Électronique et transmission de l’information
LOW PEAK CURRENT CLASS E RESONANT FULL-WAVE
LOW dv/dt RECTIFIER DRIVEN BY A VOLTAGE GENERATOR
ŞERBAN BÎRCĂ-GĂLĂŢEANU1
Key words : Power Electronics, Rectifiers, Class E, Resonant Rectifiers.
Abstract - Full-wave rectifiers allow to halve the diode peak voltages (or currents), but
these values may still be high at duty factor values far from 0.5. A new class E resonant
full-wave rectifier is proposed and analyzed, whose peak diode current is equal to the
load current. Diode voltage slopes are ideally zero after switch-off only for duty factors
larger than 0.5. The circuit contains two rectifier diodes but a single LC circuit. All
major parasitic reactive elements are included in the rectifier topology. Basic
parameters of the circuit are derived as functions of the diodes' duty factor. This allows
the user to find the optimum component values for a given application.
Experimental waveforms and measurements agree with simulations and calculations.
1. INTRODUCTION
To reduce switching losses at higher frequencies, resonant and pseudoresonant circuits have been developed, for instance Class E [7] rectifiers [1, 2].
After the half-wave rectifiers, full-wave circuits are proposed [4] for further
reduction of the transformer size, of the diode peak voltages (or currents) and of
the input current (or voltage) harmonics.
This paper aims to present the analysis of a resonant full-wave voltage-driven
low dv/dt rectifier whose peak diode current equals the load current (Fig. 1a, b).
This allows to find component values for optimum operation in a given application.
Diode voltage slopes are ideally zero after switch-off only for duty factors
larger than 0.5. The basic circuit contains two rectifier diodes, two inductors, but a
single capacitor C across the two rectifier diodes (Fig. 1a). Voltages on the two
inductances are identical, hence they may be coupled, which reduces circuit cost
and radiated EMI. Output current is almost constant due to the filter inductor Lf and
a constant current produces no voltage drop across an inductance, hence the load
can be connected at any terminal of the inductance L1, without affecting circuit
operation. Consequently, inductances L and L1 may be concentrated in a single
inductance: L (Fig. 1b) or L1. The L C circuit has the resonant frequency f0. It now
appears like a second-order low-pass filter.
1
I.U.F.M. of the Academy of Nantes, France.
Rev. Roum. Sci. Techn.– Électrotechn. et Énerg., 52, 3, p. 331–342, Bucarest, 2007
332
Şerban Bîrcă-Gălăţeanu
D
L
+ vD
+
~
iD
_
Lf
iD1_
D1 +
C
Io
+
vD1
v
2
Vo RL
_
L1
a
+
~
D
L
v
+ vD
C
iD
D1
_
iD1
_
Lf
vD1
Vo RL
+
Io
_
+
+
~
L
v
C
D1
b
+
~
L
v
D
Lf
iD1
+
Io
Vo RL
_
c
_
iD
Lf
vD1
C
_
vD
+
+
+
Io
Vo RL
+
~
L
D
v
iD
Lf
D1
_
d
iD1
+
Io
Vo RL
_
e
Fig. 1 – The resonant full-wave voltage-driven low dv/dt rectifier with low peak diode current:
a) basic circuit diagram; b) single L circuit diagram; c) equivalent circuit when diode D is off;
d) equivalent circuit when diode D1 is off; e) equivalent circuit when both diodes are on.
Rectifier diodes may never be both off, because there is no other current path
towards the output inductor Lf which acts as a current sink. At light loads, diodes
conduct only one at a time (Fig. 1c and d). The duty cycle D is 0.5 (Fig. 2b). At
heavy loads, there are intervals when both diodes are on (D ≥ 0.5, see Fig. 1e and
Fig. 2a). The circuit has been analyzed and measured in both operating modes, at
frequencies around f0. At the resonant frequency the input reactive current may be
very high and the voltage transfer factor – very low.
2. ANALYSIS OF OPERATION
The circuit is symmetrical in ac, that is the waveforms of the voltages and
currents on D1 are identical with the waveforms of the voltages and currents on
D with a delay of half a period. Also, the waveform of the input current i has
two identical half-waves plus a continuous component equal to half the output
current Io.
3
Low peak current class E resonant full-wave low dv/dt rectifier
333
All the circuit components are assumed ideal. Diode capacitances are
absorbed into the resonant capacitor C and isolation transformer leakage inductance is
included in the inductance L. Hence, the rectifier is especially suitable for highfrequency applications such as resonant dc-dc converters.
The inductance Lf is large enough for the output current have negligible
ripple; thus the Lf – RL circuit can be replaced by a dc current sink Io = Vo / RL.
The rectifier is fed by an ideal sinusoidal voltage source
v = Vm sin (ωt + ϕ),
(1)
where ϕ is the phase angle of the input voltage when the series diode D turns off
(Figs.2, a and b).
Notations are, as in [2–6, 9 and 10], for Fig.1.b:
ω 02 =
1
LC
, Q=
V
RL
ω
, M= o , A=
.
ωL
ω0
Vm
(2)
Q has only the form of a quality factor, but is in fact the rated load resistance.
From the continuity conditions for the capacitor voltages and inductance currents, a
system of two equations is obtained which gives sin ϕ and cos ϕ of the form
sin ϕ = –
M 2
NS
M 2
NC
(A – 1)
, cos ϕ = –
(A – 1)
,
DΦ
DΦ
Q
Q
(3)
tan ϕ = NS/NC.
The average value of the voltage v is zero, therefore the average value of vD1
and vD is Vo. From this condition we get the equation to derive the Q - factor, hence
M = Q DΦ
/(
)
(4)
A2 − 1 NS 2 + NC 2 .
Circuit parameters are given by the following expressions when D ≥ 0.5, where
d=2πD
a = (2 π – d)/A,
2
a
d
a
d

