superposition, thévenin and norton -...
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6.002 Fall 2000 Lecture 13
6.002 CIRCUITS ANDELECTRONICS
Superposition, Thévenin and Norton
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6.002 Fall 2000 Lecture 23
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
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6.002 Fall 2000 Lecture 33
Consider
Linearity
Write node equations –
VI
1R
2R+–
021
=−+− I
Re
RVe
Notice:linear in IVe ,,
VI,eVNo terms
e
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6.002 Fall 2000 Lecture 43
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 ,,
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6.002 Fall 2000 Lecture 53
Linearity
or IRRRRV
RRRe
21
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!
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6.002 Fall 2000 Lecture 63
Linearity HomogeneitySuperposition⇒
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6.002 Fall 2000 Lecture 73
Linearity HomogeneitySuperposition
Homogeneity
1x2x y...
1xα2xα yα...
⇓
⇒
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6.002 Fall 2000 Lecture 83
Linearity HomogeneitySuperposition
Superposition
ax1ax2 ay... ...
bx1bx2 by
⇒
ba xx 11 +ba xx 22 +
ba yy +
⇓
...
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6.002 Fall 2000 Lecture 93
Linearity HomogeneitySuperposition
Specific superposition example:
1V0 1y
02V 2y
01 +V20 V+ 21 yy +
⇓
⇒
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6.002 Fall 2000 Lecture 103
Method 4: Superposition methodThe output of a circuit isdetermined by summing theresponses to each sourceacting alone.
independent sources
only
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6.002 Fall 2000 Lecture 113
i
+–0=V+
-v
i
short
+
-v
i
0=I+
-v
i
open
+
-v
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6.002 Fall 2000 Lecture 123
Back to the exampleUse superposition method
VI
1R
2R+–
e
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6.002 Fall 2000 Lecture 133
Back to the exampleUse superposition method
V acting alone
V0=I2R+–
e
1R
I acting alone
0=VI
1R
2R
e
VRR
ReV21
2
+=
IRRRReI
21
21
+=
IRRRRV
RRReee IV
21
21
21
2
++
+=+=
sum superposition
Voilà !
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6.002 Fall 2000 Lecture 143
saltwater
output showssuperposition
Demo
constant+–
sinusoid
+–?
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6.002 Fall 2000 Lecture 153
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
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6.002 Fall 2000 Lecture 163
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
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6.002 Fall 2000 Lecture 173
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
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6.002 Fall 2000 Lecture 183
Example:1R
V+–
1i
1R
V+–
1i
TH
TH
RRVVi
+−
=1
1
I2R
ITHR
THV +–
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6.002 Fall 2000 Lecture 193
Example:
:THR
:THV
2IRVTH =
2RRTH =
+
-THV 2R I
+
-THR 2R
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6.002 Fall 2000 Lecture 203
Graphically, iRvv THTH +=
i
Open circuit( )0≡i
THvv = OCV
Short circuit( )0≡v TH
TH
Rvi −
=SCI−
v
THR1
THv
SCI−
OCV“ ”
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6.002 Fall 2000 Lecture 213
Method 5:
The Norton Method
in recitation,see text
+–+–
i+
-v
Nortonequivalent
TH
THN R
VI =
NTH RR =NI
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6.002 Fall 2000 Lecture 223
Summary
… 101100 …
Discretize matterLMD LCA
Physics EE
R, I, V Linear networks
Analysis methods (linear)KVL, KCL, I — VCombination rulesNode methodSuperpositionThéveninNorton
NextNonlinear analysisDiscretize voltage