Four-pole Parameters Research Articles

Theory of a two source-location method for direct experimental evaluation of the four-pole parameters of an aeroacoustic element

A new method, called the two source-location method, is described for measurement of the four-pole or the transfer matrix parameters for an acoustic element or a subsystem of elements by means of the four-microphone technique and use of the transfer function approach. Theoretical expressions are derived for the two source-location method, and these are shown to be identical to those for the two-load method. However, the two methods are conceptually different. The two-source method is shown to be functionally much more stable and entirely independent of the loading terminations. An uncertainty analysis is presented and used for estimation of possible nonequalization errors in the transfer matrix parameters. Finally, several examples of successful measurement use of the new method are included.

Derivation of four-pole parameters including higher order mode effects for expansion chamber mufflers with extended inlet and outlet

A derivation of the four-pole parameters including higher order mode effects, but with flow-acoustic interactions neglected, is presented for expansion chamber mufflers with extended inlet and outlet. The solution of the problem is based on a standard mode-matching technique. The solution is derived for the general case of straight cylindrical ducts with rigid walls. The end walls in the chamber are described as uniform and locally reacting surfaces. The validity of the solution is tested for the case of circular and concentric ducts.

The transfer matrix of a uniform duct with a linear temperature gradient

Self-consistent expressions for the four-pole parameters of a uniform duct with a linear temperature gradient have been derived for the general case of a superimposed mean flow. These expressions, and their simplified derivatives for the case of no mean flow, differ from those in earlier literature in first order terms of the temperature gradient. Experimental measurements validate the approximations made and indicate little practical benefit in detailed modelling of temperature variations.

A simple numerical method for three-dimensional analysis of simple expansion chamber mufflers of rectangular as well as circular cross-section with a stationary medium

Three-dimensional effects are a primary source of discrepancy between the measured values of automotive muffler performance and those predicted by the plane wave theory at higher frequencies. The basically exact method of (truncated) eigenfunction expansions for simple expansion chambers involves very complicated algebra, and the numerical finite element method requires large computation time and core storage. A simple numerical method is presented in this paper. It makes use of compatibility conditions for acoustic pressure and particle velocity at a number of equally spaced points in the planes of the junctions (or area discontinuities) to generate the required number of algebraic equations for evaluation of the relative amplitudes of the various modes (eigenfunctions), the total number of which is proportional to the area ratio. The method is demonstrated for evaluation of the four-pole parameters of rigid-walled, simple expansion chambers of rectangular as well as circular cross-section for the case of a stationary medium. Computed values of transmission loss are compared with those computed by means of the plane wave theory, in order to highlight the onset (cutting-on) of various higher order modes and the effect thereof on transmission loss of the muffler. These are also compared with predictions of the finite element methods (FEM) and the exact methods involving eigenfunction expansions, in order to demonstrate the accuracy of the simple method presented here.

Theoretical prediction of the transmission loss of circular reversing chamber mufflers

A mathematical expression is derived for the prediction of the transmission loss of a reversing chamber muffler with circular cross-section. The chamber is modelled as a piston-driven circular rigid tube with no losses and the resultant actual velocity potential in the chamber is obtained by the superposition of the two velocity potentials due to each of the two input pistons. All the geometrical parameters such as the inlet/outlet offset distances from the center of chamber, the relative separation angle between the inlet and outlet, the length-to-diameter ratio and the area expansion ratio are taken into consideration in the formulation as well as the acoustic higher order modes generated at the area discontinuities. By using the derived four-pole parameters, the transmission loss of a reversing chamber muffler can be easily predicted at any frequency and for any configuration. The theoretical results of the present study are compared with the measured transmission losses and they agree very well.

