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A design procedure is described whereby multiple resistance networks can be synthesized from a single layer, monolithic, resistive structure. The method is illustrated by detailing the design of attenuators operating on an image-impedance basis. The single sheet geometry used is a rectangle with conducting tabs placed on its boundary. Conformal transformation techniques are applied to facilitate the computation of the required open-circuit resistance values which characterize a given rectangular geometry. Two SchwarzChristoffel transformations and a bilinear transformation permit the rapid design of a monolithic resistive network meeting prescribed specifications. The mapping of critical points is diagrammed to help the reader clearly visualize the transformations involved. Specific analytical examples are carried out, and curves are presented showing the dependence of network characteristics upon the parameters of sheet resistivity, geometric ratios, and position and size of the conducting tabs. The synthesis of two pads is described to illustrate graphical techniques involved. Both attenuators are designed to operate between 75 Omega characteristic impedances, and to have 2 db of insertion loss. The first pad is realized by holding tab size constant and deriving the necessary rectangular geometry and resistivity. In the second case, the resistivity is held constant while the required geometry is determined. In general, alternative physical embodiments which are electrically equivalent afford a desirable flexibility from the viewpoint of network fabrication. The above synthesis techniques have been substantiated by the construction and measurement of laboratory models. Experimental data are given which represent the results obtained using tantalum film designs. Excellent correlation with theory is noted. Possible advantages which monolithic single-sheet networks offer over the discrete multiple resistor circuits which they replace are reduction of the number of components, better HF performance, ease of manufacture, and higher reliability due to the need for fewer interconnections. The single-sheet resistive components are notable for their compatibility with current thin-film fabrication techniques.