emulatd cur mode cntrl for buck regs using sample and ...abstract – while naturally sampled peak...

LM3495

Emulated Current Mode Control for Buck Regulators Using Sample and Hold

Technique

Literature Number: SNVA537

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS USING SAMPLE AND HOLD TECHNIQUE

Small Signal Linear Analysis and Comparison to Peak and Valley

Methods

by

Robert Sheehan Principal Applications Engineer

National Semiconductor Corporation Santa Clara, CA

PES02 Tuesday, October 24, 2006

8:30am – 9:30am

Power Electronics Technology Exhibition and Conference October 24-26, 2006

Long Beach Convention Center Long Beach, CA


Small Signal Linear Analysis and Comparison to Peak and Valley Methods

by

Robert Sheehan

Principal Applications Engineer National Semiconductor Corporation

Santa Clara, CA

Abstract – While naturally sampled peak and valley current mode control methods have been widely used, other control architectures are possible using gated sampling techniques. Theory for an emulated peak current mode control method using a gated sample and hold of the valley current is developed. This gated sampling technique removes the duty cycle dependence of the slope compensating ramp, stabilizing the modulator gain over changes in line voltage. A general solution for current mode buck regulator small signal linear equations is presented. This allows the modulator gain for any control method to be introduced into the equations, including peak, valley, average and gated sampling methods. Comparison to peak and valley is made using switching, linear and LaPlace spice models. Sub-harmonic stability bounds are demonstrated using graphical spreadsheet calculators. Theory is verified with frequency response measurements of an actual circuit.

Figure 1: Naturally sampled peak or valley current mode buck regulator.

Figure 2: Emulated peak current mode buck regulator using valley sample and hold.

1

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan 1. Introduction There are a lot of misconceptions and misinformation about current mode control in the industry. Papers that have been written at the graduate or PhD level are hard to understand. Concepts are difficult to put into practical use. Basically, an ideal current mode converter is only dependent on the dc or average inductor current. The inner current loop turns the inductor into a voltage controlled current source, effectively removing the inductor from the outer voltage control loop at dc and low frequencies. The current loop gain splits the complex conjugate pole of the output filter into two real poles, so that the characteristic of the output filter is set by the capacitor and load resistor. Only when the impedance of the output inductor equals the current loop gain does the inductor pole reappear at higher frequencies. Whether the current mode converter is peak, valley, average, or sample and hold is secondary to the operation of the current loop. As long as the dc current is sampled, current mode operation is maintained. The modulator gain is dependent on the effective slope of the ramp presented to the modulating comparator input. Each operating mode will have a unique characteristic equation for the modulator gain. The requirement for slope compensation is dependent on the relationship of the average current to the value of current at the time when the sample is taken. To understand the theory and follow the derivations presented in this paper, a good working knowledge of the references is needed. For the practical designer, simplified transfer functions along with tabulated general gain parameters provide the basic tools for design analysis. The primary application for emulated current mode is high input voltage to low output voltage operating at a narrow duty cycle. In any practical design, device capacitance may cause a significant leading edge spike on the current sense waveform. By sampling the inductor current at the end of the switching cycle and adding an external ramp, the minimum on time can be significantly reduced, without the need for blanking or filtering which is normally required for peak current mode control. 2. Linear Modeling

Figure 3: Buck regulator linear models.

2

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan

Averaged Model Reference [1] has been one of the most popular papers covering current mode control. The analysis presented here refers the inductor current to the control voltage as the basis for writing the transfer functions. Starting with , write the transfer function in terms of voltages: Ov

LO

OSWO ZZ

Zvv

+⋅= (1)

dVDvv ININSW ⋅+⋅= (2)

)KvKvRiv(Fd OOIINiLCm ⋅+⋅−⋅−⋅= (3)

O

OL Z

vi = (4)

Combining equations 1 through 4 yields:

LO

OOOIIN

O

iOCmININO ZZ

ZKvKv

ZR

vvFVDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+⋅−⋅−⋅⋅+⋅= (5)

Define Kmp as:

mINmp FVK ⋅= (6) Setting to zero allows the control to output gain to be found: INv

)ZKR(KZZZK

vv

OOimpLO

Omp

C

O

⋅−⋅++

⋅= (7)

Setting to zero allows the line to output gain to be found: Cv

)ZKR(KZZZ)KKD(

vv

OOimpLO

OImp

IN

O

⋅−⋅++

⋅⋅−= (8)

Define ZO and ZL:

)RR(Cs1)RCs1(R

R||RCs1Z

COO

COOOC

OO +⋅⋅+

⋅⋅+⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

⋅= (9)

SLL RRLsZ ++⋅= (10)

Equations 7 and 8 can be plotted without further analysis using MATHCAD, SPICE or other application. Define terms for: Kmp Modulator gain coefficient KI Line to modulator gain block KO Output to modulator gain block Ri Current sense amplifier gain = GI · RS

3


DC Transfer Functions Control to Output Let ZO=RO, ZL=RL+RS. From equation 7:

)RKR(KRRRRK

)dc(vv

OOimpSLO

Omp

C

O

⋅−⋅+++

⋅= (11)

By factoring this becomes:

imp

OmpO

imp

SLi

O

C

O

RKKK1

RRKRR

1

1RR

)dc(vv

⋅

⋅−⋅+

⋅+

+⋅= (12)

Define Km as the modulator gain where:

Omp

mK

K1

1K−

= (13)

This allows the control to output gain to be expressed as:

im

O

imp

SLi

O

C

O

RKR

RKRR

1

1RR

)dc(vv

⋅+

⋅+

+⋅= (14)

If RO>>RL+RS the simplified expression is:

im

Oi

O

C

O

RKR

1

1RR

)dc(vv

⋅+

⋅≈ (15)

Line to Output In similar fashion:

)RKR(KRRRR)KKD(

)dc(vv

OOimpSLO

OImp

IN

O

⋅−⋅+++

⋅⋅−= (16)


imp

OmpO

imp

SL

I

mp

i

O

IN

O

RKKK1

RRKRR

1

DK

K1

RDR

)dc(vv

⋅

⋅−⋅+

⋅+

+

−

⋅⋅

= (17)

Define Kn as the audio susceptibility coefficient:

DK

K1K I

mpn −= (18)

4

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan This allows the line to output gain to be expressed as:

im

O

imp

SL

n

i

O

IN

O

RKR

RKRR

1

KR

DR)dc(

vv

⋅+

⋅+

+⋅

⋅= (19)

If RO>>RL+RS the simplified expression is:

im

O

n

i

O

IN

O

RKR

1

KR

DR)dc(

vv

⋅+

⋅⋅

≈ (20)

Continuous-Time Model

References [2] and [3] cover this model. The sampling gain is incorporated into the current loop as He(s). Derivation of the gain blocks for the on voltage and off voltage requires differentiation of the inductor current with respect to each voltage. Here, a straightforward algebraic method is used to relate the continuous-time model to the averaged model. This provides a simple means to derive the equations for different control modes. Starting with , write the transfer function in terms of voltages: d

)KvKv)s(HRiv(Fd rOFFfONeiLCm ′⋅+′⋅+⋅⋅−⋅′= (21) Combining with equations 1, 2 and 4 for the power stage yields:

LO

OrOFFfON

O

eiOCmININO ZZ

ZKvKv

Z)s(HR

vvFVDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅+′⋅+

⋅⋅−⋅′⋅+⋅= (22)

Define K′mp as:

mINmp FVK ′⋅=′ (23) Let , , ZOINON vvv −= OOFF vv = O=RO, ZL=0, He(s)=1. Solve for the dc gain equation.

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅+′⋅−′⋅+⋅−⋅′+⋅≈ rOfOfIN

O

iOCmpINO KvKvKv

RR

vvKDv)dc(v (24)

Setting to zero allows the control to output gain to be found: INv

imp

rmpfmpO

i

O

C

O

RKKKKK1

R1

1RR

)dc(vv

⋅′

′⋅′−′⋅′+⋅+

⋅≈ (25)


imp

rmpfmpO

f

mp

i

O

IN

O

RKKKKK1

R1

DK

K1

RDR

)dc(vv

⋅′

′⋅′−′⋅′+⋅+

′+

′⋅

⋅≈ (26)

5

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan Define K′m and K′n where:

rfmp

mKK

K1

1K′−′+

′

=′ D

KK

1K f

mpn

′+

′=′ (27)

Equate Km to K′m and Kn to K′n. Solve for K′f and K′r.

