Bounded Frequency-Dependent Elements

The bounded Warburg element (BW) describes linear diffusion in a homogeneous layer with finite thickness. Its impedance is written as 

At low frequencies (m — 0). it tends to a finite value. The complex-plane impedance of the BW will be shown in Section 4.2. 

A bounded CPE (BCPE) describes the impedance of a bounded homogeneous layer with CPE behaviour. Its impedance is written as 

In summary, the parameters and units of the commonly used equivalent circuit elements described above are listed in Table 4.2. 

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Figure 37-7. Transcription elements and binding factors in the herpes simplex virus thymidine kinase ffW gene. DNA-dependent RNA polymerase II binds to the region of the TATA box (which is bound by transcription factor TEND) to form a multicomponent preinitiation complex capable of initiating transcription at a single nucleotide (+1).The frequency of this event is increased by the presence of upstream c/s-acting elements (the GC and CAAT boxes). These elements bind frans-acting transcription factors, in this example Spl and CTF (also called C/EBP, NF1, NFY). These cis elements can function independently of orientation (arrows).

The maximum impact parameter has to be estimated from different considerations. The basis of this process is that the ion rapidly moves past the electron and delivers a sharp impulse to the electron. The electrons are bound in atoms and thus are orbiting with their own characteristic frequencies or time scales. Thus, the time for the ion to cross the atom should be less than the average time for an electron orbit otherwise the collision will not be adiabatic or rapid. The time for the ion to move past can be estimated as the ratio of the impact parameter to the ion s velocity, the average orbital time for an electron will clearly depend on the chemical element, as there will be an average radius and velocity, thus... 

But how do we explain that surface waves are present only in a limited frequency range, namely 6.3-8.3GHz This is explained in Fig. 4.16 in a qualitative way. It shows the scan impedance plotted earher in Fig. 4.2 but here plotted at three frequencies, namely 6, 7.7, and 10 GHz. Furthermore, these scan impedance curves are shown a bit more realistic by the fact that they for end-fire condition do no go to infinity but just to a large value depending on how large the array actually is. This point is easy to see by application of the mutual impedance concept. It simply tells that the magnitude Za of the scan impedance can never exceed Ylq=-Q Zo, , where Zo,q is the mutual impedance between the reference element in column 0 and all the elements in column q (see Chapter 3 for details). Since Zo, j and Q are bounded, so is the finite sum Za. As already shown in Fig. 4.2 and repeated in Fig. 4.16 for easy comparison, we... 

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