Function Generator

a device whose output signal y depends on one or more input signals x_i (where i = 1, 2, …) in accordance with an operation algorithm. Function generators are classified into types for one, two, or more variables, depending on the number of input variables. The functional dependence of the output signals—of the single output signal if there is one input signal or of each output signal when there are several input signals—on the input signals can be given in the form of tables, graphs, or analytic expressions. The dynamic response of the function generator y(x₁, x₂, …, x_n, t) is described by a differential equation whose right-hand side includes the input signal and its derivatives with respect to time (in the general case) and whose left-hand side includes the output signal and its derivatives with respect to time (in the general case). For engineering calculations the dynamic response is most conveniently characterized by transfer functions for the corresponding channels (input signals).

Based on the type of operation algorithm within the prospective area of application, function generators are divided into linear types, in which the functional dependence is described with sufficient accuracy by a straight line, and nonlinear types, in which the functional dependence is curvilinear; piecewise linear function generators are classified with the nonlinear types. Mechanical, electric, pneumatic, hydraulic, and mixed (electromechanical, electrohydraulic, and pneumoelectric) function generators are distinguished according to the physical nature of the input and output signals. Function generators are classed as analog, digital, or hybrid, depending on the manner of representation of the initial values. Hybrid units use digital and analog representation at the same time. In this case the input signal is usually divided into two parts: one is represented in analog form, and the other in digital form. Such function generators therefore contain digital-to-analog and analog-to-digital converters.

The most common and important function generators are for one input variable. Depending on the operation algorithm, they are classed as dynamic or shaping function generators. In the dynamic type the input signal is changed in time; for example integration, differentiation, or time delay may be carried out. In the shaping type the input signal is changed in scale or in shape of action. Proportional function generators are an example in which the input signal is changed in scale. Change in shape of action may involve the conversion of a continuous into a discrete signal—as in pulse, modulation, and coding function generators—or, vice versa, of a discrete signal into a continuous one—as in discrete analog function generators.

Both simple and complex conversions can be performed in function generators. In simple conversions the output quantity is physically inseparable from the input quantity. An example is the conversion of temperature into thermoelectromotive force or into effective resistance. Complex conversions involve at least two simple conversions. For example, the conversion of effective resistance into the attractive force of an electromagnet requires two simple conversions: effective resistance into magnetic flux and magnetic flux into attractive force of the core.

An important characteristic of function generators is conversion error, which may be random or systematic. Random errors usually have a normal distribution law. For several successive conversions the total error is , where Δ_i is the error of the individual conversions. Systematic conversion errors are added algebraically with signs being taken into account. An equally important characteristic is the sensitivity of the function generator—that is, the ratio of a very small change in the output signal to the also very small variation in the input signal that caused the output change. Feedback is introduced to vary the sensitivity, and a distinction is accordingly made between function generators with open and with closed control loops.

Function generators are used in automatic control and regulation systems, analog and hybrid computers, coding and decoding devices, remote control systems, and measuring devices.

In analog computer technology. A function generator in an analog computer is a device whose output involves a quantity that is nonlinearly dependent on the input signal. According to the nature of the dependence, there are distinguished types for representing discontinuous functions, discontinuous nonsinglevalued functions, and continuous functions of one or more independent variables. Function generators are classed as general-purpose or specialized generators, depending on the possibility of changing from one nonlinear dependence to another. Units with a linear functional dependence constitute the separate class of linear decision elements.

In single-variable function generators the prescribed nonlinear dependence is ordinarily attained through its approximation on individual sections of the change of the input signal by certain polynomials of the same degree (Newton or Lagrange polynomials). The degree of the interpolating polynomial distinguishes piecewise linear, piecewise constant, and piecewise quadratic approximations.

Three methods are used to construct multi-variable function generators: (1) the creation of a physical model of a two-dimensional surface (conoids), (2) the replacement of the complex multidimensional surface by a certain number of elementary surfaces of the same dimensionality, and (3) the exact or approximate representation of the multi-variable functions prescribed for representation by means of functions of one variable and arithmetic operations such as addition and multiplication. The first two methods require the construction of special devices; the third involves a synthesis of standard (for analog computers) linear and nonlinear decision elements. Function generators of two variables that reproduce the operations of multiplication and division are regarded as a separate class.

The error of most function generators lies between a few hundredths of a percent and a few percent.