Pt. 2: Controlling DC-DC Converters
In front of these two blocks is a
Fig.2: this classic
compensator, which we will dedepiction of the
scribe in more detail below. The
DC-DC converter
input to the compensator is an error,
closed-loop
e, that is the difference between the
system is useful
reference setpoint vr and the feedto understand
back signal vfb. The feedback signal
the circuit
is derived from the output voltage
blocks we need
we want to control by some sensor,
to model.
with a gain of H.
Fig.3: the DCThese blocks are shown in a more
DC converter
familiar way in Fig.3. The modarranged
ulator includes the Mosfet and
according to the
diode switches, plus the circuitry
block diagram
that converts a control voltage to
in Fig.2 is the
pulse-width-modulation to drive
starting point for
the Mosfet. At its simplest, this is
the development
just a comparator with an output
of a control
that is high whenever the control
model of the
voltage exceeds a ramp waveform
converter. The
compensator is
at the switching frequency.
the missing piece
The modulator therefore takes a
of the puzzle at
control voltage, vc, and produces the
this stage.
average voltage, ‹vx›. The modulator has a gain of Gm and it is easy
to see that this must be the unitless negative input for the feedback. This
quantity Gm = Vsrc/Vramp. If we assume anticipates that the compensator will
the ramp voltage is 1V, the modulator probably involve op amps.
gain simplifies to the source voltage,
Simplifying the
which is 12 in our example.
It is worth noting that there is no complex-frequency domain
Describing the filter and compenneed for any switching to occur in this
model of the modulator. The modula- sator gains requires us to dip our toes
tor produces an output equal to the into something called the complex-
average voltage over one switching frequency domain, also known as the
s-domain. Analysis of the behaviour
period. This voltage may vary with
time, but the time scale we are con- of capacitors and inductors involves
cerned about is much longer than the differential equations, which we usually describe in the time domain – for
2µs switching period.
Because this modulator controls the example, the relationship between
average voltage applied to the filter, voltage and current in an inductor is
this converter is said to be employ- v = L di/dt.
We can describe the same equations
ing voltage mode control. There is
another way to do it, which we will in the s-domain, which makes the mathematics much simpler, because differcome to later.
The sensor gain is trivial to calcu- entiation is reduced to multiplication
late. It is simply the divider ratio rep- and integration is reduced to division.
resented by the two resistors, so Hs = The conversion from the time domain
R1 ÷ (R1 + R2), or 0.204 in this case. to the s-domain requires something
The buffer is there to isolate the com- called a Laplace transform.
You don’t need to know how to do
pensator from the divider.
We will ignore the compensator for this (unless you are an electrical enthe moment, but notice that I have gineering student) because you can
absorbed the error summing junc- write the s-domain expression for most
tion into the black box and added a common components very easily withpositive input for the reference and a out having to do any maths.
Table 1 – passive component complex impedances
Component
Time Domain
s-Domain
Complex Impedance Z(s)
Inductor
v = L di/dt
v = iLs
Ls
Capacitor
i = C dv/dt
i = vCs
1/Cs
Resistor
v = Ri
v = ir
R
Practical Electronics | May | 2026
The “s” in s-domain is a complex
frequency, which is just a way of describing a unit-amplitude sinewave
with a given frequency and phase. We
won’t, however, have to deal with any
complex numbers in this explanation
– we can simply treat s like any other
algebraic variable.
To transform the inductor mentioned
above to the s-domain, the time differential (d/dt) is simply replaced by s to
get the s-domain equivalent relationship v = i • L • s, which I hope you
can see is kind of analogous to Ohm’s
law. The complex impedance of the
inductor, indicated by Z(s), is therefore simply Ls.
Using a similar logic, we can arrive at
the s-domain expressions for the voltage/current relationships of common
parts, and hence their complex impedances, shown in Table 1.
To show just how useful this transformation is, we will use these complex impedances to calculate the transfer function of a simple RC low-pass
filter shown at the top of Fig.4. If you
substitute in the complex impedances from the table and use the normal
voltage divider equation, you get the
s-domain transfer function 1 ÷ (RCs +
1) after a little bit of algebraic manipulation. It’s that easy.
Poles and zeros
You can see that there must be a
particular value of complex frequency
s that makes the denominator of the
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