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Problem 2.16, P. T. Krein,
Elements of Power Electronics
.
New York:
Oxford, 1998.
This problem defines integral cycle control.
The switching function q(t) is on for a cycle of Vin,
then off for a cycle, and so on.
If the switching function is sketched by hand, we notice three
things:
1.
The frequency of q(t) is 30 Hz.
2.
The duty ratio is 1/2.
3.
The phase is not zero.
Instead, the center of the pulse occurs at t = t
0
= 1/240 s.
Since the switching function radian frequency is 2
π
30 rad/s, and since the phase is defined as
φ
0
=
ω
t
0
, we can compute that
φ
0
= 2
π
30/240 =
π
/4 radians, or a 45 degree delay.
This can be
checked by using it to plot the results:
V0
5
:=
(Set to 5 for easier-to-read plots.)
vin t
( )
V0 cos 2
π
⋅
60
⋅
t
⋅
(
)
⋅
:=
q t
( )
if
cos 2
π
⋅
30
⋅
t
⋅
π
4
−
⎛
⎜
⎝
⎞
⎟
⎠
0
>
1
,
0
,
⎛
⎜
⎝
⎞
⎟
⎠
:=
vout t
( )
vin t
( ) q t
( )
⋅
:=
Now, some plots:
tlast
0.1
:=
i
0
2000
..
:=
t
i
tlast
2000
i
⋅
:=
0
0.02
0.04
0.06
0.08
0.1
6
−
4
−
2
−
0
2
4
6
vin t
i
( )
q t
i
( )
t
i
0
0.02
0.04
0.06
0.08
0.1
6
−
4
−
2
−
0
2
4
6
vout t
i
( )
t
i