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
=1 u
Vin
Vs
L
=0
(a) Buck
L
Vo
C
R
Vin
Vs
u =0
=1
=1 u
Vo
C
R
(b) Boost
Vin
Vo
=0
Vs
L
C
R
(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), Vin and provide a
regulated voltage, vo 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
Buck converter:
dvc
= iL − vc /R − io
dt
diL
L
= uvin − vc
dt
where, io is the load current. Boost converter:
dvc
(3.2)
C
= (1 − u)iL − vc /R − io
dt
diL
L
= vin − (1 − u)vc
dt
Buck-boost converter:
dvc
C
(3.3)
= (1 − u)iL − vc /R − io
dt
diL
L
= uvin − (1 − u)vc
dt
Introduce the following state, time and load normalization:
r
√
t
LC
iL L
vc
,
τ=√
,
γ=
,
,
x2 =
(3.4)
x1 =
vin
vin C
R
LC
Then the normalized state equations of three converters are as follows:
Normalized buck model
(3.1)
C
(3.5)
x˙1
= −γx1 + x2 − d
x˙2
= −x1 + u
io
d=
vin
r
L
C
where, with an abuse of notation, ‘.’ represents the derivation with respect to the normalized time, τ .
Normalized boost model
(3.6)
x˙1
= −γx1 + (1 − u)x2 − d
x˙2
= −(1 − u)x1 + 1
x˙1
= −γx1 + (1 − u)x2 − d
x˙2
= −(1 − u)x1 + u
Normalized buck-boost model
(3.7)
4. Model with body resistors
In more general cases, a body resistor of the inductor, RL and an equivalent series resistor (ESR) of the
capacitor, Rc can be added to the above models.
Buck model with RL and Rc Since,
dvc
C
= iL − vo /R − io
dt
dvc
vo = vc + R c C
dt
diL
L
= uvin − vo − RL iL
dt
Inserting the second equation into the first leads to:
Rc dvc
dvc
= iL − vc /R −
C
− io
C
dt
R dt
Rc
dvc
1+
C
= iL − vc /R − io
R
dt
Hence,
Rvc
RRc
vo =
+
(iL − io )
R + Rc
R + Rc
2
and the overall model is
vc
R
iL −
− io
(R + Rc )
R
Rvc
RRc
RRc io
= uvin −
− RL +
iL +
R + Rc
R + Rc
R + Rc
Rvc
RRc
vo =
+
(iL − io )
R + Rc
R + Rc
Boost model with RL and Rc
dvc
(4.2)
C
= (1 − u)iL − vo /R − io
dt
diL
L
= vin − (1 − u)vo − RL iL
dt
RRc
Rvc
+
((1 − u)iL − io )
vo =
R + Rc
R + Rc
Buck-boost model with RL and Rc
dvc
(4.3)
C
= (1 − u)iL − vo /R − io
dt
diL
L
= uvin − (1 − u)vo − RL iL
dt
Rvc
RRc
vo =
+
((1 − u)iL − io )
R + Rc
R + Rc
(4.1)
dvc
dt
diL
L
dt
C
=
5. Simulink Model
These three modes of DC-DC converters have been uniformly implemented in the MATLAB/Simulink as
show in Figure 2.
Figure 2. A uniform Simulink model of DC-DC converters.
The input-output connections of the model is shown in Figure 3 The first input to the model is the switch
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Figure 3. Input and output connections of the DC-DC converter model.
signal eight 1 or 0. The second one defines the DC source voltage and internal resistance. The third input is
used to define the output current. The model has two outputs, the output voltage and the inductor current,
which are the states of the system.
The model can be configure with a number of parameters as shown in Figure 4. These parameters are:
the capacitance, C, inductance, L, the internal resistance of the capacitor and the inductor, RC and RL
respectively. Three converter modes can be selected through the pull-down menu. One can also define
either zero or non-zero value to the initial capacitor voltage by selecting or de-selecting the “zero capacitor
voltage” option. Finally, the option “Positive Inductor Current” defines whether the condition iL ≥ 0 should
be enforced or not.
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Figure 4. Parameters of the DC-DC converter model.
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