Chapter 19
Resonant Conversion
Introduction
19.1
Sinusoidal analysis of resonant converters
19.2
Examples
Series resonant converter
Parallel resonant converter
19.3
Exact characteristics of the series and parallel resonant
converters
19.4
Soft switching
Zero current switching
Zero voltage switching
The zero voltage transition converter
19.5
Load-dependent properties of resonant converters
Fundamentals of Power Electronics
1
Chapter 19: Resonant Conversion
Introduction to Resonant Conversion
Resonant power converters contain resonant L-C networks whose
voltage and current waveforms vary sinusoidally during one or more
subintervals of each switching period. These sinusoidal variations are
large in magnitude, and the small ripple approximation does not apply.
Some types of resonant converters:
• Dc-to-high-frequency-ac inverters
• Resonant dc-dc converters
• Resonant inverters or rectifiers producing line-frequency ac
Fundamentals of Power Electronics
2
Chapter 19: Resonant Conversion
A basic class of resonant inverters
NS
NT
is(t)
Basic circuit
+
dc
source
vg(t)
+
–
vs(t)
i(t)
L
Cs
+
Cp
v(t)
–
–
Switch network
Resistive
load
R
Resonant tank network
Several resonant tank networks
L
Cs
L
L
Cp
Series tank network
Fundamentals of Power Electronics
Parallel tank network
3
Cs
Cp
LCC tank network
Chapter 19: Resonant Conversion
Tank network responds only to fundamental
component of switched waveforms
Switch
output
voltage
spectrum
fs
3fs
5fs
f
Resonant
tank
response
fs
3fs
5fs
f
fs
3fs
5fs
f
Tank current and output
voltage are essentially
sinusoids at the switching
frequency fs.
Output can be controlled
by variation of switching
frequency, closer to or
away from the tank
resonant frequency
Tank
current
spectrum
Fundamentals of Power Electronics
4
Chapter 19: Resonant Conversion
Derivation of a resonant dc-dc converter
Rectify and filter the output of a dc-high-frequency-ac inverter
Transfer function
H(s)
is(t)
+
+
dc
source +
–
vg(t)
L
Cs
+
vR(t)
vs(t)
v(t)
R
–
–
NS
Switch network
i(t)
iR(t)
–
NT
Resonant tank network
NR
NF
Rectifier network Low-pass dc
filter
load
network
The series resonant dc-dc converter
Fundamentals of Power Electronics
5
Chapter 19: Resonant Conversion
A series resonant link inverter
Same as dc-dc series resonant converter, except output rectifiers are
replaced with four-quadrant switches:
i(t)
+
L
Cs
dc
source +
–
vg(t)
v(t)
R
–
Switch network
Fundamentals of Power Electronics
Resonant tank network
6
Switch network
Low-pass ac
filter
load
network
Chapter 19: Resonant Conversion
Quasi-resonant converters
In a conventional PWM
converter, replace the
PWM switch network
with a switch network
containing resonant
elements.
Buck converter example
i1(t)
+
vg(t) +
–
v1(t)
Two
switch
networks:
+
Switch
network
C
v2(t)
R
v(t)
–
–
ZCS quasi-resonant
switch network
PWM switch network
i2(t)
i1(t)
+
+
+
v1(t)
v2(t)
v1(t)
–
–
–
Fundamentals of Power Electronics
i(t)
+
–
i1(t)
L
i2(t)
7
Lr
Cr
i2(t)
+
v2(t)
–
Chapter 19: Resonant Conversion
Resonant conversion: advantages
The chief advantage of resonant converters: reduced switching loss
Zero-current switching
Zero-voltage switching
Turn-on or turn-off transitions of semiconductor devices can occur at
zero crossings of tank voltage or current waveforms, thereby reducing
or eliminating some of the switching loss mechanisms. Hence
resonant converters can operate at higher switching frequencies than
comparable PWM converters
Zero-voltage switching also reduces converter-generated EMI
Zero-current switching can be used to commutate SCRs
In specialized applications, resonant networks may be unavoidable
High voltage converters: significant transformer leakage
inductance and winding capacitance leads to resonant network
Fundamentals of Power Electronics
8
Chapter 19: Resonant Conversion
Resonant conversion: disadvantages
Can optimize performance at one operating point, but not with wide
range of input voltage and load power variations
Significant currents may circulate through the tank elements, even
when the load is disconnected, leading to poor efficiency at light load
Quasi-sinusoidal waveforms exhibit higher peak values than
equivalent rectangular waveforms
These considerations lead to increased conduction losses, which can
offset the reduction in switching loss
Resonant converters are usually controlled by variation of switching
frequency. In some schemes, the range of switching frequencies can
be very large
Complexity of analysis
Fundamentals of Power Electronics
9
Chapter 19: Resonant Conversion
Resonant conversion: Outline of discussion
• Simple steady-state analysis via sinusoidal approximation
• Simple and exact results for the series and parallel resonant
converters
• Mechanisms of soft switching
• Circulating currents, and the dependence (or lack thereof) of
conduction loss on load power
• Quasi-resonant converter topologies
• Steady-state analysis of quasi-resonant converters
• Ac modeling of quasi-resonant converters via averaged switch
modeling
Fundamentals of Power Electronics
10
Chapter 19: Resonant Conversion
19.1 Sinusoidal analysis of resonant converters
A resonant dc-dc converter:
Transfer function
H(s)
is(t)
+
+
dc
source +
–
vg(t)
L
Cs
+
vR(t)
vs(t)
v(t)
R
–
–
NS
Switch network
i(t)
iR(t)
–
NT
Resonant tank network
NR
NF
Rectifier network Low-pass dc
filter
load
network
If tank responds primarily to fundamental component of switch
network output voltage waveform, then harmonics can be neglected.
Let us model all ac waveforms by their fundamental components.
Fundamentals of Power Electronics
11
Chapter 19: Resonant Conversion
The sinusoidal approximation
Switch
output
voltage
spectrum
fs
3fs
5fs
f
Resonant
tank
response
fs
3fs
5fs
f
Tank
current
spectrum
Tank current and output
voltage are essentially
sinusoids at the switching
frequency fs.
