There is only one forward path, identified in Figure \(\PageIndex{15}\)(b), from source \(a_{1}\) to sink \(b_{4}\), given by, \[\label{eq:27}T_{1}=S_{21}S_{32}S_{43}\], Since all the loops of the graph have nodes that touch \(T_{1}\), then \(\lambda_{1} = 1\) and, \[\label{eq:28}\sum_{k}T_{k}\lambda_{k}=T_{1}\lambda_{1}=S_{21}S_{32}S_{43}\], \[\label{eq:29}\frac{b_{4}}{a_{1}}=T=\frac{S_{21}S_{32}S_{43}}{1− S_{21}S_{32}S_{13} − S_{32}S_{43}S_{24} − S_{21}S_{32}S_{43}S_{14}}\]. Edges and nodes are used in graph theory, a superset of SFG theory. <> Neural Computing: New Challenges and Perspectives for the New Millennium, 2004 12th European Signal Processing Conference, ISCAS 2001. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Finally, the relations for transducer power gain, available power gain, and operating power gain are formulated in this chapter. We want to hear from you. Figure \(\PageIndex{3}\) depicts SFG addition. The relationships between the incident and reflected waves can be readily derived. The circuit equations for Figure \(\PageIndex{12}\)(b) are, \[\label{eq:14}V_{1}=E-I_{S}Z_{S}=V_{1}^{+}+V_{1}^{-}\quad\text{and}\quad I_{S}=\frac{V_{1}^{+}}{Z_{0}}-\frac{V_{1}^{-}}{Z_{0}}\], Now, the traveling wave voltages are related to the a and b root power waves as (see Equation \(\eqref{eq:15}\)), \[\label{eq:15}a_{1}=\frac{V_{1}^{+}}{\sqrt{\Re\{Z_{0}\}}}\quad\text{and}\quad b_{1}=\frac{V_{1}^{-}}{\sqrt{\Re\{Z_{0}\}}}\], Then combining Equations \(\eqref{eq:14}\) and \(\eqref{eq:15}\) yields, \[\begin{align}E&=V_{1}^{+}\frac{Z_{S}+Z_{0}}{Z_{0}}-V_{1}^{-}\frac{Z_{S}-Z_{0}}{Z_{0}}\nonumber \\ \frac{Z_{0}}{\sqrt{\Re\{Z_{0}\}}}\frac{1}{Z_{S}+Z_{0}}E&=\frac{V_{1}^{+}}{\sqrt{\Re\{Z_{0}\}}}\left(\frac{Z_{S}+Z_{0}}{Z_{S}+Z_{0}}\right)-\frac{V_{1}^{-}}{\sqrt{\Re\{Z_{0}\}}}\left(\frac{Z_{S}-Z_{0}}{Z_{S}+Z_{0}}\right)\nonumber \\ \label{eq:16}\frac{Z_{0}}{\sqrt{\Re\{Z_{0}\}}}\frac{1}{Z_{S}+Z_{0}}E&=a_{1}-b_{1}\Gamma_{S}\end{align}\], Comparing Equations \(\eqref{eq:13}\) and \(\eqref{eq:16}\) it is seen that, \[\label{eq:17}b_{s}=\frac{Z_{0}}{\sqrt{\Re\{Z_{0}\}}}\frac{1}{Z_{S}+Z_{0}}E\]. Signal Flow Graph. The app also can be used for demonstrating the concept of loops and cycle finding involved in … The loop attached to node \(b_{3}\) is called a self-loop. Use the link below to share a full-text version of this article with your friends and colleagues. If you do not receive an email within 10 minutes, your email address may not be registered, The sequence of SFG manipulations is shown in Figure \(\PageIndex{10}\) beginning with the SFG in the top left-hand corner. 