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# Introduction to Time - Varying Electrical Networks

## Overview

The course is intended as a gentle introduction for analog, mixed-signal and RF circuit designers to the area of time-varying circuits and systems. The pre-requisites are minimal. After a refresher and building background in linear-time invariant networks, it introduces linear time-varying (LTV) and linear periodically time-varying (LPTV) circuits. The applications of the theory are illustrated with practical examples.
INTENDED AUDIENCE :
M.Tech/M.S/Ph.D students, advanced undergraduatesPREREQUISITES : Engineering mathematics, a basic course on electric network analysis at the UG level, signals and systems.INDUSTRIES SUPPORT :Companies engaged in analog, mixed-signal and RF design

## Syllabus

### COURSE LAYOUT

Week 1:Motivation for the topics covered in the course, review of linearity and time-variance;Review of electrical network basics, incidence matrix, Tellegen's theorem;Tellegen's theorem (cntd), its use to prove reciprocity in bilateral networks, reciprocity in networks with controlled sourcesWeek 2:Reciprocity in networks with controlled sources (contd), inter-reciprocal networks;Modified Nodal Analysis (MNA) formulation to write network equations;MNA formulation (contd), MNA stamps of circuit elements, Reciprocity and inter-reciprocity revisitedWeek 3:Reciprocity and inter-reciprocity (contd), the adjoint network.;Introduction to noise in electronic circuits;Noise in RLC circuits, Nyquist's theorem, Bode's Noise TheoremWeek 4:Bode's noise theorem (contd), input referred noise sources in networks;Input-referred noise sources (contd) - equivalent noise voltage and current sources;Equivalent noise sources, noise factor
Week 5:Need to study time-varying circuits and systems;Linear time-varying (LTV) system basics;Linear Periodically Time-Varying (LPTV) systems basicsWeek 6:Harmonic transfer functions, the Zadeh expansion;MNA equations in LPTV networks with Harmonic transfer matrices;LPTV circuit example : the sampling mixerWeek 7:Impedance and admittance in LPTV networks, Norton and Thevenin equivalents;The N-path principle;N-path circuits (contd) - the time-interleaved ADCWeek 8:N-path circuits (contd) - the multiphase dc-dc converter, introduction to the N-path filter;N-path filters (contd) - input impedance and gain;N-path filters (contd)
Week 9:Reciprocity and inter-reciprocity in LPTV networks - time-reversal to generate the adjoint;Inter-reciprocity (contd), transfer-function theorem;Inter-reciprocity (contd), the frequency-reversal theoremWeek 10:Inter-reciprocal signal-flow graphs. Example : chopped amplifiers;Chopped-amplifiers with sinusoidal and square-wave modulation;Adjoint networks - the switched-RC kernel example; time-domain implications of adjoint networksWeek 11:Time-domain implications of the adjoint - example of a switched-RC network. Sampled LPTV networks;Equivalent LTI filter of a sampled LPTV system; derivation of the equivalent impulse response, switched-RC network example ;Cont. time delta-sigma as a sampled LPTV systemWeek 12:Response of LPTV systems to modulated inputs; equivalent LTI filter;Introduction to noise in LPTV networks, noise in switched-RLC networks, the Bode noise theorem applied to LPTV networks;Course summary and recap

### Taught by

Prof. Shanthi Pavan

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