ECEN5807 Power Electronics 2 1 Chapter 18: Low harmonic rectifier modeling and control Chapter 18 Low Harmonic Rectifier Modeling and Control 18.1 Modeling losses and efficiency in CCM high-quality rectifiers Expression for controller duty cycle d(t) Expression for the dc load current Solution for converter efficiency η Design example 18.2 Controller schemes Average current control Feedforward Current programmed control Hysteretic control Nonlinear carrier control 18.3 Control system modeling Modeling the outer low-bandwidth control system Modeling the inner wide-bandwidth average current controller
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ECEN5807 Power Electronics 2 1 Chapter 18: Low harmonic rectifier modeling and control
Chapter 18
Low Harmonic Rectifier Modeling and Control
18.1 Modeling losses and efficiency in CCM high-quality rectifiersExpression for controller duty cycle d(t)
Expression for the dc load currentSolution for converter efficiency ηDesign example
18.2 Controller schemesAverage current control
FeedforwardCurrent programmed controlHysteretic control
Nonlinear carrier control
18.3 Control system modelingModeling the outer low-bandwidth control system
Modeling the inner wide-bandwidth average current controller
ECEN5807 Power Electronics 2 2 Chapter 18: Low harmonic rectifier modeling and control
18.1 Modeling losses and efficiencyin CCM high-quality rectifiers
Objective: extend procedure of Chapter 3, to predict the outputvoltage, duty cycle variations, and efficiency, of PWM CCM lowharmonic rectifiers.
Approach: Use the models developed in Chapter 3. Integrate overone ac line cycle to determine steady-state waveforms and averagepower.
Boost example
+– Q1
L
C R
+
v(t)
–
D1
vg(t)
ig(t)RL i(t)
+– R
+
v(t)
–
vg(t)
ig(t)RL
i(t)DRon
+ –
D' : 1VF
Dc-dc boost converter circuit Averaged dc model
ECEN5807 Power Electronics 2 3 Chapter 18: Low harmonic rectifier modeling and control
Modeling the ac-dc boost rectifier
Rvac(t)
iac(t)+
vg(t)
–
ig(t)
+
v(t)
–
id(t)
Q1
L
C
D1
controller
i(t)
RL
+– R
+
v(t) = V
–
vg(t)
ig(t)RL
i(t) = Id(t) Ron
+ –
d'(t) : 1VF
id(t)
C(large)
Boostrectifiercircuit
Averagedmodel
ECEN5807 Power Electronics 2 4 Chapter 18: Low harmonic rectifier modeling and control
Boost rectifier waveforms
0
2
4
6
8
10
0
100
200
300
vg(t)
vg(t)
ig(t)
ig(t)
0° 30° 60° 90° 120° 150° 180°
d(t)
0
0.2
0.4
0.6
0.8
1
0° 30° 60° 90° 120° 150° 180°
0
1
2
3
4
5
6
id(t)
i(t) = I
ωt0° 30° 60° 90° 120° 150° 180°
Typical waveforms
(low frequency components)
ig(t) =vg(t)Re
ECEN5807 Power Electronics 2 5 Chapter 18: Low harmonic rectifier modeling and control
Example: boost rectifierwith MOSFET on-resistance
+– R
+
v(t) = V
–
vg(t)
ig(t) i(t) = I
d(t) Ron
d'(t) : 1
id(t)
C(large)
Averaged model
Inductor dynamics are neglected, a good approximation when the acline variations are slow compared to the converter natural frequencies
ECEN5807 Power Electronics 2 6 Chapter 18: Low harmonic rectifier modeling and control
18.1.1 Expression for controller duty cycle d(t)
+– R
+
v(t) = V
–
vg(t)
ig(t) i(t) = I
d(t) Ron
d'(t) : 1
id(t)
C(large)
Solve input side ofmodel:
ig(t)d(t)Ron = vg(t) – d'(t)v
with ig(t) =vg(t)Re
eliminate ig(t):
vg(t)Re
d(t)Ron = vg(t) – d'(t)v
vg(t) = VM sin ωt
solve for d(t):
d(t) =v – vg(t)
v – vg(t)Ron
Re
Again, these expressions neglect converter dynamics, and assumethat the converter always operates in CCM.
ECEN5807 Power Electronics 2 7 Chapter 18: Low harmonic rectifier modeling and control
18.1.2 Expression for the dc load current
+– R
+
v(t) = V
–
vg(t)
ig(t) i(t) = I
d(t) Ron
d'(t) : 1
id(t)
C(large)
Solve output side ofmodel, using chargebalance on capacitor C:
I = id Tac
id(t) = d'(t)ig(t) = d'(t)vg(t)
Re
Butd’(t) is:
d'(t) =vg(t) 1 –
Ron
Re
v – vg(t)Ron
Re
hence id(t) can be expressed as
id(t) =vg
2(t)Re
1 –Ron
Re
v – vg(t)Ron
Re
Next, average id(t) over an ac line period, to find the dc load current I.
