A Frequency Transformation for Commensurate Transmission-Line Networks

Frequency transformations are commonly used in lumped-element network theory to convert a given filter network into a related filter network. For example, an often used frequency transformation is s\´/spl rarr/ As , where the symbol /spl rarr/ stands for "is replaced by," A is a constant, the primed variable is that of the original network, and the unprimed variable is that of the transformed network. Transformation is used to scale the bandwidth of the existing network to another preferred value. Other commonly used frequency transformations in lumped filter theory are s\´/spl rarr/ As , (Iowpass to highpass transformation) s\´ /spl rarr//spl omega/ (s//spl omega//sub o/ )+( /spl omega//sub o/ /s )(Iowpass to bandpass transformatian) s\´/spl rarr/ /spl lineover/ /spl omega/ (s//spl omega//sub o/ )+( /spl omega//sub o/ /s) (Iowpass to bandstop transformation) It is emphasized that in all cases the usefulness af these transformations lies in the fact that their effects on the responses of the network are easily related to changes in the element values of the network. Because such frequency transformations are available, a given Iowpass filter may function as a prototype for a number of different types of filters, obviating the compilation of a multitude of designs for Iowpass, highpass, bandpass, and bandstop filters.