Frequency-independent Elements Research Articles

Inverse modeling of dog airway and respiratory system impedances.

Mechanical parameters of the respiratory system are often estimated from respiratory impedances using lumped-element inverse models. One such six-element model is composed of an airway branch [with a resistance (Raw) and inertance (Iaw)] separated from a tissue branch [with a resistance (Rt), inertance (It), and compliance (Ct)] by a shunt compliance representing alveolar gas compression (Cg). Even though the airways are known to have frequency-dependent resistance and inertance, these inverse models have been composed of linear frequency-independent elements. In this study we investigated the use of inverse models where the airway branch was represented by a frequency-independent Raw and Iaw, a Raw that is linearly related to frequency and an Iaw that is independent of frequency, and a system of identical parallel tubes the impedance of which was computed from the tube radius and length. These inverse models were used to analyze airway and respiratory impedances between 2 and 1,024 Hz that were predicted from an anatomically detailed forward model. The forward model represented the airways by an asymmetrically branched network with a terminal impedance representative of known Cg, Rt, It, and Ct. For respiratory impedances between 2 and 128 Hz, all models fit the data reasonably well, and reasonably accurate estimates of Cg, Rt, It, and Ct were extracted from these data. For data above 200 Hz, however, only the multiple-tube model accurately fitted respiratory impedances (Zrs). This model fitted the Zrs data best when composed of 27 tubes, each having a radius of 0.148 cm and a length of 16.5 cm.

Microwave Wide-Band Model of GaAs Dual Gate MESFET's

A multioctave model of GaAs dual gate MESFET's is presented. It consists of 28 frequency independent elements and is valid between 2 and 11 GHz. Dual gate FET's with and without intergate ohmic contact have been considered. The modeling method utilized consists in separate dc and HF characterization and equivalent circuit determination of the active device parts in their actual bias conditions. Thereby, two goals are obtained : a) The topology of the overall model can be derived from well-known simpler, partial ones; and b) the starting values of optimization are precise enough to allow reliable physical solutions.

Compact and diffuse double layer interaction in unsupported system small-signal response

Previous exact results for the small-signal impedance of an unsupported electrode/material/electrode system which include effects of the finite size of charge carriers are simplified and discussed. The material contains non-recombining charges of opposite sign with the positive one immobile and uniformly distributed. General boundary conditions which encompass the range from no electrode reaction to ohmic electrode behavior are employed. In the presence of an electrode reaction, the interaction of the compact and diffuse double layers leads to considerably more complexity in the equivalent circuit than might appear in simple treatments of the supported case, in which the diffuse double layer capacitance is neglected or the compact double layer and diffuse double layer capacitances are placed in series. Two different approximate equivalent circuits made up of frequency-independent elements are found which yield remarkable agreement with the exact results over the entire frequency range of interest. The first involves the ordinary approximate circuit (OAC) previously found in the absence of compact layer effects plus a series compact layer contribution involving a parallel resonant circuit with quality factor at resonance which may approach unity. Pseudo-inductance effects are found to be extremely significant in this representation. The second approximate equivalent circuit, simpler and almost as accurate as the first, has the same form as the original OAC but with its reaction element values altered by the presence of the compact layer. For non-Butler-Volmer electrode kinetics an upper limit is found for the experimentally determinable apparent reaction rate constant, a feature of practical importance for thin films or membranes. The response of thin films and membranes, including compact layer effects, can very readily be erroneously confused with pure bulk response, yielding entirely incorrect values for the geometrical capacitance and bulk resistance of the material.

Electrical response of materials with recombining space charge

The small-signal transient and sinusoidal steady-state responses of a material with generation and recombination (G/R) of charge are considered. Immobile positive charge and mobile negative charge arise from the dissociation of intrinsic or donor centres and recombine bimolecularly. The mobile charge may react at electrodes. An exact solution for the steady-state admittance of the system is presented, along with exact and approximate equivalent circuits. The simple approximate circuits, made up entirely of frequency-independent elements are found to approximate the exact admittance to better than 1% for all frequencies and for realistic G/R conditions in the completely blocking situation. Similar accuracy is obtained for partly blocking conditions provided G/R and electrode reaction time constants are well separated.

Simplified Model for Building-Foundation Interaction

Recently several papers have appeared concerning the structure-foundation interaction problem during earthquake motions in which the frequency domain analysis or the modified excitations has been taken due to the frequency-dependent nature of the substructure. To avoid the computational difficulties involved in this problem, the idea of simple discrete-type modeling for this part is very attractive. Adopting certain simple models of frequency-independent elements that can yield the frequency-dependent interaction forces developed by Veletsos and Verbic to fit the numerical data from the continuum mechanics approach, the writer shows a promising formulation for the responses analysis of total interaction system. This enables the direct time domain analysis.

