Kirchoff's Laws Research Articles

A functional model of a simplified clarinet

A functional model (computer simulation) of a simplified clarinet has been implemented on a digital computer. The simplified clarinet consists of a standard clarinet mouthpiece and reed attached to a straight cylindrical tube without tone holes. The tube and mouthpiece are represented in the model by a lumped element approximation to a transmission line. Application of Kirchoff's circuit laws to this circuit gives a coupled set of differential equations which were solved numerically on a digital computer to yield volume velocities and pressures in the mouthpiece and tube. The pressure in the mouthpiece drives the reed which is represented in the model as a tapered bar clamped at one end. The partial differential equation describing vibrations of such a bar was solved numerically to obtain the size and shape of the reed aperture. The reed aperture area governs the volume velocity into the tube which in turn drives the air column vibrations. The model exhibits self‐sustained oscillations, threshold blowing pressures, and “lipping” of the tone. The spectra of the mouthpiece and radiated pressures are in general agreement with actual clarinet spectra. Some interesting features of the model are the inclusion of a frequency‐dependent viscous loss in the tube and a dependence of the aperture volume velocity on the initial or rest opening of the aperture.

An Improved Gas Transmission System Simulator

Abstract CASPR (acronym for Capability And Simulation Program) is a new computer program that simulates Program) is a new computer program that simulates steady-state gas transmission networks based on the Newton-Raphson balance of nodes method. The program contains several refinements, such as an program contains several refinements, such as an optimal node-numbering scheme to eliminate pivoting of the coefficient matrix and to reduce matrix an implicit compressor-fuel gas calculation, the ability to prorate equally gas volumes entering the network system, and the optional calculation of a flowing-gas temperature profile. Introduction The simulation of gas transmission systems requires increasingly sophisticated computer software as these systems grow in size and complexity. The major complication of system modeling is the need to accommodate systems with closed loops. Other factors also contribute to the need for improved system modeling, such asaccurate accountability of more expensive compressor-station fuel gas consumption,proper proration of gas supply from many collection points proration of gas supply from many collection points to assure adequate transmission-system capacity for a given demand, anda detailed knowledge of the gas-flowing temperature through accurate simulation of heat-transfer effects. CASPR was developed to meet these requirements with improved computational efficiency through the use of optimal node numbering. THE DEVELOPMENT OF CLOSED-LOOP MODELS The need to simulate closed-loop systems has resulted in implementation of techniques evolving from the original work of Cross, to that of Epp and Fowler, Shamir and Howard, and the adaption of the last study to the gas-distribution industry by Stoner. These closed-loop models all satisfy one or more of Kirchoff's laws; (1) the sum of the potentials around a closed loop is zero, and (2) the flow into a point equals the flow out of that point. Methods satisfying one or both of these laws can be broadly classified as either explicit or implicit. Explicit models were formulated by several authors, however, the lack of mathematical rigor prevents the modeling of sensitive systems or systems with many closed loops. Implicit methods involve writing a series of nonlinear algebraic equations and solving them, using a technique such as the n-dimensional Newton-Raphson method. This method requires solving for an n x n coefficient matrix, which for large n can require large amounts of computer time and storage. Several workers have taken advantage of the usually sparse structure of the coefficient matrix to minimize this problem, although the sparse matrix formulation can be cumbersome to set up and update, especially if pivoting is required to ensure diagonal fill-in and reduce round-off error. FORMULATION OF THE NODAL PROGRAM In the well known nodal formulation, a pipeline system (Fig. 1) is represented by a network of nodes and node connecting elements (NCE's) q1? (prorate) q2? (prorate) Node Node Node1 2 1 3 P1 P2? Pc? Pc qf1? q3 q1 . unknown prorated receipt point volume at node 1 p1 . fixed pressure at node 1 q2 . unknown prorated receipt point volume at node 2 P2 . unknown pressure at node 2 (compressor station suction) qf1 . compressor station 1 fuel gas (function of horsepower) PC . unknown horsepower at compressor station 1 . compressor station 1 P3 . fixed pressure at node 3 (compressor station discharge) q3 . fixed delivery point volume at node 3 FIG. 1 - SCHEMATIC OF NODES AND NCE'S. SPEJ P. 389

Analysis of a Four Quadrant Ring Multiplier with General No-Memory Bridge Arm Impedances

An analysis is made of a four-quadrant ring multiplier in which the currents in the bridge arm inpedances are represented as general, no-memory functions of the arm impedance voltages. Application of Kirchoff's current and voltage laws to the ring circuit yields an implicit functional relationship between the output and drive voltages of the bridge that involves the bridge load resistor and the general arm impedance functions. The form of the Taylor's series expressing the output voltage as a function of the drive voltages is developed for the special but important case in which the ring consists of four identical arm impedance functions. General formulas are derived for the first three nonzero coefficients occurring in this series. An evaluation is made of circuit performance for the case in which 1N72 junction diodes serve as the bridge arm impedances. A direct comparison is made between predicted and measured bridge output voltage for this case with the ring circuit used to square a sinusoidal signal. Finally, an analysis is made of the bridge drive voltage rejection capabilities for mismatched arm impedances. Numerical values of rejection (in db) are computed for each drive voltage from measured parameter values of the 1N72 junction diodes used in the tests.

