Graphics Programs Reference
In-Depth Information
Differentiating Eq. (b) and substituting the charge-current relationship
dq
2
/
=
dt
i
2
, we get
di
1
dt
=
−
3
Ri
1
−
2
Ri
2
+
E
(
t
)
(c)
L
di
2
dt
=−
2
3
di
1
dt
−
i
2
3
RC
+
1
3
R
dE
dt
(d)
Wecould substitute
di
1
/
dt
fromEq. (c) into Eq. (d), so that the latterwouldassume
the usualform
di
2
/
i
2
), but it ismoreconvenienttoleave the equations
as theyare.Assuming that the voltagesource isturned onattime
t
dt
=
f
(
t
,
i
1
,
=
0, plot the
loop currents
i
1
and
i
2
from
t
=
0to0
.
05 s. Use
E
(
t
)
=
240 sin(120
π
t
)V,
R
=
1
.
0
,
L
=
0
.
2
×
10
−
3
H and
C
=
3
.
5
×
10
−
3
F.
22.
L
L
i
1
i
2
E
C
C
i
1
i
2
R
R
The constant voltagesource
E
of the circuit shown isturned onat
t
=
0, causing
transientcurrents
i
1
and
i
2
inthe two loopsthatlast about0
.
05 s. Plot these currents
10
−
3
H
from
t
=
0to0
.
05 s, using the followingdata:
E
=
9 V,
R
=
0
.
25
,
L
=
1
.
2
×
10
−
3
F. Kirchoff'sequationsfor the two loops are
and
C
=
5
×
L
di
1
q
1
−
q
2
dt
+
Ri
1
+
=
E
C
L
di
2
dt
q
2
−
q
1
q
2
C
=
+
Ri
2
+
+
0
C
Additionaltwoequations are the current-charge relationships
dq
1
dt
=
di
2
dt
=
i
1
i
2
7.4
Stability and Stiffness
Loosely speaking, amethod of numerical integrationissaid to bestable if the effects
of localerrors do not accumulate catastrophically; that is, if the globalerrorremains
bounded. If the methodis unstable, the globalerrorwill increase exponentially,even-
tually causing numericaloverflow. Stability has nothing to do with accuracy; in fact,
an inaccurate method can be very stable.
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