Application of the Braun Tube for the Solution of Differential Equations by Electrical Means

This is an English translation of the original German article by Hans Kleinwächter, published in Archiv für Elektrotechnik, Vol. XXXIII, No. 2 (1939).

Application of the Braun Tube for the Solution of Differential Equations by Electrical Means

By Hans Kleinwächter, Reichenberg/Sudetenland. (Report from the Gewerbefördergungsinstitut in Reichenberg.) (Received 2 July 1938.) DK 621.385.832 : 518.5

Abstract. An attempt is made to exploit the properties of an inductance coil — in which the applied voltage is the differential quotient of the current with respect to time — for the solution of differential equations. Each differential equation corresponds to a specific circuit configuration. Changes in the magnitude of individual circuit elements correspond to changes in the parameters of the associated differential equation.

A large number of extremely simply constructed differential equations have no elementary functions as solutions. One must therefore resort either to a computational approximation method or a graphical method. The former often presents convergence difficulties, and furthermore the approach taken depends on the magnitude of the parameters of the differential equation. The graphical method is limited to first- and second-order differential equations with the aid of curvature circles. In both cases one obtains a single solution that depends on the respective initial conditions.

A method is therefore described here that makes it possible, through simple adjustment of an apparatus, to display on the luminescent screen of a Braun tube the solution of certain differential equations for arbitrary initial conditions and parameters instantaneously.

This method consists in the application of an electrical circuit whose individual components interact electrically at every instant in the same way that the relationship between the dependent variable and the second, independent variable (time) and the differential quotients — as prescribed by the given differential equation — requires. The defining component of this device is a specially constructed Braun tube I with one-dimensional deflection and a resistance rod that intercepts the electron beam directed perpendicularly to it. One end of this rod is grounded via a bias voltage, so that the other end, owing to the small capacitance that the rod possesses, charges instantaneously to a potential v₁ corresponding to the ohmic voltage drop and thus to the magnitude y.

y represents the sought time-dependent variable (Figure 1).

To represent the quantities y′, y″, etc. electrically as well, v₁ is applied to the grid of a screen-grid tube II operating in the linear portion of its characteristic curve, in whose anode circuit an inductance L is located. The self-induction voltage v₂ that develops across L is proportional to the time derivative of the anode current and thus to y′. For the representation of higher differential quotients, v₂ is in turn applied to the grid of a further tube, and so on.

For the solution of a linear first-order differential equation with constant coefficients of the form:

y′ + ay = b

the circuit diagram of Figure 1 results. The voltages v₁ and v₂ are applied to the deflection plates of the main tube I. Thereby the instantaneous magnitude of y is linked to the value of y′ by the following calculation:

v₁ = voltage at the end of the resistance rod
y = deflection of the electron beam from the null position
y₀ = length of the rod to the null point
i = beam current
ϱ = resistance of the rod per centimeter
v₁₀ = bias of the rod
iₐ = anode current of tube II
S = transconductance of tube II
iₐ₀ = anode current at grid voltage 0
v₂ = voltage across inductance L
A = deflection of the electron beam per volt of deflection voltage
v₂₀ = bias of the inductance coil L

The parameter relations yield:

v₁ = v₁₀ − (y + y₀)ϱi
iₐ = Sv₁ + iₐ₀
v₂ = −L·(diₐ/dt) → v₂ = y′·(term)
y = A(v₁ + v₂)

Substituting gives the differential equation:

y′ + y·(ALS̃ϱi)⁻¹ = (constant)

or in canonical form:

y′ + ay = b, solution: y = (b/a) + C·e^(−t/τ)

The magnitudes of the parameters a and b are determined from the electrical data of the devices according to the calculation above; the parameters can thus be varied. The initial condition is determined by the originally present deflection y₀ of the cathode ray, which is produced by an auxiliary voltage that is switched off at the moment t = 0.

That this deflection y₀ equals the initial value of the subsequent continuous beam motion can be seen from the fact that the change of y in the first time element must not be finite, as this would correspond to an infinitely large y′ and thus an infinitely large v₂.

For differential equations of nth order, the same reasoning yields the initial condition:

y = y₀, y₀′ = 0, y₀″ = 0, … y₀^(n−1) = 0

To solve differential equations in which the product of two variables appears, a device is needed that performs this multiplication. The Braun tube described above proves suitable if one variable (t, y, y′, y″,…, y^(n)) is applied to the Wehnelt cylinder and the other to the deflection system. The voltage at the end of the collector rod iR is then proportional to the product of the two control voltages (Figure 2). By repeating this circuit, higher powers can also be achieved.

Example:

ay″ + by′x + cy′ + dy = e (Figure 2)

The circuit relations for y, v₁, v₂, v₃, v₄ and the parameter correspondences to a, b, c, d, e are derived in terms of the device constants (transconductances S, inductances L, beam currents i, deflection sensitivities A, rod resistances ϱ, bias voltages v₀).

It is also mentioned in conclusion that differential equations with terms of the form φ(t, y, y′,…y^(n))^(1/2) or more complex functional forms can also be represented, since the deflection of the cathode ray is inversely proportional to the square root of the anode voltage.

In all cases, the main tube I is connected in parallel with an imaging tube that takes over the control voltages from I; on its luminescent screen the imaging tube displays the solution y as a function of time.

Summary.

It is shown that the method makes it possible to display individual solutions of differential equations of higher order and degree on the screen of a Braun tube. Only purely electrical apparatus was used, which represents a self-imposed restriction. As examples, the circuit diagrams for the differential equations ay′ + by = c and ay″ + by′x + cy′ + dy = e are shown, and approaches for more complicated cases are indicated.

[Translation covers the first 3 pages (the complete article); no further content remains.]