   Journal of Applied Physics 16, 172-186 (1945)

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Abstract: Numerical methods are developed to solve certain types of linear and nonlinear partial differential equations to any desired degree of accuracy with the aid of equivalent electrical networks. The methods of solution of ordinary differential equations, both linear and nonlinear, follow as special cases. Three types of problems are considered:

1. Initial-value problems. If the field quantities are known along a surface, the networks may be solved by a straight-forward step-by-step calculation The networks may also be looked upon as supplying a 'schedule' of operations that can be put on a digital calculating machine.

For time-varying problems new types of networks are developed in which time appears as an extra spatial variable. Examples of new networks for the elastic field and for the general nonlinear wave equation are given. Sample calculations and theoretical checks of a transient heat flow problem and of an ordinary differential equation are also included.

2. Boundary-value problems. Four methods of solution are given, the first three being cut and try processes.
(a) The method of weighted averages;
(b) The method of unbalanced currents and voltages;
(c) The 'relaxation' method;
(d) The 'diffusion' method, that changes the boundary-value problem into an initial-value problem by adding to the original partial differential equation a time variable of the form A∂φ/∂t, allowing the unbalanced currents to 'diffuse' in time.

These numerical methods may also be used to improve the accuracy of the results found on the Network Analyzer. Examples of calculations are given for the electromagnetic and the elastic fields.

3. Characteristic-value problems. Their methods of solution are similar to those of boundary-value problems. An additional method of unbalanced admittances is also indicated. It is shown that by calculating the power in the network, the characteristic value of the assumed function is 'found'.

An improved characteristic value of the linear harmonic oscillator, solved initially on a Network Analyzer, is calculated as an example.

In general the electrical networks may-be used to check the consistency and correctness of solutions arrived at by other methods, approximate or exact. The unbalanced currents at the junctions (easily calculated) give a quantitative measure of the deviation from the correct solution.

NETS

The method of solving partial differential equations numerically by the use of equivalent circuits is by no means new. It has long been known (1,2) that Laplace equation ∇2φ = 0 in rectangular coordinates may be solved to any desired degree of accuracy by (considering the two-dimensional case) drawing a net of squares, Fig. 1, each with area ∆x∆y, and associating with each line the real positive number, unity. If with each junction (representing a point of space) the correct value of the dependent variable φ is associated, then each φ must be the average of the four neighboring φ's. An extension of this method to Poissons equation ∇2φ = k, where k is an arbitrary function, has also been in use.

The equivalent circuits presented here may be looked upon as generalizations to more general partial differential equations of the type of net of Fig. 1a used by mathematicians to represent Laplaces equation. In particular (considering Fig. 1b): 1. The square net with two lines crossing at right angles is replaced by nets with more complex configurations. Several lines may now meet at a junction and at arbitrary angles.

2 With each line is associated not unity but either a real, or an imaginary or a complex number a, b or c+id, positive or negative.

3. The junctions are associated not with one, but several dependent variables.

In these more general nets the variables represent line-, surface-, and volume integrals of field quantities on a set of interspersed small cubes into which the field is subdivided. Accordingly, not just any haphazard set of partial differential equations can be represented by a net. This physical interpretation requires that the equations should be in a tensor-density form.

It is a rather arbitrary but still valuable procedure to associate the numbers at the junctions by absolute potentials and the numbers along the lines by impedances, in particular. (Fig. 2):

1. a positive real number by a positive resistor;
2. a negative real number by a negative resistor;
3. a positive imaginary number byan inductor;
4. a negative imaginary number by a capacitor. This physical interpretation of the mathematical net allows additional physical concepts such as current, power,- flux, etc., to be associated with the network, thereby simplifying ifs analysis and solution. It is true that in topology these additional concepts are also associated with the mathematical net, in particular current as "1-chain", power as "intersection-index", etc.,(see ref. (1), p.213) but apparently these concepts have not yet been used for the solution of nets representing the Laplace-Poisson equations.

In transient equivalent circuits not only numbers are associated with the lines of the net, but also operators, such as ∂/∂t. In nonlinear circuits the line numbers are functions of the junction numbers. When only positive and negative real numbers exist, it is customary to replace a positive resistance by an inductance and a negative resistance by a capacitor (since none or only a few negative resistances exist on practical network analyzers). Of course the algebraic symbols may be replaced by a mechanical model (masses and springs) or an acoustical model (pipes and chambers) instead of an electrical circuit.

