DC/DC Converters

1.1 DC motors

1.1.1 Electromechanical model

A direct current (DC) machine consists of two distinct elements:

– an armature containing a coil, located at the rotor;

– a field coil (or magnets fulfilling the same function) at the stator.

The presence of the field coil means that the armature coil operates within a magnetic flux of ψf. When a current ia is supplied to the armature, is enabled electromechanical energy conversion by creating a motor torque of the form:

m=k.ψf.ia

where k is a constant which is characteristic of the machine. Furthermore, we note that the flux ψf is:

– either a function of the current iexc circulating in the field coil, if this exists (more precisely, a linear function for a non-saturated machine: ψf = Kϕ.iexc);

– or a constant ψf = Ψf in the case of an inductor using permanent magnets.

Due to energy conversion, it is possible to establish that the mechanical and electrical instantaneous powers are identical. Consequently, the machine holds an electromotive force (e.m.f.) ea such that:

a.ia=tm.ω

where ω is the rotation speed of the machine expressed in rad/s. Thus, we obtain:

a=k.ψf.ω

As the armature contains a wound coil, we inevitably encounter a resistance Ra and an inductance La: consequently, the equivalent electrical model of the DC motor is a series circuit (Ra, La, ea). For the purposes of our study in the remainder of this chapter, we will use the hypothesis of negligible resistance; consequently, the model of the machine is reduced to a series circuit (La, ea), and the mechanical inertia Jm of the motor is such that, on the scale of the switching period Td of the converter (i.e. operating period of the switch), the speed ω may be considered constant (slow evolution across a high number of switching periods – typically, the mechanical time constant of the machine will be around 100 times larger than Td). Finally, all of our converter studies will be carried out in a steady state, i.e. all of the electrical values (currents and voltages) will be periodical. This hypothesis allows us to postulate that as the voltage VL at the terminals of inductance L is written as a function of current IL as follows:

L(t)=LdILdt

then we must have

VL〉Td=1Td∫(Td)VL(t)dt=0.

1.1.2 Applications

Mechanical applications are characterized in terms of mechanical couple and speed reversibility; operations are qualified by the number of quadrants used in the torque/speed reference frame. As we see from equations [1.1] and [1.3], there is a direct correspondence between the torque/speed and current/voltage reference frames.

The cases encountered in practice correspond to five different types of converters (choppers):

– one-quadrant chopper, with a single rotation direction in a motor function (step-down chopper);

– one-quadrant chopper, with a single rotation direction in a generating function (step-up chopper);

– two-quadrant chopper, with a single rotation direction for motor or generator functions (current-reversible two-quadrant chopper);

– two-quadrant chopper, with two rotation directions but only one torque – motor or generator direction (voltage-reversible two-quadrant chopper);

– four-quadrant chopper, with two rotation directions and two torques – motor or generator directions (four quadrant, i.e. full bridge chopper).

1.2 Step-down chopper

1.2.1 Structure and general equation model

The structure of a step-down chopper is shown in Figure 1.4. This is a single quadrant converter, used to operate a DC motor in rotor mode with a single rotation direction.

The equation model of the circuit is based on two independent loops and one node:

=VT−Vd−Vd=VloadIload=IT+Id

1.2.2 Continuous conduction

In the case of continuous conduction, the operation of the converter over a switching period Td is split into two distinct phases, with durations of α.Td (T ON, D OFF) and (1 − α).Td (T OFF, D ON), respectively.

During the first phase, we control transistor switch-off. Consequently,

T=0

hence:

d=−E

As the voltage at the diode terminals is negative, the diode is switched off:

d=0

hence:

T=Iload

This also gives us:

load=E

In the case of an (La, ea) modeling of the DC motor, supposing that ea = Ea = cte, we may note:

adIloaddt=E−Ea

hence:

load(t)=Iload(0)+E−EaLa(t)

At the end of this phase, we may write:

load(α.Td)=Iload(0)+E−EaLa(α.Td)

and the value of Iload(0) is, in accordance with the definition of continuous conduction, non-null.

During the second phase, we control transistor switch-on. Hence:

T=0

At the start of this phase, as Iload is non-null and cannot be subject to discontinuity, the diode is switched on:

d=Iload

We, therefore, write:

d=0

and, in the load:

load=0

and:

T=E

The evolution of the current in the load is, therefore, written as:

load(t)=Iload(α.Td)−EaLa(t−α.Td)

Given that the circuit is operating in permanent mode, we note that the voltage at the terminals of the inductance La of the machine Vload − Ea presents an average value of zero for the switching period Td; hence:

Vload−Ea〉Td=1Td(∫0α.Td(E−Ea)dt+∫α.TdTd(−Ea)dt)=1Td(α.Td.(E−Ea)−(1−α).Td.(Ea))=0

hence:

a=α.E

Once this equation is established, we may rewrite the evolution of current Iload over the two phases of the switching period. For the first phase, the current...