EE 255 - Chapter 3c - B1 = VM and 1 2 R 2C 2 B2 = RC VM 1 2...

86
1
2 2 2 1
M
V
B
R C ω
=
+
and
2
2 2 2 1
M
RC V
B
R C
ω
ω
-
=
+
Therefore, the forced solution to the equation is
( )
,
2 2 2 2 2 2
sin cos
1 1
M M
C f
V RC V
v t t t
R C R C
ω
ω ω
ω ω
= -
+ +
and the total solution is
( ) ( ) ( )
, , C C n C f
v t v t v t = +
( )
2 2 2 2 2 2
sin cos
1 1
t
M M
RC
C
V RC V
v t Ae t t
R C R C
ω
ω ω
ω ω
-
= + -
+ +
The initial condition for this problem is that the voltage on the capacitor is zero at the beginning of the half-
cycle:
( )
0
2 2 2 2 2 2
0 sin 0 cos 0 0
1 1
M M
RC
C
V RC V
v Ae
R C R C
ω
ω ω
-
= + - =
+ +
2 2 2
0
1
M
RC V
A
R C
ω
ω
- =
+
2 2 2 1
M
RC V
A
R C
ω
ω
=
+
Therefore, the voltage across the capacitor as a function of time before the PNPN diode fires is
( )
2 2 2 2 2 2 2 2 2
sin cos
1 1 1
t
M M M
RC
C
RC V V RC V
v t e t t
R C R C R C
ω ω
ω ω
ω ω ω
-
= + -
+ + +
If we substitute the known values for R , C , ω , and V
M
, this equation becomes
( )
83.3 35.76 7.91 sin 35.76 cos t
C
v t e t t ω ω - = + -
This equation is plotted below:

87
It reaches a voltage of 40 V at a time of 4.8 ms. Since the frequency of the waveform is 60 Hz, the
waveform there are 360 ° in 1/60 s, and the firing angle α is
( )
360
4.8 ms 103.7
1/60 s
α
°
= = ° or 1.810 radians
Note : This problem could also have been solved using Laplace Transforms, if desired.
(b) The rms voltage applied to the load is
/
2 2 2
rms
1
( ) sin
M
V v t dt V t dt
T
π ω
α
ω
ω
π
= =
/
2
rms
1 1
sin2
2 4
M
V
V t t
π ω
α
ω ω
π
= -
( ) ( )
g187
g188
g186
g171
g172
g170
- - - = α π α π
π
2 sin 2 sin
4
1
2
1
2
rms
M
V
V
Since α = 1.180 radians, the rms voltage is
rms
0.1753 0.419 71.0 V
M M
V V V = = =
3-14. Figure P3-9 shows a three-phase full-wave rectifier circuit supplying power to a dc load. The circuit uses
SCRs instead of diodes as the rectifying elements.
(a) What will the load voltage and ripple be if each SCR is triggered as soon as it becomes forward biased?
At what phase angle should the SCRs be triggered in order to operate this way? Sketch or plot the
output voltage for this case.
(b) What will the rms load voltage and ripple be if each SCR is triggered at a phase angle of 90 ° (that is,
half way through the half-cycle in which it is forward biased)? Sketch or plot the output voltage for
this case.

88
S
OLUTION
Assume that the three voltages applied to this circuit are:
( )
sin
A M
v t V t ω =
( ) ( )
sin 2 /3
B M
v t V t ω π = -
( ) ( )
sin 2 /3
C M
v t V t ω π = +
The period of the input waveforms is T , where 2 / T π ω = . For the purpose of the calculations in this
problem, we will assume that ω is 377 rad/s (60 Hz).
(a) The when the SCRs start to conduct as soon as they are forward biased, this circuit is just a three-
phase full-wave bridge, and the output voltage is identical to that in Problem 3-2. The sketch of output
voltage is reproduced below, and the ripple is 4.2%. The following table shows which SCRs must conduct
in what order to create the output voltage shown below. The times are expressed as multiples of the period
T of the input waveforms, and the firing angle is in degrees relative to time zero.