ECEN5807 Power Electronics 2 8 Chapter 18: Low harmonic rectifier modeling and control
Dc load current I
I = id Tac= 2
Tac
V M2
Re
1 –Ron
Resin2 ωt
v –VM Ron
Resin ωt
dt
0
Tac/2
Now substitute vg (t) = VM sin ωt, and integrate to find ⟨ id(t)⟩Tac:
This can be written in the normalized form
I = 2Tac
V M2
VRe1 –
Ron
Re
sin2 ωt
1 – a sin ωtdt
0
Tac/2
with a =VM
VRon
Re
ECEN5807 Power Electronics 2 9 Chapter 18: Low harmonic rectifier modeling and control
Integration
By waveform symmetry, we need only integrate from 0 to Tac/4. Also,make the substitution θ = ωt:
I =V M
2
VRe1 –
Ron
Re
2π
sin2 θ1 – a sin θ
dθ0
π/2
This integral is obtained not only in the boost rectifier, but also in thebuck-boost and other rectifier topologies. The solution is
4π
sin2 θ1 – a sin θ
dθ0
π/2
= F(a) = 2a2π – 2a – π +
4 sin– 1 a + 2 cos– 1 a
1 – a2
• Result is in closed form
• a is a measure of the lossresistance relative to Re
• a is typically much smaller thanunity
ECEN5807 Power Electronics 2 10 Chapter 18: Low harmonic rectifier modeling and control
The integral F(a)
F(a)
a
–0.15 –0.10 –0.05 0.00 0.05 0.10 0.150.85
0.9
0.95
1
1.05
1.1
1.15
4π
sin2 θ1 – a sin θ
dθ0
π/2
= F(a) = 2a2π – 2a – π +
4 sin– 1 a + 2 cos– 1 a
1 – a2
F(a) ≈ 1 + 0.862a + 0.78a2
Approximation viapolynomial:
For | a | ≤ 0.15, thisapproximate expression iswithin 0.1% of the exactvalue. If the a2 term isomitted, then the accuracydrops to ±2% for | a | ≤ 0.15.The accuracy of F(a)coincides with the accuracyof the rectifier efficiency η.
ECEN5807 Power Electronics 2 11 Chapter 18: Low harmonic rectifier modeling and control
18.1.4 Solution for converter efficiency η
Converter average input power is
Pin = pin(t) Tac=
V M2
2Re
Average load power is
Pout = VI = VV M
2
VRe1 –
Ron
Re
F(a)2 with a =
VM
VRon
Re
So the efficiency is
η =Pout
Pin= 1 –
Ron
ReF(a)
Polynomial approximation:
η ≈ 1 –Ron
Re1 + 0.862
VM
VRon
Re+ 0.78
VM
VRon
Re
2
ECEN5807 Power Electronics 2 12 Chapter 18: Low harmonic rectifier modeling and control
Boost rectifier efficiency
η =Pout
Pin= 1 –
Ron
ReF(a)
0.0 0.2 0.4 0.6 0.8 1.00.75
0.8
0.85
0.9
0.95
1
VM /V
η
Ron/Re
= 0.05
Ron/Re
= 0.1
Ron/Re
= 0.15
Ron/Re
= 0.2
• To obtain highefficiency, choose Vslightly larger than VM
• Efficiencies in the range90% to 95% can then beobtained, even with Ron
as high as 0.2Re
• Losses other thanMOSFET on-resistanceare not included here
ECEN5807 Power Electronics 2 13 Chapter 18: Low harmonic rectifier modeling and control
18.1.5 Design example
Let us design for a given efficiency. Consider the followingspecifications:
Output voltage 390 VOutput power 500 Wrms input voltage 120 VEfficiency 95%
Assume that losses other than the MOSFET conduction loss arenegligible.
