Page 157 - Electric Drives and Electromechanical Systems

150 Electric Drives and Electromechanical Systems

The solution to these equations results in the triangular waveform which is shown in
Fig. 5.9 and is expressed as,

V s E m
i a ðtÞ¼ I a ð0Þþ t for t t t 1 (5.14a)
L a

V s E m
i a ðtÞ¼ I a ðt 1 Þþ ðt t 1 Þ for t 1 t t f (5.14b)
L a
During steady-state operations, the current is periodic and I(0) ¼ I(t f ), hence,

V s E m V s þ E m
t 1 ðt f t 1Þ (5.15)
L a L a
If Eq. (5.15) is combined with Eq. (5.6) and with,
E m
r ¼ (5.16)
V s
then the total current variation, D, can be determined to be,

V s t f 2
D ¼ I a ðt 1 I a ð0Þ¼ 1 r (5.17)
2L a
It should be recognised that this equation is equally applicable to both motor and
non-motor loads. The peak-current variation will occur when r ¼ 0, and it is given by

V s t f
D max ¼ [5.18]
2L a
where L a is the armature inductance, plus any addition inductance used to reduce the
form factor. If the amplifier is driving a non-motor load, L a is solely the load inductance.
Since the motor’s torque is a function of the average armature current, while armature
heating is a function of the r.m.s. value of the armature current, it is important to note
that with a bipolar servo amplifier, even at zero mean current, there is current flowing
through the motor, leading to armature heating. This has to be minimised to allow the
best possible frame size of motor to be selected. The quality of the waveform is measured
by the form factor, which is given by,

I rms
Form factor ¼ (5.19)
I average
For a switching amplifier with output waveforms as shown in Fig. 5.9:
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
D
I 2 þ
average 12 (5.20)
Form factor ¼
I average
and on substituting for D in Eq. (5.18) gives,
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2

V s (5.21)
Form factor ¼ 1 þ
6:9 L a f s I average