lcc converter small signal modeling psds seminar taiwan...lcc converter small signal modeling brent...

LCC Converter Small Signal Modeling

Brent McDonald

7-2 Texas Instruments – 2014/15 Power Supply Design Seminar

Introduction

•  Demanding efficiency requirements are driving engineers to the LLC resonant converter

•  How do we identify and verify a robust set of compensation values for this converter?

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Discussion Outline

•  LLC Converter •  Modeling Process •  Case Study •  Tools •  Practical Limitations •  Conclusion

Operating States

Texas Instruments – 2014/15 Power Supply Design Seminar 7-4

State Q1 Q2 Q3 Q4 1 ON OFF OFF ON 2 ON OFF ON OFF 3 ON OFF OFF OFF 4 OFF ON OFF ON 5 OFF ON ON OFF 6 OFF ON OFF OFF

State Variables

ILR(t)

ILM(t)

VCR(t)

VCO(t)

Transformer

Gate

Drive

Gate

Drive

Gate

Drive

VIN

ILR(t)

ISEC(t)

VCO(t)CO

VOUT(t)

LR

ILM(t)

LK

LM NP

NS

NS

Q1

Q2

VCR(t) CR

Q4

Q3

esr

7-5

Mode: Resonance

Texas Instruments – 2014/15 Power Supply Design Seminar


Mode State Sequence: 1→5


Mode: Below Resonance


Mode State Sequence: 1→3→5→6


Mode: Resonance


Mode State Sequence: 1→4→5→2

Modeling Process Overview


Fundamental Steps 1.  Identify the states and

modes used 2.  Average the states for

the identified mode 3.  Calculate DC operating

point 4.  Linearize the result

Controller

D(z)

Controller

D(z)

Operating Point

Plant

G(s)

VIN(t)

VIN(s)

VE(s)

VE(t)

VREF(s)

VREF(t)

CE(s)

CE(t)

VOUT(t)

VOUT(s)

IOUT(s)

IOUT(t)

σ

σ

?

Plant

Non-linear,

Time-varying

System

ControlEffort

LLC Power Stagef(x(t), u(t),t)

OutputSquare Wave Harmonic Content

Composite Plot

Output

Control Effort

Time (µs)

Time (µs)Time (µs)

Time (µs)0 2 4 6 8 10

0 2 4 6 8 100.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

1.0

0.5

0.0

-0.5

-1.0

1.0

0.5

0.0

-0.5

-1.0

1.0

0.5

0.0

-0.5

-1.0

1.0

0.5

0.0

-0.5

-1.0

Volta

ge (m

V)Vo

ltage

(mV)A

mpl

itude

(V)

Am

plitu

de (V

)

Raw OutputDescribing Function Output

1ST

3RD

5TH

7-9

Fourier Series


ILR(t)ILM(t)

VCR(t)

VOUT(t)

ISEC(t)

LLC Resonant Tank Waveforms

Cur

rent

(A)

Cur

rent

(A)

Volta

ge (V

)Vo

ltage

(V)

420

-2-4

60

40

20

0

-20

300250200150100500

12.2012.1512.1012.0512.0011.9511.9011.8511.80

Time (µs)0 5 10 15 20 25

Time (µs)0 5 10 15 20 25

Time (µs)0 5 10 15 20 25

Time (µs)0 5 10 15 20 25

10010

10.1

0.010.001

0.00011E-051E-06

1000

100

10

1

0.1

0.01

10

1

0.1

0.01

0.001

0.0001

Time (µs)0 5 10 15 20 25

1

0.1

0.01

0.001

0.0001

1E-05-1 1 3 5 7 9 11 13 15

0 2 4 6 8 10 12 14 16

-1 1 3 5 7 9 11 13 150 2 4 6 8 10 12 14 16

-1 1 3 5 7 9 11 13 150 2 4 6 8 10 12 14 16

Mag

nitu

de [A

]M

agni

tude

[A]

Mag

nitu

de [A

]

Harmonic

Harmonic

Harmonic

Harmonic

State Variable Spectral Content

Resonant Current Magnetizing Current Tesonant Capacitor Voltage Couput Capacitor Voltage

Some harmonics may be significant

Output

-1 1 3 5 7 9 11 13 150 2 4 6 8 10 12 14 16

Mag

nitu

de [A

]

Spectral Considerations


State Variable Harmonic’s Included ILR(t) 1, 3, 5, 7, 9, 11 ILM(t) 1, 3, 5, 7 VCR(t) 0, 1, 3, 5, 7

VCOUT(t) 0

7-11

Linearization


1.0

0.8

0.6

0.4

0.2

0.0

0 2 4 6 8 10

Input Vector

Ou

tpu

t V

ec

tor

Function

Linearized Function

Operating Point

Model Linearization

f(x0)+ (x− x0)∂f(x0)∂x0

+O((x− x0)2)