Neglect harmonics of
switch output voltage
waveform, and model only
the fundamental
component.
Remaining ac waveforms
can be found via phasor
analysis.
fs
Fundamentals of Power Electronics
3fs
5fs
12
f
Chapter 19: Resonant Conversion
19.1.1 Controlled switch network model
4
π Vg
NS
Vg
is(t)
1
vg
+
–
Fundamental component
vs1(t)
vs(t)
+
t
2
2
vs(t)
–
1
– Vg
Switch network
If the switch network produces a
square wave, then its output
voltage has the following Fourier
series:
4Vg
vs(t) = π
Σ 1n sin (nωst)
The fundamental component is
4Vg
vs1(t) = π sin (ωst) = Vs1 sin (ωst)
So model switch network output port
with voltage source of value vs1(t)
n = 1, 3, 5,...
Fundamentals of Power Electronics
13
Chapter 19: Resonant Conversion
Model of switch network input port
Is1
NS
is(t)
1
vg
+
–
2
ig(t)
+
2
vs(t)
ω st
–
is(t)
1
ϕs
Switch network
Assume that switch network
output current is
i g(t) T = 2
Ts
s
i s(t) ≈ I s1 sin (ωst – ϕ s)
i g(τ)dτ
0
T /2
s
≈ 2
I s1 sin (ωsτ – ϕ s)dτ
Ts 0
2 I cos (ϕ )
=π
s1
s
It is desired to model the dc
component (average value)
of the switch network input
current.
Fundamentals of Power Electronics
T s/2
14
Chapter 19: Resonant Conversion
Switch network: equivalent circuit
+
vg
2I s1
π cos (ϕ s)
vs1(t) =
4Vg
π sin (ωst)
is1(t) =
Is1 sin (ωst – ϕs)
+
–
–
• Switch network converts dc to ac
• Dc components of input port waveforms are modeled
• Fundamental ac components of output port waveforms are modeled
• Model is power conservative: predicted average input and output
powers are equal
Fundamentals of Power Electronics
15
Chapter 19: Resonant Conversion
19.1.2 Modeling the rectifier and capacitive
filter networks
| iR(t) |
iR(t)
+
i(t)
+
vR(t)
v(t)
–
–
NR
Rectifier network
V
vR(t)
ωst
iR(t)
R
NF
Low-pass
filter
network
dc
load
–V
ϕR
Assume large output filter
capacitor, having small ripple.
If iR(t) is a sinusoid:
vR(t) is a square wave, having
zero crossings in phase with tank
output current iR(t).
Then vR(t) has the following
Fourier series:
∞
4V
1 sin (nω t – ϕ )
vR(t) = π
Σ
s
R
n
n = 1, 3, 5,
Fundamentals of Power Electronics
i R(t) = I R1 sin (ωst – ϕ R)
16
Chapter 19: Resonant Conversion
Sinusoidal approximation: rectifier
Again, since tank responds only to fundamental components of applied
waveforms, harmonics in vR(t) can be neglected. vR(t) becomes
vR1(t) = 4V
π sin (ωst – ϕ R) = V R1 sin (ωst – ϕ R)
Actual waveforms
V
with harmonics ignored
4
πV
vR(t)
ωst
iR(t)
vR1(t)
fundamental
ωst
iR1(t)
vR1(t)
Re
Re = 82 R
π
i R1(t) =
–V
ϕR
ϕR
Fundamentals of Power Electronics
17
Chapter 19: Resonant Conversion
Rectifier dc output port model
| iR(t) |
iR(t)
+
i(t)
+
vR(t)
v(t)
–
R
Output capacitor charge balance: dc
load current is equal to average
rectified tank output current
i R(t)
–
NR
NF
Rectifier network
V
Low-pass
filter
network
dc
load
vR(t)
Ts
=I
Hence
T s/2
I= 2
TS 0
2I
=π
R1
I R1 sin (ωst – ϕ R) dt
ωst
iR(t)
–V
ϕR
Fundamentals of Power Electronics
18
Chapter 19: Resonant Conversion
Equivalent circuit of rectifier
iR1(t)
Rectifier input port:
+
Fundamental components of
current and voltage are
sinusoids that are in phase
vR1(t)
Hence rectifier presents a
resistive load to tank network
+
Re
–
2
π I R1
V
R
–
Re = 82 R
π
Effective resistance Re is
Re =
I
vR1(t) 8 V
=
i R(t) π 2 I
Rectifier equivalent circuit
With a resistive load R, this becomes
Re = 82 R = 0.8106R
π
Fundamentals of Power Electronics
19
Chapter 19: Resonant Conversion
19.1.3 Resonant tank network
Transfer function
H(s)
is1(t)
vs1(t)
+
–
Zi
iR1(t)
+
Resonant
network
vR1(t)
Re
–
Model of ac waveforms is now reduced to a linear circuit. Tank
network is excited by effective sinusoidal voltage (switch network
output port), and is load by effective resistive load (rectifier input port).
Can solve for transfer function via conventional linear circuit analysis.
Fundamentals of Power Electronics
20
Chapter 19: Resonant Conversion
Solution of tank network waveforms
Transfer function:
Transfer function
H(s)
vR1(s)
= H(s)
vs1(s)
is1(t)
Ratio of peak values of input and
output voltages:
VR1
= H(s)
Vs1
vs1(t)
+
–
Zi
iR1(t)
+
Resonant
network
vR1(t)
Re
–
s = jω s
Solution for tank output current:
i R(s) =
vR1(s) H(s)
=
v (s)
Re
Re s1
which has peak magnitude
H(s) s = jω
s
I R1 =
Vs1
Re
Fundamentals of Power Electronics
21
Chapter 19: Resonant Conversion
19.1.4 Solution of converter
voltage conversion ratio M = V/Vg
Transfer function
H(s)
is1(t)
Vg
+
–
+
–
iR1(t)
+
Resonant
network
Zi
vR1(t)
I
+
Re
2
π I R1
–
2I s1
π cos (ϕ s)
M= V = R
Vg
V
I
2
π
I
I R1
Fundamentals of Power Electronics
vs1(t) =
I R1
VR1
R
–
Re = 82 R
π
4Vg
π sin (ωst)
1
Re
V
H(s)
s = jω s
VR1
Vs1
22
4
π
Vs1
Vg
Eliminate Re:
V = H(s)
Vg
s = jω s
Chapter 19: Resonant Conversion
Conversion ratio M
V = H(s)
Vg
s = jω s
So we have shown that the conversion ratio of a resonant converter,
having switch and rectifier networks as in previous slides, is equal to
the magnitude of the tank network transfer function. This transfer
function is evaluated with the tank loaded by the effective rectifier
input resistance Re.
