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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

Design example

Also, VM

V = 120 2 V390 V

= 0.435

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

95% efficiency withVM/V = 0.435 occurswith Ron/Re ≈ 0.075.

So we require aMOSFET with onresistance of

Ron ≤ (0.075) Re

= (0.075) (27.4 Ω) = 2 Ω

ECEN5807 Power Electronics 2 15 Chapter 18: Low harmonic rectifier modeling and control

18.2 Controller schemes

Average current control

Feedforward

Current programmed control

Hysteretic control

Nonlinear carrier control

ECEN5807 Power Electronics 2 16 Chapter 18: Low harmonic rectifier modeling and control

18.2.1 Average current control

+–

+

v(t)

–

vg(t)

ig(t)

Gatedriver

Pulse widthmodulator

CompensatorGc(s)

+–

+–

Currentreference

vr(t)

va(t)≈ Rs ⟨ ig(t)⟩Ts

LBoost example

Low frequency(average) componentof input current iscontrolled to followinput voltage

ECEN5807 Power Electronics 2 17 Chapter 18: Low harmonic rectifier modeling and control

Use of multiplier to control average power

+–

+

v(t)

–

vg(t)

ig(t)

Gatedriver

Pulse widthmodulator

CompensatorGc(s)

+–

+–vref1(t)kx xy

xy

Multiplier

vg(t)

vcontrol(t) Gcv(s)+

–

Voltage reference

C

vref2(t)

v(t)

verr(t)

va(t)

Pav =V g,rms

2

Re= Pload

As discussed in Chapter17, an output voltagefeedback loop adjusts theemulated resistance Re

such that the rectifierpower equals the dc loadpower:

An analog multiplierintroduces thedependence of Re

on v(t).

ECEN5807 Power Electronics 2 18 Chapter 18: Low harmonic rectifier modeling and control

18.2.2 Feedforward

+–

+

v(t)

–

vg(t)

ig(t)

Gatedriver

Pulse widthmodulator

CompensatorGc(s)

+–

+–vref1(t)

x

y

multiplier

vg(t)

vcontrol(t) Gcv(s)+

–

Voltage reference

k vxyz2z

Peakdetector VM

vref2(t)

va(t)

Feedforward is sometimesused to cancel outdisturbances in the inputvoltage vg(t).

To maintain a given powerthroughput Pav, the referencevoltage vref1(t) should be

vref 1(t) =Pavvg(t)Rs

V g,rms2

ECEN5807 Power Electronics 2 19 Chapter 18: Low harmonic rectifier modeling and control

Feedforward, continued

+–

+

v(t)

–

vg(t)

ig(t)

Gatedriver

Pulse widthmodulator

CompensatorGc(s)

+–

+–vref1(t)

x

y

multiplier

vg(t)

vcontrol(t) Gcv(s)+

–

Voltage reference

k vxyz2z

Peakdetector VM

vref2(t)

va(t)

vref 1(t) =k vvcontrol(t)vg(t)

V M2

Pav =k vvcontrol(t)

2Rs

Controller with feedforwardproduces the following reference:

The average power is thengiven by

ECEN5807 Power Electronics 2 20 Chapter 18: Low harmonic rectifier modeling and control

18.2.3 Current programmed control

Boost converter

Current-programmed controller

Rvg(t)

ig(t)

is(t)vg(t)

+

v(t)

–

i2(t)

Q1

L

C

D1

vcontrol(t)

Multiplier Xic(t)

= kx vg(t) vcontrol(t)

+–

+ +

+

–

Comparator Latch

ia(t)Ts0

S

R

Q

ma

Clock

Current programmedcontrol is a naturalapproach to obtain inputresistor emulation:

Peak transistor current isprogrammed to followinput voltage.

Peak transistor currentdiffers from averageinductor current,because of inductorcurrent ripple andartificial ramp. This leadsto significant inputcurrent waveformdistortion.