Some aspects of polarization in ionic crystals with electrode reactions

Space charge polarization involving an electrode reaction with surface adsorption is examined. As recently demonstrated by Lányi, the presence of such a process may be represented using a complex, frequency-dependent rate constant. Lányi's rate constant is shown to be normally a special case of a more general complex rate constant following from earlier work of Armstrong and others on electrode reactions with an adsorbed intermediate species. An exact expression, based on earlier work of one of the authors, is given for the impedance of a uni-univalent extrinsic-conduction system with mobile charges of only a single sign and for which the electrode processes involve such adsorbed intermediate species. Conditions are then found which reduce the exact results to the approximate expressions for series capacitance and resistance derived by Lányi for his special case of calcium-doped NaCl. These conditions are well satisfied for his specific choices of parameter values. Curves showing the exact frequency response of various series and parallel quantities are presented, together with complex impedance plane results. Finally, it is shown how under conditions that include the range of usual experimental interest the entire space-charge, electron reaction/adsorption system can be well represented by a simple equivalent circuit of frequency-independent elements.

Synthesis of broad-band arrays with arbitrary frequency-independent elements

The application of dynamic programming to the synthesis of broad-band antenna arrays with arbitrary frequency-independent elements is investigated. Four criteria are proposed for the optimization of the array. By employing these criteria it is shown that the optimum array is the one with monotonic spacings. For such arrays the principle of optimality is proved and the limits, within which dynamic programming is applied, are analytically calculated. A 35-element array with ratio of upper-to-lower frequencies equal to 3 is synthesized by this method.

Small-signal characteristics of lateral transistors

This paper considers the small-signal characteristics of lateral transistors made on epitaxial layers of finite thickness, as is the case in most monolithic integrated circuits. For tractability and ease of comprehension, one-dimensional approximations are made, but these are usually adequate for predicting the essential features of the frequency response. Various circuit models are proposed, including the low-frequency charge control model whose validity can be extended to higher frequencies by the use of frequency-dependent elements. Alternatively, multilump models with frequency-independent elements can be used at the higher frequencies. Experimental data on small-signal common-emitter current gain is presented in support of the theory.

Electrical Response of Materials Containing Space Charge with Discharge at the Electrodes

The system considered consists of a solid or liquid material containing mobile positive and negative charges between two identical plane electrodes separated by a distance l. The results obtained apply also for a single working electrode, without specific adsorption, and an indifferent electrode. Uni-univalent, nonrecombining positive and negative charges, usually of equal mobility, are assumed. Frequency and transient responses are compared in the linear regime for two different boundary conditions. One condition requires complete blocking of positive and negative charges at the electrodes; the other requires complete blocking for charges of one sign but allows free discharge of those of opposite sign. The working electrode is thus reversible, or Ohmic, for charges of one sign. The method of solution includes electromigration effects, does not require electroneutrality anywhere, and leads to results satisfying Poisson's equation exactly everywhere for all times and frequencies. These results apply to solids and liquids with either electronic or ionic conduction. They are relevant to low-conductivity liquids, as in Kerr and liquid-crystal-display cells, to ordinary solid and liquid electrolytes, and at least to some extent to fused salts. Although the frequency and transient responses involve continuous distributions of relaxation times, three main relaxation times, τD, MτD, and M2τD, occur. Here τD is the dielectric relaxation time of the charge-containing material and M ≡ l / 2LD, where LD is the appropriate Debye length. Results are obtained for any value of M but are summarized here for the usual experimental condition M ≫ 1. At high normalized frequencies, Ω ≡ ωτD ≫ 1, the system parallel capacitance associated with mobile charge is proportional to ω−3/2 for both blocking and discharge conditions. The series capacitance is proportional to ω−1/2 for these conditions. It is shown, for the first time in the completely blocking situation, that this high-frequency response is associated with a Warburg, or diffusion, impedance. An equivalent circuit is obtained for the discharge case with a maximum of frequency-independent elements and a minimum frequency dependence of remaining elements. It involves a dc path and two distinct Warburg impedances, occurring in different frequency ranges. Cole–Cole plots of the effective complex dielectric constant associated with the “interface” elements of the equivalent circuit yield, for both boundary conditions, curves very close to that arising from a Davidson–Cole continuous distribution of relaxation times with distribution parameter β = 0.5. In the limit of large M, blocking frequency response becomes identical with such Davidson–Cole response. Interface transient response is found in both cases to involve regions where the current decay is proportional to t−1/2, but that in the discharge case can occur over a very long time range extending to t ∼ 5 × 10−4 M2τD. For large M, this limiting time may be measured in days or months. The long-time limit of the transient charge is finite and consistent with the low-frequency limiting capacitance of the system. Although the Ω → 0 space-charge capacitance in the blocking case is independent of l, the corresponding dominant capacitance in the discharge case is extrinsic and directly proportional to l, making potentiostatic measurements inapplicable at very long times. It is about M / 12 times larger than the blocking-case capacitance and arises from charge diffusion (not space charge) in a finite length, the entire bulk region. Tremendously large capacitances may thus appear at ultra-low frequencies since M may often be as large as 106. Some applications and limitations of the present results to electrolyte situations are discussed, and the important effects of a finite l are emphasized. Faradaic and non-Faradaic processes do not separate clearly in the present discharge case, but it is suggested that when an excess of indifferent electrolyte is present in addition to the discharging ion such separation may be a good approximation. An approximate equivalent circuit is proposed for this situation which differs in important ways from those heretofore employed by electrochemists.

