Information Technology Reference
In-Depth Information
Where:
I
L
(
s
)and
U
C
(
s
), respectively, refer to the inductor currents and capaci-
tor voltage of virtual LC circuit. The input signal of virtual LC circuit's control
function towards inductance current and capacitor voltage can push forward that
the control function of the resonant controller to sinusoidal signal is equivalent
to the control function of integral controller to DC signal [11-13].
As
ω
n
is the resonance angular frequency, 1
/ω
n
can be regarded as a factor
of resonant coecient
K
r
, the output equation of the virtual LC circuit (4) can
be rewritten for:
R
o
(
s
)=
I
L
(
s
)cos
θ
n
−
U
C
(
s
)sin
θ
n
(5)
To discrete above continuous-time output equation may have:
R
o
(
k
)=
C
I
L
(
k
)
U
C
(
k
)
(6)
Where:
C
=[cos
θ
n
−
sin
θ
n
].
The continuous time state equation of virtual LC circuit is:
⎡
⎤
⎦
=
0
−ω
n
ω
n
0
I
L
(
t
)
U
C
(
t
)
+
ω
n
0
R
i
(
t
)
d
I
L
(
t
)
d
t
d
U
C
(
t
)
d
t
⎣
(7)
To discrete above continuous-time state equation may have:
I
L
(
k
+1)
U
C
(
k
+1)
=
A
I
L
(
k
)
U
C
(
k
)
+
BR
i
(
k
)
(8)
Where:
A
=
cos(
ω
n
T
)
;
B
=
sin(
ω
n
T
)
1
.
sin(
ω
n
T
)
sin(
ω
n
T
)cos(
ω
n
T
)
−
−
cos(
ω
n
T
)
Take inductor current and capacitor voltage of virtual LC circuit as output,
the Z-domain transfer function of input and output can be obtained by formulas
(6) and (8) as:
R
(
z
)=
C
(
zI−A
)
−
1
B
=
(
k
1
n
−
k
2
n
)
z
−
k
1
n
−
k
2
n
(9)
z
2
−
2
z
cos(
ω
n
T
)+1
Where:
k
1
n
=cos
θ
n
sin(
ω
n
T
);
k
2
n
=sin
θ
n
[1
cos(
ω
n
T
)].
If directly discrete the formula (3) by step response invariant method can get
the following formula (10):
−
A
)
−
1
B
=
(
k
1
n
−
k
2
n
)
z
−
k
1
n
−
k
2
n
1
ω
n
R
(
z
)=
C
(
zI
−
2
z
cos(
ω
n
T
)+1
·
(10)
z
2
−
Compared discrete time transfer function (9) with (10), (9) expands
ω
n
times,
this is because in formula (4), 1
/ω
n
is treated as a factor of resonant coecient
K
r
in order to facilitate the discussion of virtual LC discretization method.
Search WWH ::
Custom Search