### Lecture 11 slide deck

```R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder The Modeling Approach
Extension of Chapter 3 Methods
Sketch the converter waveforms
– Including the switching transitions (idealizing assumptions
– In particular, sketch inductor voltage, capacitor current, and
input current waveforms
vL = 0, iC = 0, ig = Ig
Use the resulting equations to construct an equivalent
circuit model, as usual
2
Buck Converter Example
•
•
•
•
Ideal MOSFET, p–n diode with reverse recovery
Neglect semiconductor device capacitances, MOSFET
switching times, etc.
Neglect conduction losses
Neglect ripple in inductor current and capacitor voltage
3
Assumed
waveforms
Diode recovered charge Qr,
reverse recovery time tr
These waveforms assume
that the diode voltage
changes at the end of the
reverse recovery transient
• a “snappy” diode
• Voltage of soft-recovery
diodes changes sooner
estimate of induced
switching loss
4
Combine for complete model
The two independent current sources consume power
Vg (trIL /Ts + Qr /Ts)
equal to the switching loss induced by diode reverse recovery
9
Solution of model
Output:
V = DVg
Efﬁciency:
= Pout / Pin
Pout = VIL
Pin = Vg (DIL + trIL /Ts + Qr /Ts)
Combine and simplify:
= 1 / [1 + fs (tr /D + Qr R /D2Vg )]
10
Predicted Efficiency vs Duty Cycle
Switching frequency 100 kHz
Input voltage 24 V
Recovered charge 0.75 µCoul
Reverse recovery time 75 nsec
Buck converter with diode reverse recovery
100.00%
90.00%
80.00%
(no attempt is made here to
model how the reverse
recovery process varies with
inductor current)
efﬁciency
• Poor efﬁciency at low duty
cycle
Efficiency
70.00%
60.00%
50.00%
40.00%
30.00%
20.00%
10.00%
0.00%
0
0.2
0.4
0.6
Duty cycle
11
0.8
1
Boost Converter Example
Model same effects as in previous buck converter example:
• Ideal MOSFET, p–n diode with reverse recovery
• Neglect semiconductor device capacitances, MOSFET
switching times, etc.
• Neglect conduction losses
• Neglect ripple in inductor current and capacitor voltage
12
Boost
converter
Transistor and diode
waveforms have same
shapes as in buck
example, but depend
on different quantities
13
Inductor volt-second balance
and average input current
As usual: vL = 0 = Vg – D V
Also as usual: ig = IL
14
Capacitor
charge balance
iC = id – V/R = 0
= – V/R + IL(D Ts – tr)/Ts – Qr /Ts
Collect terms: V/R = IL(D Ts – tr)/Ts – Qr /Ts
15
Construct model
The result is:
The two independent current sources consume power
V (trIL /Ts + Qr /Ts)
equal to the switching loss induced by diode reverse recovery
16
Predicted V/Vg vs duty cycle
Boost converter with diode reverse recovery
8
7
With RL only
6
5
V/Vg
Switching frequency 100 kHz
Input voltage 24 V
Recovered charge 5 µCoul
Reverse recovery time 100 nsec
Inductor resistance RL = 0.3
(inductor resistance also inserted
into averaged model here)
4
3
2
1
With RL and diode reverse recovery
0
0
0.2
0.4
0.6
Duty cycle
17
0.8
1
Summary
The averaged modeling approach can be extended to
include effects of switching loss
Transistor and diode waveforms are constructed,
including the switching transitions. The effects of the
switching transitions on the inductor, capacitor, and
input current waveforms can then be determined
Inductor volt-second balance and capacitor charge
balance are applied
Converter input current is averaged
Equivalent circuit corresponding to the the averaged
equations is constructed
18
4.2.1. Power diodes!
A power diode, under reverse-biased conditions:!
v
+
–
p
+
+
+
{ {
low doping concentration
n-
–
E
v
+
n
–
–
–
depletion region, reverse-biased
Fundamentals of Power Electronics!
9!
Chapter 4: Switch realization!
Forward-biased power diode!
v
+
–
i
{
conductivity modulation
n-
p
+
–
+
+
+
n
+
–
+
–
minority carrier injection
Fundamentals of Power Electronics!
10!
Chapter 4: Switch realization!
Diode in OFF state:"
reversed-biased, blocking voltage!
v(t)
v
+
–
t
n–
p
n
E
v
i(t)
+
0
{
–
t
Depletion region, reverse-biased
•  Diode is reverse-biased!
•  No stored minority charge: q = 0!
(1)
•  Depletion region blocks applied
reverse voltage; charge is stored in
capacitance of depletion region!
