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Circuits & Electronics Lecture

This document summarizes key concepts from a lecture on circuit analysis, including linearity, superposition, and Thévenin and Norton equivalent circuits. The key points are: - Circuit analysis uses Kirchhoff's laws and node analysis methods for linear circuits. - Linear circuits exhibit properties of linearity, superposition, and homogeneity that allow their analysis. - The superposition principle allows solving for circuits using individual sources separately and summing responses. - Thévenin and Norton equivalents allow replacing complex networks with simplified single-source models to analyze external circuits.

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813 views22 pages

Circuits & Electronics Lecture

This document summarizes key concepts from a lecture on circuit analysis, including linearity, superposition, and Thévenin and Norton equivalent circuits. The key points are: - Circuit analysis uses Kirchhoff's laws and node analysis methods for linear circuits. - Linear circuits exhibit properties of linearity, superposition, and homogeneity that allow their analysis. - The superposition principle allows solving for circuits using individual sources separately and summing responses. - Thévenin and Norton equivalents allow replacing complex networks with simplified single-source models to analyze external circuits.

Uploaded by

etasuresh
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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6.

002

CIRCUITS AND ELECTRONICS

Superposition, Thvenin and Norton

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Review
Circuit Analysis Methods
z KVL:
loop

Vi = 0

KCL: Ii = 0
node

VI

z Circuit composition rules z Node method the workhorse of 6.002

KCL at nodes using V s referenced from ground (KVL implicit in (ei e j ) G )

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Linearity
Consider
R1

e
J

R2

Write node equations

e V e + I =0 R1 R2
Notice: linear in e,V , I No eV ,VI terms

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Linearity
Consider

R1

R2

J
V + I R1

Write node equations -e V e + I =0 R1 R2 Rearrange -1 1 R + R e 1 2 conductance matrix


=

linear in e,V , I

node linear sum voltages of sources

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Linearity
Write node equations -e V e + I =0 R1 R2 Rearrange -1 1 R + R e 1 2 conductance matrix
=

linear in e,V , I

V + I R1

node linear sum voltages of sources

G
or

R2 R1 R2 e= V+ I R1 + R2 R1 + R2
e = a1V1 + a2V2 + + b1 I1 + b2 I 2 +

Linear!
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Linearity

Homogeneity Superposition

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Linearity

Homogeneity Superposition

Homogeneity
x1 x2 . . .

x1 x2 . . .

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Linearity

Homogeneity Superposition

Superposition

x1a x2 a . . .

ya

x1b x2 b . . .

yb

x1a + x1b x2 a + x2 b . . . y a + yb

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Linearity

Homogeneity Superposition

Specific superposition example:

V1 0

y1

0 V2

y2

V1 + 0 0 + V2 y1 + y2

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Method 4: Superposition method The output of a circuit is determined by summing the responses to each source acting alone.
es c r u o s t n e nd e p e i nd only

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

i + v short + v -

V =0 +

i + v open + v -

I =0

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Back to the example


Use superposition method

R1

e
J

R2

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Back to the example


Use superposition method V acting alone
e
R1

R2

R2 V I = 0 eV = R1 + R2

I acting alone
R1

e R1 R2 eI = I R1 + R2
J

V =0

R2

sum

superposition

R2 R1 R2 e = eV + eI = V+ I R1 + R2 R1 + R2

Voil !
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Demo
salt water

constant +

sinusoid

output shows superposition

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Consider

Yet another method


y network r a r t i N Arb resistors Vm In + J
i

+ v -

By superposition
v =

mVm + n I n + Ri
m n

no resistance units units By setting n I n = 0, mVm = 0, i = 0 i = 0

also independent of external excitement & behaves like a resistor

All n I n = 0, mVm = 0

independent of external excitation and behaves like a voltage vTH


Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Or

v = vTH + RTH i

As far as the external world is concerned (for the purpose of I-V relation), Arbitrary network N is indistinguishable from: RTH

Thvenin equivalent network

+ vTH

+ v -

vTH RTH

open circuit voltage at terminal pair (a.k.a. port) resistance of network seen from port ( Vm s, I n s set to 0)

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Method 4: The Thvenin Method


J + N + i + v -

Thvenin equivalent
RTH

i + v -

+ vTH

Replace network N with its Thvenin equivalent, then solve external network E.
Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Example:
i1 R1

+ V

R2

i1 R1

+ V

RTH VTH

+ I

V VTH i1 = R1 + RTH

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Example:

VTH : VTH = IR2

+ VTH -

R2

RTH : RTH = R2

+ RTH -

R2

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Graphically,
i

v = vTH + RTH i

1 RTH

v
vTH

V
OC

I SC

Open circuit (i 0) Short circuit (v 0)

v = vTH
vTH i = RTH

VOC
I SC

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Method 5:

in recitation, see text

The Norton Method


i J + + + v -

IN

RTH = RN

Norton equivalent

IN =

VTH RTH

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

6.002 Fall 2000

Lecture 3

Summary
Discretize matter LMD Physics

LCA EE

R, I, V

Linear networks

Analysis methods (linear) KVL, KCL, I V Combination rules Node method Superposition Thvenin Norton Next Nonlinear analysis Discretize voltage

6.002 Fall 2000 Lecture 3

101100

Cite as: Anant Agarwal and Jeffrey Lang, course materials for 6.002 Circuits and Electronics, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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