Fundamentals of Power Electronics
23
Chapter 19: Resonant Conversion
19.2 Examples
19.2.1 Series resonant converter
transfer function
H(s)
is(t)
+
dc
source +
–
vg(t)
L
+
Cs
vR(t)
vs(t)
Fundamentals of Power Electronics
v(t)
R
–
–
NS
switch network
i(t)
+
iR(t)
–
NT
resonant tank network
24
NR
NF
rectifier network low-pass dc
filter
load
network
Chapter 19: Resonant Conversion
Model: series resonant converter
transfer function H(s)
L
is1(t)
Vg
+
–
+
–
C
Zi
iR1(t)
+
vR1(t)
I
+
Re
2
π I R1
V
–
2I s1
π cos (ϕ s)
vs1(t) =
4Vg
π sin (ωst)
Re
Re
=
Z i(s) R + sL + 1
e
sC
s
Q eω 0
=
2
s
1+
+ ωs
Q eω 0
0
H(s) =
Fundamentals of Power Electronics
series tank network
R
–
Re = 82 R
π
1 = 2π f
0
LC
L
R0 =
M = H( jωs) =
C
R
Qe = 0
Re
ω0 =
25
1
1+Q
2
e
1 –F
F
2
Chapter 19: Resonant Conversion
Construction of Zi
|| Zi ||
1
ωC
ωL
f0
R0
Qe = R0 / Re
Re
Fundamentals of Power Electronics
26
Chapter 19: Resonant Conversion
Construction of H
|| H ||
1
Qe = Re / R0
Re / R0
f0
C
ωR e
Fundamentals of Power Electronics
27
R /
e ω
L
Chapter 19: Resonant Conversion
19.2.2 Subharmonic modes of the SRC
switch
output
voltage
spectrum
Example: excitation of
tank by third harmonic of
switching frequency
fs
3fs
5fs
f
resonant
tank
response
Can now approximate vs(t)
by its third harmonic:
4Vg
vs(t) ≈ vsn(t) = nπ sin (nωst)
fs
3fs
5fs
f
tank
current
spectrum
Result of analysis:
H( jnωs)
V
M=
=
n
Vg
fs
Fundamentals of Power Electronics
3fs
5fs
28
f
Chapter 19: Resonant Conversion
Subharmonic modes of SRC
M
1
1
3
1
5
etc.
1 f
5 0
Fundamentals of Power Electronics
1 f
3 0
29
f0
fs
Chapter 19: Resonant Conversion
19.2.3 Parallel resonant dc-dc converter
is(t)
+
L
+
dc
source +
–
vg(t)
+
Cp
vs(t)
vR(t)
v(t)
R
–
–
NS
switch network
i(t)
iR(t)
–
NT
resonant tank network
NR
NF
rectifier network
low-pass filter
network
dc
load
Differs from series resonant converter as follows:
Different tank network
Rectifier is driven by sinusoidal voltage, and is connected to
inductive-input low-pass filter
Need a new model for rectifier and filter networks
Fundamentals of Power Electronics
30
Chapter 19: Resonant Conversion
Model of uncontrolled rectifier
with inductive filter network
I
iR(t)
i(t)
iR(t)
+
+
ωst
vR(t)
vR(t)
v(t)
R
–
–
–I
NR
k
ϕR
4
πI
NF
rectifier network
low-pass filter
network
dc
load
iR1(t)
fundamental
Fundamental component of iR(t):
vR1(t)
i R1(t) = 4I
π sin (ωst – ϕ R)
ωst
vR1(t)
Re
2
Re = π R
8
i R1(t) =
ϕR
Fundamentals of Power Electronics
31
Chapter 19: Resonant Conversion
Effective resistance Re
Again define
Re =
vR1(t) πVR1
=
4I
i R1(t)
In steady state, the dc output voltage V is equal to the average value
of | vR |:
V= 2
TS
T s/2
0
2V
VR1 sin (ωst – ϕ R) dt = π
R1
For a resistive load, V = IR. The effective resistance Re can then be
expressed
2
π
Re =
R = 1.2337R
8
Fundamentals of Power Electronics
32
Chapter 19: Resonant Conversion
Equivalent circuit model of uncontrolled rectifier
with inductive filter network
iR1(t)
I
+
+
2V
π R1
Re
vR1(t)
–
+
–
V
R
–
2
π
Re =
R
8
Fundamentals of Power Electronics
33
Chapter 19: Resonant Conversion
Equivalent circuit model
Parallel resonant dc-dc converter
transfer function H(s)
is1(t)
iR1(t)
+
L
Vg
+
–
+
–
Zi
C
vR1(t)
I
+
Re
2
π V R1
+
–
–
2I s1
π cos (ϕ s)
vs1(t) =
4Vg
π sin (ωst)
M = V = 82 H(s)
Vg π
parallel tank network
R
–
2
Re = π R
8
H(s) =
s = jω s
V
Z o(s)
sL
Z o(s) = sL || 1 || Re
sC
Fundamentals of Power Electronics
34
Chapter 19: Resonant Conversion
Construction of Zo
|| Zo ||
Re
Qe = Re / R0
R0
f0
ωL
