0195136047.pdf

762 BASIC CONTROL SYSTEMS

Only thecontinuous conductionmode of operation of the drive for motoring and regenerative

braking will be considered here. The angleαcan be greater or less thanγ, which is the angle at

which the source voltagevsis equal to the back emfE,

γ=sin−^1

(

E

Vm

)

(16.1.21)

whereVmis the peak value of the supply voltage. For the case ofα<γ, waveforms are shown in

Figure 16.1.13(a) for the motoring operation. It is possible to turn on thyristorsT 1 andT 3 because

ia>0, even thoughvs<E. The same is true for thyristorsT 2 andT 4.

WhenT 1 andT 3 conduct during the intervalα≤ωt≤(π+α), the following volt–ampere

equation holds:

vs=E+iaRa+La

dia

dt

(16.1.22)

Multiplying both sides byiat, wheretis a small time interval, we obtain

vsiat=Eiat+i^2 aRat+Laia

(

dia

dt

)

t (16.1.23)

in which the terms can be identified as the energy supplied or consumed by the respective elements.

Figure 16.1.13(b) corresponds to the regenerative braking operation, in which

γ′=π−γ=π−sin−^1

(

|E|

Vm

)

(16.1.24)

andαcan be greater or less thanγ′.

WhenT 1 andT 3 conduct, Equation (16.1.22) describes the motor operation withE=Kmωm.

Thus,

va=La

dia

dt

+Raia+Kmωm=Vmsinωt (16.1.25)

Similarly, whenT 2 andT 4 conduct,

va=La

dia

dt

+Raia+Kmωm=−Vmsinωt (16.1.26)

whereωis the supply frequency,ωmis the motor speed, andKmis the motor back emf constant.

(a)

ωt

va

T 1 , T 3 T 2 , T 4

E

va

ia ia

ia

−vs vs

(^0) αγ ππ + α 2 π
(b)
ωt
vs
T 2 , T 4 T 1 , T 3 T 2 , T 4
E
−vs
ia
va va
(^02) π
γ′ γ
α π
Figure 16.1.13Continuous conduction mode of operation.(a)Motoring (case shown forα<γ).(b)re-
generative braking (case shown forα<γ′).