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Synthesis of LCM and RC One-Port Networks Wai-Kai Chen
| Content Provider | Scilit |
|---|---|
| Copyright Year | 2002 |
| Description | Therefore, the driving-point immittance of a lossless network is always the quotient of even to odd or odd to even polynomials. Its zeros and poles must occur in quadrantal symmetry, being symmetric with respect to both axes. As a result, they are simple and purely imaginary from stability considerations, orZ(s) can be explicitly written as(75.6)whereω≥0. This equation can be expanded in partial fraction as(75.7)whereω=ω, and the residuesH,KandKare all real and positive. Substitutings=jωand writingZ(jω) = ReZ(jω) +jImZ(jω) results in an odd function known as thereactance functionX(ω):(75.8)Taking the derivatives on both sides yields(75.9)Since every factor in this equation is positive for all positive and negative values ofω, we conclude that(75.10)It states that the slope of the reactance function versus frequency curve is always positive, as depicted in Fig. 75.1. Consequently, the poles and zeros ofZ(s) alternate along thejω-axis. This is known as theseparation propertyfor reactance function credited to Foster [9]. Because of this, the pole and zero frequencies of (75.6) are related by(75.11)We now consider the realization ofZ(s). If each term on the right-hand side of (75.7) can be identified as the input impedance of the LC one-port, the series connection of these one-ports would yield the desired realization. The first term is the impedance of an inductor of inductanceH, and the second term corresponds to a capacitor of capacitance 1/K. Each of the remaining term can be realized as a parallel combination of an inductor of inductance 2K/ωand a capacitor of capacitance 1/2K. The resulting realization is shown in Fig. 75.2 known as thefirst Foster canonical form. Likewise, if we consider the admittance functionY(s) = 1/Z(s) and expanded it in partial fraction, we obtain(75.12)Z s m nm nnm ( ) + ==Z s H s s ss s s( ) = +( ) +( ) +( ) +( ) +( )ω ω ωω ω... Book Name: The Circuits and Filters Handbook |
| Related Links | https://api.taylorfrancis.com/content/chapters/edit/download?identifierName=doi&identifierValue=10.1201/9781420041408-78&type=chapterpdf |
| Ending Page | 2382 |
| Page Count | 12 |
| Starting Page | 2371 |
| DOI | 10.1201/9781420041408-78 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2002-12-23 |
| Access Restriction | Open |
| Subject Keyword | Book Name: The Circuits and Filters Handbook Hardware and Architecturee Frequencies Function Inductor Impedance Port Expanded It in Partial Expanded in Partial Fraction |
| Content Type | Text |
| Resource Type | Chapter |