Fundamentals of Power Electronics
1
ωC
35
Chapter 19: Resonant Conversion
Construction of H
|| H ||
Re / R0
Qe = Re / R0
1
f0
1
ω 2LC
Fundamentals of Power Electronics
36
Chapter 19: Resonant Conversion
Dc conversion ratio of the PRC
Z o(s)
8
M= 2
sL
π
= 82
π
= 82
π 1+
s = jω s
1
s + s
ω0
Q eω 0
2
s = jω s
1
1–F
2 2
+ F
Qe
2
Re R
8
M= 2
=
π R0 R0
At resonance, this becomes
• PRC can step up the voltage, provided R > R0
• PRC can produce M approaching infinity, provided output current is
limited to value less than Vg / R0
Fundamentals of Power Electronics
37
Chapter 19: Resonant Conversion
19.3 Exact characteristics of the
series and parallel resonant dc-dc converters
Define
f0
f0
< fs <
k+1
k
1 <F< 1
k+1
k
or
1 + ( – 1) k
ξ=k+
2
subharmonic index ξ
ξ=3
etc. k = 3
mode index k
ξ=1
k=2
f0 / 3
Fundamentals of Power Electronics
k=1
f0 / 2
k=0
fs
f0
38
Chapter 19: Resonant Conversion
19.3.1 Exact characteristics of the
series resonant converter
Q1
D1
Q3
L
C
D3
1:n
+
Vg
+
–
R
Q2
D2
Q4
V
–
D4
Normalized load voltage and current:
M= V
nVg
Fundamentals of Power Electronics
J=
39
InR0
Vg
Chapter 19: Resonant Conversion
Continuous conduction mode, SRC
Tank current rings continuously for entire length of switching period
Waveforms for type k CCM, odd k :
vs(t)
Vg
– Vg
iL(t)
Q1
π
Q1
Q1
π
π
D1
ωst
D1
Q2
(k – 1) complete half-cycles
γ
Fundamentals of Power Electronics
40
Chapter 19: Resonant Conversion
Series resonant converter
Waveforms for type k CCM, even k :
vs(t)
Vg
– Vg
iL(t)
Q1
D1
π
π
π
D1
Q1
ωst
D2
Q2
D1
k complete half-cycles
γ
Fundamentals of Power Electronics
41
Chapter 19: Resonant Conversion
Exact steady-state solution, CCM
Series resonant converter
M ξ sin
2 2
2
γ
Jγ
1
+ 2
+ (– 1) k
2
ξ 2
2
cos
2
γ
=1
2
where
M= V
nVg
γ=
InR0
J=
Vg
ω0Ts π
=
F
2
• Output characteristic, i.e., the relation between M and J, is elliptical
• M is restricted to the range
0≤M ≤ 1
ξ
Fundamentals of Power Electronics
42
Chapter 19: Resonant Conversion
Control-plane characteristics
For a resistive load, eliminate J and solve for M vs. γ
M=
ξ tan
4
2
Qγ
2
γ
Qγ
+
2
2
γ
ξ – cos
2
2
2
(–1) k+1 +
1+
Qγ
γ
ξ tan
+
2
2
4
2
Qγ
2
2
2
2
cos 2
γ
2
Exact, closed-form, valid for any CCM
Fundamentals of Power Electronics
43
Chapter 19: Resonant Conversion
Discontinuous conduction mode
Type k DCM: during each half-switching-period, the tank rings for k
complete half-cycles. The output diodes then become reverse-biased
for the remainder of the half-switching-period.
vs(t)
Vg
– Vg
iL(t)
Q1
π
Q1
π
π
D1
ωst
X
Q2
k complete half-cycles
γ
Fundamentals of Power Electronics
44
Chapter 19: Resonant Conversion
Steady-state solution: type k DCM, odd k
M=1
k
Conditions for operation in type k DCM, odd k :
f0
fs <
k
2(k – 1)
2(k + 1)
>J>
γ
γ
Fundamentals of Power Electronics
45
Chapter 19: Resonant Conversion
Steady-state solution: type k DCM, even k
J = 2k
γ
Conditions for operation in type k DCM, even k :
f0
fs <
k
1 >M> 1
k–1
k+1
Ig = gV
gyrator model, SRC
operating in an even
DCM:
+
g
Vg
46
+
V
g = 2k
γR0
–
Fundamentals of Power Electronics
Ig = gVg
–
Chapter 19: Resonant Conversion
Control plane characteristics, SRC
1
Q = 0.2
0.9
Q = 0.2
0.8
0.35
M = V / Vg
0.7
0.5
0.35
0.6
0.75
0.5
0.3
0.2
0.1
0
1
0.5
0.4
0.75
1
1.5
2
3.5
5
10
Q = 20
0
1.5
2
3.5
5
10
Q = 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F = f s / f0
Fundamentals of Power Electronics
47
Chapter 19: Resonant Conversion
Mode boundaries, SRC
k = 1 DCM
1
0.9
0.8
0.7
0.4
k = 3 DCM
0.3
k=4
DCM
0.2
etc.