ECEN5807 Power Electronics 2 21 Chapter 18: Low harmonic rectifier modeling and control

CPM boost converter: Static input characteristics

ig(t) Ts=

vg(t)Lic

2(t) fs

V – vg(t)vg(t)V +

maLV

in DCM

ic(t) + maT s

vg(t)V

– 1 +vg

2(t)Ts

2LVin CCM

Mode boundary: CCM occurs when

ig(t) Ts>

TsV2L

vg(t)V 1 –

vg(t)V

or, ic(t) >TsVL

maLV +

vg(t)V 1 –

vg(t)V

It is desired that ic(t) =vg(t)Re

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0vg(t)

V

j g(t)

=i g

(t)

Ts

Rba

se

VCCM

DCM

R e =

Rba

se

R e =

4R ba

se

R e =

0.3

3Rba

seR e

= 0

.5R ba

se

R e =

2R ba

se

R e =

0.2

R base

Re

= 0

.1R

base

Re= 10

R base

ma = V2L

Rbase = 2LTs

Static input characteristics ofCPM boost, with minimumslope compensation:

Minimum slope compensation:ma = V

2L

ECEN5807 Power Electronics 2 22 Chapter 18: Low harmonic rectifier modeling and control

18.3 Control system modelingof high quality rectifiers

Two loops:

Outer low-bandwidth controller

Inner wide-bandwidth controller

Boost converter

Wide-bandwidth input current controller

vac(t)

iac(t)+

vg(t)

–

ig(t)

ig(t)vg(t)

+

vC(t)

–

i2(t)

Q1

L

C

D1

vcontrol(t)

Multiplier X

+–vref1(t)

= kxvg(t)vcontrol(t)

Rsva(t)

Gc(s)

PWM

Compensator

verr(t)

DC–DCConverter Load

+

v(t)

–

i(t)

d(t)

+–Compensatorand modulator

vref3

Wide-bandwidth output voltage controller

+–Compensatorvref2

Low-bandwidth energy-storage capacitor voltage controller

vC(t)

v(t)

ECEN5807 Power Electronics 2 23 Chapter 18: Low harmonic rectifier modeling and control

18.3.1 Modeling the outer low-bandwidthcontrol system

This loop maintains power balance, stabilizing the rectifier outputvoltage against variations in load power, ac line voltage, andcomponent values

The loop must be slow, to avoid introducing variations in Re at theharmonics of the ac line frequency

Objective of our modeling efforts: low-frequency small-signal modelthat predicts transfer functions at frequencies below the ac linefrequency

ECEN5807 Power Electronics 2 24 Chapter 18: Low harmonic rectifier modeling and control

Large signal modelaveraged over switching period Ts

Re(vcontrol)⟨ vg(t)⟩Ts

vcontrol

+

–

Ideal rectifier (LFR)

acinput

dcoutput

+–

⟨ ig(t)⟩Ts

⟨ p(t)⟩Ts

⟨ i2(t)⟩Ts

⟨ v(t)⟩TsC Load

Ideal rectifier model, assuming that inner wide-bandwidth loopoperates ideally

High-frequency switching harmonics are removed via averaging

Ac line-frequency harmonics are included in model

Nonlinear and time-varying

ECEN5807 Power Electronics 2 25 Chapter 18: Low harmonic rectifier modeling and control

Predictions of large-signal model

Re(vcontrol)⟨ vg(t)⟩Ts

vcontrol

+

–

Ideal rectifier (LFR)

acinput

dcoutput

+–

⟨ ig(t)⟩Ts

⟨ p(t)⟩Ts

⟨ i2(t)⟩Ts

⟨ v(t)⟩TsC Load

vg(t) = 2 vg,rms sin ωt

If the input voltage is

Then theinstantaneous poweris:

p(t)Ts

=vg(t) Ts

2

Re(vcontrol(t))=

vg,rms2

Re(vcontrol(t))1 – cos 2ωt

which contains a constant term plus a second-harmonic term

ECEN5807 Power Electronics 2 26 Chapter 18: Low harmonic rectifier modeling and control

