1. Review of Circuit Theory Concepts

ECE 65, Winter 2013, F. Najmabadi

Lecture notes: Section 1

Circuit Theory is an Approximation to Maxwell’s Electromagnetic Equations

F. Najmabadi, ECE 65, Winter2013, Intro (2/15)

A circuit is made of a bunch of “elements” connected with “ideal (i.e., no resistance) wires”.

Circuit Theory is an Approximation to Maxwell’s Electromagnetic Equations by assuming o Speed of light is infinite (or dimension of the circuit is much smaller

than wave-length of voltage/current waveforms).

o Electric and magnetic fields are confined within each element: 1) Internal of an element manifests itself as an iv characteristic eq. 2) Elements communicates with each other only through the wires!

Since the rest of the circuit only sees the iv characteristics of an element, different physical elements with similar iv characteristics are identical!

Linear circuits have many desirable properties

F. Najmabadi, ECE 65, Winter2013, Intro (3/15)

A linear circuit element has a linear iv characteristic equation, Av + B i + C = 0 (either in time or frequency domain)

If all elements in a circuit are linear, the circuit would be linear and has many desirable properties (e.g., proportionality and superposition) which are essential for many functional circuits.

Circuit theory has “symbols” for ideal linear elements: o Five “two-terminal elements”: resistors, capacitors, inductors,

independent voltage and independent current sources

o Four “four-terminal elements”: controlled voltage and current sources.

It is essential to remember that the above ideal elements are NOT representative of physical devices. Rather they are representative of elements with a certain iv characteristic equation.

Practical elements are only approximated by “ideal” circuit theory elements

F. Najmabadi, ECE 65, Winter2013, Intro (4/15)

i

v

i

v

At high enough current, the resistor “burns” up

As the current increases, resistor heats up and its resistance increases

A Lab resistor can be approximated as an ideal circuit theory resistor for a range of current or voltage (identified by its rated maximum power)

Real resistor

“Ideal” circuit theory elements are NOT representatives of physical devices!

F. Najmabadi, ECE 65, Winter2013, Intro (5/15)

Is a symbol for

Is NOT representative of this

“Ideal” circuit theory elements are representative of elements with a certain iv characteristic equation.

F. Najmabadi, ECE 65, Winter2013, Intro (6/15)

Can be approximated with this

Can be approximated with this (for small signals)

In fact, in integrated circuit we usually configure transistors to act as resistors (to save space among other benefits).

Currents and voltages are circuit variables

F. Najmabadi, ECE 65, Winter2013, Intro (7/15)

Equations governing the circuits are: o Internal of each element: iv characteristic equation of each element: v = f(i) o How the elements are connected: KCL: (conservation of charge), and KVL: (topology)

A circuit with N two-terminal element has 2N variables and need 2N equations: o N iv characteristic equation

o N KCL/KVL

Node-voltage (or mesh current) method reduces the number of equations to be solved by automatically satisfying all KVLs (or KCLs).

o Use node-voltage methods unless circuit is very simple!

We will analyze many functional circuits

F. Najmabadi, ECE 65, Winter2013, Intro (8/15)

Two-terminal Networks

Function is defined by the iv equation

Two-port Networks

Function is defined by the transfer function (e.g., vo in terms of vi)

If the network only contains linear elements, its function can be characterized by several parameters (or numbers) instead of an algebraic function

A linear two-terminal network can be represented by its Thevenin Equivalent

F. Najmabadi, ECE 65, Winter2013, Intro (9/15)

Thevenin Theorem: If all elements inside a two-terminal network are linear, the iv equation of the two-terminal network would be linear: Av + B i + C = 0

o A linear two-terminal network can be modeled with two ideal circuit theory elements (vT = −C/A, RT = −B/A)

o If the two-terminal network does NOT contain an independent source, vT = 0 and it reduces to a resistor.

o See Lecture note for examples of computing/measuring Thevenin equivalent circuit

iRvv TT −=

A Functional circuit contains several two-terminal and two-port networks

F. Najmabadi, ECE 65, Winter2013, Intro (10/15)

Two-terminal network containing an independent source

Two-terminal network containing NO independent source

We divide the circuit into building blocks to simplify analysis and design

Source only sees a load resistor

F. Najmabadi, ECE 65, Winter2013, Intro (11/15)

A two-terminal network containing NO independent source

We only need to analyze the response of a source ONCE with RL as a parameter.

For a linear source, we only find the Thevenin parameters of the source.

Two-port network

F. Najmabadi, ECE 65, Winter2013, Intro (12/15)

A two-terminal network containing NO independent source

Transfer function of a two-port network can be found by solving the above circuit once.

A two-terminal network containing AN independent source

Accuracy

Mathematical precision is neither possible nor required in practical systems!

Accuracy (or tolerance) in practical systems

F. Najmabadi, ECE 65, Winter2013, Intro (14/15)

Measurement Accuracy: o Measuring instruments have a finite accuracy.

o When a scope with an 2% read a voltage of 1.352 V, it means that the real voltage is in the range of 1.352 ± 0.02 × 1.352 (or between 1.325 and 1.379 V).

Component Accuracy: o Components are manufactured with a finite accuracy (tolerance).

o A 1k resistor with 5% accuracy has a resistance between 0.950 and 1.050k.

Modeling Accuracy o We “approximate” practical circuit elements with ideal circuit theory

elements. (we will see this throughout the course for non-linear elements)

Analysis Accuracy: o We make approximation in the analysis by ignoring terms. (next Slide)

How accuracy affect analysis:

F. Najmabadi, ECE 65, Winter2013, Intro (15/15)

When a number has, A, has a relative accuracy of ε, it means that its value is between A (1 – ε) and A (1 + ε).

Alternatively, we are saying that all numbers in that range are approximately equal to each other.

When we assume a << A, we mean:

)1( )1( εε +≤≤−⇔≈ ABAAB

)1( )1(

AaAAAaAAA

AaAAAaA

εεεεεε

≤≤−+≤+≤−+≤+≤−⇒≈+

|| || AaAa ε≤⇒<<