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Chapter 11
ANALYSIS OF A SIMPLE DRIVE SYSTEM
11.1 Introduction
In this chapter we will look at some basic drive implementations with a DC
machine as discussed in section 10.5 starting at page 272. Our aim is to arrive
at generic models of all major drive components, (excluding the DC machine
which has already been discussed) which we can then transpose to a Simulink
and/or Caspoc type environment in the tutorials at the end of this chapter.
In the sequel of the chapter a so-called ‘predictive dead-beat’ current control
algorithm will be presented [Svensson, 1988] which will allow precise torque
control of the DC machine used. The techniques described here are fundamental
not only to the DC machine but to all the machines discussed in this book.
11.2 Basic single phase uni-polar drive circuit
An elementary drive model as shown in figure 11.1 has almost the same
structure as the general drive model given in figure 1.2 on page 4. In this exam-
Figure 11.1.
Basic electrical drive
296
FUNDAMENTALS OF ELECTRICAL DRIVES
ple the mechanical ‘load’ module has been removed. Furthermore, the power
source is now shown in a two-wire configuration (+
,
) which is helpful here
because a symbolic implementation of the ‘converter’ module is shown. The
DC motor is represented as an
R
-
L
-
e
network as was discussed in section 10.5.2
on page 276. The purpose of the modulator is to control the converter switch
shown in figure 11.1 on the basis of a set-point given by the controller. A simple
controller structure is also given in figure 11.1 which shows a micro-processor
(
µ
P or DSP), which is a digital computational element that implements the con-
trol algorithm of the drive. The input to the control module is the load current
i
(
t
) which is obtained via a current sensor which measures the load current,
i.e. the armature current of the DC machine in this case. A user input value
i
∗
is also shown which represents the reference current level.
The aim of this drive circuit is to control the current in the motor in such a
way that the reference current value matches the actual load current under all
circumstances, i.e. transient as well as in steady-state. A typical situation to be
discussed is to apply a step change to the reference current and our aim is to
ensure that the load current will match this step change, within the limits of the
system. To achieve this aim we will need to initially discuss in some detail the
functioning of the modules shown in figure 11.1. Afterwards we will develop
a control structure which can be implemented in the micro-processor (
µ
Por
DSP) as to realize our task.
−
11.2.1 Power source
A DC voltage source is assumed here which has a value of
u
DC
. The bottom
side is set to 0V which means that the upper wire (red) shown in figure 11.1 has
a potential of
u
DC
. The voltage source is ‘uni-polar’ which means that there is
only one voltage level other than zero. At a later stage in this chapter we will
replace the power source by a bipolar power source which gives us a positive
and a negative supply voltage level with respect to 0V. The term ‘uni-polar
drive’ reflects the ability to operate with a variable but single positive supply
value.
11.2.2 Converter module
The converter module shown in figure 11.1 consists of a single two-way
switch. In reality such a switch is formed by two switches as shown in figure 11.2
which also gives the power source module. The switches are controlled by two
logic signals
Sw
t
,
Sw
b
where logic 1 corresponds to a ‘closed’ switch state
and logic 0 to an ‘open’ switch state. In this case there are four possible switch
combinations of switch states:
Sw
t
=1,
Sw
b
=1(both switches closed;
‘shoot-through’ mode, this state should always be avoided) and
Sw
t
=0,
Sw
b
=0(both switches open; ’idle’ mode, normally used to disable the inverter
297
Analysis of a simple Drive System
Figure 11.2.
Two switch
converter with power source
output), both are not considered in the following part. The remaining two states
are:
Sw
t
=1,
Sw
b
=0(top switch closed/bottom switch open),
Sw
t
=0,
Sw
b
=1(top switch open/bottom switch closed). In the first case (
Sw
t
=1,
Sw
b
=0), the converter output is connected to the positive supply line, i.e.
u
=
u
DC
, while in the second case
Sw
t
=0,
Sw
b
=1, the output line is
connected to the lower supply line, i.e.
u
=0. The two switches can therefore in
symbolic form be replaced by a single two-way switch as shown in figure 11.1,
where the logic signal
Sw
is used to control its state. The state
Sw
=1
corresponds to the switch in the ‘up’ state, i.e. the output voltage is given as
u
=
u
DC
. As expected, the switch state
Sw
=0corresponds to the switch in
the ‘down’ state, i.e. the output voltage is given as
u
=0.
