Aerospace and Electronic Systems Magazine March 2017 - 31

tance L and capacitor with capacitance C. While the transistor is
on, the inductor current iL (t) increases since the input voltage is
higher than the output voltage when the transistor is turned off;
the diode must start to conduct since the inductor current cannot
stop instantly. The voltage across the diode is zero when it is conducting and the inductor current will decrease. Figure 2(a) shows
the waveform of the control signal. The converter is usually designed with a low magnitude of ripples in the output voltage. For
the resistance and inductance (RL) circuit, the current increases
and decreases exponentially as shown in Figure 2(b). The voltage
across the diode is equal to the input voltage or equal to zero.
The output filter of the converter filters this voltage waveform
and the magnitude of the ripple in the output voltage depends
on the filter design. If the inductor current becomes zero before
the transistor is turned on, it will stay at zero until the transistor
is turned on since the diode can only convey in one way. If the
converter is controlled so that the inductor current is zero during
some portion of the switching period, it is supposed to be operating in discontinuous conduction mode. Otherwise, it operates in
continuous conduction mode.
The switching period Ts of the converter is determined by
the control signal δ (t), as indicated in Figure 2(a). In this fig-

Figure 1.

The buck converter with a voltage controller.

MARCH 2017

ure, the switching period is held constant. The average output
voltage is controlled by changing the width of the pulses. In
Figure 2(b). the falling edge is controlled, i.e., when the transistor should turn off. The duty cycle d(t) is a real value in the
interval 0 to 1 and it is equal to the ratio of the width of a
pulse ton to the switching period Ts. The duty cycle is actually a
discrete-time signal.
In this section, a linear time-invariant model of the buck
converter is derived by means of state-space averaging during
MOSFET on as in Figure 3(a) and during MOSFET off as in
Figure 3(b). The converter can be described as switching between different time-invariant systems and the state-space description of each one of these systems is first derived. These
state-space descriptions are used as a starting point in the
method of state-space averaging. This method is applied to the
buck converter and the result is a linear time-invariant model in
state-space description. Eventually, several transfer functions
are drawn out from this model until reaching the control-tooutput transfer function held by
G (s) =

RVg (1 + sR cC )
vˆo ( s )
=
2
ˆ
d ( s ) R + s ( L + RR cC ) + s ( R + R c ) LC

(1)

Figure 2.

(a) The waveform of the control signal. (b) The waveform of the inductor current.

IEEE A&E SYSTEMS MAGAZINE

31



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