ECE Department, University of Illinois ECE 552 Numerical Circuit Analysis I. Hajj Spring 2015 Lecture One INTRODUCTION Copyright © I. Hajj 2012 All rights reserved

Introduction System analysis is a basic step in system design Computer-Aided analysis or simulation helps in the design of complex systems before the systems are built or manufactured

Four steps of system simulation 1.System modeling, including component and device modeling 2.System equation formulation 3.Equation solution 4.Display and interpretation of solution

Assumption To start, we consider Electric Circuits that are modeled as interconnections of lumped elements (as opposed to distributed elements): Resistors; capacitors; inductors; independent sources. We will not consider the derivation of device models in this course. We will assume that device models are provided. We will concentrate on equation formulation and equation solution techniques.

Circuit equations include two components: Element characteristics Topological equations or how the elements are interconnected

Types of Equations to be Solved Linear equations Nonlinear algebraic equations Differential-algebraic equations Partial differential equations (distributed elements)

Element Characteristics (Chapter 1) Resistors Characterized by an algebraic relation between voltage v and current i (Note the associated reference directions of v and i)

Linear two-terminal resistor (Ohm's Law) v = ri i = gv r, g are constant => time-invariant r(t), g(t) => time-varying

Independent Sources Current Sources Examples: i = 5 A, i = k sin ωt A

Independent Sources Voltage Sources Examples: v = 5 V, v = B cos(ωt + Φ) V Remark: Independent sources are characterized by an algebraic relationship

Note: We will allow a 2-terminal or one-port resistor to be specified as

Linear Multiterminal Resistors i = [G]v, v = [R]i or If G, R, or H are constant matrices, then resistor is time-invariant; if they are functions of time, then resistor is time-variant.

"Controlled" or "Dependent" sources, e.g., i l =αv 2, v 2 =βv l can be considered as part of a two-port representation Example

OR

General form of linear resistor characteristics, including dependent and independent sources is: [G]v + [R]i = s

Examples: 3-Terminal Resistor 2-Port-Resistor

OR Hybrid Representation i 1 = a 11 v 1 + a 12 i 2 + s 1 v 2 = a 21 v 1 + a 22 i 2 + s 2 Matrix Form

Circuit Diagram 3-Terminal Remark: Terminal equations are sufficient. There is no need to represent multiterminal or multiport as interconnection of 2-Terminal elements (see above) 2-Port

Macromodeling (e.g., Op-Amp) Resistive macromodel Relation between voltages and currents at terminals or ports are derived from the internal equations. Internal voltages and currents of macromodel can be computed later, if desired. This leads the Hierarchical Analysis.

Nonlinear Resistors Two terminals i = g(v) voltage-controlled v v i v f(i,v) = 0 v = r(i) current-controlled v

or f ( i, v ) = 0 Multiterminal Nonlinear Resistor or v = r ( i ) i = g ( v ) i and v are vectors

Linear, time-invariant Time-varying linear capacitor In steady-state sinusoidal analysis I c = ( jωC ) V c Capacitors

Multiterminal Linear Capacitors Nonlinear Capacitors v c = f(q c ), f(v c,q c )=0

I 1 = sC 11 + sC 12 I 2 = sC 21 + sC 22

Symmetric case. However, there is no need to generate an equivalent circuit

Multiterminal Nonlinear q v

Inductors Linear, time-invariant Φ L = Li L In sinusoidal steady-state analysis

Inductors Time-varying linear inductor Nonlinear Inductor i L = f( ϕ L ), f(i L, ϕ L )=0

Multiterminal Inductor

Linear Two-Port Inductor (Transformer)

Symmetric case. However, there is no need to generate an equivalent circuit

Mem-Devices Charge-Controlled Memristor: ϕ M (t) = f M (q M ) i M = dq M /dt, v M = dϕ M /dt Flux-Controlled Memristor: q M (t) = f M (ϕ M ) i M = dq M /dt, v M = dϕ M /dt

Mem Systems – Current-Controlled Memristive System: v M = f 1 (x,i M,t)i M (t) dx/dt= f 2 (x,i M,t) – Voltage-Controlled Memristive System: i M = f 1 (x,v M,t)v M (t) dx/dt= f 2 (x,v M,t)

Memcapacitive Systems Voltage-Controlled Memcapacitive System: q M = f 1 (x,v M,t)v M (t) i M = dq M /dt dx/dt= f 2 (x,v M,t) Charge-Controlled Memcapacitive System: v M = f 1 (x,q M,t)q M (t) i M = dq M /dt dx/dt= f 2 (x,q M,t)

Meminductive Systems Current-Controlled Meminductive System: ϕ M = f 1 (x,i M,t)i M (t) v M = d ϕ M /dt dx/dt= f 2 (x,i M,t) Flux-Controlled Meminductive System: v M = f 1 (x, ϕ M,t) ϕ M (t) i M = dq M /dt dx/dt= f 2 (x, ϕ M,t)

Memdevices Symbols

f(i, v, q, φ, σ, ρ,x, ˙x, t) = 0, f(i, v, q, φ,x, ˙x, t) = 0, General Element