11.2.3 Controller module
Today, the controller is in most cases digital. This means that the analog
input variables, here in the form of the measured current
i
(
t
) and user defined
reference current
i
∗
(
t
), need to be converted to a digital form. We have therefore
introduced in figure 11.1 a new building block in the form of an ‘analog-digital’
(A/D) converter.
The function of the unit is readily shown with the aid of figure 11.3: an input
function
x
(
t
) to the A/D converter. The diagram shows an example waveform
together with a set of discrete time points
t
k−
1
,
t
k
,
t
k
+1
where
k
can be any
integer value. The difference in time between any two time points is constant
and equal to the ‘sampling interval period’
T
s
. For drive systems the sampling
time is in the order of 100
µ
s, 1ms. The output of the converter module is such
that at these time points the input is ‘sampled’ i.e. the output is then set to
equal the input value. Hence, at these ‘sampling points’ the output changes to
match the instantaneous value found at the input of the converter. The output
is therefore held constant during the sampling time. For example, the output
x
(
t
k
)) represents the value of the input variable as sampled at the time mark
t
k
.
 298
FUNDAMENTALS OF ELECTRICAL DRIVES
Figure 11.3.
A/D converter unit with example input/output waveforms
The A/D units are used to sample the measured current and reference current
values. These inputs, at for example
t
k
, are then used by the micro-processor or
DSP to calculate an output variable known as the ‘reference incremental flux’
∆Ψ
∗
(
t
k
), which acts as an input to the modulator. We will define the variable
∆Ψ
∗
(
t
k
) in the next section.
11.2.4 Modulator module
The basic task of the modulator module is to control the switch or switches
of the converter module in such a way that the condition according to equa-
tion (11.1) is met (within the constraint of this unit) for each sampling interval.
∆Ψ
∗
(
t
k
)=∆Ψ(
t
k
)
(11.1)
where ∆Ψ (
t
k
) is known as the incremental flux level which is defined as
∆Ψ (
t
k
)=
t
k
+1
t
k
u
(
τ
)
dτ
(11.2)
The term
u
(
t
) shown in equation (11.2) represents the instantaneous voltage
across the load (output from the converter) within a sample period, in this
case between sample points
t
k
,t
k
+1
. We have in the past (see equation (2.7))
commented on the fact that it is the incremental flux which controls the current
in an inductance. The inductance forms a key element for our machine models
and it is therefore appropriate to work with the ‘incremental flux’ as a control
variable [Svensson, 1988]. Condition (11.1) in fact states that the modulator
should set the converter switches during each sampling interval in such a way
as to ensure that the reference incremental flux value at, for example, time
t
k
(as provided by the controller) matches the converter incremental flux value (as
defined by equation (11.2)).
We will now consider two basic ‘single edged’ modulation strategies by
examining the converter incremental flux as a function of the switch on/off
 299
Analysis of a simple Drive System
Figure 11.4.
Incremental
flux and output wave forms:
rising edge modulation
time within a sample interval
t
k
...t
k
+1
. We will in the first instance make use
of the converter configuration as shown in figure 11.1.
The so-called ‘rising edge’ type modulation strategy calls for the switch
Sw
to be placed in the ‘up’ (switch logical control level 1) position after a time
t
r
measured from the start of the sampling interval. The switch is placed in
the ‘down’ (switch logical control level 0) position at the end of each sampling
interval. An example of the output voltage waveform which appears as a re-
sult of this modulation strategy is given in figure 11.4 for the sampling interval
t
k
...t
k
+1
. Also shown in figure 11.4 is the incremental flux value as a func-
tion of the rise time
t
r
. This variable can change between zero and
T
s
.For
a particular value
t
r
we can evaluate the incremental flux by making use of
equation 11.2 which in this case gives the function ∆Ψ (
t
r
)=
u
DC
(
T
s
−
t
r
),
which is illustrated in figure 11.4. It is noted that this function repeats each
sample interval and this gives us the possibility to find the required
t
r
value for
a given incremental flux reference value.
The basic algorithm for finding the rise time
t
r
is based on the use of equa-
tion (11.1). Basically, we compare for each sample interval the required refer-
ence value with the incremental flux function ∆Ψ (
t
r
) and move the converter
switch to the ‘up’ position when the condition ∆Ψ
∗
≥
∆Ψ (
t
r
) is met. An
example as given in figure 11.5, shows two consecutive sampling intervals
where the reference incremental flux levels (as provided by the controller) are
takentobe∆Ψ
∗
(
t
k−
1
) and ∆Ψ
∗
(
t
k
) respectively.
The switching point for
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