Impmp

f KK

1K

1DK −⎟⎟⎠

⎞⎜⎜⎝

⎛

′−⋅=′ Of

mpmpr KK

K1

K1K +′+−′

=′ (28)

Figure 4: Simplified continuous-time model using general gain parameters. This model is valid for control to output transfer functions of all operating modes. The line to output transfer function is valid at dc, but diverges from the

actual response over frequency.

⎟⎟⎠

⎞⎜⎜⎝

⎛−

′⋅+=

mmpi

Le K

1K

1RZ)s(H)s(H (29)

Where:

2n

2

zne

ωs

Qωs1)s(H +⋅

+= Tπωn =

π2Q z −= (30)

Unified Model

The unified model is presented in references [4] and [5]. This uses a single pole in series with the modulator to account for the sampling gain. Starting with , write the transfer function in terms of voltages: d

)KvRiv()s(Fd INiLCm ⋅−⋅−⋅= (31) Combining with equations 1, 2 and 4 for the power stage yields:

LO

OIN

O

iOCmININO ZZ

ZKv

ZR

vv)s(FVDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅−⋅⋅+⋅= (32)

Define:

)s(HK)s(FV pmmIN ⋅=⋅ (33) Let ZO=RO, ZL=0, Hp(s)=1. Solve for the dc gain equation.

6


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅−⋅+⋅≈ Kv

RR

vvKDv)dc(v INO

iOCmINO (34)


im

Oi

O

C

O

RKR

1

1RR

)dc(vv

⋅+

⋅≈ (35)


im

O

m

i

O

IN

O

RKR

1

DK

K1

RDR

)dc(vv

⋅+

−⋅

⋅≈ (36)

Comparing this to the averaged and continuous-time models:

DK

K1Km

n −= Therefore: ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅= n

mK

K1DK (37)

Figure 5: Unified model uses the high frequency asymptote for a single pole in series with the modulator. This accurately models the current loop and control to output transfer functions, but is limited to PCM1, VCM1 and

VCM3 operating modes.

Hp

p

ωs1

1)s(H+

= Where: Qω

ω nHp = (38)

Algebraic manipulation of the closed loop continuous-time expression for H(s) to find a general open loop expression for Hp(s) would appear to be straightforward. For peak or valley current mode with a fixed slope

compensating ramp, ⎟⎟⎠

⎞⎜⎜⎝

⎛−

′⋅−≅

⋅ mmpi

L

zn K1

K1

RZ

Qωs . H(s) reduces to

2n

2

ωs1+ , which leads to the simplification of

a single pole for Hp(s). This is not the case for other operating modes. Further work is needed to develop a general expression for Hp(s). Though the unified model shows the potential to accurately model the line to output transfer function, this has not been validated. Comparisons of line to output bode plots from the linear model to SPICE results from the switched model do not match over frequency.

7


Simplified Transfer Functions

The simplified transfer functions assume poles that are well separated by the current loop gain. Expressions for the averaged model do not show the additional phase shift due to the sampling effect. The control to output gain of the continuous-time model accurately represents the circuit’s behavior to half the switching frequency. The line to output expressions for audio susceptibility are accurate at dc, but diverge from the actual response over frequency. Averaged Model:

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+

+⋅

⋅+

⋅≈

LP

Z

im

Oi

O

C

O

ωs1

ωs1

ωs1

RKR

1

1RR

vv

(39)

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+

+⋅

⋅+

⋅⋅

≈

LP

Z

im

O

n

i

O

IN

O

ωs1

ωs1

ωs1

RKR

1

KR

DRvv

(40)

Continuous-Time Model:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+

+⋅

⋅+

⋅≈

2n

2

nP

Z

im

Oi

O

C

O

ωs

Qωs1

ωs1

ωs1

RKR

1

1RR

vv

(41)

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+

+⋅

⋅+

⋅⋅

≈

2n

2

nP

Z

im

O

n

i

O

IN

O

ωs

Qωs1

ωs1

ωs1

RKR

1

KR

DRvv

(42)

Where: CO

Z RC1ω⋅

= ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

+⋅≈imOO

P RK1

R1

C1ω

LRK

ω impL

⋅≈

Tπωn = (43)

Vr

Vo

Vramp

Vc

100p IC=0C2

E110

S4

G2

1G3

5u

G1

1

1R5

5V3

1mR4V2

1KR8

1m

R1

S3

100u IC=5C1

L1

5u IC=5

1R3

10V1

S110m

R2

U1Q

QNSR

S2

250p IC=0C3

1

R7

100

R6

U2 U3

U4

S5

0.885V4

Vslope

Vin

Vcs

Figure 6: SIMPLIS SPICE schematic - emulated peak current mode synchronous buck.

8


TABLE 1A

SUMMARY OF GENERAL GAIN PARAMETERS TSV eSLOPE ⋅= Tπωn =

Mode eS , nS cm , Q mK , nKPCM1

TV

S SLe =

LR)VV(

S iOINn

⋅−=

n

eC S

S1m +=

)5.0Dm(π1Q

C −′⋅⋅= IN

SLi

m

VV

LTR)D5.0(

1K+⋅⋅−

=

DLTR5.0

VV

K iIN

SLn ⋅⋅⋅−=

PCM2 TKV

S SLOe

⋅=

LR)VV(

S iOINn

⋅−=

n

eC S

S1m +=

)5.0Dm(π1Q

C −′⋅⋅=

DK2LTR)D5.0(

1KSLi

m⋅⋅+⋅⋅−

=

DLTR5.0KK iSLn ⋅⎟⎠⎞

⎜⎝⎛ ⋅⋅−=

VCM1 T

VS SL

e =

LRV

S iOn

⋅=

n

eC S

S1m +=

)5.0Dm(π1Q

C −⋅⋅= IN

SLi

m

VV

LTR)5.0D(

1K+⋅⋅−

=

IN

SLin V

VD

LTR5.0K +⋅⋅⋅=

VCM2 T

K)VV(S SLOIN

e⋅−

=

LRV

S iOn

⋅=

n

eC S

S1m +=

)5.0Dm(π1Q

C −⋅⋅=

DK2LTR)5.0D(

1KSLi

m′⋅⋅+⋅⋅−

=

DKD

KD

LTR5.0K SL

SLin ⋅−+⋅⋅⋅=

EPCM1 T

VS SL

e =

LRV

S iINn

⋅=

n

eC S

Sm =

)5.0m(π1QC −⋅

= IN

SLi

m

VV

LTR)5.0D(

1K+⋅⋅−

=

IN

SLin V

VD

LTR5.0K +⋅⋅⋅=

EPCM2 T

KVS SLIN

e⋅

=

LRV

S iINn

⋅=

n

eC S

Sm =

)5.0m(π1QC −⋅

= SLi

mK

LTR)5.0D(

1K+⋅⋅−

=

DLTR5.0K in ⋅⋅⋅=

EVCM1 T

VS SL

e =

LRV

S iINn

⋅=

n

eC S

Sm =

)5.0m(π1QC −⋅

= IN

SLi

m

VV

LTR)D5.0(

1K+⋅⋅−

=

DLTR5.0

VV

K iIN

SLn ⋅⋅⋅−=

EVCM2 T

KVS SLIN

e⋅

=

LRV

S iINn

⋅=

n

eC S

Sm =

)5.0m(π1QC −⋅

= SLi

mK

LTR)D5.0(

1K+⋅⋅−

=

DLTR5.0

DK

K iSL

n ⋅⋅⋅−=

9


TABLE 1B

SUMMARY OF GENERAL GAIN PARAMETERS TSV eSLOPE ⋅= Tπωn =

Mode eS , nS cm , Q mK , nKVCM3

TKV

S SLINe

⋅=

LRV

S iOn

⋅=

n

eC S

S1m +=

)5.0Dm(π1Q

C −⋅⋅=

SLi

mK

LTR)5.0D(

1K+⋅⋅−

=

DK

DLTR5.0K SL

in +⋅⋅⋅=

EPCM3 T

VK)VV(S SLSLOIN

e+⋅−

=

LRV

S iINn

⋅=

n

eC S

Sm =

)5.0m(π1QC −⋅

= IN

SLSLi

m

VV

K)D21(LTR)5.0D(

1K+⋅⋅−+⋅⋅−

=

IN

SLSLin V

VD)K

LTR5.0(K +⋅−⋅⋅=

EPCM4 T

VKVS SLSLIN

e+⋅

=

LRV

S iINn

⋅=

n

eC S

Sm =

)5.0m(π1QC −⋅

= IN

SLSLi

m

VV

KLTR)5.0D(

1K++⋅⋅−

=

IN

SLin V

VD

LTR5.0K +⋅⋅⋅=

(a) PCM1 Vo=5 Q=0.637 Plot Vo/Vc

-30

-20

-10

0

10

20

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-300

-200

-100

0

100

200

Phas

e / d

egre

es

Gain Vin=6 Gain Vin=10 Gain Vin=50 Phase Vin=6 Phase Vin=10 Phase Vin=50

(b) EPCM1 Vo=5 Q=0.637 Plot Vo/Vc

-30

-20

-10

0

10

20

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-300

-200

-100

0

100

200

Phas

e / d

egre

es


(c) PCM2 Vo=5 Q=0.637 Plot Vo/Vc

-30

-20

-10

0

10

20

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-300

-200

-100

0

100

200

Phas

e / d

egre

es


(d) EPCM2 Vo=5 Q=0.637 Plot Vo/Vc

-30

-20

-10

0

10

20

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-300

-200

-100

0

100

200

Phas

e / d

egre

es


Figure 7: SIMPLIS SPICE results of control to output gain.

