Image Processing Reference
In-Depth Information
where
t
m
(k) stands for the mass of the toner at cycle k in grams
m
d
(k) is the mass of toner dispensed from the dispenser at cycle k in grams
m
i
(k) is the mass of toner which is used for the image (based on the consump-
tion pro
le at cycle k in grams)
m
is the transport delay in toner dispense (in number of cycles) that includes
the delay in the mixing system
The index k denotes the iteration number (or measurement-process-actuation cycle).
For simplicity, we assume that the TC cycle is synchronous with the measurement-
process-actuation cycle of the level 2 control system. This assumption is unrealistic
for most of the practical development systems.
The mass dispensed is calculated by using the duty cycle which will be derived
by processing the measured TC signal inside the feedback controller.
m
d
(
k
) ¼
d
(
k
)
R
max
t
(
:
)
9
60
where
R
max
is the maximum dispense rate at which the toner can be dispensed from
the toner reservoir
t
is the measurement-process-actuation period in seconds
d(k) is the duty cycle at cycle k
Generally, a pixel counter is included in many modern printing systems that can be
used to calculate the total area coverage of the image in which the toner will be
developed. The m
i
(k) is calculated by using the area coverage a(k) of the image. It is
given by
m
i
(
k
) ¼
a
(
k
)
R
max
t
(
9
:
61
)
From the de
nition of TC, if m
c
is the carrier mass, then we have
t
m
(
k
)
m
c
t
c
(
k
) ¼
(
9
:
62
)
Rewriting Equation 9.59 in terms of TC
t
c
(
k
þ
1
) ¼
t
c
(
k
) þ
g
[
u
(
k
m)
v
(
k
)]
(
9
:
63
)
where u(k) is the dispensed mass, m
d
(
k
)
, and v(k) is the developed mass, m
i
(
k
)
, and g
is a scale factor given by
1
m
c
g
¼
(
9
:
64
)
The Equation 9.63 can be transformed to the z-domain by applying the z-transform.
After the transformation we get
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