Models and ApplicationsUniversity of Southern California213–740–8677[USC Fax]Original: December 20021.0.INTRODUCTION2.0.UNIFORM TRANSMISSION LINE2.1.TWO PORT MODEL OF DISTRIBUTED LINE2.1.1.Self-Resonant Frequency2.1.2.Characteristic Impedance2.1.3.Propagation Coefficient3.0.TERMINATED TRANSMISSION LINES3.1.MATCH-TERMINATED LINE3.2.SHORT-CIRCUITED LINE3.3.CAPACITIVE LOAD3.4.LOSSLESS LINES3.4.1.Quarter Wavelength Line3.4.2.Half Wavelength Line4.0.DISTRIBUTED ACTIVE NETWORKS5.0.REFERENCESEE 541, Fall 2006: Course Notes #6 Distributed Circuit Models and Applications Dr. John Choma Professor of Electrical & Systems Architecture Engineering University of Southern California Department of Electrical Engineering-Electrophysics University Park: Mail Code: 0271 Los Angeles, California 90089–0271 213–740–4692 [USC Office] 213–740–8677 [USC Fax] [email protected] ABSTRACT: This report addresses the models, electrical properties, and fundamental applica-tions of distributed transmission lines. The concepts underpinning such metrics as characteristic impedance and propagation coefficient are defined and assiduously scrutinized. The report concluded with a brief introduction of active distributed structures. Original: December 2002Course Notes #6 University of Southern California John Choma 1.0. INTRODUCTION When the signal frequencies imposed on an active or passive network are small so that their associated wavelengths are large, the classic lumped circuit approximation applies to all circuit level analyses conducted on the network. This lumped circuit presumption is comforting from several perspectives. For example, it allows branch elements of the circuit undergoing study to be identified unambiguously, and it permits a straightforward analytical definition of the volt-ampere properties of these elements. As a corollary to this branch identification attribute, the interconnection of these branches pinpoints the junctions and nodes of the circuit, thereby enabling a systematic application of the Kirchhoff laws to determine circuit equilibrium. More-over and virtually by definition, the lumped circuit approximation supports the tacit presumption that the current flowing into one terminal of a two-terminal branch element is, at any instant of time, precisely the same as the current that resultantly flows out of the branch. In effect, the large signal wavelengths associated with modest frequencies permeate the branch uniformly so that said branch can be viewed as a kind of a “giant node,” for which the algebraic sum of cur-rents is necessarily zero. A final and related engineering comfort level is that the voltages with respect to any reference node along any length of interconnect are, at any instant of time, identi-cal and independent of the actual line length. Subcircuit#1+−V2I2I2+−V1I2I2PackagingSubcircuit#2+−V4I4I4+−V3I3I3(a).Subcircuit#1+−V2I2I2+−V1I2I2PackagingSubcircuit#2+−V4I4I4+−V3I3I3(b).Interconnect ModelInterconnect Model Fig. (1). (a). An Abstraction Of Electrical Subcircuits And Packaging Connected Together With Wir-ing Or Metallization For Which Distributed Resistance, Inductance, And Capacitance Cannot Be Ignored. (b). An Abstraction Of Two Port Models Interposed To Account For The Elec-trical Effects Of Distributed Interconnect Impedances. Analytical complexities accrue when, in contrast to the situation overviewed above, the signal frequencies of interest are so large that their wavelengths are comparable to, or even smaller than, the feature sizes of the elements embedded in a circuit. In this case, branch ele-ments and their connective junctions and nodes are blurred because of distributed resistances, August 2006 192 Distributed CircuitsCourse Notes #6 University of Southern California John Choma inductances, and capacitances of requisite electrical wiring or integrated circuit metalization. These parasitic impedances can significantly impact the high frequency equilibrium state of the circuit. The problem can be conceptualized with the aid of Figure (1a), which shows the electri-cal terminals of a packaged network connected to the input terminals of a first subcircuit. The output terminals of this first subcircuit are then connected to drive the input terminals of a second subcircuit, and so on. Lumped circuit wisdom constrains voltage V1 at the package terminals to be identical to voltage V2 at the input of the first subcircuit. Likewise, voltage V3 at the output port of the first subcircuit is the same as the voltage, V4, developed at the input of the second subcircuit in response to the diagrammed interconnection between these two circuits. Similarly, the presumptions, I1 ≡ I2 and I3 ≡ I4, as regards the currents indicated in the diagram are com-monplace in a lumped circuit environment. Because the incident and return lines have unavoid-able distributed series resistance, series inductance, shunt conductance, and shunt capacitance, these voltage and current presumptions are invalid at very high frequencies. An engineering explanation and an analytical account of these voltage and current variances hinges on the devel-opment of meaningful interconnect two port models inserted in appropriate signal flow paths, as is suggested in Figure (1b). These models, and the analyses that surround them, comprise the major thrust of this report. Equally important is the attention given herewith to gainfully exploiting distributed circuits as special purpose filters in very high frequency analog signal processors. 2.0. UNIFORM TRANSMISSION LINE Figure (2a) offers a two port transmission line abstraction of a pair of electrical lines of length L. Figure (2b) is the corresponding schematic symbol of this interconnect system, where Zo, represents the characteristic impedance of the line. The concept underlying characteristic impedances is addressed shortly. In Figure (2a), the lines are drawn on a Cartesian scale in which x = 0 designates the input port of the transmission medium, while x = L corresponds to the output port. Voltages V1 and V2 respectively symbolize the input and output port voltages, while currents I1 and I2 are the corresponding port currents. The propriety of modeling the subject transmission system as a lumped equivalent cir-cuit rests exclusively on a comparison of the line length to the wavelength of the signals trans-mitted by the line between its input and output ports. To