10

TABLE 2A CONTROL VOLTAGE AND DUTY CYCLE EQUATIONS FOR AVERAGED MODEL Mode eS , nS

Cv , dPCM1

TV

S SLe =

LR)VV(

S iOINn

⋅−=

CSLiOINiL vdVRLTd)vv(5.0Ri =⋅+⋅⋅⋅−⋅+⋅

SLiOIN

iOiINiLC

VLTR)VV(5.0

DLTR5.0vD

LTR5.0vRiv

d+⋅⋅−⋅

⎟⎠⎞

⎜⎝⎛ ⋅⋅⋅⋅+⎟

⎠⎞

⎜⎝⎛ ⋅⋅⋅⋅−⋅−

=

PCM2 TKV

S SLOe

⋅=

LR)VV(

S iOINn

⋅−=

CSLOiOINiL vdKvRLTd)vv(5.0Ri =⋅⋅+⋅⋅⋅−⋅+⋅

SLOiOIN

SLiOiINiLC

KVLTR)VV(5.0

DKLTR5.0vD

LTR5.0vRiv

d⋅+⋅⋅−⋅

⋅⎟⎠⎞

⎜⎝⎛ −⋅⋅⋅+⎟

⎠⎞

⎜⎝⎛ ⋅⋅⋅⋅−⋅−

=

VCM1 T

VS SL

e =

LRV

S iOn

⋅=

CSLiOiL v)d1(VRLT)d1(v5.0Ri =−⋅−⋅⋅−⋅⋅−⋅

SLiO

iOiLC

VLTRV5.0

)D1(LTR5.0vRiv

d+⋅⋅⋅

⎟⎠⎞

⎜⎝⎛ −⋅⋅⋅⋅+⋅−

=

VCM2 T

K)VV(S SLOIN

e⋅−

=

LRV

S iOn

⋅=

CSLOINiOiL v)d1(K)vv(RLT)d1(v5.0Ri =−⋅⋅−−⋅⋅−⋅⋅−⋅

( )

SLOINiO

SLiOSLINiLC

K)VV(LTRV5.0

)D1(KLTR5.0v)D1(KvRiv

d⋅−+⋅⋅⋅

−⋅⎟⎠⎞

⎜⎝⎛ −⋅⋅⋅+−⋅−⋅−⋅−

=

EPCM1 T

VS SL

e =

LRV

S iINn

⋅=

CSLiOINiL vdVRLTd)vv(5.0Ri =⋅+⋅⋅⋅−⋅−⋅

SLiINO

iOiINiLC

VLTR)VV(5.0

DLTR5.0vD

LTR5.0vRiv

d+⋅⋅−⋅

⎟⎠⎞

⎜⎝⎛ ⋅⋅⋅−⋅+⎟

⎠⎞

⎜⎝⎛ ⋅⋅⋅−⋅−⋅−

=

EPCM2 T

KVS SLIN

e⋅

=

LRV

S iINn

⋅=

CSLINiOINiL vdKvRLTd)vv(5.0Ri =⋅⋅+⋅⋅⋅−⋅−⋅

SLINiINO

iOiSLINiLC

KVLTR)VV(5.0

DLTR5.0vD

LTR5.0KvRiv

d⋅+⋅⋅−⋅

⎟⎠⎞

⎜⎝⎛ ⋅⋅⋅−⋅+⋅⎟

⎠⎞

⎜⎝⎛ ⋅⋅−⋅−⋅−

=

EVCM1 T

VS SL

e =

LRV

S iINn

⋅=

CSLiOiL v)d1(VRLT)d1(v5.0Ri =−−⋅⋅−⋅⋅+⋅

LTRV5.0V

)D1(LTR5.0vRiv

diOSL

iOiLC

⋅⋅⋅−

⎟⎠⎞

⎜⎝⎛ −⋅⋅⋅−⋅+⋅−

=

EVCM2 T

KVS SLIN

e⋅

=

LRV

S iINn

⋅=

CSLINiOiL v)d1(KvRLT)d1(v5.0Ri =−⋅⋅−⋅⋅−⋅⋅+⋅

( )

LTRV5.0KV

)D1(LTR5.0v)D1(KvRiv

diOSLIN

iOSLINiLC

⋅⋅⋅−⋅

⎟⎠⎞

⎜⎝⎛ −⋅⋅⋅−⋅+−⋅−⋅−⋅−

=

11


TABLE 2B CONTROL VOLTAGE AND DUTY CYCLE EQUATIONS FOR AVERAGED MODEL Mode eS , nS

Cv , dVCM3

TKV

S SLINe

⋅=

LRV

S iOn

⋅=

CSLINiOiL v)d1(KvRLT)d1(v5.0Ri =−⋅⋅−⋅⋅−⋅⋅−⋅

( )

SLINiO

iOSLINiLC

KVLTRV5.0

)D1(LTR5.0v)D1(KvRiv

d⋅+⋅⋅⋅

−⋅⎟⎠⎞

⎜⎝⎛ ⋅⋅⋅+−⋅−⋅−⋅−

=

EPCM3 T

VK)VV(S SLSLOIN

e+⋅−

=

LRV

S iINn

⋅=

CSLSLOINiOINiL vdVdK)vv(RLTd)vv(5.0Ri =⋅+⋅⋅−+⋅⋅⋅−⋅−⋅

SLiSLOIN

iSLOiSLINiLC

V)LTR5.0K()VV(

DLTR5.0KvD

LTR5.0KvRiv

d+⋅⋅−⋅−

⋅⎟⎠⎞

⎜⎝⎛ ⋅⋅−⋅+⋅⎟

⎠⎞

⎜⎝⎛ ⋅⋅−⋅−⋅−

=

EPCM4 T

VKVS SLSLIN

e+⋅

=

LRV

S iINn

⋅=

CSLSLINiOINiL vdVdKvRLTd)vv(5.0Ri =⋅+⋅⋅+⋅⋅⋅−⋅−⋅

SLSLINiINO

iOiSLINiLC

VKVLTR)VV(5.0

DLTR5.0vD

LTR5.0KvRiv

d+⋅+⋅⋅−⋅

⎟⎠⎞

⎜⎝⎛ ⋅⋅⋅−⋅+⋅⎟

⎠⎞

⎜⎝⎛ ⋅⋅−⋅−⋅−

=

(a) PCM1 Vo=5 Q=0.637 Plot Vo/Vin

-120

-100

-80

-60

-40

-20

0

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-180

-120

-60

0

60

120

180

Phas

e / d

egre

es

Gain Vin=6 Gain Vin=10 Gain Vin=50Phase Vin=6 Phase Vin=10 Phase Vin=50

(b) EPCM1 Vo=5 Q=0.637 Plot Vo/Vin

-120

-100

-80

-60

-40

-20

0

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-240

-180

-120

-60

0

60

120

Phas

e / d

egre

esGain Vin=6 Gain Vin=10 Gain Vin=50 Phase Vin=6 Phase Vin=10 Phase Vin=50

(c) PCM2 Vo=5 Q=0.637 Plot Vo/Vin

-120

-100

-80

-60

-40

-20

0

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-180

-120

-60

0

60

120

180

Phas

e / d

egre

es


(d) EPCM2 Vo=5 Q=0.637 Plot Vo/Vin

-120

-100

-80

-60

-40

-20

0

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-180

-120

-60

0

60

120

180

Phas

e / d

egre

es


Figure 8: SIMPLIS SPICE results of line to output gain.

