superposition theorem.pdf
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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
6.002 CIRCUITS ANDELECTRONICS
Superposition, Thévenin 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
0=∑loop
iV
ReviewCircuit Analysis Methods
Circuit composition rules
Node method – the workhorse of 6.002KCL at nodes using V ’s referenced from ground(KVL implicit in “ ”) ( )ji ee − G
KVL: KCL:0=∑
nodeiI
VI
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
Linearity
Write node equations –
VI
1R
2R+–
021
=−+− I
Re
RVe
Notice:linear in IVe ,,
VI,eVNo terms
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
Consider
Linearity
Write node equations --
Rearrange --
VI
1R
2R+–
021
=−+− I
Re
RVe
IRVe
RR+=⎥
⎦
⎤⎢⎣
⎡+
121
11
G e S=
conductancematrix
nodevoltages
linear sumof sources
linear in IVe ,,
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
or IRR
RRVRR
Re21
21
21
2
++
+=
…… +++++= 22112211 IbIbVaVae
Write node equations --
Rearrange --
021
=−+− I
Re
RVe
IRVe
RR+=⎥
⎦
⎤⎢⎣
⎡+
121
11
G e S=
conductancematrix
nodevoltages
linear sumof sources
linear in IVe ,,
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 HomogeneitySuperposition⇒
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 HomogeneitySuperposition
Homogeneity
1x2x ... y
1xα2xα yα...
⇓
⇒
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 HomogeneitySuperposition
Superposition
ax1ax2 ay... ...
bx1bx2 by
⇒
ba xx 11 +ba xx 22 +
ba yy +
⇓
...
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 HomogeneitySuperposition
Specific superposition example:
1V0 1y
02V 2y
01 +V20 V+ 21 yy +
⇓
⇒
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 methodThe output of a circuit isdetermined by summing theresponses to each sourceacting alone.
independent sources
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
+–0=V+
-v
i
short
+
-v
i
0=I+
-v
i
open
+
-v
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 exampleUse superposition method
VI
1R
2R+–
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
Back to the exampleUse superposition methodV acting alone
V0=I2R
+–
e
1R
I acting alone
0=VI
1R
2R
e
VRR
ReV21
2
+=
IRR
RReI21
21
+=
IRR
RRVRR
Reee IV21
21
21
2
++
+=+=
sum superposition
Voilà !
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
saltwater
output showssuperposition
Demo
constant+–
sinusoid
+–?
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
ConsiderYet another method…
resistors
nounits
By setting
0,0
==∀
iInn
0,0
==∀
iVmm
All
0,0
=∀=∀
mm
nnVI
+–mV
nI
Arbitrary network N
By superpositionRiIVv n
nnm
mm ++= ∑∑ βα
+
-v
i
i
resistanceunits
independent of external excitation and behaves like a voltage “ ”THv
alsoindependentof externalexcitement &behaves likea resistor
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
OriRvv THTH +=
As far as the external world is concerned (for the purpose of I-V relation), “Arbitrary network N” is indistinguishable from:
i+–
THR
THv+
-
vThéveninequivalentnetwork
THR
THv open circuit voltageat terminal pair (a.k.a. port)resistance of network seenfrom port( ’s, ’s set to 0)mV nI
N
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 Thévenin Method
Replace network N with its Théveninequivalent, then solve external network E.
E
Thévenin equivalent
+–
THR
THv+
-
v
i
E
+–
+–
i
+
-v
N
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:1R
V+–
1i
1R
V+–
1i
TH
TH
RRVVi
+−
=1
1
I2R
ITHR
THV +–
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:
:THR
:THV
2IRVTH =
2RRTH =
+
-THV 2R I
+
-THR 2R
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, iRvv THTH +=
i
Open circuit( )0≡i
THvv = OCV
Short circuit( )0≡v TH
TH
Rvi −
=SCI−
v
THR1
THv
SCI−
OCV“ ”
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:
The Norton Method
in recitation,see text
+–
+–
i
+
-v
Nortonequivalent
TH
THN R
VI =
NTH RR =NI
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
… 101100 …
Discretize matterLMD LCA
Physics EE
R, I, V Linear networks
Analysis methods (linear)KVL, KCL, I — VCombination rulesNode methodSuperpositionThéveninNorton
NextNonlinear analysisDiscretize voltage