The equations of small oscillations superimposed upon a state of steady motion may usually be represented by a set of linear algebraic equations ZαβΔxβ = Δfα. When a complex dynamical system is given, such as a servomechanism, it is possible to establish the resultant equations by first removing all interconnections between the various amplification stages and finding the impedance matrix Zαβ of each isolated stage. The effect of their interconnection is representable by a matrix of transformation Cαα′. The impedance matrix Zαβ′ of the resultant system may be found without any physical analysis by a simple routine matrix manipulation involving Zαβ, Cαα′, its partial derivatives ∂Cαα′/∂xα and the steady-state force vector fα. (In oscillatory problems the law of transformation of the impedance matrix Zαβ is not that of a tensor.) The method is illustrated by setting up the differential equations of various types of speed-, pressure-, and position-governing systems used in central-station work.

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