PART I. INITIAL-VALUE PROBLEMS

Time-Dependeut Equations

Partial differential equations containing time differ usually from other in the manner in whith the boundary conditions are specified. In this part it will be assumed that at time t = 0, the values of the dependent variables and their derivatives are known. Considering time as an extra spatial variable, the equivalent circuit consists of the original network (in which time does not enter) arranged in layers (one-, two-, or three-dimensional), separated by distances ∆t and interconnected by suitable impedances. Figures 35 represent the nonlinear wave equation, for various numbers of independent variables. In orthogonal curvilinear coordinates where the f s are arbitrary functions of u1, u2, u3, and t. When time enters as a second-order derivative, the networks are shown in Figs. 6 and 7. When time enters in a more complex form, different arrangements must be used. Figure 8 shows the time equivalent circuit of the elastic field. Since time enters as ∂2 / ∂t2 for each dependent variable, all component networks are identical and for all spatial reference frames they are the same as those given in "Equivalent Circuits of the Elastic Field." (3) It should be noted that while in the elastic and fluid flow fields the time coordinate may be left out or included at will, such is not the case in the electromagnetic field. In the latter the non-privileged role of the time coordinate necessitates a radically new treatment.

Time as a Spatial Variable

Let in a linear problem at t = 0 an instantaneous impulse be impressed on the physical system. On the network this impulse is impressed by generators at t = 0 and the voltage distribution gives the effect of the impulse in time and in space. (In the absence of a Network Analyzer the voltage distribution may be found by the methods to be shown presently.) Since the network extends to infinity along time, the network beyond any specified time t0 may be replaced by a characteristic impedance.

If the effects of an impulse are known, then the effect of any other force applied within any time interval may be found by simple superposition of the known voltage distribution.

It should be noted that the effect of more than one impulse cannot be studied simultaneously on the Networkl Analyzer, since a generator impressed, say, at t = 5∆t produces a voltage distribution at points corresponding to previous times, a situation impossible in classical dynamical problems. The numerical method to be shown prevents this "backsliding" of the impulses impressed at various times.

The Numerical Calculations

The numerical method is based on the fact that if all the potentials on the network are defined at t = n∆t and at t = (n+1)∆t (and also the boundary conditions along time), then all the potentials and currents'càn be calculatçd for the rest of the hetwork in a straightforward manner up to any desired number of ∆t's.

The numerical work consists of calculating all the admittances and currents on the second or (n+1)∆t network, (Since in nonlinear problems the admittances are functions of the potentials, the former can all be calculated.) Adding up at a junction all the currents, induding the one coming from the lower network, the resultant current must flow upward to the next network (n+2)∆t, Fig. 4. Since the admittance of the connecting coil is known (even in nonlinear

problems) the absolute potentials of the corresponding points on the next network (n+2)∆t are thereby determined. The work of calculating currents is again repeated to proceed to the next network at (n+3)∆t.

Starting Assumptions

It should be noted that in order to get started, it is necessary to know not only all the currents on the network say at t = 0 but also the currents coming from the next lower network, t = - ∆t. However, in most problems the boundary conditions are so defined that at t = 0 the currents coming from the lower network are not known, although the sum of the lower and upper connecting currents, namely ∆t(∂φ / ∂t) is known at every point. At the start of each problem thus the question arises how to divide the resultant currents into two parts at t = 0. Several methods are available:

(1) Half the current may be assumed to flow up, the other down. This assumes the slope of the curve φ = f(t) to be constant in the neighborhood of t = 0. After proceeding through two or three layers of networks it is possible to discern a more logical division than halving and to start all over again with the new divisions.

(2) Standard methods of approximations given in textbooks (4) may be used. For instance, after finding the potentials on the t = ∆t network, the differences of potentials between the two networks define the slope of the chord between the points t = 0 and t = ∆t. Using this slope instead of that of the tangent ∆t(∂φ / ∂t) at t = 0, a better division is accomplished. This step may be repeated several times between the first two networks.