Average input power is
Pin =Poutη = 500 W
0.95= 526 W
Then the emulated resistance is
Re =V g, rms
2
Pin=
(120 V)2
526 W= 27.4 Ω
ECEN5807 Power Electronics 2 14 Chapter 18: Low harmonic rectifier modeling and control
ECEN5807 Power Electronics 2 33 Chapter 18: Low harmonic rectifier modeling and control
Constant power load
vac(t)
iac(t)
Re
+
–
Ideal rectifier (LFR)
C
i2(t)ig(t)
vg(t)
i(t)
load
+
v(t)
–
pload(t) = VI = Pload
Energy storagecapacitor
vC(t)
+
–
Dc-dcconverter
+–Pload V
⟨ pac(t)⟩Ts
Rectifier and dc-dc converter operate with same average power
Incremental resistance R of constant power load is negative, and is
R = – V 2
Pav
which is equal in magnitude and opposite in polarity to rectifierincremental output resistance r2 for all controllers except NLC
ECEN5807 Power Electronics 2 34 Chapter 18: Low harmonic rectifier modeling and control
Transfer functions with constant power load
v(s)vcontrol(s)
=j2
sC
v(s)vg,rms(s)
=g2
sC
When r2 = –R, the parallel combination r2 || R becomes equal to zero.The small-signal transfer functions then reduce to
ECEN5807 Power Electronics 2 35 Chapter 18: Low harmonic rectifier modeling and control
18.3.2 Modeling the inner wide-bandwidthaverage current controller
+–
+–
L
C R
+
⟨v(t)⟩Ts
–
⟨vg(t)⟩Ts⟨v1(t)⟩Ts
⟨i2(t)⟩Ts
⟨i(t)⟩Ts
+
⟨v2(t)⟩Ts
–
⟨i1(t)⟩Ts
Averaged switch network
Averaged (but not linearized) boost converter model, Fig. 7.42:
vg(t) Ts= Vg + vg(t)
d(t) = D + d(t) ⇒ d'(t) = D' – d(t)
i(t)Ts
= i1(t) Ts= I + i(t)
v(t)Ts
= v2(t) Ts= V + v(t)
v1(t) Ts= V1 + v1(t)
i2(t) Ts= I2 + i2(t)
In Chapter 7,we perturbedand linearizedusing theassumptions
Problem: variations in vg,i1 , and d are not small.
So we are faced with thedesign of a controlsystem that exhibitssignificant nonlineartime-varying behavior.
ECEN5807 Power Electronics 2 36 Chapter 18: Low harmonic rectifier modeling and control
Linearizing the equations of the boost rectifier
When the rectifier operates near steady-state, it is true that
v(t)Ts
= V + v(t)
with
v(t) << V
In the special case of the boost rectifier, this is sufficient to linearizethe equations of the average current controller.
The boost converter average inductor voltage is
Ld ig(t) Ts
dt= vg(t) Ts
– d'(t)V – d'(t)v(t)
substitute:
Ld ig(t) Ts
dt= vg(t) Ts
– d'(t)V – d'(t)v(t)
ECEN5807 Power Electronics 2 37 Chapter 18: Low harmonic rectifier modeling and control
Linearized boost rectifier model
Ld ig(t) Ts
dt= vg(t) Ts
– d'(t)V – d'(t)v(t)
The nonlinear term is much smaller than the linear ac term. Hence, itcan be discarded to obtain
Ld ig(t) Ts
dt= vg(t) Ts
– d'(t)V
Equivalent circuit:
+–
L
+– d'(t)Vvg(t) Ts
ig(t) Ts
ig(s)d(s)
= VsL
ECEN5807 Power Electronics 2 38 Chapter 18: Low harmonic rectifier modeling and control
The quasi-static approximation
The above approach is not sufficient to linearize the equations needed todesign the rectifier averaged current controllers of buck-boost, Cuk,SEPIC, and other converter topologies. These are truly nonlinear time-varying systems.
An approximate approach that is sometimes used in these cases: thequasi-static approximation
Assume that the ac line variations are much slower than the converterdynamics, so that the rectifier always operates near equilibrium. Thequiescent operating point changes slowly along the input sinusoid, and wecan find the slowly-varying “equilibrium” duty ratio as in Section 18.1.
The converter small-signal transfer functions derived in Chapters 7 and 8are evaluated, using the time-varying operating point. The poles, zeroes,and gains vary slowly as the operating point varies. An average currentcontroller is designed, that has a positive phase margin at each operatingpoint.
ECEN5807 Power Electronics 2 39 Chapter 18: Low harmonic rectifier modeling and control
Quasi-static approximation: discussion
In the literature, several authors have reported success using thismethod
Should be valid provided that the converter dynamics are suffieientlyfast, such that the converter always operates near the assumedoperating points
No good condition on system parameters, which can justify theapproximation, is presently known for the basic converter topologies
It is well-understood in the field of control systems that, when theconverter dynamics are not sufficiently fast, then the quasi-staticapproximation yields neither necessary nor sufficient conditions forstability. Such behavior can be observed in rectifier systems. Worst-case analysis to prove stability should employ simulations.