Separation of power source into its constant andtime-varying components

+

–

⟨ i2(t)⟩Ts

⟨ v(t)⟩TsC Load

V g,rms2

Re–

V g,rms2

Recos2 2ωt

Rectifier output port

The second-harmonic variation in power leads to second-harmonicvariations in the output voltage and current

ECEN5807 Power Electronics 2 27 Chapter 18: Low harmonic rectifier modeling and control

Removal of even harmonics via averaging

t

v(t)

⟨ v(t)⟩T2L

⟨ v(t)⟩Ts

T2L = 12

2πω = π

ω

ECEN5807 Power Electronics 2 28 Chapter 18: Low harmonic rectifier modeling and control

Resulting averaged model

+

–

⟨ i2(t)⟩T2L

⟨ v(t)⟩T2LC Load

V g,rms2

Re

Rectifier output port

Time invariant model

Power source is nonlinear

ECEN5807 Power Electronics 2 29 Chapter 18: Low harmonic rectifier modeling and control

Perturbation and linearization

v(t)T2L

= V + v(t)

i2(t) T2L= I2 + i2(t)

vg,rms = Vg,rms + vg,rms(t)

vcontrol(t) = Vcontrol + vcontrol(t)

V >> v(t)

I2 >> i2(t)

Vg,rms >> vg,rms(t)

Vcontrol >> vcontrol(t)

Let with

The averaged model predicts that the rectifier output current is

i2(t) T2L=

p(t)T2L

v(t)T2L

=vg,rms

2 (t)

Re(vcontrol(t)) v(t)T2L

= f vg,rms(t), v(t)T2L

, vcontrol(t))

ECEN5807 Power Electronics 2 30 Chapter 18: Low harmonic rectifier modeling and control

Linearized result

I2 + i2(t) = g2vg,rms(t) + j2v(t) –vcontrol(t)

r2

g2 =df vg,rms, V, Vcontrol)

dvg,rmsvg,rms = Vg,rms

= 2Re(Vcontrol)

Vg,rms

V

where

– 1r2

=df Vg,rms, v

T2L, Vcontrol)

d vT2L

v T2L= V

= –I2

V

j2 =df Vg,rms, V, vcontrol)

dvcontrolvcontrol = Vcontrol

= –V g,rms

2

VRe2(Vcontrol)

dRe(vcontrol)

dvcontrolvcontrol = Vcontrol

ECEN5807 Power Electronics 2 31 Chapter 18: Low harmonic rectifier modeling and control

Small-signal equivalent circuit

C

Rectifier output port

r2g2 vg,rms j2 vcontrol R

i2

+

–

v

v(s)vcontrol(s)

= j2 R||r21

1 + sC R||r2

v(s)vg,rms(s)

= g2 R||r21

1 + sC R||r2

Predicted transfer functions

Control-to-output

Line-to-output

ECEN5807 Power Electronics 2 32 Chapter 18: Low harmonic rectifier modeling and control

Model parameters

Table 18. 1 Small-signal model parameters for several types of rectifier control schemes

Controller type g2 j2 r2

Average current control withfeedforward, Fig. 18.9

0 Pav

VVcontrol

V 2

Pav

Current-programmed control,Fig. 18.10

2Pav

VVg,rms

Pav

VVcontrol

V 2

Pav

Nonlinear-carrier charge controlof boost rectifier, Fig. 18.14

2Pav

VVg,rms

Pav

VVcontrol

V 2

2Pav

Boost with hysteretic control,Fig. 18.13(b)

2Pav

VVg,rms

Pav

VTon

V 2

Pav

DCM buck–boost, flyback,SEPIC, or Cuk converters

2Pav

VVg,rms

2Pav

VD V 2

Pav

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.