0.1
k = 1 CCM
k = 0 CCM
k = 2 CCM
k = 2 DCM
0.5
k = 3 CCM
M
0.6
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
F
Fundamentals of Power Electronics
48
Chapter 19: Resonant Conversion
Output characteristics, SRC above resonance
6
F = 1.05
F = 1.07
5
F = 1.10
4
J
F = 1.01
3
F = 1.15
2
F = 1.30
1
0
0
0.2
0.4
0.6
0.8
1
M
Fundamentals of Power Electronics
49
Chapter 19: Resonant Conversion
Output characteristics, SRC below resonance
F = 1.0
J
F = .93
F = .96
3
F = .90
2.5
F = .85
2
1.5
F = .75
4
π
k = 1 CCM
F = .5
2
π
k = 1 DCM
1
k = 2 DCM
F = .25
F = .1
0
0
0.2
0.4
0.6
0.8
1
M
Fundamentals of Power Electronics
50
Chapter 19: Resonant Conversion
19.3.2 Exact characteristics of
the parallel resonant converter
Q1
D1
Q3
L
1:n
D3
D7
Vg
+
–
C
R
D8
Q2
D2
Q4
+
D5
D6
V
–
D4
Normalized load voltage and current:
InR0
J=
Vg
M= V
nVg
Fundamentals of Power Electronics
51
Chapter 19: Resonant Conversion
Parallel resonant converter in CCM
vs(t)
CCM closed-form solution
Vg
γ
sin (ϕ)
M = 2γ ϕ –
γ
cos
2
γ
ω0t
– Vg
iL(t)
– cos – 1 cos
γ
γ
+ J sin
2
2
for 0 < γ < π (
+ cos – 1 cos
γ
γ
+ J sin
2
2
for π < γ < 2π
ϕ=
vC(t)
vC(t)
V=
Fundamentals of Power Electronics
52
vC(t)
Ts
Chapter 19: Resonant Conversion
Parallel resonant converter in DCM
vC(t)
Mode boundary
J > J crit(γ)
J < J crit(γ)
for DCM
for CCM
J crit(γ) = – 1 sin (γ) +
2
sin 2
γ
+ 1 sin 2 γ
2
4
DCM equations
ω0t
D5 D8
D6 D7
D6 D7
α
M C0 = 1 – cos (β)
J L0 = J + sin (β)
cos (α + β) – 2 cos (α) = –1
iL(t)
– sin (α + β) + 2 sin (α) + (δ – α) = 2J
β+δ=γ
M = 1 + 2γ (J – δ)
δ
D5 D8
D5 D8
D5 D8
D6 D7
D6 D7
β
γ
I
ω0t
(require iteration)
–I
Fundamentals of Power Electronics
53
Chapter 19: Resonant Conversion
Output characteristics of the PRC
3.0
2.5
F = 0.51
0.6
2.0
0.7
J
1.5
0.8
0.9
1.0
1.0
0.5
1.5
1.3
1.2
1.1
F=2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
M
Solid curves: CCM
Fundamentals of Power Electronics
Shaded curves: DCM
54
Chapter 19: Resonant Conversion
Control characteristics of the PRC
with resistive load
3.0
Q=5
2.5
M = V/Vg
2.0
1.5
Q=2
1.0
Q=1
Q = 0.5
0.5
Q = 0.2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
fs /f0
Fundamentals of Power Electronics
55
Chapter 19: Resonant Conversion
19.4 Soft switching
Soft switching can mitigate some of the mechanisms of switching loss
and possibly reduce the generation of EMI
Semiconductor devices are switched on or off at the zero crossing of
their voltage or current waveforms:
Zero-current switching: transistor turn-off transition occurs at zero
current. Zero-current switching eliminates the switching loss
caused by IGBT current tailing and by stray inductances. It can
also be used to commutate SCR’s.
Zero-voltage switching: transistor turn-on transition occurs at
zero voltage. Diodes may also operate with zero-voltage
switching. Zero-voltage switching eliminates the switching loss
induced by diode stored charge and device output capacitances.
Zero-voltage switching is usually preferred in modern converters.
Zero-voltage transition converters are modified PWM converters, in
which an inductor charges and discharges the device capacitances.
Zero-voltage switching is then obtained.
Fundamentals of Power Electronics
56
Chapter 19: Resonant Conversion
19.4.1 Operation of the full bridge below
resonance: Zero-current switching
Series resonant converter example
+
Q1
vds1(t)
D1
Q3
L
+
iQ1(t) –
Vg
C
D3
+
–
vs(t)
Q2
D2
Q4
–
is(t)
D4
Operation below resonance: input tank current leads voltage
Zero-current switching (ZCS) occurs
Fundamentals of Power Electronics
57
Chapter 19: Resonant Conversion
Tank input impedance
Operation below
resonance: tank input
impedance Zi is
dominated by tank
capacitor.
∠Zi is positive, and
tank input current
leads tank input
voltage.
|| Zi ||
1
ωC
ωL
R0
Re
f0
Qe = R0 /Re
Zero crossing of the
tank input current
waveform is(t) occurs
before the zero
crossing of the voltage
vs(t).
Fundamentals of Power Electronics
58
Chapter 19: Resonant Conversion
Switch network waveforms, below resonance
Zero-current switching
vs1(t)
Vg
vs(t)
+
t
Q1
vds1(t)
D1
Q3
L
C
D3
+
iQ1(t) –
– Vg
vs(t)
is(t)
Q2
Ts
+ tβ
2
tβ
D2
Q4
–
is(t)
D4
t
Ts
2
Conduction sequence: Q1–D1–Q2–D2
Conducting
devices:
Q1
Q4
D1
D4
Q2
Q3
“Soft”
“Hard”
“Hard”
turn-on of turn-off of turn-on of
Q 1, Q 4
Q 2, Q 3
Q 1, Q 4
Fundamentals of Power Electronics
Q1 is turned off during D1 conduction
interval, without loss
D2
D3
“Soft”
turn-off of
Q2, Q3
59
Chapter 19: Resonant Conversion
ZCS turn-on transition: hard switching
vds1(t)
Vg
+
Q1
vds1(t)
D1
Q3
L
+
iQ1(t) –
t
ids(t)
vs(t)
Q2
Ts
+ tβ
2
tβ
Conducting
devices:
C
D3
Q1
Q4
D1
D4
Ts
2
“Soft”
“Hard”
turn-on of turn-off of
Q1, Q4
Q1, Q4
Fundamentals of Power Electronics
D2
Q4
–
is(t)
D4
t
Q2
Q3
D2
D3
Q1 turns on while D2 is conducting. Stored
charge of D2 and of semiconductor output
capacitances must be removed. Transistor
turn-on transition is identical to hardswitched PWM, and switching loss occurs.
60
Chapter 19: Resonant Conversion
19.4.2 Operation of the full bridge below
resonance: Zero-voltage switching
Series resonant converter example
+
Q1
vds1(t)
D1
Q3
L
+
iQ1(t) –
Vg
C
D3
+
–
vs(t)
Q2
D2
Q4
–
is(t)
D4
Operation above resonance: input tank current lags voltage
Zero-voltage switching (ZVS) occurs
Fundamentals of Power Electronics
61
Chapter 19: Resonant Conversion
Tank input impedance
Operation above
resonance: tank input
impedance Zi is
dominated by tank
inductor.
∠Zi is negative, and
tank input current lags
tank input voltage.
|| Zi ||
1
ωC
ωL
R0
Re
f0
Qe = R0 /Re
Zero crossing of the
tank input current
waveform is(t) occurs
after the zero crossing
of the voltage vs(t).