(3) ∆t maybe subdivided into 5 or more subdivisions. Because of the finer divisions the results arrived at by the 5th ∆t may be assumed to be close enough approximations and may be used as starting values on the coarser network.

Other Assumptions

Since the starting conditions are rarely known exactly, it will often be found that the currents or voltages in successive steps do not lie on smooth curves but oscillate around them in an attempt to correct themselves. Such overshooting and undershooting of smooth curves may be minimized by judicious changing at intervals of the division of currents that flow "up" and "down" between the layers. Most numerical methods do not possess this self-correcting property of the equivalent circuits.

When some of the admittances lying between two successive layers (between two ∆t's) are functions of the dependent variables (hence, of time) their value has to be found by extrapolating from the known values of the dependent variables at two previous times.

Ordinary Differential Equations

In ordin~ary differential equations the independent variable may be either a spatial variable x or time t. If it happens to be the time, then usually sufficient information is available at t = 0 to get the calculations started as above.

When the independent variable x is a spatial variable then usually only part of the boundary conditions (say the value of φ) are defined at x = 0. Some boundary conditions (say the slope, of φ) may, however, be defined only at x = a. In such cases it is possible to start out with a tentative slope at x = 0 and find with the above step-by-step method the slope at x = a. Several new starts should eventually match the boundary conditions at the two different values of x. An example,

di/dt = 10 - 2i

with initial conditions i = 0 and t = 0 (also di/dt =10 at t=0) is worked out in Fig. 9 in steps of ∆t= 0.1. The numerals in parenthesis give the results found in Doherty and Keller (5) for the same problem, found by a standard method requiring a cut and try procedure not only at the first step (between t = 0 and t = ∆t) but at every step.

Other worked out examples may be found in "The Application of Network Analysis to Some Electron Optical Problems." (13)

Example of a Transient Heat Flow Problem

Let the temperature at the edge of a plate be raised from zero to unit temperature at t = 0 and be kept at that same temperature. The problem is to find the distribution of temperature along the plate (at distance x from the edge) at various times.

The equation of flow is Also ∆x=0.1, a∆t=0.0025. Hence ∆t=0.0024/a

The boundary conditions are: Since the known data at t = 0 are rather meager (T and ∂T / ∂ t are zero at every point along x except near x = 0) for a preliminary calculation the assumed ∆t is subdivided into five parts, so that a∆t = 0.0005. The corresponding network to be used to get started is shown in Fig. 10. The layer at t = 0 only contains one current 5 flowing into the ground. (The existence of a ground is a boundary condition.) Hence this current cannot very well be assumed to divide into two equal parts. Instead, let it be assumed that a similar current in the next layer at t=a∆t/2 also has a value 5. Now half of that may flow up and half down. Assuming slightly less than half, say 2.45 flowing up, one potential on the next layer is 24.5. Its neighboring potential on the same layer is still undetermined but it must be small and could as well be assumed to be zero for the purpose of calcijilating the current flowing into this junction. The latter (24.5 - 0) 0.005 = 0.1225 is assumed again to flow half up, half down.

Continuing the calculation through five layers, the voltages on the fifth layer can be assumed to be approximately correct. Since the first layer contains too few informations, the calculation is continued to the tenth layer in order to be able to use the fifth and tenth layers as the first two layers of the coarser network of Fig. 11. Fig. 11. Transient heat-flow calculation.

The calculations on Fig. 11 are straightforward. The numbers in parenthesis give the theoretically correct temperatures, showing satisfactory agreement with the approximate method. The farther the calculations are continued from the crudely approximated first layer, the better is the agreement with the analytical solution. It should be remembered that the set of vertical currents, by

their deviation from a smooth curve, do show that some of the temperatures are too high and some are too low. Hence, they afford a means of correcting the results at intervals.

Equations Not Containing Time

When an equation does not contain time, it still may happen that all field quantities are known upon a layer so that the next layer may be calculated. A class of such problems occurs in the design of magnetic or dielectric equipotential surfaces where a definite flux density (magnetic or electrostatic) is desired across a given surface. The problem is to shape the pole faces or electrodes to produce this desired flux distribution. For instance, in a betatron (6) a known axially symmetric flux distribution is desired at the median plane of the accelerated electron. The equivalent circuit for the vector potential ψθ is shown in Fig. 12. Here the following quantities are known:

1. As all flux lines are perpendicular to the median plane, H(r) is zero and the vertical currents along the median plane are zero.
2. The flux distribution B(z) along the median plane are given, hence the differences of potentials along the corresponding coils are also known.
3. The current density in the current carrying region are known, hence all currents entering the junctions at the corresponding region are also known.