12


TABLE 3A

SUMMARY OF GAIN PARAMETERS FOR AVERAGED MODEL L

RVS iO

f⋅

=

Mode eS , nS mF , mpK IK , OK

PCM1 T

VS SL

e =

LR)VV(

S iOINn

⋅−=

T)SS5.0(1F

enm ⋅+⋅=

IN

SLi

mp

VV

DLTR5.0

1K+′⋅⋅⋅

=

DLTR5.0K iI ⋅⋅⋅=

DLTR5.0K iO ⋅⋅⋅=

PCM2 TKV

S SLOe

⋅=

LR)VV(

S iOINn

⋅−=

T)SS5.0(1F

enm ⋅+⋅=

DKDLTR5.0

1KSLi

mp⋅+′⋅⋅⋅

=

DLTR5.0K iI ⋅⋅⋅=

DKLTR5.0K SLiO ⋅⎟

⎠⎞

⎜⎝⎛ −⋅⋅=

VCM1 T

VS SL

e =

LRV

S iOn

⋅=

T)SS5.0(1F

enm ⋅+⋅=

IN

SLi

mp

VV

DLTR5.0

1K+⋅⋅⋅

=

0KI =

DLTR5.0K iO ′⋅⋅⋅=

VCM2 T

K)VV(S SLOIN

e⋅−

=

LRV

S iOn

⋅=

T)SS5.0(1F

enm ⋅+⋅=

DKDLTR5.0

1KSLi

mp′⋅+⋅⋅⋅

=

DKK SLI ′⋅−=

DKLTR5.0K SLiO ′⋅⎟

⎠⎞

⎜⎝⎛ −⋅⋅=

EPCM1 T

VS SL

e =

LRV

S iINn

⋅=

T)S)SS(5.0(1F

enfm ⋅+−⋅=

DLTR5.0

VV

1Ki

IN

SLmp

′⋅⋅⋅−=

DLTR5.0K iI ⋅⋅⋅−=

DLTR5.0K iO ⋅⋅⋅−=

EPCM2 T

KVS SLIN

e⋅

=

LRV

S iINn

⋅=

T)S)SS(5.0(1F

enfm ⋅+−⋅=

DLTR5.0K

1KiSL

mp′⋅⋅⋅−

=

DLTR5.0KK iSLI ⋅⎟⎠⎞

⎜⎝⎛ ⋅⋅−=


EVCM1 T

VS SL

e =

LRV

S iINn

⋅=

T)S5.0S(1F

fem ⋅⋅−=

DLTR5.0

VV

1K

iIN

SLmp

⋅⋅⋅−=

0KI =

DLTR5.0K iO ′⋅⋅⋅−=

EVCM2 T

KVS SLIN

e⋅

=

LRV

S iINn

⋅=

T)S5.0S(1F

fem ⋅⋅−=

DLTR5.0K

1KiSL

mp⋅⋅⋅−

=

DKK SLI ′⋅−=

DLTR5.0K iO ′⋅⋅⋅−=

13


TABLE 3B

SUMMARY OF GAIN PARAMETERS FOR AVERAGED MODEL L

RVS iO

f⋅

=

Mode eS , nS mF , mpK IK , OK

VCM3 T

KVS SLIN

e⋅

=

LRV

S iOn

⋅=

T)SS5.0(1F

enm ⋅+⋅=

SLi

mpK

LTRD5.0

1K+⋅⋅⋅

=

DKK SLI ′⋅−=

DLTR5.0K iO ′⋅⋅⋅=

EPCM3 T

VK)VV(S SLSLOIN

e+⋅−

=

LRV

S iINn

⋅=

T)S)SS(5.0(1F

enfm ⋅+−⋅=

IN

SLiSL

mp

VV

D)LTR5.0K(

1K+′⋅⋅⋅−

=


⎜⎝⎛ ⋅⋅−=

DLTR5.0KK iSLO ⋅⎟⎠⎞

⎜⎝⎛ ⋅⋅−=

EPCM4 T

VKVS SLSLIN

e+⋅

=

LRV

S iINn

⋅=

T)S)SS(5.0(1F

enfm ⋅+−⋅=

DLTR5.0

VV

K

1K

iIN

SLSL

mp′⋅⋅⋅−+

=


⎜⎝⎛ ⋅⋅−=


(a) PCM1 Vo=5 Q=0.637 Current Loop

-40-30-20-10

010203040

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-400-300-200-1000100200300400

Phas

e / d

egre

es


(b) EPCM1 Vo=5 Q=0.637 Current Loop

-50-40-30-20-10

0102030

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-500-400-300-200-1000100200300

Phas

e / d

egre

esGain Vin=6 Gain Vin=10 Gain Vin=50 Phase Vin=6 Phase Vin=10 Phase Vin=50

(c) PCM2 Vo=5 Q=0.637 Current Loop

-40-30-20-10

010203040

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-400-300-200-1000100200300400

Phas

e / d

egre

es


(d) EPCM2 Vo=5 Q=0.637 Current Loop

-50-40-30-20-10

0102030

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-500-400-300-200-1000100200300

Phas

e / d

egre

es


Figure 9: SIMPLIS SPICE results of current loop gain.