Fundamentals of Power Electronics
62
Chapter 19: Resonant Conversion
Switch network waveforms, above resonance
Zero-voltage switching
vs1(t)
Vg
vs(t)
+
Q1
t
vds1(t)
D1
Q3
L
C
D3
+
iQ1(t) –
vs(t)
– Vg
is(t)
Q2
tα
D2
Q4
–
is(t)
D4
t
Ts
2
Conduction sequence: D1–Q1–D2–Q2
Conducting D1
devices: D
4
“Soft”
turn-on of
Q1, Q4
Q1
Q4
D2
D3
Q1 is turned on during D1 conduction
interval, without loss
Q2
Q3
“Hard”
“Soft”
“Hard”
turn-off of turn-on of turn-off of
Q1, Q4
Q2, Q3
Q2, Q3
Fundamentals of Power Electronics
63
Chapter 19: Resonant Conversion
ZVS turn-off transition: hard switching?
vds1(t)
Vg
+
Q1
vds1(t)
D1
Q3
L
+
iQ1(t) –
t
C
D3
vs(t)
ids(t)
Q2
tα
Conducting D1
devices: D
4
“Soft”
turn-on of
Q1, Q4
Q1
Q4
Ts
2
D2
Q4
–
is(t)
D4
t
D2
D3
“Hard”
turn-off of
Q1, Q4
Fundamentals of Power Electronics
Q2
Q3
When Q1 turns off, D2 must begin
conducting. Voltage across Q1 must
increase to Vg. Transistor turn-off
transition is identical to hard-switched
PWM. Switching loss may occur (but see
next slide).
64
Chapter 19: Resonant Conversion
Soft switching at the ZVS turn-off transition
+
Q1
D1 C
leg
Vg
Q3
vds1(t)
Cleg
–
D3
is(t)
+
+
–
vs(t)
Q2
D2
Cleg
Cleg
D4
–
Q4
vds1(t)
Conducting
devices:
Turn off
Q1, Q4
X D2
D3
t
• Introduce delay
between turn-off of Q1
and turn-on of Q2.
So zero-voltage switching exhibits low
switching loss: losses due to diode
stored charge and device output
capacitances are eliminated.
Commutation
interval
Fundamentals of Power Electronics
to remainder
of converter
• Introduce small
capacitors Cleg across
each device (or use
device output
capacitances).
Tank current is(t) charges and
discharges Cleg. Turn-off transition
becomes lossless. During commutation
interval, no devices conduct.
Vg
Q1
Q4
L
65
Chapter 19: Resonant Conversion
19.4.3 The zero-voltage transition converter
Basic version based on full-bridge PWM buck converter
Q3
Q1
D1 C
leg
Vg
+
–
Cleg
ic(t)
Lc
D3
+
Q2
D2
Cleg
v2(t)
Cleg
D4
Q4
–
v2(t)
Vg
• Can obtain ZVS of all primaryside MOSFETs and diodes
Can turn on
Q1 at zero voltage
• Secondary-side diodes switch at
zero-current, with loss
• Phase-shift control
Fundamentals of Power Electronics
Conducting
devices:
Q2
Turn off
Q2
66
X D1
t
Commutation
interval
Chapter 19: Resonant Conversion
19.5 Load-dependent properties
of resonant converters
Resonant inverter design objectives:
1. Operate with a specified load characteristic and range of operating
points
• With a nonlinear load, must properly match inverter output
characteristic to load characteristic
2. Obtain zero-voltage switching or zero-current switching
• Preferably, obtain these properties at all loads
• Could allow ZVS property to be lost at light load, if necessary
3. Minimize transistor currents and conduction losses
• To obtain good efficiency at light load, the transistor current should
scale proportionally to load current (in resonant converters, it often
doesn’t!)
Fundamentals of Power Electronics
67
Chapter 19: Resonant Conversion
Topics of Discussion
Section 19.5
Inverter output i-v characteristics
Two theorems
• Dependence of transistor current on load current
• Dependence of zero-voltage/zero-current switching on load
resistance
• Simple, intuitive frequency-domain approach to design of resonant
converter
Examples and interpretation
• Series
• Parallel
• LCC
Fundamentals of Power Electronics
68
Chapter 19: Resonant Conversion
Inverter output characteristics
transfer function
H(s)
Let H∞ be the open-circuit (RÕ∞)
transfer function:
vo( jω)
vi( jω)
io(t)
ii(t)
sinusoidal
source
+
vi(t) –
= H ∞( jω)
resonant
network
+
Zo v (t)
o
Zi
purely reactive
–
resistive
load
R
R→∞
and let Zo0 be the output impedance
(with vi Õ short-circuit). Then,
R
vo( jω) = H ∞( jω) vi( jω)
R + Z o0( jω)
This result can be rearranged to obtain
vo
2
+ io
2
Z o0
2
= H∞
2
vi
2
The output voltage magnitude is:
vo
2
= vov *o =
H∞
2
1 + Z o0
with
vi
2
Hence, at a given frequency, the
output characteristic (i.e., the relation
between ||vo|| and ||io||) of any
resonant inverter of this class is
elliptical.
2
/ R2
R = vo / io
Fundamentals of Power Electronics
69
Chapter 19: Resonant Conversion
Inverter output characteristics
General resonant inverter
output characteristics are
elliptical, of the form
2
|| io ||
I sc =
vi
d
oa
l
ed ||
tch | Z o0
a
m =|
R
Z o0
2
vo
io
+ 2 =1
V 2oc
I sc
with
H∞
inverter output
characteristic
Voc = H ∞
I sc =
H∞
I sc
2
vi
Voc
2
vi
Voc = H ∞
vi
|| vo ||
Z o0
This result is valid provided that (i) the resonant network is purely reactive,
and (ii) the load is purely resistive.