DΦ = 4  A cos sin + sin cos  ,
2
2
2
2

NS = A sin a + sin d,
NC = cos d – cos a,
(5)
2
1 
a
d 
d
a 
Q=
 A tan + tan   A tan + tan   .

2
2 
2
2 
2 π 
(6)
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Şerban Bîrcă-Gălăţeanu
4
We have noticed that the expression of the normalized load resistance Q as a
function of the diode conduction duty factor D can always be brought to a
quadratic form [6]. This one can be written as a transcendental equation for D,
with Q as a parameter. It also has been demonstrated [6] and checked for this
circuit that the normalized start-up diode current h = iD ON /Io = A
2πQ.
The input voltage being sinusoidal, the input power of the rectifier is fed only
by the fundamental component of the input current i. At the operating frequency,
the rectifier has the input impedance of a parallel Ri – Li circuit.
By equating the input and the output power, one obtains
Ri / RL = 1 / (2 M2).
(7)
The equivalent input inductance Li at the operating frequency is obtained from the
reactive component of the input current.
Li /L = π (A2 – 1) / DL,
(8)
DL = A2π /2 + (A sin a cos d + cos a sin d) A2/(A2 – 1) –
– A sin a sin ϕ sin(ϕ– d) + [π – d + sin d cos(d – 2ϕ)]/2.
The rectifier has the following parameters for D = 0.5 :
a = π/A
NS = A sin a
tan ϕ = NS/NC = – A tan(a/2)
M=1
/(
DΦ = 2 A2 (1 + cos a)
NC = – (1 + cos a)
Q = 1/(2 π A2)
A2 − 1 1 + A2 tan 2 (a / 2)
)
 A2 − 1   1
Li
2
1
A tan (a / 2) 
=
.
  −
2
2
π A − 1 1 + [ A tan (a / 2)]2 
L
 A  2
(9)
(10)
Parameters ϕ, Q, M, VDRM /Vo and Li (for A ≥ 1) (Figs. 3, 4, 5, 6) are
monotonic functions of D: ϕ , VDRM /Vo and Li increase, Q and M decrease when D
increases. For D > 0.7, Li /L is close to 1. For D close to 0.5 and A < 0.8, Li has very
high values, hence reactive currents are very low. Operating frequencies close to f0
must be avoided, because M is very low. For any load value, the voltage transfer
factor M is low and strongly dependent on the load, which is an important
drawback of the class E rectifiers. Diode peak voltage is almost independent of the
operating frequency.
When D = 0.5, d = π = constant (a = π/A). Circuit parameters may no longer
be expressed and plotted as functions of D, but as functions of Q. In order to have a
limited range for the x-axis, ϕ, Li /L and M are plotted as functions of the auxiliar
parameter q = 1/(2 π A2Q), q = 0 to 1.
5
Low peak current class E resonant full-wave low dv/dt rectifier
a)
b)
Fig. 2 – Key waveforms of the ideal rectifier, when: a) D ≥ 0.5; b) D = 0.5.
335
336
Şerban Bîrcă-Gălăţeanu
6
Circuit operation has been simulated using OrCAD Capture version 9.1.
Waveforms show no ringing, because there are no parasitics in simulation models:
DΦ = 2 [(1 + cos a) A2 (2/π + π/2) – A sin a (A2 – 1)],
NS = (1 + cos a) A2 (1 – 2 π Q) + A sin a (4 Q A2 + π/2) – 2 (A2 – 1),
(11)
NC = – (1 + cos a)(4 Q A2 + π/2) + A sin a (1 – 2 π Q).
The voltage transfer factor M is still given by expression (4):
Li /L = π (A2 – 1)/(A2 DL),
DL =
π
2
–
A sin a 
2