Small-signal model with frequency-independent elements for the avalanche region of a microwave negative-resistance diode

An experimentally derived four-element incremental model for the avalanche region of a microwave negative-resistance diode and a simplified measuring procedure for determining the element values are presented. The element values depend on bias but not on frequency. The four-element model predicts the same frequency behavior as the theoretically derived Gilden-Hines model when one degree of freedom is eliminated. Comparison with theory predicts a simple low-frequency (1-MHz) indirect measurement of device drift region carrier transit time.

Double layer capacitance and relaxation in electrolytes and solids

After a review of the idealizations usually made in theories of space-charge polarization in liquids and solids, published theoretical and experimental work dealing with the a.c. response of two-electrode systems is discussed and various corrections pointed out. A calculation of the equilibrium space charge capacitance of two blocking electrodes separated by material containing mobile positive and negative charge of arbitrary valences is presented for the limit of vanishingly small applied static potential. The result of this calculation may be used to obtain the frequency dependence of diffuse layer parallel capacitance and conductance in the Debye dispersion frequency range, where the motion of charges in space charge regions leads to an admittance involving only a single time constant. Expression given previously by Baker and Buckle for this time constant and for the low-frequency limiting capacitance are corrected, along with their conditions for the extent of the Debye dispersion range. The relaxation time constant is G∞(r–1)τD/G0, where G0 and G∞ are the low-and high-frequency limiting values of the series conductance, τD is the dielectric relaxation time of the material containing mobile charges, rM cotanh M, and M is the ratio of the separation between electrodes to twice the Debye length. Finally, deviations from Debye dispersion behaviour for various 1 : 1 valence theories are calculated and compared; for the two-blocking-electrodes situation, Debye behaviour and a simple equivalent circuit of frequency-independent elements extend up to nearly ωτD∼1. Beyond this range, the parallel space-charge capacitance shows ω– limiting behaviour, similar to that previously found only with partly blocking electrodes, and the equivalent series space-charge capacitance eventually decreases as ω–½.

Electrical Impedance of Isolated Amnion

The electrical impedance of the guinea pig amniotic membrane was measured, under standardized conditions, over the frequency range of 20 to 7000 cycles/second. This impedance can be represented analytically by a simple frequency-dependent function which is precisely of the form of the Debye relaxation equation. The observed data exhibit a broad dispersion centered at a frequency of 1050 cycles/second and a narrow distribution of time constants centered about 152 microseconds, both effects being due to the polydisperse nature of amniotic tissue. If the narrow time-constant distribution is approximated by a single time constant, amnion impedance can be simulated by a simple electrical circuit of frequency-independent elements. The Maxwell-Wagner interfacial treatment, although successfully adapted for cell suspensions, is shown to lose its quantitative significance in the case of the tightly structured amnion. In addition, determinations were made on the chemical composition of amniotic fluid, fetal blood and urine, and maternal blood and urine; the DC potential across the amniotic membrane was also measured.

Theory of ac Space-Charge Polarization Effects in Photoconductors, Semiconductors, and Electrolytes

A linear theory is developed of the ac behavior of solid or liquid materials containing charge carriers which can move freely within the material but cannot leave it through the electrodes. The theory applies for any degree of dissociation of neutral centers and recombination of positive and negative charge carriers, but these carriers are assumed to have been produced by dissociation from only one species of neutral center. The mobile carriers may be electrons, positive holes, positive ions, negative ions, positive ion vacancies, or negative ion vacancies. The general solution for the admittance of the material is obtained for an arbitrary ratio between the mobilities of positive and negative carriers, but, because of the complexity of the result, it is only discussed in detail in the present paper for the following special cases: (a) charge carriers of only one sign mobile, arbitrary recombination time; (b) charge carriers of both signs mobile with the same mobility, arbitrary recombination time; and (c) charge carriers of both signs mobile with unequal mobilities and very short recombination time. In case (a), two dispersion regions may appear, with that at lower frequencies arising from recombination and the other from the finite mobility of the carriers. Both regions follow Debye dispersion curves accurately over a wide frequency range, making it possible to represent the electrical behavior of the material for any recombination time by means of a simple equivalent circuit containing only frequency-independent elements. In cases (b) and (c), only the motional dispersion region appears, and it again follows Debye curves. Finally, the results of the present theory are compared with those of other theories of ac space-charge effects in semiconductors and electrolytes.