Fundamentals of Power Electronics
70
Chapter 19: Resonant Conversion
Matching ellipse
to application requirements
Electrosurgical generator
|| io ||
|| io ||
50Ω
Electronic ballast
inverter characteristic
inverter characteristic
2A
40
ed
tch
ma
lamp characteristic
d
loa
2kV
|| vo ||
Fundamentals of Power Electronics
0W
71
|| vo ||
Chapter 19: Resonant Conversion
Input impedance of the resonant tank network
Transfer function
H(s)
Z (s)
R
1 + o0
R
Z o0(s)
Z i(s) = Z i0(s)
= Z i∞(s)
Z o∞(s)
1+ R
1
+
Z o∞(s)
R
1+
where
v
Z i0 = i
ii
R→0
Z i∞ =
Fundamentals of Power Electronics
vi
ii
Effective
sinusoidal
source
+
vs1(t) –
72
Resonant
network
Zi
+
Zo
Purely reactive
Z o0 =
R→∞
i(t)
is(t)
vo
– io
vi → short circuit
Z o∞ =
v(t)
–
vo
– io
Effective
resistive
load
R
vi → open circuit
Chapter 19: Resonant Conversion
Other relations
If the tank network is purely reactive,
then each of its impedances and
transfer functions have zero real
parts:
Z = – Z*
Reciprocity
Z i0 Z o0
=
Z i∞ Z o∞
i0
Z i∞ = – Z
Z o0 = – Z
Z o∞ = – Z
H∞ = – H
Tank transfer function
H(s) =
where
H ∞(s)
1+ R
Z o0
H∞ =
H∞
2
vo(s)
vi(s)
= Z o0
Fundamentals of Power Electronics
i0
*
i∞
*
o0
*
o∞
*
∞
Hence, the input impedance
magnitude is
2
R
1+
Z o0
2
2
*
Z i = Z iZ i = Z i0
2
R
1+
Z o∞
R→∞
1 – 1
Z i0 Z i∞
73
2
2
Chapter 19: Resonant Conversion
Zi0 and Zi∞ for 3 common inverters
Series
L
1
ωC
Cs
Z i0(s) = sL + 1
sC s
|| Zi∞ ||
s
ωL
Zo
Zi
|| Zi0 ||
Z i∞(s) = ∞
f
Parallel
1
ωC
L
Z i0(s) = sL
p
ωL
Zi
Cp
|| Zi∞ ||
Zo
Z i∞(s) = sL + 1
sC p
|| Zi0 ||
f
LCC
L
1
ωC +
s
Cs
1
ωC
Zi
Cp
Z i0(s) = sL + 1
sC s
p
1
ωC
s
Zo
ωL
|| Zi∞ ||
Z i∞(s) = sL + 1 + 1
sC p sC s
|| Zi0 ||
f
Fundamentals of Power Electronics
74
Chapter 19: Resonant Conversion
A Theorem relating transistor current variations
to load resistance R
Theorem 1: If the tank network is purely reactive, then its input impedance
|| Zi || is a monotonic function of the load resistance R.
l
l
l
l
So as the load resistance R varies from 0 to ∞, the resonant network
input impedance || Zi || varies monotonically from the short-circuit value
|| Zi0 || to the open-circuit value || Zi∞ ||.
The impedances || Zi∞ || and || Zi0 || are easy to construct.
If you want to minimize the circulating tank currents at light load,
maximize || Zi∞ ||.
Note: for many inverters, || Zi∞ || < || Zi0 || ! The no-load transistor current
is therefore greater than the short-circuit transistor current.
Fundamentals of Power Electronics
75
Chapter 19: Resonant Conversion
Proof of Theorem 1
Previously shown:
Zi
2
= Z i0
2
1+ R
Z o0
1+ R
Z o∞
á Differentiate:
2
d Zi
dR
2
á Derivative has roots at:
= 2 Z i0
2
2
–
1
Z o∞
2
R
1+
Z o∞
2
R
2
2
So the resonant network input
impedance is a monotonic function
of R, over the range 0 < R < ∞.
(i) R = 0
(ii) R = ∞
(iii) Z o0 = Z o∞ , or Z i0 = Z i∞
Fundamentals of Power Electronics
2
1
Z o0
In the special case || Zi0 || = || Zi∞ ||,
|| Zi || is independent of R.
76
Chapter 19: Resonant Conversion
Example: || Zi || of LCC
|| Zi ||
1
ωC +
s
1
ωC
f0
f∞
p
rea
1
ωC
inc
rea
s
ing
R
ωL
inc
s
gR
sin
• for f < f m, || Zi || increases
with increasing R .
• for f > f m, || Zi || decreases
with increasing R .
• at a given frequency f, || Zi ||
is a monotonic function of R.
• It’s not necessary to draw
the entire plot: just construct
|| Zi0 || and || Zi∞ ||.
fm
f
Fundamentals of Power Electronics
77
Chapter 19: Resonant Conversion
Discussion: LCC
|| Zi0 || and || Zi∞ || both represent
series resonant impedances,
whose Bode diagrams are easily
constructed.
|| Zi0 || and || Zi∞ || intersect at
frequency fm.
1
ωC +
s
|| Zi ||
LCC example
f0
1
ωC
f∞
p
ωL
1
ωC
s
|| Zi∞ ||
|| Zi0 ||
For f < fm
1
2π LC s
1
f∞ =
2π LC s||C p
1
fm =
2π LC s||2C p
f0 =
fm
then || Zi0 || < || Zi∞ || ; hence
transistor current decreases as
load current decreases
For f > fm
f
then || Zi0 || > || Zi∞ || ; hence
transistor current increases as
load current decreases, and
transistor current is greater
than or equal to short-circuit
current for all R
Fundamentals of Power Electronics
L
Zi∞
78
Cs
Cp
L
Zi0
Cs
Cp
Chapter 19: Resonant Conversion
Discussion -series and parallel
Series
L
1
ωC
Cs
• No-load transistor current = 0, both above
and below resonance.
|| Zi∞ ||
s
ωL
Zo
Zi
|| Zi0 ||
f
Parallel
• Above resonance: no-load transistor current
is greater than short-circuit transistor
current. ZVS.
1
ωC
L
p
ωL
Zi
Cp
|| Zi∞ ||
Zo
|| Zi0 ||
f
LCC
L
1
ωC +
s
Cs
• ZCS below resonance, ZVS above
resonance
1
ωC
• Below resonance: no-load transistor current
is less than short-circuit current (for f <fm),
but determined by || Zi∞ ||. ZCS.
p
1
ωC
ωL
s
Zi
Cp
Zo
|| Zi∞ ||
|| Zi0 ||
f
Fundamentals of Power Electronics
79
Chapter 19: Resonant Conversion
A Theorem relating the ZVS/ZCS boundary to
load resistance R
Theorem 2: If the tank network is purely reactive, then the boundary between
zero-current switching and zero-voltage switching occurs when the load
resistance R is equal to the critical value Rcrit, given by
Rcrit = Z o0
– Z i∞
Z i0
It is assumed that zero-current switching (ZCS) occurs when the tank input
impedance is capacitive in nature, while zero-voltage switching (ZVS) occurs when
the tank is inductive in nature. This assumption gives a necessary but not sufficient
condition for ZVS when significant semiconductor output capacitance is present.
