Date of Award

Fall 1963

Document Type

Thesis - Restricted

Degree Name

Master of Science (MS)


Electrical Engineering and Computer Science

First Advisor

Horgan, J. D.

Second Advisor

Mertz, Robert L.


Analog computers having a high repetition rate require as auxiliary equipment an analog multiplier which can, without time delay, produce the time-varying product of two rapidly varying input signals. At present the only commercially available and widely used multiplier design is the so-called "quarter-square diode multiplier." The basic component of this multiplier is a squarer whose output voltage is linearly proportional to the square of the input voltage. The squaring circuit consists of a number of parallel branches, each consisting of a diode and a resistor in series. The output-input characteristic of the circuit is linear as long as the same number of branches are conducting. Once another diode opens, more resistance is added in parallel, and the output - input characteristic abruptly changes its slope. The input voltage at which another diode opens is called a breakpoint. The locations of the breakpoints are controlled by the diode bias voltages, and the slope of the output-input characteristic between successive breakpoints is controlled by the resistance in the branches. Thus, within a fairly large range, the breakpoints and the slopes between them can be independently set by adjusting potentiometers and variable resistors. This makes possible a straight-line approximation to the ideal parabolic, or squaring, output-input characteristic. The problem of determining the breakpoints and slopes for an optimum approximation to the ideal parabolic, or squaring, characteristic has been solved analytically by minimizing the integral of error-squared. However, the design based on this solution permits a large rercent error in, the range of small inputs. This thesis presents the least-square solution, further attempts that have been made to improve it by using a weighting, function, and a critique of these efforts. Then a new solution is presented which, by means of a digital computer method of systematic search and successive optimization called dynamic programming, minimizes the square - root of the integral of per-unit error squared. The resulting design greatly reduced the percent error in the lower range of input, without losing the principal advantage of the least - squared- error solution, namely rather small maximum error over the entire range.



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