Download The Effect of Ripple Steering on Control Loop Stability for a CCM

The Effect of Ripple Steering on Control Loop Stability
for a CCM PFC Boost Converter
Fariborz Musavi, Murray Edington
Department of Research, Engineering
Delta-Q Technologies Corp.
Burnaby, BC, Canada
[email protected], [email protected]
Abstract—In this paper, an average switch model approach to
the power stage modeling, feedback compensation and
dynamic analysis of PFC boost converters with coupled
magnetic filter is presented. The model is expressed by
derivation of power stage transfer functions for a conventional
boost converter, and then followed by the power stage transfer
functions for a PFC boost converter with coupled magnetic
filter. Experimental and simulation results of a prototype boost
converter converting universal AC input voltage to 400 V DC
at 1.8 kW are given to verify the proof of concept, and
analytical work reported. The experimental results
demonstrate that the model can correctly predict the steadystate and large signal dynamic behavior of a CCM PFC boost
converter with coupled magnetic filter.
I.
INTRODUCTION
Coupled magnetics filter techniques, known also as
ripple steering techniques [1], have existed for many years,
and have been applied to several topologies [2, 3]. As shown
in Figure 1, this technique replaces a series smoothing
inductor with a pair of coupled inductors and a blocking
capacitor. The circuit works by allowing the DC component
of the input current to pass through the series winding, at the
input, while the high frequency ripple component is filtered
by steering it through the additional coupled inductor and
capacitor combination.
1
Wilson Eberle, 2 William G. Dunford
Department of Electrical and Computer Engineering
University of British Columbia | 1 Okanagan | 2 Vancouver
1
Kelowna, BC, Canada | 2 Vancouver, BC, Canada
1
[email protected] | 2 [email protected]
complexity, especially in its differential filtering section,
which includes Cx capacitors and differential mode
inductors. Reducing the Cx capacitors to a minimal size has
an additional benefit for applications with tight specifications
on standby power consumption. Cx capacitors cause a
considerable reactive current to flow through the filter,
which is a source of additional and unwanted loss.
Furthermore, with ripple steering, the discharge resistor that
must be placed in parallel to Cx for safety reasons can be
higher. As a result, both losses are minimized.
With ripple steering, the control strategy for the PFC
stage is similar to that of a conventional boost converter, but
the power stage transfer functions are different. This paper
proposes a model to verify the effect of the added coupled
filter to the power stage transfer functions, which can be
used to design the feedback loop compensation.
Figure 2. Modified PFC boost converter with coupled inductors.
II.
Figure 1. Smoothing transformer in a coupled filter.
AVERAGE SWITCH MODEL
A Model for the PWM-switch was first introduced in [6,
7]. It was then adopted for different converters in [8-12]. For
the average switch model of a converter, the active and
passive switches shown in Figure 3 are replaced with the
PWM-switch equivalent circuits shown in Figures 4 and 5.
Ripple steering for PFC boost converters has been
presented in [4, 5]. Figure 2 illustrates the modified PFC
boost converter with a coupled inductor. The ripple-steering
technique has several advantages in a PFC boost converter.
Since it eliminates most of the differential-mode conducted
noise, it enables the reduction in EMI filter size and
Figure 3. One active switch and one passive switch.
This work has been sponsored and supported by Delta-Q Technologies
Corporation.
Figure 7. Circuit of coupled inductors.
Figure 4. PWM-Switch.
(4)
(5)
(6)
From above equations, Figure 7 can be replaced with its
equivalent circuit shown in Figure 8 [2].
Figure 5. PWM-Switch with dependant voltage and current sources.
Applying this model to a modified boost converter with
coupled inductors, any of the transfer functions between the
output variables (output voltage and inductor current) and the
input variables (input voltage and duty ration) can be
derived.
Applying the PWM switch model results in the
equivalent circuit shown in Figure 6, and by application of
superposition and circuit theory, the power stage transfer
functions can be derived:
.
.
(1)
.
Figure 8. Equivalent circuit of coupled inductors.
The equivalent circuit of coupled inductor can be
modeled as:
(7)
(8)
.
.
.
(2)
.
The low voltage loop cut off frequency of around 10 Hz
is well below the LB×Co resonant frequency (around 125
Hz), so the voltage loop transfer function can be
approximated by:
(3)
.∆
.
.
Figure 9. Modified boost converter with coupled inductors.
Figure 9 shows the equivalent circuit of boost converter
with a coupled inductor, where active and passive switches
are replaced by the averaged model of PWM switch. By
application of superposition and other circuit theory, the
power stage transfer functions can be derived:
.
Figure 6. Conventional boost converter with PWM switch.
III.
.
.
