2.2.
Inductor volt-second balance, capacitor charge
balance, and the small ripple approximation
Actual output voltage waveform, buck converter
iL(t)
1
Buck converter
containing practical
low-pass filter
L
+ vL(t) –
Vg
+
iC(t)
2
+
–
C
R
v(t)
–
Actual output voltage
waveform
Actual waveform
v(t) = V + vripple(t)
v(t)
V
v(t) = V + vripple(t)
dc component V
0
t
Fundamentals of Power Electronics
7
Chapter 2: Principles of steady-state converter analysis
The small ripple approximation
Actual waveform
v(t) = V + vripple(t)
v(t)
v(t) = V + vripple(t)
V
dc component V
0
t
In a well-designed converter, the output voltage ripple is small. Hence,
the waveforms can be easily determined by ignoring the ripple:
vripple < V
v(t) ≈ V
Fundamentals of Power Electronics
8
Chapter 2: Principles of steady-state converter analysis
Buck converter analysis:
inductor current waveform
iL(t)
1
L
+ vL(t) –
original
converter
2
+
–
Vg
+
iC(t)
C
R
v(t)
–
switch in position 1
iL(t)
L
L
+ vL(t) –
Vg
+
–
switch in position 2
+
iC(t)
C
R
Vg
v(t)
+
–
iL(t)
iC(t)
C
R
v(t)
–
–
Fundamentals of Power Electronics
+
+ vL(t) –
9
Chapter 2: Principles of steady-state converter analysis
Inductor voltage and current
Subinterval 1: switch in position 1
iL(t)
Inductor voltage
L
+ vL(t) –
iC(t)
vL = Vg – v(t)
Vg
Small ripple approximation:
+
–
+
C
R
v(t)
–
vL ≈ Vg – V
Knowing the inductor voltage, we can now find the inductor current via
vL(t) = L
diL(t)
dt
Solve for the slope:
diL(t) vL(t) Vg – V
=
≈
L
L
dt
Fundamentals of Power Electronics
⇒ The inductor current changes with an
essentially constant slope
10
Chapter 2: Principles of steady-state converter analysis
Inductor voltage and current
Subinterval 2: switch in position 2
L
Inductor voltage
+
+ vL(t) –
vL(t) = – v(t)
Small ripple approximation:
Vg
+
–
iL(t)
iC(t)
C
R
v(t)
–
vL(t) ≈ – V
Knowing the inductor voltage, we can again find the inductor current via
vL(t) = L
diL(t)
dt
Solve for the slope:
diL(t)
≈– V
L
dt
Fundamentals of Power Electronics
⇒ The inductor current changes with an
essentially constant slope
11
Chapter 2: Principles of steady-state converter analysis
Inductor voltage and current waveforms
vL(t)
Vg – V
DTs
D'Ts
t
–V
Switch
position:
iL(t)
1
2
1
iL(DTs)
I
iL(0)
0
Fundamentals of Power Electronics
vL(t) = L
diL(t)
dt
∆iL
Vg – V
L
–V
L
DTs
12
Ts
t
Chapter 2: Principles of steady-state converter analysis
Determination of inductor current ripple magnitude
iL(t)
iL(DTs)
I
iL(0)
∆iL
Vg – V
L
0
–V
L
DTs
Ts
t
(change in iL) = (slope)(length of subinterval)
Vg – V
DTs
2∆iL =
L
⇒
Vg – V
L=
DTs
2∆iL
Vg – V
∆iL =
DTs
2L
Fundamentals of Power Electronics
13
Chapter 2: Principles of steady-state converter analysis
Inductor current waveform
during turn-on transient
iL(t)
Vg – v(t)
L
– v(t)
L
iL(Ts)
iL(0) = 0
0 DTs Ts
iL(nTs)
2Ts
nTs
iL((n + 1)Ts)
(n + 1)Ts
t
When the converter operates in equilibrium:
i L((n + 1)Ts) = i L(nTs)
Fundamentals of Power Electronics
14
Chapter 2: Principles of steady-state converter analysis
The principle of inductor volt-second balance:
Derivation
Inductor defining relation:
di (t)
vL(t) = L L
dt
Integrate over one complete switching period:
iL(Ts) – iL(0) = 1
L
Ts
vL(t) dt
0
In periodic steady state, the net change in inductor current is zero:
Ts
0=
vL(t) dt
0
Hence, the total area (or volt-seconds) under the inductor voltage
waveform is zero whenever the converter operates in steady state.
An equivalent form:
T
s
1
0=
v (t) dt = vL
Ts 0 L
The average inductor voltage is zero in steady state.
Fundamentals of Power Electronics
15
Chapter 2: Principles of steady-state converter analysis
Inductor volt-second balance:
Buck converter example
vL(t)
Vg – V
Inductor voltage waveform,
previously derived:
Total area λ
t
DTs
–V
Integral of voltage waveform is area of rectangles:
Ts
λ=
vL(t) dt = (Vg – V)(DTs) + ( – V)(D'Ts)
0
Average voltage is
vL = λ = D(Vg – V) + D'( – V)
Ts
Equate to zero and solve for V:
0 = DVg – (D + D')V = DVg – V
Fundamentals of Power Electronics
16
⇒
V = DVg
Chapter 2: Principles of steady-state converter analysis
The principle of capacitor charge balance:
Derivation
Capacitor defining relation:
dv (t)
iC(t) = C C
dt
Integrate over one complete switching period:
vC(Ts) – vC(0) = 1
C
Ts
iC(t) dt
0
In periodic steady state, the net change in capacitor voltage is zero:
0= 1
Ts
Ts
iC(t) dt = iC
0
Hence, the total area (or charge) under the capacitor current
waveform is zero whenever the converter operates in steady state.
The average capacitor current is then zero.
Fundamentals of Power Electronics
17
Chapter 2: Principles of steady-state converter analysis