Frequency Scaling of a Linear Time-Invariant Network by a Time-varying Function

Author

Belal, A. ; Shenoi, B.A.

Volume

Issue

fYear

1982

fDate

1/1/1982 12:00:00 AM

Firstpage

Lastpage

Abstract

Given a linear time-invariant $RLC$ network, with input $x(t)$ and output $y(t)$ , then the well-known frequency scaling theorem states that multiplication of all $L$ \´s and $C$ \´s by some constant $a^{-1}$ is equivalent to changing the input to $ax(at)$ and the output to $ay(at)$ . We show here that when the multiplier is a time-varying function $a^{-1}(t)$ , the equivalent result is to change the input from $x(t)$ to $a(\\gamma ^{-1}(t))x(\\gamma ^{-1}(t))$ and the output from $y(t)$ to $a(\\gamma ^{-1}(t))y(\\gamma ^{-1}(t))$ where $\\gamma (t)= \\int_{0}^{t}frac{d \\tau }{a(t)}$ . Some illustrative examples are footnote[1]{given}. (1)In this correspondence $a^{-1}$ means $frac{1}{a} a^{-1}(t)= 1/a(t)$ ; but $u^{-1}(t), \\gamma ^{-1}(t),$ are inverse functions.

Keywords

Linear circuits, time-invariant; RLC circuits; Capacitance; Capacitors; Differential equations; Frequency; Impedance; Inductors; Kernel; Laplace equations; RLC circuits; Transforms;

fLanguage

English

Journal_Title

Circuits and Systems, IEEE Transactions on

Publisher

ieee

ISSN

0098-4094

Type

jour

DOI

10.1109/TCS.1982.1085072

Filename

1085072

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=1186245