PWM SWITCH MODEL OF BOOST CONVERTER WITH
COUPLED INDUCTOR
In order to model the boost converter with coupled
inductors, the winding arrangements shown in Figure 7 can
be described mathematically by:
.
.
.
.
.
(9)
.
.
1
.
(10)
3
2 .
2
1
.
2
2 .
1
3
(11)
The voltage loop transfer function approximation is also
valid here, enabling (10) to be simplified as (3). Figure 10
shows the Bode plot of current loop transfer functions for the
modified boost converter with coupled inductor and a
conventional boost converter.
75
The open loop bode plot is given in Figure 13. As it can
be seen, at cross over frequency, the open loop has a phase
margin of 65°, providing for a stable design.
75
0
50
− 25
25
− 60
0
− 90
− 25
− 120
− 50
Conventional Boost
Plant Transfer Function
Phase and Magnitude
− 75
3
1×10
4
Plant Transfer Function
Phase and Magnitude
− 150
− 150
− 75
10
5
1×10
− 120
Phase (Degree)
− 90
0
Magnitude (dB)
− 60
25
− 50
− 30
− 30
Modified Boost Plant
Transfer Function
Phase and Magnitude
Phase (Degree)
Magnitude (dB)
50
0
Controller Type II
Phase and Magnitude
6
1×10
1×10
100
3
1×10
− 180
7
1×10
4
1×10
5
1×10
6
1×10
− 180
7
1×10
Frequency (Hz)
Frequency (Hz)
Figure 10. Current loop plant Bode plots. LB = 400 µH, Co = 1000 µF, RL =
88 Ω, Ldc = 400 µH, Lac = 250 µH, M = 260 µH, Cs = 1 µF.
Figure 11. Current loop plant and Compensator Type II Bode plots.
LB = 400 µH, Co = 1000 µF, RL = 88 Ω, R2 = 11.5 kΩ, C2 = 4.7 nF and C3 =
230 pF.
As it can be noted that the phase margins for both
converters are the same and steady at -90 °. But, the main
difference is at the crossover frequencies, where for the
conventional boost it is significantly lower. If the same
feedback compensation network were to be used for both
converters, the steady state and large signal responses would
be inadequate for the coupled inductor boost.
FEEDBACK COMPENSATION DESIGN
A. Current Loop Compensation for the Conventional Boost
Converter
Figure 11 illustrates the current loop plant and a type II
compensator Bode plots. The current loop power stage has a
cross over frequency at 10 kHz, so a type II compensator
was chosen with a pole and zero, as shown in Figure 12.
The compensator has a pole at zero frequency, and
another pole at:
(12)
f
C
R C
and one zero at:
f
C
R C
C
(13)
Figure 12. Type II compensator network.
− 90
100
80
− 108
60
40
− 126
20
0
− 20
− 144
Phase (Degree)
The compensation is selected so that the open-loop
transfer function achieves the following criteria:
1. A high gain at low frequency in order to compensate
the steady state error.
2. A gain slope maintained at -20 dB/dec around the
crossover frequency in order to ensure enough phase
margin and, therefore, closed-loop stability.
3. A very small gain at high frequency in order to
reduce the influence of the switching harmonics and
overall noise.
Magnitude (dB)
IV.
− 40
− 60
− 80
− 100
3
1×10
Compensated Current Loop
Plant (Open Loop)
Phase and Magnitude
4
1×10
− 162
5
1×10
− 180
6
1×10
Frequency (Hz)
Figure 13. Open loop Bode plot for current loop. LB = 400 µH, Co = 1000
µF, RL = 88 Ω, R2 = 11.5 kΩ, C2 = 4.7 nF and C3 = 230 pF.
B. Voltag Loop Compensation for the Conventional Boost
Converter
The voltage loop transfer function and compensator
network Bode plot is given in Figure 14. The compensator
100
− 36
Plant Transfer Function
Phase and Magnitude
− 72
0
Controller Type II
Phase and Magnitude
− 20
− 108
Phase (Degree)
− 60
25
Controller Type II
Phase and Magnitude
Modified boost Plant
Transfer Function
Phase and Magnitude
0
− 40
− 60
− 120
− 150
3
4
1×10
1×10
5
6
1×10
− 180
7
1×10
Frequency (Hz)
Figure 16. Current loop plant and compensator Bode plots for boost
converter with coupled inductor. RL = 88 Ω, Ldc = 400 µH, Lac = 250 µH, M
= 260 µH, Cs = 1 µF, R2 = 22.1 kΩ, C2 = 1 nF and C3 = 150 pF.
