Discontinuous Conduction
When the conduction is discontinuous, the voltage across the inductor is
zero for part of the cycle since there is no current through the inductor.
Let D1T be the time for which the switch is ON in one cycle and
let D2T be the period for which the diode conducts. Since the conduction
is discontinuous,
An expression for the output voltage can be obtained in terms of source voltage,
duty cycle D1 of the switch and duty cycle D2 of the
diode. Since the net change in inductor current is over a cycle, the net volt-seconds
area associated with the inductor is zero. The waveforms relevant to the inductor
when the conduction is discontinuous are shown in Fig. 7. From Fig. 7,
On simplifying, an expression for Vo can be obtained. Then
The value of D1, the duty cycle of the switch, is usually known,
but the period for which the diode conducts is an unknown quantity depending
on the other circuit parameters. The value of D2 can be determined
in several ways. Here it is determined using the power balance between the
input and output. When the circuit is ideal, the input power equals output
power. Let the average source current be IS and the average output
current be Io. Then
Using equation (28), we get that
The average source current be IS can be obtained from Fig. 7.
The average source current is the same as the average inductor current. Let
the peak inductor current be DIL and
the period for which this current flows is (D1T + D2T).
This period is the base of the triangle that defines the inductor current.
The average inductor current is obtained as the area of this triangle divided
by the cycle period. We have that
Equating equations(30) and (31),
From equation (3),
Substituting for DIL from equation
(33) in equation (32), we get that
Equation (34) can be re-written as:
Solving for D2,
Equation (36) states how D2 varies as a function of R, D1
, f and L. Once D2 is known, Vo can be obtained from
equation (28).
It is possible to get an expression for Vo as a function of R,
D1 , f and L. For this, we equate the average load current with
the average diode current. The average output current can be obtained from
the average output voltage and the load resistor. The average diode current
is:
Using the expression for DIL from equation
(33), and replacing the L.H.S. by the average load current,
Hence we obtain that
By substituting for D2 from equation (36) in the above equation,
we can get an expression for Vo/VS. Alternatively, equation
(28) can be re-written as:
Using the expression for D2 from equation (39) in equation (40),
That is,
Solving for the ratio of output to source voltage and taking the positive
root of the expression on the R.H.S. of equation (42),
Equation (43) states how (Vo/VS) varies as a function
of R, D1 , f and L
Two applets are presented below, the first applet simulates the behaviour
of the ideal circuit in open loop, whereas the second applet is about the
discontinuous mode of operation.
click here to open the applet
The first applet presents four types of responses. When the circuit is switched
on at a fixed duty cycle with no energy stored in either the inductor or capacitor
initially, the transient inductor current happens to be large. If the input
voltage has any ripple content, its effect can be seen by selecting either
the periodic response or the transient response over one input cycle.
The second applet displays two sets of curves. The first set illustrates
how the ratio of output voltage to input voltage varies as a function of the
ratio of load resistance to critical resistance for different duty cycles
of the switch, where the critical resistance is calculated from equation (25).
The critical resistance at a given duty cycle can be stated to be:
The second set of curves illustrates how the duty cycle of the diode varies
as a function of the ratio of load resistance to critical resistance for different
duty cycles of the switch. The values of critical resistance at various values
of duty cycle are also displayed. The parameters to be set initially are the
frequency of operation and the value of inductor.
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