the selection over the data. Let’s narrow it to two cases:
Case 1: Your voltage data grows before peaking and then decaying to zero.
Case 2: Your data looks simply like an exponential decay.
Either way, use the following set of equations:
V
(
t
) =
V
0
e

t
/
2
τ
sinh
ωt
τ
=
L
R
ω
=
r
R
2
4
L
2

1
LC
>>
modelOver
=
@
(
p
,
x
)
p
(1) .
exp
(

x
./(2
p
(2) ) ) .
sinh
(
p
(3) .
x
+
p
(4) )
+
p
(5) ;
p(1):
V
0
; in case 1 you can probably just guess the value of the peak. Case 2: use your
initial voltage point.
p(2):
τ
, and p(3):
ω
; these are confusing to interpret for most people, the safest thing is
to use your predicted values from the equations above.
p(4): This is the phase
φ
. In case 1,
φ
≈
0 so start there and slowly increase it until it
fits. For case 2,
φ
≈
π/
2.
p(5): A vertical shift in the data that is approximately zero.
4
Frequency Domain Amplitudes
For the frequency domain, we are modelling the following equation:

V
(
ω
)

=
V
0
R/R
t
r
1 +
Q
2
ω
ω
0

ω
0
ω
2
2
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Also, be aware that these are angular frequencies, so you’ll need to convert your linear frequen
cies (in Hz) by
ω
= 2
πf
.
>>
x
= 2
pi
x
;
% convert
your
f r e q .
to
angular
f r e q . ,
only
i f
you
haven
t
already
>>
modelFrequency
=
@
(
p
,
x
)
p
(1) . /
s q rt
(1+
p
(2) ˆ2.
(
x
. /
p
(3)

p
(3) . /
x
) . ˆ 2 )

V
(
ω
0
)

=
V
0
R/R
t
, so find the peak in your data; its yvalue is p(1) and its xvalue is
p(3).
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 Spring '11
 shpyrko
 Statistics, Frequency, Time series analysis, Exponential decay

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