Noise reduction with perforated three-duct muffler components

This paper describes the authors’ distributed parameter approach for derivation of closed-form expressions for the four-pole parameters of the perforated three-duct muffler components. In this method, three simultaneous second-order partial differential equations are first reduced to a set of six first-order ordinary differential equations. These equations are then uncoupled by means of a modal matrix. The resulting 6 × 6 matrix is reduced to the 2 × 2 transfer matrix using the relevant boundary conditions. This is combined with transfer matrices of other elements (upstream and downstream of this perforated element) to predict muffler performance like noise reduction, which is also measured. The correlation between experimental and theoretical values of noise reduction is shown to be satisfactory.

Analysis of higher-order mode effects in the circular expansion chamber with mean flow

The effects of higher-order acoustic modes produced by the areal discontinuities of the simple expansion chamber with mean flow on the acoustic performance are studied. The chamber is modeled as a piston-driven circular rigid tube with no losses and, by using the Fourier–Bessel expansion, a general expression of the output pressure to the given input uniform volume velocity is obtained for a whole chamber. Quantitative estimation for the transmission loss can be performed using the derived four-pole parameters, whereas the characteristics of the chamber, as a result of the interactions between the plane wave and the transverse waves, can be easily investigated with respect to the relative locations of the inlet/outlet, the chamber length, and the mean flow. The expression as a form of C in four-pole parameters is adopted to describe the chamber characteristics for computational convenience. All the parameters involved in the experimental works by Eriksson [L. J. Eriksson, J. Acoust. Soc. Am. 72, 1208–1211 (1982)] are taken in the examples. Using the derived four-pole parameters, the transmission losses for both the concentric and offset inlet/outlet configurations are predicted and compared with the experimental results. The theoretical results of the present study are in good agreement with the experiments of the authors and the previous works.

A note on various formulations for four-pole parameters of a pipe with mean flow

This note is concerned with an examination of various formulations for the determination of four-pole parameters of a uniform pipe containing a moving medium. The correct four-pole parameters are identified. Their limitations in applications to the acoustic simulation of exhaust mufflers and piping systems are discussed.

Evaluation of four-pole parameters for ducts with flow by the finite element method

A finite element formulation of the equations of acoustic wave propagation in the presence of a mean flow of low Mach number has been obtained. The resulting equation system has been solved with two different sets of boundary conditions in order to obtain the four-pole parameters of a specified section of duct. The results have been compared with those of existing one-dimensional models of the flow. The finite element formulation has been developed as it enables one to determine accurately the wave propagation in a duct of any given geometry. The work is intended to aid in the prediction of the insertion loss of intake silencers on internal combustion engines.

Evaluation of four-pole parameters for a straight pipe with a mean flow and a linear temperature gradient

Four-pole parameters for a straight pipe system in the presence of a uniform mean flow and a linear temperature gradient are presented. The system was analyzed using first-order perturbation theory and a Green’s function approach. The general four-pole matrix derived reduces exactly to several other simpler cases already in the literature. The use of the matrix in engine exhaust system predictions is also given.

Mechanical four-pole parameters and their application

Analysis of the sinusoidal vibration of damped mechanical systems by the use of four-pole parameter theory is described. Values of the four-pole parameters are derived or stated for lumped systems, such as a dynamic absorber, and for distributed systems, such as a uniform rod in longitudinal vibration. Also stated are the parameters that describe the bending vibrations of a Bernoulli-Euler beam and a thin circular plate, both of which may be envisaged as four-pole systems when they are driven and terminated so that only symmetrical vibrations about their mid-points are excited. Numerical results are plotted for two examples in which four-pole theory is used: (i) to predict the force transmissibility to the clamped boundary of a center-driven circular plate when a dynamic absorber with optimum tuning and damping is attached to its mid-point; (ii) to predict the response ratio and force transmissibility across a two-stage or compound mounting system having a central impedance which ceases to remain masslike at high frequencies.