+ 1 +
sin ϕ  [A sin a cos ϕ + (1 + cos a) sin ϕ] +
2
A −1 
π

+ sin ϕ (cos ϕ + sin ϕ sin a / A).
(12)
(13)
At D = 0.5, maximum point phase grows from 90 to 120 degrees and VDRM/Vo
– from 3.2 to 4.08 when A decreases from 2 to 0.5 (Fig. 10) and q increases from 0
to 1. For A = 1, VDRM/Vo grows rapidly when q approaches 1.
Two resonances take place at D = 0.5, at frequencies between f0 /2 and 1.2 f0.
A resonance has already been noticed for other Class E rectifiers operating at
frequencies different from f0 [5]. At frequencies close to f0, the input equivalent
inductance Li is almost zero in a large range of load values (Figs. 7 and 8). The
circuit then has a very high input current. Voltage transfer factor M has much
higher values than for D > 0.5.
When considering all circuit parameters, f = f0 / 2 seems to be a good
compromise (Fig. 9). Operation at D = 0.5 still reduces switching losses because
dv/dt is low, even if it is not zero after turn-off. This mode is a good choice because
M is relatively high, VDRM/Vo is low and both are only slightly dependent on the
load (on parameter q ). For q = 0.9 to 1, Li /L >> 1, hence the reactive current is
very low. Derive 5 has been used to calculate and draw parameters as functions of
the duty factor D, because this was the only way to have explicit expressions only.
3. EXPERIMENTAL RESULTS
A circuit of the full-wave rectifier of Fig. 1a was constructed with the
resonant frequency f0 = 1 MHz, the output voltage Vo = 5 V and the duty factor
D = 0.5 when the output current Io = 50 mA (RL = 100 Ω). It follows L = L1 = 20.5
µH (8/9 IF220 fixed inductors from Affero S.A.) and C = C1 = 1.2 nF ceramic
capacitors, as in [5], to allow a direct comparison. Cf = 470 nF is a multilayer
ceramic capacitor. D and D1 are Motorola 1N5817 (or ST Semi BYV10-20A).
Current and voltage waveforms have been obtained with Tektronix 7613
oscilloscope and P6022 current probe. Waveforms are presented for the operating
7
Low peak current class E resonant full-wave low dv/dt rectifier
337
frequency f = f0 = 1 MHz and load resistance of 25 Ω (Fig.11). Waveforms at the
frequency f0 / 2 = 707 kHz are not significantly different. One may note that
Fig. 3 – Initial phase angle ϕ of the input voltage, as a function of D.
Fig. 4 – Normalized load resistance Q, as a function of D.
338
Şerban Bîrcă-Gălăţeanu
Fig. 5 – Ac to dc voltage transfer factor M, as a function of D.
Fig. 6 – Li /L versus diodes’ duty ratio D.
8
9
Low peak current class E resonant full-wave low dv/dt rectifier
Fig. 7 – D = 0.5. Initial phase angle ϕ of the input current, as a function of q.
Fig. 8 – D = 0.5. Li /L versus parameter q.
339
340
Şerban Bîrcă-Gălăţeanu
10
Fig. 9 – D = 0.5. Ac to dc voltage transfer factor M, as a function of parameter q.
Fig. 10 – VDRM/Vo versus diodes' duty ratio D.
ringing is always present on the diode and capacitor current waveforms but not on
the inductance current waveforms. This ringing is mainly due to the inductance
introduced by the current probe itself and should be much lower in a short-leads
11
Low peak current class E resonant full-wave low dv/dt rectifier
341
operational circuit. Ringing is, proportionally, larger when RL value is larger,
especially on the input voltage v.
The input voltage has very low
harmonics. Even harmonics are absent
due to the half-wave symmetry.
Input voltage rms value was
measured with a Rohde & Schwarz
wide-band voltmeter.
The normalized output-power
capability is :
cp = Io Vo /(IDM VDRM) = Vo / VDRM, (14)
because IDM = Io. Peak reverse diode
voltage VDRM is almost half as high as in
the half-wave low dv/dt rectifier [3, 5,
9, 10]. Lower VDRM allow the use of
Schottky diodes, which are, generally,
low-voltage devices.
As in the other Class E rectifiers,
using D ≈ 0.5 is an optimum choice in
order to have low diode peak voltage. If
the input reactive current is to be
minimized, the rectifier is to be driven
a
b
at frequencies lower than f0 / 2 [11].
4. CONCLUSIONS
A class E resonant full-wave
voltage-driven low dv/dt rectifier has
been analyzed. Its peak diode current is
always equal to the load current. Peak
diode reverse voltage is almost half as
high as in half-wave rectifiers. Major
parasitic reactive elements are included
in the rectifier topology. Basic
parameters of the circuit were derived
as functions of the diodes' duty factor,
using the time-domain analysis.
c
Fig. 11 – Current and voltage waveforms
(RL = 25 Ω): a) iD (200 mA/div) and vD (5 V/div);
b) iD1 (200 mA/div) and vD1 (10 V/div);
c) i (100 mA/div) and v (20 V/div).
342
Şerban Bîrcă-Gălăţeanu
12
Input current has low level odd harmonics only. When D ≥ 0.5, normalized
load resistance Q and the voltage transfer factor M have lower values and the input
inductance has values closer to 1 as the duty factor is higher.
Best parameter values are obtained at f around f0 / 2 , for D = 0.5 – 0.6 or
D = 0.5 and parameter q = 0.8 to 1, that is around the boundary between the two
operating modes. The user must choose D value according to the application constraints, then find the other parameters from the graphs. Component values follow.
Experimental waveforms and measurements agree with simulations and
calculations.
Received on 11 October, 2006
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