14


TABLE 4A SUMMARY OF GAIN PARAMETERS FOR CONTINUOUS-TIME MODEL Mode eS , nS mF′ , mpK′ fK′ , rK ′

PCM1 T

VS SL

e =

LR)VV(

S iOINn

⋅−=

T)SS(1F

enm ⋅+=′

IN

SLi

mp

VV

DLTR

1K+′⋅⋅

=′

)1D5.0(DLTRK if −⋅⋅⋅⋅=′

2ir )D(

LTR5.0K ′⋅⋅⋅=′

PCM2 TKV

S SLOe

⋅=

LR)VV(

S iOINn

⋅−=

T)SS(1F

enm ⋅+=′

DKDLTR

1KSLi

mp⋅+′⋅⋅

=′

)1D5.0(DLTRK if −⋅⋅⋅⋅=′

DK)D(LTR5.0K SL

2ir ⋅−′⋅⋅⋅=′

VCM1 T

VS SL

e =

LRV

S iOn

⋅=

T)SS(1F

enm ⋅+=′

IN

SLi

mp

VV

DLTR

1K+⋅⋅

=′

2if D

LTR5.0K ⋅⋅⋅−=′

)D1(LTR5.0K 2

ir −⋅⋅⋅=′

VCM2 T

K)VV(S SLOIN

e⋅−

=

LRV

S iOn

⋅=

T)SS(1F

enm ⋅+=′

DKDLTR

1KSLi

mp′⋅+⋅⋅

=′

DKDLTR5.0K SL

2if ′⋅+⋅⋅⋅−=′

)D1(LTR5.0K 2

ir −⋅⋅⋅=′

EPCM1 T

VS SL

e =

LRV

S iINn

⋅=

TS1F

em ⋅=′

INSLmp VV

1K =′

2if D

LTR5.0K ⋅⋅⋅=′

2ir )D(

LTR5.0K ′⋅⋅⋅=′

EPCM2 T

KVS SLIN

e⋅

=

LRV

S iINn

⋅=

TS1F

em ⋅=′

SLmp K

1K =′

DKDLTR5.0K SL

2if ⋅−⋅⋅⋅=′

DK)D(LTR5.0K SL

2ir ⋅−′⋅⋅⋅=′

EVCM1 T

VS SL

e =

LRV

S iINn

⋅=

TS1F

em ⋅=′

INSLmp VV

1K =′

2if D

LTR5.0K ⋅⋅⋅−=′

2ir )D(

LTR5.0K ′⋅⋅⋅−=′

EVCM2 T

KVS SLIN

e⋅

=

LRV

S iINn

⋅=

TS1F

em ⋅=′

SLmp K

1K =′

DKDLTR5.0K SL

2if ′⋅+⋅⋅⋅−=′

DK)D(LTR5.0K SL

2ir ′⋅+′⋅⋅⋅−=′

15


TABLE 4B SUMMARY OF GAIN PARAMETERS FOR CONTINUOUS-TIME MODEL Mode eS , nS mF′ , mpK′ fK′ , rK ′

VCM3 T

KVS SLIN

e⋅

=

LRV

S iOn

⋅=

T)SS(1F

enm ⋅+=′

SLi

mpKD

LTR

1K+⋅⋅

=′

DKDLTR5.0K SL

2if ′⋅+⋅⋅⋅−=′

DK)D1(LTR5.0K SL

2ir ′⋅+−⋅⋅⋅=′

EPCM3 T

VK)VV(S SLSLOIN

e+⋅−

=

LRV

S iINn

⋅=

TS1F

em ⋅=′

IN

SLSL

mp

VV

DK

1K+′⋅

=′

DKDLTR5.0K SL

2if ⋅−⋅⋅⋅=′

2ir )D(

LTR5.0K ′⋅⋅⋅=′

EPCM4 T

VKVS SLSLIN

e+⋅

=

LRV

S iINn

⋅=

TS1F

em ⋅=′

IN

SLSL

mp

VV

K

1K+

=′

DKDLTR5.0K SL

2if ⋅−⋅⋅⋅=′

DK)D(LTR5.0K SL

2ir ⋅−′⋅⋅⋅=′

TABLE 5 Calculated General Gain Parameters VIN=10 VO=5 D=0.5 RO=1 Ri=0.1 T=5μ L=5μ MODE VSLOPE KSL VSL Km Kn mc Q PCM1 VSL 0.10 0.50 20.00 0.025 2.000 0.637 PCM2 VO·KSL 0.10 --- 10.00 0.025 2.000 0.637 VCM1 VSL 0.10 0.50 20.00 0.075 2.000 0.637 VCM2 (VIN-VO)·KSL 0.10 --- 10.00 0.175 2.000 0.637 EPCM1 VSL 0.10 1.00 10.00 0.125 1.000 0.637 EPCM2 VIN·KSL 0.10 --- 10.00 0.025 1.000 0.637 EVCM1 VSL 0.10 1.00 10.00 0.075 1.000 0.637 EVCM2 VIN·KSL 0.10 --- 10.00 0.175 1.000 0.637 VCM3 VIN·KSL 0.10 --- 10.00 0.225 3.000 0.318 EPCM3 (VIN-VO)·KSL +VSL 0.10 0.50 20.00 0.025 1.000 0.637 EPCM4 VIN·KSL +VSL 0.05 0.50 10.00 0.075 1.000 0.637 PCM2 0.5·VO·KSL 0.10 --- 20.00 0.000 1.250 2.546 VCM2 0.5·(VIN-VO)·KSL 0.10 --- 20.00 0.100 1.250 2.546

Table Notation: PCM – Peak Current Mode VCM – Valley Current Mode EPCM – Emulated Peak Current Mode EVCM – Emulated Valley Current Mode 1 – Fixed slope compensation VSL 2 – Optimal slope compensation proportional to KSL3, 4 – Other fixed or proportional slope compensation implementations In the last section of Table 5 for PCM2, peak current mode with a slope compensating ramp equal to one half the down slope of the inductor current represents a special case. Kn=0 which corresponds to infinite line rejection. This is difficult to achieve due to finite resistance and parasitic elements in a practical circuit. This is not optimal since sub-harmonic oscillation is possible at D>0.5 with feedback control. See section 4 on general slope compensation requirements.

16

17

TABLE 6 Calculated Values and Measured PSPICE Switching Circuit Data VIN=6 VO=5 D=0.83 RO=1 Ri=0.1 T=5μ L=5μ Calculated Measured MODE VSLOPE KSL VSL VO/VC VO/VIN VO/VC VO/VIN

PCM1 VSL 0.10 0.50 6.67 0.231 6.66 0.240 PCM2 VO·KSL 0.10 --- 4.29 0.149 4.25 0.154 VCM1 VSL 0.10 0.10 6.67 0.324 6.55 0.325 VCM2 (VIN-VO)·KSL 0.10 --- 6.00 0.392 5.98 0.383 EPCM1 VSL 0.10 0.60 4.29 0.506 4.24 0.502 EPCM2 VIN·KSL 0.10 --- 4.29 0.149 4.25 0.148 EVCM1 VSL 0.10 0.60 6.00 0.292 6.00 0.300 EVCM2 VIN·KSL 0.10 --- 6.00 0.392 5.96 0.388 VCM3 VIN·KSL 0.10 --- 4.29 0.577 4.23 0.571 EPCM3 (VIN-VO)·KSL +VSL 0.10 0.50 6.67 0.231 6.61 0.231 EPCM4 VIN·KSL +VSL 0.05 0.50 3.75 0.391 3.71 0.389






(a) EPCM2 Vin=10 Vo=5 Plot Vo/Vc

-50-40-30-20-10

01020

100 1000 10000 100000 1000000Frequency / Hz

Gai

n / d

B

-300-200-1000100200300400

Phas

e / d

egre

es

Gain Q=6.37 Gain Q=1.27Gain Q=0.637 Gain Q=0.318Phase Q=6.37 Phase Q=1.27Phase Q=0.637 Phase Q=0.318

(b) EPCM2 Vin=10 Vo=5 Plot Vo/Vin

-100-90-80-70-60-50-40-30-20

100 1000 10000 100000 1000000Frequency / Hz

Gai

n / d

B

-140-120-100-80-60-40-20020

Phas

e / d

egre

es

Gain Q=6.37 Gain Q=1.27Gain Q=0.637 Gain Q=0.318Phase Q=6.37 Phase Q=1.27Phase Q=0.637 Phase Q=0.318

(c) EPCM2 Vin=10 Vo=5 Current Loop

-50-40-30-20-10

0102030

100 1000 10000 100000 1000000Frequency / Hz

Gai

n / d

B

-500-400-300-200-1000100200300

Phas

e / d

egre

esGain Q=6.37 Gain Q=1.27Gain Q=0.637 Gain Q=0.318Phase Q=6.37 Phase Q=1.27Phase Q=0.637 Phase Q=0.318

Figure 10: SIMPLIS SPICE results for EPCM2 with varying Q.

3. Discrete Time Analysis for Slope Compensation

(a) Peak current mode (b) Valley current mode

Figure 11: Slope compensation for peak and valley current mode buck.

18


Peak Current Mode Trailing Edge Modulation

For peak current mode control, the peak current must be accounted for in its relationship to the average current. By perturbing the duty cycle, we can see the effect on VC. Write the equation for the voltage at VC:

CSLOPEiOINiL vdVRLTd)vv(5.0Ri =⋅+⋅⋅⋅−⋅+⋅ (44)

VIN, VO, and VSLOPE are considered to be fixed with respect to the period T. The duty cycle is perturbed, which shows the effect on the control voltage with respect to the inductor current. The quantity , and .

)T(iI LL =Δ )T(dD =Δ)T(vV CC =Δ

CSLOPEiOINiL VDVRLTD)VV(5.0RI Δ=Δ⋅+⋅⋅Δ⋅−⋅+⋅Δ (45)

To maintain constant average inductor current, the control voltage must change by ½ the ripple current times the current sense gain:

iPKC RI5.0V ⋅Δ⋅−=Δ (46)

LTDV

LT)D1(VI OOPK ⋅Δ⋅−=⋅Δ−⋅=Δ (47)

Combine equations and solve for ∆D/∆IL:

INSLOPEiIN

i

L VVLTR)D5.0(

1VR

ID

+⋅⋅−⋅−=

ΔΔ (48)

The term INSLOPEi VV

LTR)D5.0(

1

+⋅⋅− is equivalent to the peak current mode modulator gain Km with fixed

VSLOPE. By factoring, this becomes:

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅−

+⋅

⋅−=ΔΔ

5.0)D1(L/R)VV(

T/V1

LT

1V

1ID

iOIN

SLOPEINL (49)

This can be expressed as:

)5.0Dm(LT

1V

1ID

cINL −′⋅⋅

⋅−=ΔΔ (50)

Where: D1D −=′n

ec S

S1m +=

Slope compensating ramp: T/VS SLOPEe =Positive current sense ramp: L/R)VV(S iOINn ⋅−=

19

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan Reviewing the equation for ∆D/∆IL, when D>0.5, the T/L term goes negative. VSLOPE must be large enough to compensate, or this will manifest itself in the time domain as a sub-harmonic oscillation of the current loop. If the condition is set such that:

iIN

SLOPE

RVV

LTD

⋅=⋅ (51)

Then mC·D′ = 1, so the current loop will be stable for any duty cycle. To meet this condition, solve for VSLOPE:

LTRVV iOSLOPE ⋅⋅= (52)

Peak Current ModeD=0.6 Q=6.37

0.60.70.80.9

11.1

0 1E-05 2E-05 3E-05 4E-05 5E-05

T

V VrampI(L)*Gi*Rs

Peak Current ModeD=0.4 Q=6.37

0.60.70.80.9

11.1

0 1E-05 2E-05 3E-05 4E-05 5E-05

T

V VrampI(L)*Gi*Rs

Figure 12: Peak current mode sub-harmonic oscillation. For D<0.5, sub-harmonic oscillation is damped. For D>0.5,

sub-harmonic oscillation builds with insufficient slope compensation.