Describing Function Analysis

Control

Effort

LLC Power Stage

f(x(t), u(t),t)Output

•  Linear System

•  Non-Linear System

Fsk

ss = 2Ts

⋅ (AiTi−1

Ti∫i=1

Q

∑ ⋅x(t)ss + Bi ⋅U0 ) ⋅sin(k ⋅ωs ⋅ t) ⋅dt

xss(t) = X0ss + Xck

ss ⋅cos(k ⋅ωs ⋅ t)+ Xckss ⋅sin(k ⋅ωs ⋅ t)( )

k=1

∞

∑

Fck

ss = 2Ts

⋅ (AiTi−1

Ti∫i=1

Q

∑ ⋅x(t)ss + Bi ⋅U0 ) ⋅cos(k ⋅ωs ⋅ t) ⋅dt

− x(t) = A ⋅x(t) + B⋅u(t)− y(t) = C ⋅x(t) + D ⋅u(t)

− x(t) = f(x(t) + u(t),t)− y(t) = g(x(t) + u(t),t)


Steady State Operating Point

•  LLC + COUT fourth order system

•  Piecewise linear simulation:

–

–

•  Lightning fast, highly accurate results

⇒

xn(t) = A ⋅xn(t)+B⋅U

xn(ti ) = (eA⋅Δt − I) ⋅A−1 ⋅B⋅U + eA⋅Δt ⋅xn(ti−1)

LLC Small Signal ModelLLC Small Signal Model

Frequency (Hz)

Frequency (Hz)

10 100 1000 104

10 100 1000 104

Frequency (Hz)

Frequency (Hz)

10 100 1000 104

105

10 100 1000 104

110

105

100

95

90

85

80

75

180

135

90

45

0

-45

-90

-135

-180

180

135

90

45

0

-45

-90

-135

-180

Gain

(d

B)

110

105

100

95

90

85

80

75

Gain

(d

B)

Ph

ase (

°)

Ph

ase (

°)

Model

Measurement

Model

Simulation

Model Validation


Model vs. Circuit Simulation Model vs. Measurement


Compensation Objectives

•  Stability –  How do we achieve stability? –  How do we ensure that we have sufficient stability

margin for all operating points?

•  Performance –  Reference tracking –  Load transient response –  Input voltage transient rejection


Plant Analysis

•  4TH order system, 2ND order response -  G0 ~ 85 dB -  Q ~ 1.35 -  fp ~ 4 kHz

•  Stability Objectives -  Φm ≥ 45 ° -  gm ≥ 10 dB

Frequency (Hz)

Ph

as

e (

°)

Plant Response

Phase

Ga

in (

dB

)

90

80

70

60

50

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

Q

G0

fP

Compensation


Frequency (Hz)

Ph

as

e (

°)

Compensation

Phase

Plant

Compensator

Ga

in (

dB

)

140

120

100

80

60

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

•  1/s term is required to eliminate DC error

•  2 zeros are required for stability -  QZ = 1.35 -  fZ = 4 kHz

G0 ⋅

s2(2 ⋅π ⋅fZ)

2 +s

2 ⋅π ⋅fZ ⋅QZ+1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

s

Overall Stability


• 

•  Stability Margins

Frequency (Hz)

Ph

as

e (

°)

Compensation

Phase

Compensation

Plant

Loop

Ga

in (

dB

)

50

0

-50

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

gmfbw

Om

G0 ⋅

s2(2 ⋅π ⋅fZ)

2 +s

2 ⋅π ⋅fZ ⋅QZ+1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

s

– G0 = 0 dB

– Qz = 1.35

– fz = 4 kHz

– φm ≅ 95°

– gm ≅ 12 dB

– fbw ≅ 3 kHz

Overall Stability with an Extra Pole


Frequency (Hz)

Ph

ase (

°)

Compensation

Phase

Compensation

Plant

Loop

Gain

(d

B)

50

0

-50

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

gmfbw

Om

•  Stability Margins

G0 ⋅

s2(2 ⋅π ⋅fZ)

2 +s

2 ⋅π ⋅fZ ⋅QZ+1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

s⋅ s2 ⋅π ⋅ fp

+1⎛

⎝⎜⎜

⎞

⎠⎟⎟

– φm ≅ 90°

– gm ≅ 20 dB

– fbw ≅ 3 kHz

– G0 = 0 dB

– Qz = 1.35

– fz = 4 kHz

– fp = 15 kHz

Frequency (Hz)

Phas

e (°

)

Compensation ZOUT(s)

Phase

CompensationPlantLoop

Open LoopClosed LoopZCOUT

(s)

Gai

n (d

B)

,PSHGDQFH��ȍ�

100

50

0

-50

-100

0.1

0.01

0.001

10-4

10-5

10-6

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

Phas

e (°

)

Phase180

135

90

45

0

-45

-90

-135

-18010 100 1000 104

Frequency (Hz)10 100 1000 104

Frequency (Hz)

