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ModelingLinear Feedback Networks with Multiple Dependent Sources
Eugene Paperno
Abstract—Linear feedback networks with multiple dependent sources are described analytically in terms of return ratios. In addition to the return ratios,representing the contribution of each dependent source to its control terminals, cross return ratios are introduced,representing the contributions of each dependent source to the control terminals of other dependent sources. Analytical modelsand functional diagrams are developed for the closed-loop gain of generic feedback networks with one, two, and three dependent sources. The obtained analytical models and functional diagrams can easily be extended tofeedback networks with a greater number of dependent sources.
Index Terms—Linear feedback networks, multiple dependent sources, network theory, return ratio, cross return ratio.
I.INTRODUCTION
M
ODELINGfeedback networks is very important for revealing the effect of feedback on the network transfer function. Analytical and functional models of a feedback network not only provide better intuitive insight into its operation but also allow applying control theory tools to better analyze its behavior.
Although, any linear feedback network can easily be simulated, the obtained solution provides no insight into the network functional structure.
Unfortunately, existing literature[1]-[19] suggests exact analytical or functional modeling only for linear feedback networks with a single dependent source. In some cases, feedback networks with two or more dependent sources can be approximated by models based on a single dependent source. It is not always clear, however, how accurate this approximation is.
The aim of the present work is to develop exact analytical and functional models for linear feedback networks with multiple dependent sources. The proposed approach is based on the 'return ratio' concept [18], includes no assumptions or approximations and can be applicable to linear feedback networks with any number of dependent sources and any number of feedback loops.
Return ratios, Ti, describe the contribution of dependent sources to their own control terminals. To consider contributions of each dependent source to the control terminals of other dependent sources in a network, we introduce cross return ratios, Tij, i≠j. This allows us to extend the canonical feedback model to linear networks with multiple dependent sources.
Moreover, we show that the return ratios, Ti, and cross return ratios, Tij, can be combined into a generalized return ratio, T. Obtaining T is very important for analyzing the network stability. To the best of our knowledge, the present work shows for the first time how to obtain T for linear networks with multiple dependent sources.
Below, we first revisit the modeling of linear feedback networks with a single dependent source, and then develop models for linear feedback networks with two and three dependent sources. Following the proposedapproach, the developed models can easily be extended to feedback networks with multiple dependent sources.
II. Feedback Networks with a Single Dependent Source
Let us consider a generic linear feedback network with a single dependent source shown in Fig. 1 (a).
To find the closed-loop gain of the network, ACL≡so/ss, we first define the following open-lop partial gains:
input transmission,(1)
feedforward transmission , (2)
direct transmission ,(3)
open-loop gain ,(4)
feedback transmission,(5)
return ratio .(6)
We thenfind by superposition the control and output signals:
,(7)
.(8)
From (7) and (8),
.(9)
Based on Fig. 1 and (9), a functional model of the generic feedback network can be developed(see Fig. 2). It is important to note that the functional block B, combining the feedback and feedforward transmissions, represents a bidirectional feedback network.
III.Feedback Networks with Multiple Dependent Sources
A.Feedback Networks with Two Dependent Sources
A generic linear feedback network with two dependent sources is shown in Fig. 3. The control and output signals of this networkcan be found as follows:
,(10)
,(11)
,(12)
where
input transmission,(13)
input transmission,(14)
cross return ratio ,(15)
Fig. 1. Finding the partial open-lop gains of a generic feedback network with a single dependent source. (a) Original network. (b) The network, where the independent source is the only active one. (c) The network, where the equivalent independent sourceas, replacing the dependent source, is the only active one.
Fig. 2. Functional model of afeedback network with a single dependent source. The block B represents a bidirectional feedback network.
cross return ratio .(16)
Equations (10) and (11) can be solved for the independent source value:
,(17)
.(18)
Considering (13)(18), the closed-loop gain can be found as follows:
,(19)
where
,(20)
,(21)
.(22)
Equations (10)(12) and (19) can be represented as the feedback functional model shown in Fig. 4.
B.Feedback Networks with Three Dependent Sources
The above analysis of linear feedback networks with two dependent sources can easily be extended to feedback networks with multiple dependent sources. For example, for a generic feedback network with three dependent sources, the control and output signals can be found as follows:
,(23)
,(24)
,(25)
.(26)
From (23)(26), the closed loop gain can be found as follows:
,(27)
Fig. 3. Definingthe partial open-lop gains for a generic feedback network with two dependent sources.
where
,(28)
,(29)
,(30)
.(31)
Equations (27)(31) can be representedas a feedback functional model shown in Fig. 5.
III.Conclusion
Analytical and functional models for linear feedback networks with one, two, and three dependent sources are developed. The proposed approach includes no approximations and can be applicable to linear feedback networks with any number of dependent sources and any number of feedback loops.
Acknowledgment
The author wishes to express his deepest gratitude to Prof. Shmuel (Sam) Ben-Yaakov for very fruitful and inspiring discussions.
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Fig. 4. Functional model of a feedback network with two dependent sources. Note that AOL112=T12, and AOL221=T21.
Fig. 5. Functional model of a feedback network with three dependent sources.
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