Sufficient data are available to continue the calculation of potentials and currents without any trial and error. The lines orthogonal to the equipotential lines represent the possible pole shapes to produce the desired flux distribution. When no current-carrying regions exist, a modification of the network, representing the scalar potential φ may be used, e.g., as shown in Fig. 10 of "Equivalent Circuits of Compressible and Incompressible Flow Fields. " (7)

PART II. BOUNDARY VALUE PROBLEMS

The Problem

The equivalent circuits consist of a configuration of coils repeated along one, two, or three axes. The final shape of the network is the same as the shape of the field to be analyzed. While the illustrations will refer to two-dimensional networks, all rules and formulae are equally valid for one- and three-dimensional networks.

In analyzing a field, at first it should be divided into a very few subdivisions only. After its field distribution has been calculated, it should be subdivided into a finer mesh then, if needed, into still finer meshes. In most problems it will be necessary to introduce fine meshes only in certain portions, usually around corners.

The problem in all cases is the following:

(1) Given the boundary conditions in the form of known currents flowing in some of the coils and in the form of known absolute potentials at certain junctions.

(2) Find the absolute potentials of all junctions and the currents flowing in all conductors.

In linear problems the impedances of all coils are also known. In non-linear problems only some of the impedances are known, while the value of others are functions of the as yet unknown absolute, potentials and their derivatives. Some of the junction currents also may be functions of the potentials.

A Preliminary Step

In most networks the absolute potentials of the junctions have a physical meaning. In particular: (1) In the electromagnetic field it is Ey, the field- intensity, along the y-direction, or Hy.
(2) In elasticity it is the total displacement u or v.

(3) In hydrodynamics it may be the stream- function ψ or velocity potential φ etc.

In all methods of attack as a preliminary step a tentative potential distribution at the junctions is assumed. Usually an intelligent guess may be made and if the guess is far off, that only entails more labor, without, however, influencing the correctness of the final answer. (Instead of the potential distribution, a guess may be made at the current-distribution.) Of course in linear problems this preliminary step may be done on a Network Analyzer, even though only a very few units are available, or the results of an approximate or exact solution may be used as starting points.

Once the absolute potentials at all the junctions are assumed to be known, it is always possible to calculate the currents flowing in all the coils. In non-linear problems at first the coil admittances and junction currents must be calculated from the known potentials, then the coil currents. Examples of non-linear networks are those of the potential flow of a compressible fluid in the physical plane, given in reference 7, Figs. 10 and 11.

The method of attack is based upon the fact that if the correct potential distribution is assumed and if all currents in the coils have been calculated then:
(1) The sum of the currents

entering each junction is zero.
(2) The weighted average of the potentials of neighboring junctions is equal to the potential of the central junction.

Three types of numerical calculations may be performed depending upon whether the weighted average of the currents entering a junction, or of the voltages surrounding a junction, or of both, are calculated.

Because of the bad guess, however, actually at no time will the above conditions be fulfilled at all junctions simultaneously. The purpose of the calculation is to reduce the difference in the entering and departing currents (or the difference between the surrounding and the central potentials) as close to zero as possible at all junctions.

This difference in the currents will be called the "unbalanced current" iu , and the difference in the voltages the "unbalanced vOltage" eu . It should be noted that:
(1) The "unbalanced voltages" offer a clue on how much to change the guessed-at voltages to reduce the error.
(2) The "unbalanced currents" give an indication of how much the guessed-at values differ from the correct values.

Calculation of the "Unbalanced" Currents and Voltages The unbalanced currents iu at each junction are best calculated by summing up the currents entering it. To find the unbalanced voltage. eu at a junction whose absolute potential is e0, let Fig. 13 be considered. The unbalanced current may also be found by If it is assumed that no unbalanced currents exist (iu = 0) then the correct value of the assumed potential e0would be E, the weighted average Since e0 is assumed instead of E, the unbalanced voltage is Hence the relation between and eu at every junction is is the sum of the admittances of all the coils leading to the junction.

