Lecture 1

•Course Overview–System modeling, analysis and design

•Basic Circuit Parameters•Passive Sign Convention•Related educational module:

– Section 1.1

Pre-requisite and Co-requisite requirements• Pre-requisites (recommended)

• Basic exposure to electricity and magnetism• Two semesters of Calculus

• Co-requisites (recommended)• Differential equations

• Pre- and Co-requisite requirements are rather weak• Superficial introductions to necessary topics provided at

the appropriate points during this course

Course Goals• Introduction to modeling, analysis and design of

electrical circuits• We will often use a systems-level approach:

= f{u(t)}

What are modeling, analysis and design?

• We model the system by determining the mathematical relationship between the input and the output

• System analysis often refers to determining the output from a system, for some given input

• System design involves creating a system to provide some desired output

Modeling, analysis, and design – overview

Modeling

Analysis

Design

Implement

Test results from designed system

may result in model modifications and

re-design

Comparison of model results and test data may result in model

modifications

General Modeling Approaches

Modeling Approach

Physical Criteria Types of Governing Equations

Lumped Parameters

Parameters change slowly relative to component response time

Ordinary differential equations

Modeling Approach

Physical Criteria Types of Governing Equations

Lumped Parameters

Parameters change slowly relative to component response time

Ordinary differential equations

Distributed Parameters

Parameters change rapidly relative to component response time

Partial differential equations

Modeling Approach

Physical Criteria Types of Governing Equations

Lumped Parameters

Parameters change slowly relative to component response time

Ordinary differential equations

Distributed Parameters

Parameters change rapidly relative to component response time

Partial differential equations

Linear Relationships between dependent variables are linear

Linear differential equations

Modeling Approach

Physical Criteria Types of Governing Equations

Lumped Parameters

Parameters change slowly relative to component response time

Ordinary differential equations

Distributed Parameters

Parameters change rapidly relative to component response time

Partial differential equations

Linear Relationships between dependent variables are linear

Linear differential equations

Nonlinear Relationships between dependent variables are non-linear

Nonlinear differential equations

Modeling Approach

Physical Criteria Types of Governing Equations

Lumped Parameters

Parameters change slowly relative to component response time

Ordinary differential equations

Distributed Parameters

Parameters change rapidly relative to component response time

Partial differential equations

Linear Relationships between dependent variables are linear

Linear differential equations

Nonlinear Relationships between dependent variables are non-linear

Nonlinear differential equations

Time-varying System physical parameters change with time

Differential equations whose coefficients vary with time

Circuits I modeling approach

• We will restrict our attention to lumped parameter models of linear, time-invariant systems• Governing equations will be linear, constant-coefficient,

ordinary differential equations

• Slinky demo– Linear– Nonlinear– Lumped– Distributed

Basic Circuit Parameters

• Charge (q) is the basic quantity in circuit analysis• Units are Coulombs (C) 1 Coulomb -6.241018

electrons

• Current (i) is the rate of change of charge with time:

• Units are Amperes (A) dt

dqi

A ,mperesA Second

Coulombs

Basic Circuit Parameters – continued

• Voltage (v) is the change in energy of a unit charge at two different points:

• Units are Volts (V) V ,Volts Coulomb

Joules

dq

dWv

Basic Circuit Parameters – continued

• Power (P) is the time rate of change of energy:

• Units are Watts (W)

ivdt

dq

dq

dW

dt

dWP

Passive Circuit Elements

• For a passive circuit element, the total energy delivered to the circuit element by the rest of the circuit is non-negative• The element can store energy, but it cannot create

energy

• Active circuit elements can supply energy to the circuit from external sources

Passive Sign Convention• We will assume the sign

of the current relative tovoltage for passive circuitelements

• Positive current enters thenode at the higher voltage

• Sign must be known for active circuit elements

Passive Sign Convention – continued • You can assume (arbitrarily) either the voltage

polarity or the current direction• This assumption dictates the assumed direction of the

other parameter

• These assumptions provide reference voltage polarities and current directions

• Subsequent analysis is performed based on this assumption; a negative result simply means that the assumed voltage polarity or current direction was incorrect

Passive Sign Convention – Example 1

• Provide the appropriate sign convention for the missing parameter on the passive elements represented by grey boxes.

Passive Sign Conventions – Hints• It is generally counter-productive to attempt to

determine the “correct” voltage polarities and current directions before analyzing the circuit

• Just arbitrarily choose either the assumed voltage polarity or current direction for each passive circuit element• This choice dictates the sign of the other parameter

• Perform analysis using assumed signs• Negative signs mean that the assumption was incorrect

Passive Sign Convention – Example 2

• Assign reference voltage and current directions for the passive elements represented by shaded boxes in the circuit below:

Passive Sign Convention – Example 3

• Assign reference voltage and current directions for the passive elements represented by shaded boxes in the circuit below:

Passive Sign Convention – Example 4

• For the circuit below, the sign convention shown is chosen

• After analyzing the circuit, it is determined that I1 = -3mA, I2 = 3mA, V1 = -1.5V, and V2 = 2.5V. Re-draw the circuit showing the actual voltages and currents and their directions