Fundamentals of Power Electronics!
12!
Chapter 4: Switch realization!
Turn-on transient!
v(t)
Diode conducts with low on-resistance
t
•  charge to increase
voltage across
depletion region!
Diode is forward-biased. Supply minority
charge to n– region to reduce on-resistance
Charge depletion region
i(t)
On-state current determined by converter circuit
t
(1)
(2)
Fundamentals of Power Electronics!
13!
The current i(t) is
determined by the
converter circuit. This
current supplies: !
•  charge needed to
support the on-state
current!
•  charge to reduce onresistance of n–
region!
Chapter 4: Switch realization!
Turn-off transient!
v
+
–
i (< 0)
n-
p
n
+
+
+
+
+
+
+
+
}
Removal of stored minority charge q
Fundamentals of Power Electronics!
14!
Chapter 4: Switch realization!
Diode turn-off transient"
continued!
v(t)
t
(4) Diode remains forward-biased.
Remove stored charge in n– region
(5) Diode is reverse-biased.
Charge depletion region
capacitance.
i(t)
tr
0
t
di
dt
Area
–Qr
(1)
(2)
Fundamentals of Power Electronics!
(3)
(4)
15!
(5)
(6)
Chapter 4: Switch realization!
The diode switching transients induce
switching loss in the transistor!
iA
+
vA
fast
transistor
+
–
–
+
–
Vg
iL(t)
–
vB
+
L
Qr
Vg
silicon
diode
0
0
t
diode
waveforms
•  Diode recovered stored charge
Qr flows through transistor
during transistor turn-on
transition, inducing switching
loss!
•  Qr depends on diode on-state
forward current, and on the
rate-of-change of diode current
during diode turn-off transition!
iL
vA(t)
iB
Fundamentals of Power Electronics!
see Section 4.3.2!
iA(t)
transistor
waveforms
iL
iB(t)
vB(t)
0
0
t
area
–Qr
–Vg
tr
pA(t)
= vA iA
area
~QrVg
area
~iLVgtr
16!
t0
t1 t2
t
Chapter 4: Switch realization!
Types of power diodes!
Standard recovery!
Reverse recovery time not specified, intended for 50/60Hz!
Fast recovery and ultra-fast recovery!
Reverse recovery time and recovered charge specified!
Intended for converter applications!
Schottky diode!
A majority carrier device!
Essentially no recovered charge!
Model with equilibrium i-v characteristic, in parallel with
depletion region capacitance!
Restricted to low voltage (few devices can block 100V or more)!
Fundamentals of Power Electronics!
18!
Chapter 4: Switch realization!
Paralleling diodes!
Attempts to parallel diodes, and share the
current so that i1 = i2 = i/2, generally don t
work.!
i
i1
i2
Reason: thermal instability caused by
temperature dependence of the diode
equation.!
+
+
v1
v2
current, or reduced voltage.!
–
–
One diode will hog the current.!
To get the diodes to share the current, heroic
measures are required:!
•  Select matched devices!
•  Package on common thermal substrate!
•  Build external circuitry that forces the currents to balance!
Fundamentals of Power Electronics!
20!
Chapter 4: Switch realization!
Ringing induced by diode stored charge!
see Section 4.3.3!
iL(t)
L
vi(t)
+
–
vL(t)
vi(t) +
–
silicon
diode
iB(t)
+
vB(t)
–
V1
t
0
–V2
C
iL(t)
•  Diode is forward-biased while iL(t) > 0!
•  Negative inductor current removes diode
stored charge Qr!
•  When diode becomes reverse-biased,
negative inductor current flows through
capacitor C.!
•  Ringing of L-C network is damped by
parasitic losses. Ringing energy is lost.!
Fundamentals of Power Electronics!
21!
0
t
area
– Qr
vB(t)
t
0
–V2
t1
t2
t3
Chapter 4: Switch realization!
Energy associated with ringing!
Recovered charge is!
Qr = –
t3
t2
iL(t) dt
vi(t)
V1
t
0
Energy stored in inductor during interval
t2 ≤ t ≤ t3 :!
t3
WL =
vL(t) iL(t) dt
t2
Applied inductor voltage during interval
t2 ≤ t ≤ t3 :!
di (t)
vL(t) = L L = – V2
dt
Hence,!
t3
t3
di (t)
WL =
L L iL(t) dt =
( – V2) iL(t) dt
dt
t2
t2
–V2
iL(t)
0
vB(t)
t
0
–V2
W L = 12 L i 2L(t 3) = V2 Qr
t1
Fundamentals of Power Electronics!
t
area
– Qr
22!
t2
t3
Chapter 4: Switch realization!
```