The Field Equations of Maxwell

In Laplace's equation, where y1 = y2 = y3 = y4 = 1 and yg = I = 0, In Maxwell's equation, in rectangular coordinates, Fig. 4, where y1 = y2 = y3 = y4 = yL and yg = yc (acapacitance, a negative number),  a number less than four.

That is, finding the unbalanced currents and voltages for the field equations of Maxwell in rectangular coordinates is no more complicated than finding the same quantities in Laplace's equation (i.e., the number 4 is replaced by k). It will be found in most problems that the general equations (1), (4), and (2) for finding iu,eu and E may be considerably simplified.

The Method of Unbalanced Currents and Voltages

One of the quickest ways (if not the quickest) to solve a complicated network is the following:

1. Assume (or measure by a Network Analyzer) the potentials at all junctions.
2. Calculate at all junctions the unbalanced currents iu and the unbalanced voltages eu.
3. Knowing the whole set of unbalanced quantities at the junctions assume a new set of potentials that are more correct. In general, where e u is positive, the potential should decrease and with negative e u it should increase.
4. Calculate over again the whole set of new unbalanced quantities at every junction.

Such successive guesses and calculations should reduce the unbalanced quantities below any desired amount. Plotting curves of the unbalanced currents, voltages, and of the guessed-at values at frequent intervals helps to speed up the convergence.

Examples of Unbalanced Quantity Calculations

Let a square wave guide or cavity resonator, by a conducting surface that extends to infinity in the y direction, be divided into sixteen squares. The net of coils in each square consists of two inductances (z=0.8 or y=1.25) and a ground capacitor (z=1.3 or y=0.77). The currents in the inductance represent Ex and Ez, while the voltages across the capacitors represent Hy. Fig. 14, bounded in the x-z plane on three sides.

For boundary conditions, it is assumed that at the open end (upper end) a sinusoidal current distribution in space, Ex, is impressed. As a result, a sinusoidal potential distribution Hy appears at the junctions. Neither the magnitude nor the wave-length of Hy is known.

Tentatively let the values shown on Fig. 14 be assumed as the junction potentials H y. (Actually these values were found on the Network Analyzer.) FIG. 15. Unbalanced currents and voltages on the wave guide network.

Figure 15 gives the unbalanced currents and voltages that will have to be liquidated. It should be noted that the greatest unbalanced junction current is 7.93 amperes. As the greatest coil current is 102.8 amperes, that represents an 8 percent inaccuracy.

Knowing the unbalanced currents and voltages at every junction, a new set of junction potentials may now be assumed. For an elastic stress problem of a beam, a set of junction potentials (representing displacements in the x or y directions) and a set of unbalanced currents are shown in Fig. 16 (reproduced from Carter (8)).

The Method of Weighted Averages

Instead of attempting to reduce the unbalanced quantities simultaneously at all junctions, it is possible to reduce them at one junction at a time. Two such methods will be shown .

(1)The unbalanced current and voltage are reduced to zero at a junction without compensating at the neighboring junctions (the method of weighted averages).
(2) Each time the unbalancect current or voltage is reduced at a junction, the unbalanced quantities at the neighboring junctions are compensated simultaneously (the relaxation method).

The first method is especially valuable when the guessed at value of potentials differs from the correct value only at isolated points or regions. Such is the case usually in Network Analyzer solutions when at some junctions the instrument reading is incorrect or the board unit happened to be incorrectly set. The steps are as follows:

(1) Assume (or measure) the potentials at all junctions.
(2) Calculate E, the weighted average, at any junction by Eq. (2).
(3) Replace e0 by E.

In the previous method e0 was not replaced by E. Only the corresponding eu for each e0 was calculated, leaving e0 undisturbed.

It is customary to start at one corner of the network and change in succession all e0 to E, utilizing the already corrected potentials wherever they are available. A disadvantage of this method is that it is equivalent to reducing the unbalanced potential eu at each junction immediately to zero.

A more advantageous procedure is to replace at each junction the value of e0 not by E but by a value larger than E, or smaller (depending on the value of the neighboring potentials) leaving thereby at each junction an unliquidated e u that is within the allowable limit of error.

