedance Z to the Coley admittance Y or vice versa, it can be shown that the as of the two Cole Eqs 9.26 and 9.29 are invariant, but the characteristic time constants Tc are not. In fact ... 

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