Valley Current Mode Leading Edge Modulation

For valley current mode control, the valley current must be accounted for in its relationship to the average current. By perturbing the duty cycle, we can see the effect on VC. Write the equation for the voltage at VC:

CSLOPEiOiL v)d1(VRLT)d1(v5.0Ri =−⋅−⋅⋅−⋅⋅−⋅ (53)



CSLOPEiOiL VDVRLTDV5.0RI Δ=Δ⋅+⋅⋅Δ⋅⋅+⋅Δ (54)

To maintain constant average inductor current, the control voltage must change by ½ the ripple current:

iPKC RI5.0V ⋅Δ⋅=Δ (55)

LTD)VV(I OINPK ⋅Δ⋅−=Δ (56)


20


INSLOPEiIN

i

L VVLTR)5.0D(

1VR

ID

+⋅⋅−⋅−=

ΔΔ (57)

The term INSLOPEi VV

LTR)5.0D(

1

+⋅⋅− is equivalent to the valley current mode modulator gain Km with fixed

VSLOPE. By factoring, this becomes:

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

+⋅

⋅−=ΔΔ

5.0DL/RVT/V

1LT

1V

1ID

iO

SLOPEINL (58)

This can be expressed as:

)5.0Dm(LT

1V

1ID

cINL −⋅⋅

⋅−=ΔΔ (59)

Where: n

ec S

S1m +=

Slope compensating ramp: T/VS SLOPEe =Negative current sense ramp: L/RVS iOn ⋅= Reviewing the equation for ∆D/∆IL, when D<0.5, the T/L term goes negative. VSLOPE must be large enough to compensate, or this will manifest itself in the time domain as a sub-harmonic oscillation of the current loop. If the condition is set such that:

iIN

SLOPE

RVV

LT)D1(

⋅=⋅− (60)

Then mC·D = 1, so the current loop will be stable for any duty cycle. To meet this condition, solve for VSLOPE.

LTR)VV(V iOINSLOPE ⋅⋅−= (61)

Valley Current ModeD=0.6 Q=6.37

0.60.70.80.9

11.1

0 1E-05 2E-05 3E-05 4E-05 5E-05

T

V VrampI(L)*Gi*Rs

Valley Current ModeD=0.4 Q=6.37

0.60.70.80.9

11.11.2

0 1E-05 2E-05 3E-05 4E-05 5E-05

T

V VrampI(L)*Gi*Rs

Figure 13: Valley current mode sub-harmonic oscillation. For D>0.5, sub-harmonic oscillation is damped. For D<0.5, sub-harmonic oscillation builds with insufficient slope compensation.

21


Emulated Peak Current Mode Valley Sample and Hold

The valley current is sampled on the down slope of the inductor current. This is used as the DC value of current to start the next cycle. A slope compensating ramp is added to produce VRAMP at the modulator input. Write the equation for the voltage at VC:

CSLOPEiOINiL vdVRLTd)vv(5.0Ri =⋅+⋅⋅⋅−⋅−⋅ (62)



CSLOPEiOINiL VDVRLTD)VV(5.0RI Δ=Δ⋅+⋅⋅Δ⋅−⋅−⋅Δ (63)


iPKC RI5.0V ⋅Δ⋅−=Δ (64)

LTDV

LT)D1(VI OOPK ⋅Δ⋅−=⋅Δ−⋅=Δ (65)


LTR5.0VV

1VR

ID

iINSLOPEIN

i

L ⋅⋅−⋅−=

ΔΔ (66)

By factoring, this becomes:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⋅

⋅−=ΔΔ

5.0L/RVT/V

LT

1V

1ID

iIN

SLOPEINL (67)

This can also be written as:

( )5.0mLT

1V

1ID

CINL −⋅

⋅−=ΔΔ (68)

Where: n

eC S

Sm = T/VS SLOPEe = L/RVS iINn ⋅=

Reviewing the equation for ∆D/∆IL, when mC<0.5, the T/L term goes negative. VSLOPE must be large enough to compensate, or this will manifest itself in the time domain as a sub-harmonic oscillation of the current loop. If the condition is set such that:

LRV

TV iINSLOPE ⋅

= (69)

Then mC = 1, so the current loop will be unconditionally stable. Unlike peak or valley current mode, slope compensation is independent of duty cycle. To meet this condition, solve for VSLOPE.

22


LTRVV iINSLOPE ⋅⋅= (70)

Emulated Peak Current Mode

D=0.6 Q=6.37

0.60.70.80.9

11.1

0 1E-05 2E-05 3E-05 4E-05 5E-05

T

V

VrampI(L)*Gi*Rs

Emulated Peak Current ModeD=0.4 Q=6.37

0.60.70.8

0.91

1.1

0 1E-05 2E-05 3E-05 4E-05 5E-05T

VVrampI(L)*Gi*Rs

Figure 14: Emulated peak current mode sub-harmonic oscillation. Tendency for sub-harmonic oscillation is independent of duty cycle. Even with minimal damping, it will eventually die out.

Emulated Valley Current Mode

Peak Sample and Hold The peak current is sampled on the positive slope of the inductor current. This is used as the DC value of current to start the next cycle. A slope compensating ramp is added to produce VRAMP at the modulator input. Write the equation for the voltage at VC:

CSLOPEiOiL v)d1(VRLT)d1(v5.0Ri =−⋅−⋅⋅−⋅⋅+⋅ (71)



CSLOPEiOiL VDVRLTDV5.0RI Δ=Δ⋅+⋅⋅Δ⋅⋅−⋅Δ (72)


iPKC RI5.0V ⋅Δ⋅=Δ (73)

LTD)VV(I OINPK ⋅Δ⋅−=Δ (74)


LTR5.0VV

1VR

ID

iINSLOPEIN

i

L ⋅⋅−⋅−=

ΔΔ (75)

By factoring, this becomes:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⋅

⋅−=ΔΔ

5.0L/RVT/V

LT

1V

1ID

iIN

SLOPEINL (76)

23

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan This can be expressed as:

)5.0m(LT

1V

1ID

cINL −⋅

⋅−=ΔΔ (77)

Where: n

eC S

Sm = T/VS SLOPEe = L/RVS iINn ⋅=

Reviewing the equation for ∆D/∆IL, when mC<0.5, the T/L term goes negative. VSLOPE must be large enough to compensate, or this will manifest itself in the time domain as a sub-harmonic oscillation of the current loop. If the condition is set such that:

LRV

TV iINSLOPE ⋅

= (78)

Then mC = 1, so the current loop will be unconditionally stable. Unlike peak or valley current mode, slope compensation is independent of duty cycle. To meet this condition, solve for VSLOPE.

LTRVV iINSLOPE ⋅⋅= (79)

4. General Slope Compensation Requirements The graphs in the preceding section were made by entering the discrete time expressions for the inductor current and slope compensating ramp into an Excel spreadsheet. The results are plotted on a cycle by cycle basis. This shows the current loop behavior without any feedback to the control voltage. Reference [6] discusses the voltage loop gain effect on the slope compensation requirement for peak current mode. Peaking of the closed loop gain due to insufficient slope compensation and ripple on the control voltage can cause sub-harmonic oscillation before the calculated limit, i.e. at duty cycles below 0.5 for peak current mode. For any mode of operation, when the sum of the sensed inductor current’s slope (times the current sense gain) plus the slope of the compensation ramp is proportional to VIN, any tendency toward sub-harmonic oscillation will damp in one switching cycle. This condition is represented by equations 52, 61, 70 and 79, which corresponds to a Q of 0.637. Operation is considered to be optimal, in that the effective sampled gain inductor pole is fixed in frequency with respect to changes in line voltage. This allows for the highest closed loop gain without any tendency toward sub-harmonic oscillation. Increasing the external ramp beyond this point will lower the modulator gain, consequently shifting toward a more voltage mode behavior. The effective sampled gain inductor pole frequency (45° phase shift) is given by:

⎟⎠⎞⎜

⎝⎛ −⋅+⋅

⋅⋅= 1Q41

QT41)Q(f 2

L (80)

5. Correlation of Measured Data The LM3495 standard reference design was chosen as a platform for correlation. It is an emulated peak current mode controller with MOSFET current sensing and internally generated slope compensation using mode EPCM4. Operating parameters: VIN=12 VO=1.2 T=2μs VSL=0.125 KSL=1/16 L=1μH RL=3mΩ GI=4 RS=3.4mΩ CO=200μF RC=1mΩ RO=1Ω and 0.2Ω

24

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan Measurement of control to output gain was made using an AP200 frequency response analyzer. The simplified transfer function using equation 41 was entered into SPICE as LaPlace for the calculated results. Nominal data sheet values were used for the components and operating parameters. Comparison of the results is shown in figure 15.

