§ 2-3 Linearity and superposition
Let us first define a linear element as a passive element that
has a linear voltage-current relationship.
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C
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L
Lii n d u c t o rl i n e a r ?? ?
We must also define a linear dependent source as a dependent
current or voltage source where output current or voltage is
proportional only to first power(幂 ) of some current or voltage
variable in the circuit or to the sum of such quantities.
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A linear circuit -- independent sources; linear
dependent sources; linear elements.
The response is proportional to the source or that
multiplication of all independent source voltages and
currents by a constant k increases all the current and
voltage responses by the same factor k (including the
dependent source voltage or current outputs).
Linearity theorem:
In any linear resistive network containing several
sources,The voltage across or the current through
any resistor or source may be calculated by adding
algebraically all the individual voltage or current
caused by each independent source acting alone,
Note that dependent sources are in general active
in every experiment.
Superposition theorem:
with
all other independent voltage sources replaced by
short circuit and all other independent current
sources replaced by open circuit.
It usually turns out that little if any time is saved in
analyzing a circuit containing one or more dependent
sources by use of the superposition principle,for there
must always be at least two sources in operation,one
independent source and all the dependent sources.
A group of independent sources may be made active
and inactive collectively,if we wish,For example,
suppose there are three independent sources.
Altenatively,we may find the
response due to the first and second sources operating
with the third inactive,and then add to this the response
caused by the third source acting alone.
The theorem
above staes that we may find a given response by
considering each of the three sources acting alone and
adding the three results,
A voltage source in parallel with a current source:
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Let us first define a linear element as a passive element that
has a linear voltage-current relationship.
?
q
0
C
?Cqc a p a c i t o rl i n e a r ??
i
?
0
L
Lii n d u c t o rl i n e a r ?? ?
We must also define a linear dependent source as a dependent
current or voltage source where output current or voltage is
proportional only to first power(幂 ) of some current or voltage
variable in the circuit or to the sum of such quantities.
)(6.0)(4.26.0 2121 NoiY e si ss ??? ???
0
i
R
Rir e s i s t o rl i n e a r ?? ?
?
A linear circuit -- independent sources; linear
dependent sources; linear elements.
The response is proportional to the source or that
multiplication of all independent source voltages and
currents by a constant k increases all the current and
voltage responses by the same factor k (including the
dependent source voltage or current outputs).
Linearity theorem:
In any linear resistive network containing several
sources,The voltage across or the current through
any resistor or source may be calculated by adding
algebraically all the individual voltage or current
caused by each independent source acting alone,
Note that dependent sources are in general active
in every experiment.
Superposition theorem:
with
all other independent voltage sources replaced by
short circuit and all other independent current
sources replaced by open circuit.
It usually turns out that little if any time is saved in
analyzing a circuit containing one or more dependent
sources by use of the superposition principle,for there
must always be at least two sources in operation,one
independent source and all the dependent sources.
A group of independent sources may be made active
and inactive collectively,if we wish,For example,
suppose there are three independent sources.
Altenatively,we may find the
response due to the first and second sources operating
with the third inactive,and then add to this the response
caused by the third source acting alone.
The theorem
above staes that we may find a given response by
considering each of the three sources acting alone and
adding the three results,
A voltage source in parallel with a current source:
0''/' ?? iRi s?
i
?
?
s?
si
R
"i
si
R
'i
?
?
s?
R? ?
i
?
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s?
si
i
?
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s?
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