Transcript Rectifier Concepts

DC-DC Switch-Mode Supply
• Average output voltage
• Discontinuous or continuous mode of
operation
• Ripple output voltage
DC-DC Switched Converter
µ
Vi n =
=
¶
d
dv
v
L
C c + c + vc
dt
dt
R
d2 vc L dvc
LC 2 +
+ vc
dt
R dt
vc = Vi n + k1e¸ 1 t + k2e¸ 2 t
0
0
vc(0) = vc0 , and vc(0) = vc0
The equation when the switch is open is given by:
µ
¶
d
dvc vc
0 = L
C
+
+ vc
dt
dt
R
d2 vc L dvc
= LC 2 +
+ vc
dt
R dt
We can write the solution to the above equation as:
vc = k3 e¸ 1 t + k4 e¸ 2 t
0
We can solve for k3 and k4 using initial conditions: vc(Ton ), and vc(Ton ).
0
0
Finally vc (Tof f ) = vc0 , and vc (Tof f ) = vc0 .
T his gives us six equat ions t o solve in six unknowns:
vc0
=
Vi n + k1 + k2
vc0
=
¸ 1 k1 + ¸ 1 k 2
Vi n + k1 e¸ 1 t o n + k2 e¸ 2 t o n
=
k 3 + k4
¸ 1 k1 e¸ 1 t o n + ¸ 2 k2 e¸ 2 t o n
=
k3 ¸ 1 + k4 ¸ 2
vc0
=
k3 e¸ 1 t o f f + k4 e¸ 2 t o f f
=
k3 ¸ 1 e¸ 1 t o f f + k4 ¸ 2 e¸ 2 t o f f
0
0
vc0
We can use Maple to solve for the six constants. The script can be run in
Maple with:
restart; currentdir("C:\\wattle\\courses\\PowerElectronics\\maple\\"); read
"switchedDCrippleJuly2007.ma";
The pScpice script is: Fig7-4aDC2DC.sch
## Description: Step-down dc-dc converter Figure 7-4(a), page 165 - Power Electronics - Mohan, Undeland,
and Robbins.
## The pSpice file to simulate the same circuit is Fig7-4aDC2DC.sch in pSpice/switchedMode
##restart; currentdir("C:\\wattle\\courses\\PowerElectronics\\maple\\"); read "switchedDCrippleJuly2007.ma";
Vin:=20:
Dcyc:=0.5; #Duty Cycle
T := 20*10^(-6):
ton:=T*Dcyc:
toff:= (1-Dcyc)*T:
Rl:=100: Ls:=0.001: Cf:=0.0225*10^(-6);
#
lambda[1]:= ((-Ls/Rl)+sqrt(Ls*(Ls-4*Rl^2*Cf))/Rl)/(2*Ls*Cf): #roots of the characteristic equations
lambda[2]:= ((-Ls/Rl)-sqrt(Ls*(Ls-4*Rl^2*Cf))/Rl)/(2*Ls*Cf):
vcon:=t->Vin + k[1]*exp(lambda[1]*t) + k[2]*exp(lambda[2]*t);
vcond:=unapply(diff(vcon(t),t),t);
vcoff:=t->k[3]*exp(lambda[1]*t) + k[4]*exp(lambda[2]*t);
vcoffd:=unapply(diff(vcoff(t),t),t);
eq[1]:=vcon(0)=vc0;
eq[2]:=vcond(0)=vc0d;
eq[3]:=vcon(ton)=vcoff(0);
eq[4]:=vcond(ton)=vcoffd(0);
eq[5]:=vcoff(toff)=vc0;
eq[6]:=vcoffd(toff)=vc0d;
sol:=solve({eq[ieq] $ieq=1..6},{vc0,vc0d,k[ik] $ik=1..4});
VoltageRipple:=subs(sol,vcon(ton)-vcon(0));
Tutorial Problems
Feel free to solve these problems using
Maple, Maxima, or pSpice.