758 BASIC CONTROL SYSTEMS

vs = Vm sin ωt

(a)

vL

vD

i D

R

+

−

+

+

−

− Figure 16.1.10Single-phase half-wave rectifier.(a)Circuit.

(b)Waveforms of voltages and current.

Vm

−Vm

ωt

vs

(b)

0

2

ππ 2 π

R

ωt

vD

(^0) π 2 π
Vm
ωt
vL
0
2
ππ 2 π
Vm
ωt
i
0
2
ππ 2 π
zero when the diode is said to be in ablocking condition. The waveforms of voltagesvS,vL,vD,
and currentiare shown in Figure 16.1.10(b).
The average voltageVdcis given by
Vdc=
1
T
∫T
0
vL(t) dt (16.1.1)
becausevL(t)=0 forT/ 2 ≤t≤T. In our case,
Vdc=
1
T
∫T/ 2
0
vmsinωt dt=−
Vm
ωT
(
cos
ωT
2
− 1
)
(16.1.2)
Using the relationshipsf= 1 /Tandω= 2 πf, we obtain
Vdc=
Vm
π
= 0. 318 Vm (16.1.3)
Idc=
Vdc
R

0. 318 Vm
   R
   (16.1.4)