MATHEMATICAL MODELS OF DC-DC CONVERTERS
YI CAO
1. DC-DC Converters
There are three kinds of switching mode DC-DC converters, buck, boost and buck-boost. The buck mode
is used to reduce output voltage, whilst the boost mode can increase the output voltage. In the buck-boost
mode, the output voltage can be maintained either higher or lower than the source but in the opposite
polarity. The simplest forms of these converters are schematically represented in Figure 1. These converters
V
in
L
C
R
V
o
V
s
u
=1
=0
(a) Buck
V
in
L
C
R
V
o
V
s
u
=0
=1
(b) Boost
V
in
L
C
R
V
o
V
s
u
=0=1
(c) Buck-boost
Figure 1. Switching-mode DC-DC converters: (a) buck, (b) boost and (c) buck-boost.
consist of the same components, an inductor, L, a capacitor, C and a switch, which has two states u = 1
and u = 0. All converters connect to a DC power source with a voltage (unregulated), V
in
and provide a
regulated voltage, v
o
to the load resistor, R by controlling the state of the switch. In some situations, the
load also could be inductive, for example a DC motor, or approximately, a current load, for example in a
cascade configuration. For simplicity, here, only current and resistive loads are to be considered.
2. Principles
The working principles of the these DC-DC converters can be explained as follows. In the buck mode,
when the switch is on position 1, the DC source supplies power to the circuit which results an output voltage
across the resistor. When the switch changes its position to 0, the energy stored in the inductor and capacitor
will discharge through the resistor. Appropriately controlling the switching position can maintain the output
voltage at a desired level lower than the source.
In the boost mode, when the switch is on position 1, the circuit is separated into two parts: on the left,
the source is charging the inductor, meanwhile the capacitor on the right maintains the output voltage using
previously stored energy. When the switch changes its position to 0, both the DC source and energy stored
in the inductor will supply power to the circuit on the right, hence boost the output voltage. Again, the
output voltage can be maintain at desired level by controlling the switching sequence.
Finally, for the buck-boost mode, switch positions 1 and 0 represents charging and discharging modes
of the inductor. Appropriately controlling the switching sequence can result in output voltage higher or
lower than the DC source. Since the inductor cannot change the direction of current, the output voltage is
opposite to the DC source.
3. Model under ideal assumptions
Under ideal assumptions: ideal switch, ideal capacitor and ideal inductor, these converters can be described
using ordinary differentiation equations as follows:
1