Fundamentals of Power Electronics
80
Chapter 19: Resonant Conversion
Proof of Theorem 2
Previously shown:
Z
1 + o0
R
Z i = Z i∞
Z
1 + o∞
R
If ZCS occurs when Zi is capacitive,
while ZVS occurs when Zi is
inductive, then the boundary is
determined by ∠Zi = 0. Hence, the
critical load Rcrit is the resistance
which causes the imaginary part of Zi
to be zero:
Note that Zi∞, Zo0, and Zo∞ have zero
real parts. Hence,
Z
1 + o0
Rcrit
Im Z i(Rcrit) = Im Z i∞ Re
Z
1 + o∞
Rcrit
1–
= Im Z i∞ Re
1+
Z o∞
2
R 2crit
Solution for Rcrit yields
Im Z i(Rcrit) = 0
Rcrit = Z o0
Fundamentals of Power Electronics
Z o0Z o∞
R 2crit
81
– Z i∞
Z i0
Chapter 19: Resonant Conversion
Discussion ÑTheorem 2
Rcrit = Z o0
l
l
l
l
l
– Z i∞
Z i0
Again, Zi∞, Zi0, and Zo0 are pure imaginary quantities.
If Zi∞ and Zi0 have the same phase (both inductive or both capacitive),
then there is no real solution for Rcrit.
Hence, if at a given frequency Zi∞ and Zi0 are both capacitive, then ZCS
occurs for all loads. If Zi∞ and Zi0 are both inductive, then ZVS occurs for
all loads.
If Zi∞ and Zi0 have opposite phase (one is capacitive and the other is
inductive), then there is a real solution for Rcrit. The boundary between
ZVS and ZCS operation is then given by R = Rcrit.
Note that R = || Zo0 || corresponds to operation at matched load with
maximum output power. The boundary is expressed in terms of this
matched load impedance, and the ratio Zi∞ / Zi0.
Fundamentals of Power Electronics
82
Chapter 19: Resonant Conversion
LCC example
l
l
l
l
For f > f∞, ZVS occurs for all R.
For f < f0, ZCS occurs for all R.
For f0 < f < f∞, ZVS occurs for
R< Rcrit, and ZCS occurs for
R> Rcrit.
Note that R = || Zo0 || corresponds
to operation at matched load with
maximum output power. The
boundary is expressed in terms of
this matched load impedance,
and the ratio Zi∞ / Zi0.
|| Zi ||
1
ωC +
s
1
ωC
ZCS ZCS: R>Rcrit ZVS
for all R ZVS: R<Rcrit for all R
ωL
p
1
ωC
s
Z i∞
|| Zi0 ||
Z i0
|| Zi∞ ||
{
f1
fm
f
Rcrit = Z o0
Fundamentals of Power Electronics
f∞
f0
83
– Z i∞
Z i0
Chapter 19: Resonant Conversion
LCC example, continued
∠Zi
R
90˚
R=0
60˚
easi
incr
ZCS
30˚
ng R
0˚
R crit
||
||
-30˚
ZVS
Z o0
-60˚
R=∞
f0
fm
-90˚
f∞
Typical dependence of Rcrit and matched-load
impedance || Zo0 || on frequency f, LCC example.
Fundamentals of Power Electronics
f
f0
84
f∞
Typical dependence of tank input impedance phase
vs. load R and frequency, LCC example.
Chapter 19: Resonant Conversion
19.6 Summary of Key Points
1.
2.
The sinusoidal approximation allows a great deal of insight to be
gained into the operation of resonant inverters and dc–dc converters.
The voltage conversion ratio of dc–dc resonant converters can be
directly related to the tank network transfer function. Other important
converter properties, such as the output characteristics, dependence
(or lack thereof) of transistor current on load current, and zero-voltageand zero-current-switching transitions, can also be understood using
this approximation. The approximation is accurate provided that the
effective Q–factor is sufficiently large, and provided that the switching
frequency is sufficiently close to resonance.
Simple equivalent circuits are derived, which represent the
fundamental components of the tank network waveforms, and the dc
components of the dc terminal waveforms.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Summary of key points
3.
4.
5.
Exact solutions of the ideal dc–dc series and parallel resonant
converters are listed here as well. These solutions correctly predict the
conversion ratios, for operation not only in the fundamental continuous
conduction mode, but in discontinuous and subharmonic modes as
well.
Zero-voltage switching mitigates the switching loss caused by diode
recovered charge and semiconductor device output capacitances.
When the objective is to minimize switching loss and EMI, it is
preferable to operate each MOSFET and diode with zero-voltage
switching.
Zero-current switching leads to natural commutation of SCRs, and can
also mitigate the switching loss due to current tailing in IGBTs.
Fundamentals of Power Electronics
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Chapter 19: Resonant Conversion
Summary of key points
6. The input impedance magnitude || Zi ||, and hence also the transistor
current magnitude, are monotonic functions of the load resistance R.
The dependence of the transistor conduction loss on the load current
can be easily understood by simply plotting || Zi || in the limiting cases as
R Õ ∞ and as R Õ 0, or || Zi∞ || and || Zi0 ||.
7. The ZVS/ZCS boundary is also a simple function of Zi∞ and Zi0. If ZVS
occurs at open-circuit and at short-circuit, then ZVS occurs for all loads.
If ZVS occurs at short-circuit, and ZCS occurs at open-circuit, then ZVS
is obtained at matched load provided that || Zi∞ || > || Zi0 ||.
8. The output characteristics of all resonant inverters considered here are
elliptical, and are described completely by the open-circuit transfer
function magnitude || H∞ ||, and the output impedance || Zo0 ||. These
quantities can be chosen to match the output characteristics to the
application requirements.
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Chapter 19: Resonant Conversion