Page 283 - Mechatronics with Experiments

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JWST499-c05
JWST499-Cetinkunt
ELECTRONIC COMPONENTS FOR MECHATRONIC SYSTEMS 269
the capacitor C and load resistor R form a complete circuit and the current flows due to
L
the non-zero voltage in the capacitor. Assume that the charge in the capacitor is zero when
the circuit is first turned on.
(i) When the diode is conducting; (V (t) − V ) ≥ V out (t)
in
FB
V (t) − V FB − R ⋅ i(t) = V out (t) (5.125)
in
i(t) = i (t) + i (t) (5.126)
C
R L
1
i (t) = ⋅ V (t) (5.127)
R
R L out
L
dV out (t)
i (t) = C ⋅ (5.128)
C
dt
∗
Let V (t) = V (t) − V , and substitute the i(t), i (t), i (t) in the first equation in order to
in in FB C R L
obtain the dynamic relationship between the input voltage and output voltage. The result
can be shown to be
( )
dV out (t) R ∗
RC + 1 + V out (t) = V (t) (5.129)
dt R in
L
where the initial value of output voltage at the begining of each cycle (when the diode first
starts to conduct) is the voltage across the capacitor, V (t ) = V (t ). During the time that
out i C i
the diode conducts, we want the output voltage to quickly track the input voltage. In order to
accomplish this we need a small time constant, which means RC should be small compared
to the input frequency, or equivalently 1∕RC should be larger than the input frequency, that
is ten times larger 1∕RC = 10 ⋅ 60 or RC = 1∕(10 ⋅ 60).
(ii) When the diode is not conducting, (V (t) − V ) < V (t), we have a capacitor C
in FB out
(with initial charge and voltage from the last instant when diode was conducting) and load
resistor R forming a closed electrical circuit. Hence, the dynamic behavior of the voltage
L
and current relationship during that period is described by
V (t ); given from previous phase or initial condition (5.130)
C i
t
1
V out (t) = V (t) = V (t ) + C ∫ i ( )d = R ⋅ i (t) (5.131)
C
C
L
C i
R L
t i
V (t)
C
i (t) =−i =− (5.132)
C
R L
R L
Evaluating this equation, it can be shown that the output voltage dynamics during the time
when the diode is not conducting is defined by
t
1
R C ∫ t i V out ( )d + V out (t) = V (t ) (5.133)
C i
L
or
dV out (t) 1
+ V out (t) = 0.0; (5.134)
dt R C
L
with the initial condition of
(t ) = V (t )
V out i C i (5.135)
where V (t ) is the voltage across the capacitor (which is same as the output voltage) the last
C i
time the diode was conducting. Notice that the time domain analytical solution of the above
®
®
equation is (which can be numerically confirmed by MATLAB /Simulink simulation)
V out (t) = V (t ) ⋅ e −(t−t i )∕(R L C) (5.136)
C i