2.12. TRANSFER FUNCTIONS OF COMMON NETWORKS
The control engineer depends heavily on passive networks to modify the transfer function of the feedback control system in order to promote stability, improve closed-loop performance, and minimize the effects of noise and other undesirable signals.
Figure 2.8a represents an electrical network which is used for integration or to provide a phase lag. This circuit obtains its integrating property from the fact that the voltage across the capacitor is proportional to the integral of the current through it. To determine the behavior of this network we must determine the transfer function relating input and output signals. The integrodifferential equations describing the behavior of this network are given by
Figure 2.8 (a) An integrating or phase-lag network. (b) An integrating network with fixed high-frequency attenuation.
These differential equations can be solved for the relation between eo(t) and ei(t) by means of the Laplace transform; eo(0+) is assumed equal to zero. Thus one finds
Eliminating I(s) results in the transfer function
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