In non-linear problems it is often possible to plot curves which give outright the value of E for given neighboring potentials (or rather for given potential-differences between neighboring junctions). If difficulties in the convergçnce arise, in place of E a value larger (or smaller) should be used.

The Relaxation Method

The relaxation method reduces the unbalanced currents or voltages at one junction at a time toward zero in a systematic manner. It is based upon the fact that whenever the absolute potential of a particular junction is changed, the unbalanced currents and voltages change only at that particular junction and at all the neighboring junctions to which a coil leads, while everywhere else the unbalanced junction currents remain unchanged. This fact may be seen by introducing a set of hypothetical generators so that (Fig. 17a):

(1) At every junction a generator with zero impedance is assumed to be connected to the ground.
(2) The generator voltage is equal to the absolute potential of the junction.
(3) The generator current to ground is equal the to unbalanced current existing at that junction.

By the introduction of these hypothetical generators the network remains unchanged and is now capable of performing new feats that it couldn't do otherwise. Since the generators have zero impedance to ground, whenever one of the generator voltage is changed, currents can flow from this generator only to the neighboring grounds. (Fig. 17b). No change of currents exist anywhere else in the network.

Hence, the unbalanced currents are reduced in two steps:

(1) Add (or subtract) a certain number of volts to the potential of a junction.
(2) Find the new unbalanced currents (or voltages) at that junction and in the neighboring junctions.

Whether the unbalanced currents, or voltages, or both should be calculated depends on the problem at hand.

The "Unbalanced-Current Patterns"

The work involved in this last step may be greatly reduced in linear problems by the device of the "unbalanced-current pattern". This device consists of calculating once for all the change in the unbalanced junction currents caused by a change of one volt at every junction. (With non- linear coils this device cannot be used.) Two cases may be distinguished: (1) If the square nets of impedances throughout the network are identical, then each junction that does not lie on the boundary, has identical current-pattern. Similarly all horizontal, also all vertical, boundary junctions have identical pattern, as well as all corners. The various patterns for the wave- guide example are given in Fig. 18 (plus current flows into a junction, minus current flows out of a junction).
(2) If the impedances vary from point to point, then for every junction a different current-pattern has to be established.

In analogy to the "unbalanced current patterns" it is possible to set up "unbalanced voltage patterns" by multiplying each iu by its respective Σy.

The Numerical Reduction

It should be noted that every time a junction- voltage is decreased, the junction currents at the neighboring junctions increase, but at the junction itself it decreases. And the decrease is always greater than the increase (4.23 amp. as against 1.25 amp.).

Hence the general procedure is to decrease (or increase) the unbalanced currents at those junctions where the unbalance is the greatest. Two columns should be established at every junction:
(1) One column showing the change in the absolute potential assumed.
(2) The other showing the new unbalanced currents (or voltages, or both).

There will be many more entries in the current column since the current varies also whenever the potential at a neighboring junction varies. Fig. 19. Reduction of unbalanced currents to one-half of their value (from 8 percent to 4 percent).

Fig. 19 shows the steps necessary to reduce in the example of Fig. 15 the unbalanced currents from 8 percent to 4 percent. By changing the voltages only at five junctions, the accuracy of the a.c. Network Analyzer results has been increased 100 percent. Fig. 20. Reduction of unbalanced currents to one-fourth of their value (from 4 percent to 2 percent).

Figure 20 shows the steps necessary to reduce the unbalanved currents from 4 percent to 2 percent. Now 24 changes had to be assumed instead of five to increase the accuracy by another 100 percent. To reduce the unbalanced currents from 2 percent to 1 percent probably the same or greater number of changes would be necessary.

It is emphasized that a practiced person could probably have made the same reduction by less than 24 steps. Also, the network is symmetrical, a fact which was not considered in the reduction.

No hard and fast rules can be established for the reduction. While it is possible to assume at a junction a change of potential that reduces its own unbalanced current immediately to zero, it will not remain so as soon as the potential at a neighboring junction is assumed to vary. Often

it is advantageous to reduce an unbalanced current not only to zero but to a negative value.

Group Relaxation

There is no limit of course to the amount and type of labor-saving devices, that can be introduced. One of them will now be considered.