(a) LM3495 Ro=1 Plot Vo/Vc

-30

-20

-10

0

10

20

30

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-180

-120

-60

0

60

120

180

Phas

e / d

egre

es

Measured Gain Calculated GainMeasured Phase Calculated Phase

(b) LM3495 Ro=0.2 Plot Vo/Vc

-30

-20

-10

0

10

20

30

100 1000 10000 100000 1000000

Frequency / Hz

Gai

n / d

B

-180

-120

-60

0

60

120

180

Phas

e / d

egre

es

Measured Gain Calculated GainMeasured Phase Calculated Phase

Figure 15: LM3495 control to output gain. 6. Conclusion This analysis has focused on current mode buck regulators with continuous inductor current. Using the methodology outlined here, general expressions for the boost and buck-boost can be developed. Preliminary work has demonstrated good correlation between switching and linear models. Using the gain coefficients from reference [2], general expressions for discontinuous conduction mode can also be developed. Limitations of existing models have been identified, with direction for further work. Regardless of the limitations, the simplified transfer functions and general gain parameters allow accurate modeling of the control to output gain. This is the most important parameter from a design standpoint. Tools like SIMPLIS provide bode plots for any transfer function directly from the switching model. REFERENCES [1] R.D. Middlebrook, “Topics in Multi-Loop Regulators and Current-Mode Programming,” IEEE PESC

Record, pp. 716–732, 1985 (also in IEEE Transactions on Power Electronics, Volume 2, Issue 2, pp. 109–124, 1987).

[2] R.B. Ridley, “A New Continuous-Time Model for Current-Mode Control with Constant Frequency,

Constant On-Time, and Constant Off-Time, in CCM and DCM,” IEEE PESC Record, pp. 382–389, 1990. [3] R.B. Ridley, “A New, Continuous-Time Model for Current-Mode Control,” IEEE Transactions on Power

Electronics, Volume 6, Issue 2, pp. 271–280, 1991. [4] F.D. Tan, R.D. Middlebrook, “Unified Modeling and Measurement of Current-Programmed Converters,”

IEEE PESC Record, pp. 380–387, 1993. [5] F.D. Tan, R.D. Middlebrook, “A Unified Model for Current-Programmed Converters,”

IEEE Transactions on Power Electronics, Volume 10, Issue 4, pp. 397–408, 1995. [6] Barney Holland, “Modeling, Analysis and Compensation of the Current-Mode Converter,” Powercon 11

Proceedings, I-2, 1984.

25

26

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan Notes:



by

Robert Sheehan


Santa Clara, CA

APPENDIX A

Figure A-1: Buck regulator linear models. Define ZO and ZL:

)RR(Cs1)RCs1(R

R||RCs1Z

COO

COOOC

OO +⋅⋅+

⋅⋅+⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

⋅= (A.1)

SLL RRLsZ ++⋅= (A.2)

A-1

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS – APPENDIX A by Robert Sheehan

Averaged Model Derivation of Transfer Functions Using General Gain Parameters

Starting with , write the transfer function in terms of voltages: Ov

LO

OSWO ZZ

Zvv

+⋅= (A.3)

dVDvv ININSW ⋅+⋅= (A.4)

)KvKvRiv(Fd OOIINiLCm ⋅+⋅−⋅−⋅= (A.5)

O

OL Z

vi = (A.6)

Combining equations A.3 through A.6 yields:

LO

OOOIIN

O

iOCmININO ZZ

ZKvKv

ZR

vvFVDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+⋅−⋅−⋅⋅+⋅= (A.7)

Define Kmp as:

mINmp FVK ⋅= (A.8) Setting to zero allows the control to output gain to be found: INv

imp

LOmp

O

O

C

O

RK

1ZKK

1Z

Zvv

+⋅+⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅

= (A.9)

Define Km as the modulator gain where:

Omp

mK

K1

1K−

= (A.10)


imp

L

m

O

O

C

O

RKZ

KZ

Zvv

++= (A.11)


imp

LOmp

O

OI

mp

IN

O

RK

1ZKK

1Z

ZD

KK

1D

vv

+⋅+⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅

= (A.12)

A-2

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS – APPENDIX A by Robert Sheehan Define Kn as the audio susceptibility coefficient:

DK

K1K I

mpn −= (A.13)

This allows the line to output gain to be expressed as:

imp

L

m

O

On

IN

O

RKZ

KZ

ZKDvv

++

⋅⋅= (A.14)

The denominator of the closed loop gain expressions is also used for the control to inductor current gain. Formal derivation is done by setting and . 0v IN = OLO Ziv ⋅=

imp

L

m

OC

L

RKZ

KZ

1vi

++= (A.15)

Continuous-Time Model

Derivation of Transfer Functions Using General Gain Parameters Starting with , write the transfer function in terms of voltages: d

)KvKv)s(HRiv(Fd rOFFfONeiLCm ′⋅+′⋅+⋅⋅−⋅′= (A.16) Combining with equations A.3, A.4 and A.6 for the power stage yields:

LO

OrOFFfON

O

eiOCmININO ZZ

ZKvKv

Z)s(HR

vvFVDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅+′⋅+

⋅⋅−⋅′⋅+⋅= (A.17)

Define K′mp as:

mINmp FVK ′⋅=′ (A.18) Let , . The transfer function becomes: OINON vvv −= OOFF vv =

LO

OrOfOfIN

O

eiOCmpINO ZZ

ZKvKvKv

Z)s(HR

vvKDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅+′⋅−′⋅+

⋅⋅−⋅′+⋅= (A.19)


)s(HRK

1ZKKK

1Z

Zvv

eimp

Lrfmp

O

O

C

O

⋅+′

⋅+⎟⎟⎠

⎞⎜⎜⎝

⎛′−′+

′⋅

= (A.20)

A-3

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS – APPENDIX A by Robert Sheehan Define Km as the modulator gain where:

rf

mp

mKK

K1

1K′−′+

′

= (A.21)


)s(HR

KZ

KZ

Zvv

eimp

L

m

O

O

C

O

⋅+′

+= (A.22)


)s(HRK

1ZKKK

1Z

DK

K1DZ

vv

eimp

Lrfmp

O

f

mpO

IN

O

⋅+′

⋅+⎟⎟⎠

⎞⎜⎜⎝

⎛′−′+

′⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛ ′+

′⋅⋅

= (A.23)

Define Kn as the audio susceptibility coefficient:

DK

K1K f

mpn

′+

′= (A.24)

This allows the line to output gain to be expressed as:

)s(HRKZ

KZ

KDZvv

eimp

L

m

O

nO

IN

O

⋅+′

+

⋅⋅= (A.25)

The denominator of the closed loop gain expressions is also used for the control to inductor current gain. Formal derivation is done by setting and . 0v IN = OLO Ziv ⋅=

)s(HR

KZ

KZ

1vi

eimp

L

m

OC

L

⋅+′

+= (A.26)

Using the closed loop denominator term, define a single sampling gain block which incorporates K′mp inside the current loop. Let:

)s(HRKZ

KZ

)s(HRKZ

KZ

eimp

L

m

Oi

m

L

m

O ⋅+′

+=⋅++ (A.27)

The sampling gain term becomes:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

′⋅+=

mmpi

Le K

1K

1RZ

)s(H)s(H (A.28)

A-4


Where: 2n

2

zne

ωs

Qωs1)s(H +⋅

+= Tπωn =

π2Q z −= (A.29)

The transfer functions can now be expressed as:

)s(HRKZZZK

vv

imLO

Om

C

O

⋅⋅++⋅

= (A.30)

)s(HRKZZZKKD

vv

imLO

Omn

IN

O

⋅⋅++⋅⋅⋅

= (A.31)

)s(HRKZZK

vi

imLO

m

C

L

⋅⋅++= (A.32)

The general transfer function is:

[ ]LO

OnINiLCmO ZZ

ZKDv)s(HRivKv

+⋅⋅⋅+⋅⋅−⋅= (A.33)

Figure A-2: Continuous-time model using general gain parameters. This model is valid for control to output transfer functions of all operating modes. The line to output transfer function is valid at dc, but diverges from the actual

response over frequency.