− 144
− 100
150
− 80
125
10
100
− 180
3
1×10
Frequency (Hz)
100
− 45
75
25
60
− 75
20
− 99
0
− 20
− 126
Phase (Degree)
− 72
− 40
− 60
− 160
0
− 25
− 50
Compensated Current Loop
Plant (Open Loop)
Phase and Magnitude
− 140
50
80
40
− 120
100
Magnitude (dB)
1
Figure 14. Voltage loop plant and compensator type II Bode plots. LB = 400
µH, Co = 1000 µF, RL = 88 Ω, R2 = 75 kΩ, C2 = 1 µF and C3 = 100 nF.
− 100
10
Compensated Current Loop
Plant (Open Loop)
Phase and Magnitude
100
1×10
3
4
1×10
1×10
5
− 180
− 200
6
1×10
Frequency (Hz)
Figure 17. Open loop plant Bode plot for boost converter with coupled
inductor. RL = 88 Ω, Ldc = 400 µH, Lac = 250 µH, M = 260 µH, Cs = 1 µF,
R2 = 22.1 kΩ, C2 = 1 nF and C3 = 150 pF.
− 153
− 80
− 100
0.1
1×10
100
− 90
Phase (Degree)
Magnitude (dB)
50
− 50
10
40
Magnitude (dB)
− 30
− 25
60
− 100
0.1
75
0
80
20
0
Phase (Degree)
100
Magnitude (dB)
for the voltage is a type II compensator. In the compensator
network, the voltage error amplifier gain is adjusted with
compensation components to attenuate the double-line
frequency ripple on the output capacitor to obtain the desired
reduction of the 3rd harmonic THD. C3 sets the reduction
level, R2 sets the phase margin to 45 degrees at cross over
frequency, and C2 sets the beginning of the phase boost. The
desired cross over frequency for voltage loop is around 10
Hz, where the line frequency varies from 50 Hz to 60 Hz.
1
10
100
− 180
3
1×10
Frequency (Hz)
Figure 15. Open loop plant Bode plot for voltage loop. LB = 400 µH, Co =
1000 µF, RL = 88 Ω, R2 = 75 kΩ, C2 = 1 µF and C3 = 100 nF.
Figure 15 shows the open loop Bode plot of compensated
voltage loop. As it can be seen, it has a high gain at low
frequencies, and at cross over frequency, the open loop has a
phase margin of 55° which is a stable design.
C. Current Loop Compensation for Boost Converter with
Coupled Inductor
Figure 16 shows the current loop plant and a type II
compensator Bode plots. The current loop power stage has a
cross over frequency at 20 KHz.
The open loop Bode plot is given in Figure 17. As it can
be seen, at the cross over frequency, the open loop has a
phase margin of 50° which is a stable design.
D. Voltag Loop Compensation for Boost Converter with
Coupled Inductor
The voltage loop transfer function approximation is also
valid here, so the voltage loop plant transfer function does
not change with coupled inductor in a boost converter. As a
result, the voltage loop compensation network is the same as
in section B.
V.
SIMULATION RESULTS
PSIM simulation was used to verify the feedback loop
design for steady state and large signal perturbation. Figure
18 shows the simulation results of boost converter with
coupled inductors, converting universal AC input voltage at
240 V to a load at 400 V DC and 1.6 kW.
Figure 19 illustrates: Top: The boost inductor current in a
conventional boost converter - No Filtering technique.
Middle: The boost inductor current with ripple steering
technique applied, and Bottom: Series capacitor current in
the coupled inductor filter.
Figure 20 shows a coupled inductor used. The inductor
values are: Ldc = 400 uH, Lac = 250 uH, M = 265 uH and the
series capacitor Cs = 1 uF.
Figure 18. PSIM simulation circuit for ripple steering ttechnique applied to
PFC boost converter.
12
10
8
6
4
2
0
Figure 21. Inductor current Idc ripple at 120 V input and 800 W output Experimental.
-2
I(M1_1)
12
10
8
6
4
2
0
-2
I(M1_2)
4
2
0
-2
-4
0.16
0.17
0.18
0.19
0.2
Figure 19. Inductor Current - No Filtering technique (T
Top) – Ripple Steering
Technique (Middle), Series cap current (Bottom).
VI.
EXPERIMENTAL RESULLTS
Prototypes of the boost converter with ccoupled inductors
and conventional boost converter were buuilt to verify the
proof-of-concept and analytical work prresented and to
benchmark the proposed compensation netw
work design.
Figure 22. Inductor current Idc ripple at 120 V input and 800 W output Simulation.
w the experimental and
Figure 21 and Figure 22 show
simulation ripple current waveformss in the dc inductor under
the following operating conditions: Vin = 120 V, Iin = 15 A,
Po = 800 W, Vo = 400 V and 70 kHzz.