Internal thermal feedback in four–poles especially in transistors

Thermal-electric interaction or internal thermal occurs in temperature dependent devices and four-poles at low frequencies, if the thermal time constants are small. An electrical equivalent circuit which describes this thermal effect is generally derived and applied to transistors in common base and common emitter configuration. It consists of current sources and of resistance-inductance circuits which can be directly related to the thermal equivalent circuit of the device. It is shown that this thermal feedback should not be neglected in the measurement of high-frequency transistors, in the design of dc or video amplifiers and voltage or current regulators. Some measurements are reported and discussed. For instance, a strong frequency dependence of some four-pole parameters, especially of the forward trans-conductance y 21 and the short circuit output admittance y 22e of high frequency transistors was found at frequencies below 1 Mc. These effects can be explained by the new equivalent circuit. Possible application of this thermal-electric interaction may include the realization of large low Q inductances for low frequency integrated circuitry, and perhaps the investigation of pinch-in and second breakdown effects. It appears that the low frequency noise figure of transistors may also be affected by this effect.

Application of Four-Pole Parameters to Torsional Vibration Problems

This paper deals with the application of the method of four-pole parameters to torsional vibrations. Results are developed from fundamental principles. The four-pole parameters for the basic rotational elements are derived. These include shafts (both lumped and distributed-parameter cases), disks, dampers, and gears. The equations which must be obeyed, when these elements are connected, are presented. The application to construction of equivalent electrical circuits is given and in particular a method for constructing the equivalent circuit of distributed-parameter systems is put forth. The torsional analogs of Thevenin’s and Norton’s theorems are given for rotational sources. The fundamentals mentioned above are then applied to the following problems: (a) The effect of substituting one four-pole for another in a torsional system. (b) The effect of opening a four-pole system and inserting a new four-pole between the separated four-poles. (c) Calculation of all the torques and angular velocities in a tandem system. (d) Calculation of natural frequencies of undamped four-pole systems.

Measurement of Four-Pole Parameters of Complex Structures

A method of measurement of the four-pole parameters of a system as a function of frequency is presented. This procedure consists of no-load dynamic measurements followed by measurements under a known load. From the foregoing measurements, the four-pole parameters of a system are derived. Application is made to an electrodynamic motion exciter with graphical presentation of the parameters with respect to amplitude, phase angle, real, and imaginary parts.

AC Temperature Compensation of Integrated Amplifiers

Abstract : Effort is expended towards an analytical basis for temperature compensation of the AC performance of transistor small-signal amplifiers. Described is a simple, accurate design theory in which this compensation is analytically determined by imposing the requirement of equality for the values of the individual small-signal fourpole parameters of the amplifier at its operating temperature limits. For values of T between -30 and +100 C in a typical noncompensated circuit, power gain, input impedance and output impedance vary by 99%, 71%, and 22%, respectively, while the comparable compensated circuit variation of each of these quantities is less than 9%. In an integrated circuit a single tapped resistance, properly selected, is the temperature sensitive element required for this compensation. (Author)

The Spacistor

A brief description is given of several Spacistor structures and the techniques of fabrication. Detailed measurements of the four-pole parameters are reported and described with the aid of physical models. The mutual conductance is shown to depend upon injector-modulator spacing, injector current and modulator bias, with little variation to 25 Mc. Improvement in both g <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> and shielding is shown to occur when the modulator is injecting charge. Values of g <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> of the order of several hundred micromhos have been observed. The input resistance depends upon the characteristics of the modulator contact in the space-charge region and upon the dc bias. Under forward-biased modulator conditions (optimum g <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> ) values of input resistance of the order of several hundred thousand ohms are obtained. Under zero-bias or reverse-bias conditions (somewhat reduced g <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> ), values of several megohms are obtained. Values of output resistance ranging to several hundred thousand ohms have been observed with possibilities of larger values being obtained with improved design. Low frequency voltage gain of the order of 10 and power gain of the order of several hundred have been exhibited. The reduction of power gain at 25 Mc is attributed mostly to the array of contact and device resistances, and device and header capacities, some of which may be reduced by appropriate design. These prevented the elimination of shunting capacities and precluded higher frequency measurements.