4 0 obj Mason’s rule is a general procedure for reducing SFGs with multiple loops and is a systematic procedure for the reduction of SFGs to a single branch. Figure \(\PageIndex{10}\): Sequence of graphical manipulations in Example \(\PageIndex{1}\) reducing a terminated two-port to a reflection only. Figures \(\PageIndex{3}\)(a and b) denote, \[\label{eq:5}x_{2}=G_{1}x_{1}+G_{2}x_{1}\]. <>/Font<>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> Provided that a few simple rules were followed, the graphical representation enabled recognition of circuit topologies and the selection of appropriate solution strategies (e.g., applying the voltage divider rule). The SFG manipulations that determine the power delivered to the load are shown in Figure \(\PageIndex{14}\) where, \[\label{eq:18}\alpha=\frac{S_{21}}{1-S_{22}\Gamma_{L}},\quad\beta=\frac{1}{1-\Gamma_{S}(S_{11}+S_{12}\Gamma_{L}\alpha)}\], \[\begin{align}b_{2}&=b_{S}\alpha\beta =b_{S}\frac{S_{21}}{(1-S_{22}\Gamma_{L})}\frac{1}{\{1 − \Gamma_{S} [S_{11} + S_{12}S_{21}\Gamma_{L}/(1 − S_{22}\Gamma_{L})]\}}\nonumber \\ &=b_{S}\frac{S_{21}}{(1-S_{22}\Gamma_{L})}\frac{(1-S_{22}\Gamma_{L})}{[(1 − S_{22}\Gamma_{L}) − S_{11}\Gamma_{S}(1 − S_{22}\Gamma_{L}) + S_{12}S_{21}\Gamma_{S}\Gamma_{L}]}\nonumber \\ &=b_{S}\frac{S_{21}}{1 − S_{22}\Gamma_{L} − S_{11}\Gamma_{S} + S_{11}S_{22}\Gamma_{S}\Gamma_{L} + S_{12}S_{21}\Gamma_{S}\Gamma_{L}}\nonumber \\ \label{eq:19}&=b_{S}\frac{S_{21}}{(1 − S_{11}\Gamma_{S})(1 − S_{22}\Gamma_{L}) − S_{12}S_{21}\Gamma_{S}\Gamma_{L}}\end{align}\], Then the power delivered to the load is (using Equation (2.4.9)), \[\label{eq:20}P_{L}=\frac{1}{2}|b_{2}|^{2}(1-|\Gamma_{L}|^{2})=\frac{\frac{1}{2}|b_{S}S_{21}|^{2}(1-|\Gamma_{L}|^{2})}{|(1-S_{11}\Gamma_{S})(1-S_{22}\Gamma_{L})-S_{12}S_{21}\Gamma_{S}\Gamma_{L}|^{2}}\], The substitution loss, \(L_{S}\), is the ratio of the power delivered to the load by an initial two-port identified by the leading superscript ‘\(i\)’, and the power delivered to the load with a final two port identified by the leading superscript ‘\(f\)’. A signal flow graph is composed of various loops and one or more paths leading from an input to an output. Please mail your requirement at hr@javatpoint.com. All rights reserved. where the first summation in Equation \(\eqref{eq:23}\) is the sum of the loop transfer functions of all the loops in the graph. A path begins at one node and traverses a number of successive edges in the direction of the arrows to arrive at the final node or sink. Two paths, open or closed, are said to be non-touching paths if they have no nodes in common. The two selfloops are added and then the graph reduces to that in Figure \(\PageIndex{9}\)(c). An SFG represents a linear operation on an input. The power of SFG analysis is that an SFG can be formulated by building up the set of equations describing a network by connecting together the SFGs of sections. 211 0 obj <>stream The two-port is shown to the right with the load attached. In the second summation, the products of the transfer functions of all pairs of non-touching loops are added. %PDF-1.5 Thus the SFG model of a source is as shown in Figure \(\PageIndex{12}\)(a) where the model is shown for a real \(Z_{0}\). Only a portion of SFG theory is considered here—the aspects relevant to manipulating scattering parameter descriptions.

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