Figure 23 and Figure 24 show the experimental current
waveforms in the ac inductor under the same test condition.
w the experimental and
Figure 25 and Figure 26 show
simulation ripple current waveformss in the dc inductor under
the following operating conditions: fsw = 70 kHz, Vin = 240
V, Iin = 15 A, Po = 1600 W and Vo = 400 V.
Figure 20. Coupled inductors used in experim
mental circuit.
Figure 23. Inductor current Iac at 120 V input and 800 W output Experimental.
Figure 26. Inductor current Idc ripple at 240 V input and 1600 W output Simulation.
ac Inductor Current
8
6
4
2
0
-2
-4
-6
-8
0.17
0.175
0.18
0.185
Time (s)
0.19
0.195
Figure 24. Inductor current Iac at 120 V input and 800 W output Simulation.
Figure 25. Inductor current Idc ripple at 240 V input and 1600 W output Experimental.
0.2
Figure 27. Inductor current Iac at 240 V input and 1600 W output Experimental.
Figure 28. Inductor current Iac at 240 V input and 1600 W output Simulation.
Figure 27 and Figure 28 show the experimental and
simulation low frequency current waveforms in the ac
inductor under the following operating conditions: fsw = 70
kHz, Vin = 240 V, Iin = 15 A, Po = 1600 W and Vo = 400 V.
VII. CONCLUSIONS
An averaged PWM model for boost converters with the
ripple steering technique has been developed and the effect
of ripple steering on the design of the current loop
compensation network has been studied. Analytical and
simulation results were compared with experimental results
for a prototype converter circuit converting the universal AC
input voltage to a load at 400 V DC and 1.6 kW. The
proposed model shows an accurate prediction of steady state
and large transient behavior for the boost converter with
coupled inductors.
REFERENCES
[1]
[2]
[3]
D.C. Hamill ;P.T.Krein, "A `zero' ripple technique applicable to any
DC converter " in IEEE Power Electronics Specialists Conference,
PESC. vol. 2, 1999, pp. 1165 - 1171.
J.W. Kolar ; H. Sree ; N. Mohan ; F.C. Zach, "Novel aspects of an
application of `zero'-ripple techniques to basic converter topologies "
in Power Electronics Specialists Conference, PESC. vol. 1, 1997, pp.
796 - 803.
N.K. Poon ; J.C.P. Liu ; C.K. Tse ; M.H. Pong, "Techniques for input
ripple current cancellation: classification and implementation," IEEE
Transactions on Power Electronics, vol. 15, pp. 1144 - 1152 2000.
[4]
J. Wang ; W.G. Dunford ; K. Mauch, "Analysis of a ripple-free inputcurrent boost converter with discontinuous conduction characteristics
" IEEE Transactions on Power Electronics, vol. 12, pp. 684 - 694,
1997.
[5] E. Chou ; F. Chen; C. Adragna ; B. Lu, "Ripple steering AC-DC
converters to minimize input filter " in IEEE Energy Conversion
Congress and Exposition, ECCE, 2009, pp. 1325 - 1330.
[6] V. Vorperian, "Simplified analysis of PWM converters using model
of PWM switch, Part I: Continuous conduction mode " IEEE
Transactions on Aerospace and Electronic Systems, vol. 26, pp. 490 496 1990.
[7] V. Vorperian, "Simplified analysis of PWM converters using model
of PWM switch, Part II: Discontinuous conduction mode," IEEE
Transactions on Aerospace and Electronic Systems, vol. 26, pp. 497 505 1990.
[8] F. Musavi ; K. Al-Haddad ; H.Y. Kanaan, "A large signal averaged
modelling and control of paralleled DC/DC converters with automatic
load sharing," in IEEE Applied Power Electronics Conference and
Exposition, APEC. vol. 2, 2005, pp. 1353 - 1358.
[9] E. Van Dijk ; J.N. Spruijt ; D.M. O'Sullivan ; J.B. Klaassens, "PWMswitch modeling of DC-DC converters " IEEE Transactions on Power
Electronics, vol. 10, pp. 659 - 665 1995.
[10] Y.W. Lu ; W. Zhang ; Yanfei Liu, "A large signal dynamic model for
single-phase AC-to-DC converters with power factor correction "
IEEE Power Electronics Specialists Conference, PESC, vol. 2, pp.
1057 - 1063, 2004.
[11] Yan-Fei Liu ; P.C. Sen, "Large-signal modeling of hysteretic currentprogrammed converters " IEEE Transactions on Power Electronics,
vol. 11, pp. 423 - 430, 1996.
[12] Byungcho Choi ; Sung-Soo Hong ; Hyokil Park, "Modeling and
small-signal analysis of controlled on-time boost power-factorcorrection circuit " IEEE Transactions on Industrial Electronics, vol.
48, pp. 136 - 142, 2001.