Power gain and bandwidth of tuned transistor amplifier stages

Relationships between operating power gain and bandwidth of an active four-pole network are developed, with particular reference to single-stage tuned transistor amplifiers. The power gain is first investigated in terms of Stern&apos;s stability factor, k, the maximum gain for a given value of k being derived by analytical-numerical means without recourse to graphical procedures. It is then shown that the power gain is more fundamentally expressed in terms of a new performance factor, n, and in particular that for large values of n (corresponding to k ≫ 1) the product of maximum power gain and n approaches an invariant, independent of the set of four-pole parameters selected for design purposes. The power gains of amplifier stages with and without unilateralizing feedback are expressed and related in terms of the performance factor, and the conditions under which the latter possibility gives higher gain are indicated.Analytical procedures and expressions for the evaluation of the bandwidth of tuned amplifier stages are presented and the conditions for maximum power gain at a given bandwidth are investigated for amplifiers with and without unilateralizing feedback.Computed results for the performance of a particular high-frequency transistor in the common-base and common-emitter configurations verify the main conclusions of the analysis and indicate the relative merits of the two configurations in regard to power gain with and without feedback applied. It is seen that for bandwidths up to a certain value a simple mismatched stage without feedback is advantageous.

Experimental Use of Electrical Network Theory in Complex Linear Mechanical Structures

Linear mechanical systems may be analyzed according to electrical network theory. The analysis may proceed from empirical mechanical impedance data without detailed knowledge of the masses, dampers, and springs involved. The analysis can then predict reduction of vibrational response due to certain experimental changes such as the addition of a tuned, damped vibration absorber. This discussion also describes how the mechanical impedance of a tuned, damped vibration absorber may be calculated using four-pole parameters.

Effects of Low Temperatures on Transistor Characteristics

The four-pole parameters of a group of similar pnp alloy junction transistors were measured in the frequency range from 0.5 to 5 mc and from room temperature down to that of liquid nitrogen. The measured parameters are translated into simple electrical networks for application in these frequency and temperature ranges. Theoretical expressions for the transistor parameters are evaluated. This evaluation utilizes the effects of temperature on surface-recombination effects, on hole mobility, and on carrier lifetime that have been reported in the literature. The transistors were subjected to grounded-emitter pulse tests in which the transistors were driven from beyond cutoff to the edge of saturation. These tests show that, for a given output pulse current, rise time decreases with temperature. The tests with the transistor in liquid nitrogen show further that rise time decreases as the output current pulse increases. Under these conditions it is possible for a 30 mw, 3 mc alloy-junction transistor to deliver a one-ampere output current pulse with a rise time of a few tenths of a microsecond and with a grounded-emitter gain of about 20.

Use of Four-Pole Parameters in Vibration Calculations

Linear elastic systems which have a single input point and a single output point can be characterized by a pair of simple linear equations, namely Finput=α11Foutput + α12Voutput,Vinput=α21Foutput + α22Voutput, where the F's are forces, the V's are velocities, and the α's are the four-pole parameters for the system. The use of four-pole parameters permits combinations of mechanical elements to be handled as a single entity, and, as the parameters belong only to the elastic system for which they are defined, they do not depend upon what precedes that particular system or follows it in a mechanical network. This permits the statement of results in a rather general form. These parameters have been used for many years in the analysis of electrical circuits, and it is the purpose of this paper to show their utility in the field of mechanical vibrations. In particular, the four-pole parameters for the basic mechanical elements mass, spring, and resistance will be obtained. Also, a description of mechanical sources will be given. The rules for the connection of four poles and a method of experimentally measuring four-pole parameters will be presented. A few specific problems illustrating the use of four poles will be set forth, namely: a generalized vibration isolation problem, the response of an elastically mounted mass on an electrodynamic shake table, shock excitation analysis, and some distributed parameter systems (i.e., helical springs and rubber in shear).