082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

474 Chapter 15 Applications of integration
y L
~
V S
y L R L
Thyristor Gate R G
V S
aT aT aT
t
Figure15.3
A thyristorfiringcircuit.
Figure15.4
Load voltage waveform.
Solution
The average value of load voltage is
∫
1 T
v
T L
dt= 1 ∫ T/2
( ) 2πt
V
T S
sin dt
T
0
= V S
T
αT
T
2π
[
−cos
( 2πt
T
)] T/2
αT
= V S
(1 +cos2πα)
2π
If α = 0, then the average value is V S
/π, the maximum value for this circuit. If
α = 0.25,thentheaveragevalueisV S
/2πwhichillustratesthatdelayingtheturning
on ofthe thyristor reduces the average value of the load voltage.
EXERCISES15.2
1 Calculate the averagevalue ofthe given functions
across the specified interval:
(a) f (t) = 1 +t across [0,2]
(b) f (x) = 2x −1 across [−1,1]
(c) f (t) =t 2 across [0,1]
(d) f (t) =t 2 across [0,2]
(e) f(z) =z 2 +zacross[1,3]
2 Calculate the averagevalue ofthe given functions
over the specified interval:
(a) f (x) =x 3 across[1,3]
(b) f(x) = 1 x across[1,2]
(c) f(t) = √ t across [0,2]
3
(d) f (z) =z 3 −1 across [−1,1]
(e) f(t) = 1 across[−3, −2]
t2 Calculate the average value ofthe following:
[ ]
(a) f (t) = sint across
0, π 2
(b) f (t) = sint across[0, π]
(c) f (t) = sin ωt across [0,π]
[ ]
(d) f (t) = cost across
0, π 2
(e) f (t) = cost across [0,π]
(f) f (t) = cos ωt across [0, π]
(g) f (t) = sin ωt +cos ωt across [0,1]
4 Calculate the average value ofthe following
functions:
(a) f(t) = √ t +1 across[0,3]
(b) f (t) = e t across[−1,1]
(c) f (t) = 1 +e t across[−1,1]