Chopper With D.C Motor As Load

Chopper With D.C Motor As Load

Let V_dc = Input Voltage,

R_a = Armature Resistance,

L = Inductance of Armature Winding,

E = Back emf generated by the motor,

Th1 = Switching Elements

D1 = Freewheeling diode.

The operation of it can be classified into two categories:

a)During On time : The gate signal is provided to the gate terminal of the thyristor and it remains in conduction in the forward biased state.Input voltage appears across the output of the motor.

b)During off time : Thyristor is turned off by force commutation using external circuitry. And it acts as the open switch and input voltage isn’t supplied to the motor.

Mathematical Calculation:

1)During On Time

The equation can be written as:

$$\eqalign{
& Vdc - i\,Ra - {{Ldi} \over {dt}} - E = 0 \cr
& {{Vdc - E} \over {Ra}} = i\, + {L \over {Ra}}{{\,di} \over {\,dt}} \cr
& Suppose\;y = {{Vdc - E} \over {Ra}} \cr
& y - i = {L \over {Ra}}{{\,di} \over {\,dt}} \cr
& or,Ra\,dt = {{Ldi} \over {y - i}} \cr
& Integrating\;this\;equation,we\;get \cr
& {{Ra} \over L}t = \; - \ln (y - i)\; + \;k \cr
& From\;the\;figure,\;we\;can\;observe\;that\;at\;time\;t = 0,\;the\;current\;is\;Ia. \cr
& Therefore,\;k = \ln (y - IA.) \cr
& Subsituting\;k\;in\;above\;equation,\;we\;get \cr
& {{Ra} \over L}t = \; - \ln (y - i)\; + \;\ln (y - Ia) \cr
& {{Ra} \over L}t = \ln ({{y - Ia} \over {y - i}}) \cr
& solving\;it,we\;get \cr
& i = \;IA\;{e^{ - {{Ra} \over L}t}} + {{Vdc - E} \over {Ra}}(1 - {e^{ - {{Ra} \over L}t}}) - - - - - - - - - - - - (i) \cr} $$

2) During off Time

The load current for mode 2 can be found by solving the equation

$$\eqalign{
& i\,Ra + {{Ldi} \over {dt}} - E = 0(with\;initial\;condition\;Ib\,) \cr
& The\;value\;of\;IB\;can\;be\;{\mathop{\rm determined}\nolimits} \;by\;subsituting\;t\; = \;t2\;in\;eqn(i), \cr
& we\;get\;the\;equation\;when\;the\;switch\;is\;open: \cr
& i = \;IB{e^{ - {{Ra} \over L}t}}\; - {E \over R}(1 - {e^{ - {{Ra} \over L}t}}) \cr
& \cr} $$

Hence, in this way the equation for charging and discharging current can be determined. And the peak-to-peak ripple current can be determined as I_diff = I_B - I_a .

References

1) Department of Radioelectronics. Power System b. n.d.—. Power System Blockset. n.d. 15 05 2016. <http://radio.feld.cvut.cz/matlab/toolbox/powersys/stud_24a.gif>

2).B Classes. CHOPERS(DC-DC Converters). n.d. 10 05 2016. <http://www.bclasses.in/images/material/eee/powerelectronics/newdcchoppers/46.jpg>