ZOUT(s), G0 = 0 dB



ZOUT(s), G0 = 9.5 dB

Frequency (Hz)

Phas

e (°

)

Compensation ZOUT(s)

Phase


Gai

n (d

B)

,PSHGDQFH��ȍ�

100

50

0

-50

-100

0.1

0.01

0.001

10-4

10-5

10-6

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

Phas

e (°

)

Phase180

135

90

45

0

-45

-90

-135

-18010 100 1000 104

Frequency (Hz)10 100 1000 104

Frequency (Hz)

Open LoopClosed LoopZCOUT

(s)


Frequency (Hz)

Phas

e (°

)

Compensation VO(s)/VIN(s)

Phase


Open LoopClosed Loop

Gai

n (d

B)

Gai

n (d

B)

100

50

0

-50

-100

-20

-30

-40

-50

-60

-70

-80

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

Phas

e (°

)

Phase180

135

90

45

0

-45

-90

-135

-18010 100 1000 104

Frequency (Hz)10 100 1000 104

Frequency (Hz)

VOUT(s)/VIN(s), G0 = 0 dB

Frequency (Hz)

Phas

e (°

)

Compensation VO(s)/VIN(s)

Phase


Open LoopClosed Loop

Gai

n (d

B)

Gai

n (d

B)

100

50

0

-50

-100

-20

-30

-40

-50

-60

-70

-80

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

Phas

e (°

)

Phase180

135

90

45

0

-45

-90

-135

-18010 100 1000 104

Frequency (Hz)10 100 1000 104

Frequency (Hz)


VOUT(s)/VIN(s), G0 = 9.5 dB

Ph

as

e (

°)

VIN = 400 V

Gain

VIN = 370 V

Gain

Phase

Increasing

Load

Increasing

Load

Increasing

Load

Increasing

Load

Ga

in (

dB

)

90

80

70

60

180

135

90

45

0

-45

-90

-135

-180

Ph

as

e (

°)

Ga

in (

dB

)

90

80

70

60

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

Frequency (Hz)

Phase

10 100 1000 104

Frequency (Hz)

10 100 1000 104

Frequency (Hz)

Legend

��ȍ��ȍ��ȍ��ȍ��ȍ��ȍ��ȍ��ȍ��ȍ��ȍ��ȍ


Load Variation


VIN Variation P

ha

se

(°)

/RDG� ��ȍGain

/RDG� ��ȍGain

Phase

Increasing

VIN

Increasing

VIN

Increasing

VIN

Increasing

Load

Ga

in (

dB

)

180

135

90

45

0

-45

-90

-135

-180

Ph

as

e (

°)

Ga

in (

dB

)

100

90

80

70

60

100

90

80

70

60

180

135

90

45

0

-45

-90

-135

-180

10 100 1000 104

10 100 1000 104

Frequency (Hz)

Frequency (Hz)

Phase

10 100 1000 104

Frequency (Hz)

10 100 1000 104

Frequency (Hz)

Legend

370 V

372 V

374 V

376 V

378 V

380 V

382 V

384 V

386 V

388 V

390 V

392 V

394 V

396 V

398 V

400 V

402 V

404 V

406 V

408 V

410 V


Fusion Digital Power Designer


Time Domain Behavior

Time Simulation Outputs Time Simulation Outputs

5 10 15 20 25

5 10 15 20 25

5 10 15 20 25

Time (µs)

Time (µs)

Time (µs)

5 10 15 20 25

5 10 15 20 25

5 10 15 20 25

Time (µs)

Time (µs)

Time (µs)

4

2

0

-2

-4

60

40

20

0

-20

40

20

0

-20

-40

300

250

200

150

100

50

0

14

12

10

8

6

4

2

0

4

3

2

1

0

-1

-2

Resonant Current

Magnetizing Current

Resonant Capacitor Voltage

Output Voltage

Input Current

Rectified Secondary Current

Output Capacitor Current

Cu

rre

nt

[A]

Cu

rre

nt

[A]

Cu

rre

nt

[A]

Cu

rre

nt

[A]

Vo

lta

ge

[V

]V

olt

ag

e [

V]


Limitations

•  Low efficiency scenarios may need additional work to achieve proper correlation

•  Does not support PWM or PSM •  Corner cases may exist which limit the accuracy due to

numerical approximations •  Additional work may be required to ensure accuracy,

especially at higher frequencies


Conclusions

•  Analytical predictions of plant pole zero behavior enables more robust compensation –  Parameter variations –  Extreme operating conditions

•  Independent validation of the DC operating point •  Instant visualization of:

–  Key system waveforms –  Harmonic content

•  Seamless integration with TI standard isolated digital

controllers


Future Work

•  DC operating point •  Performance metrics

–  ZOUT(s) –  ZIN(s) –  VOUT(s)/VIN(s)

•  Modulation methods –  FM –  PSM –  PWM

•  Additional States & Modes –  Switching transitions –  Body diode conduction

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