Studies on Galvanic Couples: II . Some Potential‐Current Relations in Galvanic Corrosion

A graphic method of expressing the electrical characteristics of a galvanic couple was developed by the application of Kirchoff's laws to a couple consisting of separate electrodes, each with an ammeter in series. When an external emf was impressed between the couple and a third electrode, also with an ammeter in series, a definite current pattern was established, depending on the relative directions and magnitudes of the impressed emf and the potential of the couple. These relations were demonstrated with the iron‐magnetite couple. Under the conditions used, a discontinuity in the potential‐current relation was observed, when the couple was cathodically polarized over a sufficient range of current. This discontinuity or “break” coincided with reduction of current, circulating within the couple, to zero. The rise in potential at this point is the logical result to be expected from the hydrogen overvoltage.

A Practical Approach to Calculating Optimum Performance of Semiconductor Rectifiers

Applying Kirchoff's circuit laws to a conventional four-arm rectifier bridge, an expression is derived for efficiency in terms of R (effective reverse resistance of one arm), F (effective forward resistance of one arm), and L (ohmic load resistance). Knowing that R and F can be kept constant by separate adjustments of input voltage and temperature of the cells enables us to maximize efficiency, E, with respect to L. The optimum value of L turns out to be ?RF, the geometrical mean of forward and reverse resistances. This value of L yields E max = [(q - 1)/(q + 1)]2, where q = ?R/F. Further analysis shows that ?R/F = V R /V F for optimum L, where V R and V F are corresponding values of reverse and forward voltage drops, so that E max is expressible in terms of V R and V F alone. Three families of curves are given to illustrate manipulation of the simple empirical data required, resulting in a final plot of E max versus temperature for various current densities in the cells. An important feature of the process depends on the interesting analytical result that for maximum efficiency the reverse and forward losses are equal, making it possible to correlate values of V R and V F . Because the empirical data involved are strictly applicable only when R and F are not appreciably affected by previous values of V F and V R respectively, a modified technique is suggested in which data are observed only when optimum L is in the circuit.

Propagation constant of a transmission line

Starting with the logarithmic variation of voltage and current along a transmission line, a derivation of the propagation constant is given in this article, depending only upon the application of Kirchoff's laws and simple hyperbolic trigonometry.

Calculating load division in distribution systems

A COMPARATIVELY simple and convenient method is available for solving the division of current in the various sections of an electrical network and obtaining the voltages at the different load points. The method consists essentially of the application of Kirchoff's laws to the solution of a network, the departure from the usual procedure being that instead of solving a number of simultaneous equations, the problem is handled in steps by the superposition of currents. The idea of a circulatory current is introduced when any change in the system is made by a closure between two points in the existing system.

Pre-Charged Condensers in Series and in Parallel

A condenser is charged from a source of direct voltage, and then is used as a booster in series with this source to charge another condenser. By repeating this process a large number of times the second condenser is finally subjected to twice the voltage of the source. This is the principle of the Delon apparatus for testing cables, and is explained in a numerical example. Then the more general case of two or more ``pre-charged'' condensers in series is considered, when these condensers are connected to some source of direct voltage; it is shown how to determine the final distribution of voltages among them. A similar problem is solved for pre-charged condensers in parallel. Finally a general network of pre-charged condensers is considered, and equations are derived similar to Kirchoff's laws, from which the final distribution of voltages and charges may be computed knowing the initial distribution.

Pre-charged condensers in series and in parallel

A condenser is charged from a source of direct voltage, and then is used as a booster in series with this source to charge another condenser. By repeating this process a large number of times the second condenser is finally subjected to twice the voltage of the source. This is the principle of the Delon apparatus for testing cables, and is explained in a numerical example. Then the more general case of two or more “pre-charged” condensers in series is considered, when these condensers are connected to some source of direct voltage; it is shown how to determine the final distribution of voltages among them. A similar problem is solved for pre-charged condensers in parallel. Finally a general network of pre-charged condensers is considered, and equations are derived similar to Kirchoff's laws, from which the final distribution of voltages and charges may be computed knowing the initial distribution.