Instead of assuming a change of voltage at one junction, a change of voltage at several junctions may be assumed simultaneously. This requires settig up "current patterns" for several junctions. An example, for three border junctions is shown in Fig. 21. FIG. 21. Current pattern for three border junction.

Networks with Complex Impedances

If both resistances and inductances occur in a network, then the unbalanced junction-currents are complex numbers i1+ji2, instead of real numbers i1. Similarly the absolute potentials are complex numbers. Then a separate current pattern has to be established for a unit change in the real and the imaginary components of junction-voltage respectively. The latter must so change that both the real and imaginary parts of unbalanced currents approach zero.

Another and perhaps a better method is to replace the equivalent circuit by a more complex circuit in which only real numbers occur. Such a step is always possible since a set of differential equations containing complex coefficients may always be replaced by a set containing no j by simply doubling the number of equations and the number of dependent variables, that is, by introducing in-phase and out-of-phase components of variables.

The "Diffusion" Method

While the relaxation and the weighted average methods change potentials at one junction at a time, the method of unbalanced currents and voltages changes the potentials at all junctions simultaneously. The first two methods are found to be effective at the beginning of the reduction, especially in the elimination of "bumps" in otherwise smooth curves; afterwards the latter method appears to be faster. When the unbalanced currents are small compared with the circuit currents but still further reduction is desired, all three methods become very slow. The following method offers a further systematization of the third method.

A partial diferential equation, not containing time, usually is considered to represent summation of currents at a junction. An unbalanced current iu has no counterpart in the equation. Let it be assumed that iu corresponds to an additional term of the form A∂φ / ∂t in the equation, adding thereby an additional independent variable t to the equation and an additional dimension to the network, such as shown in Fig. 4, for the wave equation.

From physical consideration of diffusion problems, it appears that if the initial conditions are well selected, the rate of diffusion decreases as time increases, and the currents representing A∂φ / ∂t must become in general smaller. At t = oo they all must become zero, allowing the network to represent the original equation without the time-term. By experience it has been found that at t=0, half of the unbalanced current should flow up, half down (Fig. 22 representing the one-dimensional wave equation). The coefficient A should be so selected that at first the potentials on the second layer should differ by only a small amount (say one percent or less) from the starting potentials. As time goes on, the value of A may be increased to speed up the rate of diffusion. The values of A should be so selected that the bigger currents between the layers should decrease uniformly along smooth-curves instead of oscillate around them. (Some of the smaller currents will increase.) Of course A may be an arbitrary function not only of time, but also of space, allowing thereby speedier diffusion at certain points.

In actual calculations, especially when the layers are two-dimensional, it is sufficient to have the drawing of one layer only and write the subsequent values of potentials in colums, as in the relaxation method.

PART III. CHARACTERISTIC-VALUE PROBLEMS

Oscillatory Circuits

In characteristic-value problems not only the mode of vibration (the junction-potential

distribution) is unknown, but also the frequency ω corresponding to that mode. If, however, a potential distribution is assumed, the corresponding ω may be found by the following property of the networks, a consequence of Rayleigh's Principle.

"When the network potentials correspond to a characteristic function, the algebraic sum of the positive and negative powers in the inductors and capacitors is zero."

That is, the power ΣI2XL = ΣE2YL = ΣEIL in the inductors is equal to the power ΣI2XC in the capacitors. Or the power in the variable units, that are functions of ω, is equal to the power in the constant units. This zero power in the network is utilized in finding characteristic values with the aid of a Network Analyzer. The admittances of the coils that are functions of ω are varied with w until a generator, connected in shunt with any of the coils, draws no current from the line. At such values of admittances, the network is self-supporting and the stored electrical energy oscillates between' the capacitors and the inductors.

Calculttion of Characteristic Values

Hence, for an assumed potential distribution e, the corresponding ω is found by the following steps:

1. Calculate the power ΣI2X in all the coils that are not functions of ω (positive power exists in inductors and negative in capacitors)., This power may also be found by calculating the power flowing in the coils that are functions of ω. If the latter are ground coils, then if e is the junction potential and i the current through the ground coils, power = Σei.
2. If the admittance of each of the remaining coils is the same and is equal to ω2C, then the power in these coils is e2y = Σe2ω2C = ω2CΣe2. Hence By Rayleigh's principle it is well known that if the assumed mode of vibration (potentials) is

only a rough approximation, the resultant characteristic value calculated is a better than rough approximation. By knowing the correct ω for an assumed potential distribution, the correct unbalanced currents iu and eu may now be calculated. The latter in turn may be reduced by any of the four methods shown for the solution of boundary- value problems. However, in the present problems a new and more correct ω has to be calculated occasionally as the reduction proceeds.

In the presence of coils that are functions of ω and whose admittances are usually equal, it is not necessary to introduce unbalanced currents. Instead the unbalance is redistributed by attributing different admittances to these variable coils. Hence when the correct ω has been calculated for a given potential distribution, it is assumed that with each variable coil an unbalanced admittance yu is associated. The purpose of the reduction is then to reduce the values of these unbalanced admittances.

Expressed in another way, the purpose of the reduction is to make the admittances of all variable coils the same. The calculated ω only gives a goal to aim at at the beginning. As the reduction proceeds, the common value of the admittances continuously shifts.

The use of the "unbalanced current patterns" facilitates the calculation of the unbalanced admittances at the neighboring points, when the potential at a point is changed.

Linear Harmonic Oscillator

It will be assumed that an approximate value of a characteristic function of the linear harmonic oscillator is known, Fig. 23. (Actually it has been measured on a Network Analyzer, (9) with the aid of the network of Fig. 24.) The problem is to calculate the corresponding characteristic value. Theoretically that is known to be yE=0.0880. zero generator current at yE=0.0921, a 4 percent discrepancy.

The admittance of the horizontal inductors was Yh=1.1 and those of the vertical inductors yv=O.00704x2 where x varies from 1/2 to 12 1/2 in steps of 1 as given in Table I for half the coils. The measured potentials from ground to junction are given in Table II. In the Schrödinger equation the admittance of the capacitors is proportional to ω. The energy in the horizontal coils is Σ1.1∆e2 = 2851, in the vertical coils it is fey, Σe2yv = 2890. Hence This calculated energy level differs from the theoretical value of 0.0880 by only 3 parts in 10,000. It should also be noted that the true (and lowest) energy level 0.0880 is always smaller than the approximate levels 0.0883 or 0.0921, in accordance with Rayleigh's principle, or its quantum mechanical analogue. Here ψ is the approximate wave function e. Also H∆xψ is the current i in E∆x the admittance yE.

BIBLIOGRAPHY

(1) H. W. Emmons, "Numerical methods of solving partial differential equations," Quart. App. Math. (Oct. 1944).

(2) A. Vazsonyi, "Numerical method in the theory of vibrating bodies", J. App. Phys. 15, 598 (1944)

(3) G. Kron, "Equivalent circuits of the elastic field," J. App. Mech. 11, 149-161 (1944).

(4) J.B.Scarborough, "Numerical Mathematical Analysis (Oxford University Press, 1930).

R. E. Doherty and E. Keller, Mathematics of Modern Engineering I (John Wiley & Sons, Inc, New York, 1936), p. 167.

(6) J. H. Bartlett, "The shaping of pole faces of the betatron," Phys. Rev. 63, 185 (1943).

(7) G. Kron, "Equivalent circuits of compressible and incompressible flow fields," scheduled to appear in J. Aer. Sci.

(8) G. K. Carter; "Numerical and network ~.nalyzer solutions of the equivalent circuits for the elastic field;" J. App. Mech. 11,162167 (1944).

(9) G. K. Carter. and G. Kron, "A.c. network analizer study of the Schrödinger equation," Phys. Rev. 67, 44 (1945).

(10) G. Kron, "Electric circuit models of the Schrödinger equation," Phys. Rev. 67, 39 (1945).

(11) G. Kron, "Equivalent circuit of the field equations of Maxwell," Proc. I. R. E., pp. 289 - 299 (May, 1944).

(12) R. V. Southwell, Relaxation Methods in Engineering Science (Oxford University Press, 1940).

(13) A. F. Prebus, I. Zlotowsky and G. Kron, "The application of network analysis to some electron optical problems," scheduled, to appear in Phys. Rev. http://www.quantum-chemistry-history.com