Unified Model Derivation of Transfer Functions Using General Gain Parameters

Starting with , write the transfer function in terms of voltages: d

)KvRiv()s(Fd INiLCm ⋅−⋅−⋅= (A.34)

Combining with equations A.3, A.4 and A.6 for the power stage yields:

LO

OIN

O

iOCmININO ZZ

ZKv

ZR

vv)s(FVDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅−⋅⋅+⋅= (A.35)

Define: (A.36) )s(HK)s(FV PmmIN ⋅=⋅ The transfer function becomes:

LO

OIN

O

iOCpmINO ZZ

ZKv

ZR

vv)s(HKDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅−⋅⋅+⋅= (A.37)

A-5

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS – APPENDIX A by Robert Sheehan Setting to zero allows the control to output gain to be found: INv

)s(HRKZZ)s(HZK

vv

PimLO

POm

C

O

⋅⋅++⋅⋅

= (A.38)


[ ])s(HRKZZ

Z)s(HKKDvv

PimLO

OPm

IN

O

⋅⋅++⋅⋅⋅−

= (A.39)

Setting and allows the control to inductor current gain to be found: 0v IN = OLO Ziv ⋅=

)s(HRKZZ)s(HK

vi

PimLO

Pm

C

L

⋅⋅++⋅

= (A.40)

Setting , and allows the current loop gain to be found: 0vC = 0v IN = OLO Ziv ⋅=

LO

Pim

L

L

ZZ)s(HRK

i

i+⋅⋅

−=′

(A.41)

Setting HP(s)=1 and comparing the line to output gain to the averaged and continuous-time models:

mmn KKDKKD ⋅−=⋅⋅ Therefore: ⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅= n

mK

K1DK (A.42)

The sampling gain term is defined as:

Hp

p

ωs1

1)s(H+

= Where: Qω

ω nHp = (A.43)

Figure A-3: Unified model uses the high frequency asymptote for a single pole in series with the modulator. This accurately models the current loop and control to output transfer functions, but is limited to PCM1, VCM1 and

VCM3 operating modes.

A-6


Averaged Model Derivation of Factored Pole/Zero Form

Starting with the control to output gain of equation A.11, expand ZO and ZL:

( ) imp

SL

COOm

COO

COO

COO

imp

L

m

O

O

C

O

RK

RRLs)RR(Cs1K

)RCs1(R)RR(Cs1)RCs1(R

RKZ

KZ

Zvv

+++⋅

++⋅⋅+⋅

⋅⋅+⋅+⋅⋅+⋅⋅+⋅

=++

= (A.44)

Rationalize the numerator and factor RO/Ri:

))RR(Cs1(RK

LsRKRR

1RK

RRCsRK

R

RCs1RR

vv

COOimpimp

SL

im

COO

im

O

CO

i

O

C

O

+⋅⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⋅

+⋅+

++⋅

⋅⋅⋅+

⋅

⋅⋅+⋅= (A.45)

Expand the denominator and group like terms:

))RR(Cs1(RK

LsKCsK

RCs1RR

vv

COOimp

COD

CO

i

O

C

O

+⋅⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⋅

+⋅⋅+

⋅⋅+⋅= (A.46)

Where: imp

SL

im

OD RK

RRRK

R1K

⋅+

+⋅

+= im

CO

imp

SLCOC RK

RRRKRR

1)RR(K⋅⋅

+⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅+

+⋅+= (A.47)


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅++⋅⋅+

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⋅

+⋅⎥⎦

⎤⎢⎣

⎡⋅⋅+⋅

⋅⋅+⋅=

COD

COO

impD

COD

CO

i

O

C

O

KCsK)RR(Cs1

RKLs1

KK

Cs1K

RCs1RR

vv

(A.48)

To simplify: im

OD RK

R1K

⋅+≈ OC RK ≈ 1

KCsK)RR(Cs1

COD

COO ≈⋅⋅++⋅⋅+

(A.49)

The simplified expression is:

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅⋅

+⋅⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⋅+

⋅+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

+

⋅⋅+⋅≈

imp

imO

O

im

O

CO

i

O

C

O

RKLs1

RK1

R1

Cs1

RKR

1

RCs1RR

vv

(A.50)

A-7


Continuous-Time Model Derivation of Factored Pole/Zero Form

Starting with the control to output gain of equation A.22, expand ZO and ZL:

)s(HRK

RRLs))RR(Cs1(K

)RCs1(R)RR(Cs1)RCs1(R

)s(HRKZ

KZ

Zvv

eimp

SL

COOm

COO

COO

COO

eimp

L

m

O

O

C

O

⋅+′

++⋅+

+⋅⋅+⋅⋅⋅+⋅

+⋅⋅+⋅⋅+⋅

=⋅+

′+

= (A.51)

Rationalize the numerator, factor RO/Ri and expand He(s): (A.52)

))RR(Cs1(ωs

Qωs

RKLs

RKRR

1RK

RRCsRK

R

RCs1RR

vv

COO2n

2

znimpimp

SL

im

COO

im

O

CO

i

O

C

O

+⋅⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅+

⋅′⋅

+⋅′+

++⋅

⋅⋅⋅+

⋅

⋅⋅+⋅=

Define: znimpn Qω

sRK

LsQω

s⋅

+⋅′⋅

=⋅

(A.53)

Expand the denominator and group like terms:

))RR(Cs1(ωs

QωsKCsK

RCs1RR

vv

COO2n

2

nCOD

CO

i

O

C

O

+⋅⋅+⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅+′⋅⋅+′

⋅⋅+⋅= (A.54)

Where: imp

SL

im

OD RK

RRRK

R1K

⋅′+

+⋅

+=′ im

CO

imp

SLCOC RK

RRRKRR

1)RR(K⋅⋅

+⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅′+

+⋅+=′ (A.55)


⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅⋅+′

+⋅⋅+⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛+

⋅+⋅⎥

⎦

⎤⎢⎣

⎡′′

⋅⋅+⋅′

⋅⋅+⋅=

COD

COO2

n

2

nD

COD

CO

i

O

C

O

KCsK)RR(Cs1

ωs

Qωs1

KK

Cs1K

RCs1RR

vv (A.56)

To simplify: im

OD RK

R1K

⋅+≈′ OC RK ≈′ 1

KCsK)RR(Cs1

COD

COO ≈′⋅⋅+′

+⋅⋅+ (A.57)

The simplified expression is:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅+⋅

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⋅+

⋅+⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅

+

⋅⋅+⋅≈

2n

2

n

imO

O

im

O

CO

i

O

C

O

ωs

Qωs1

RK1

R1

Cs1

RKR

1

RCs1RR

vv (A.58)

From equation A.53:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⋅′⋅

=

5.0TRK

Lπ

1Q

imp

(A.59)

A-8


Unified Linear Model Using General Gain Parameters

Figure A-4: Complete low frequency unified buck regulator linear model using general gain parameters. To model the current loop with sampling gain term, multiply Km by Hp(s) (valid for PCM1, VCM1 and VCM3 only). To

model the control to output or voltage loop with sampling gain term, multiply Gi by H(s). Derivation of model: From equations A.4 and A.34: (A.60) )KvRiv()s(FVDvv INiLCmININSW ⋅−⋅−⋅⋅+⋅= Let: (A.61) KvRivv INiLCP ⋅−⋅−= For the low frequency model let VIN·Fm(s)=Km. Equation A.60 can now be expressed as: (A.62) PmINSW vKDvv ⋅+⋅= Comparing equation A.4 to equation A.62:

therefore: PmIN vKdV ⋅=⋅ PIN

m vVK

d ⋅= (A.63)

The power stage of figure A-1 (a) shows the relationship of the input current to the inductor current: (A.64) dIDii LLIN ⋅+⋅= Since IL=VO/RO and VO =VIN·D, substitution of equation A.63 yields:

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅+⋅= P

O

mLIN v

RK

iDi (A.65)

A-9

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS – APPENDIX A by Robert Sheehan K can be found from the general gain parameters using equation A.42. Substitution of variables into equations A.10 and A.13 leads to an alternate form using averaged model parameters: (A.66) OI KDKK ⋅−=

TABLE A-1 DERIVATION OF K USING AVERAGED MODEL PARAMETERS

Mode IK OKD ⋅− K PCM1

DLTR5.0 i ⋅⋅⋅ ⎟

⎠⎞

⎜⎝⎛ ⋅⋅⋅⋅− D

LTR5.0D i DD

LTR5.0 i ′⋅⋅⋅⋅

PCM2 D

LTR5.0 i ⋅⋅⋅ DK

LTR5.0D SLi ⋅⎟

⎠⎞

⎜⎝⎛ −⋅⋅⋅− 2

SLi DKDDLTR5.0 ⋅+′⋅⋅⋅⋅

VCM1 0 ⎟⎠⎞

⎜⎝⎛ ′⋅⋅⋅⋅− D

LTR5.0D i DD

LTR5.0 i ′⋅⋅⋅⋅−

VCM2 DKSL ′⋅− DK

LTR5.0D SLi ′⋅⎟

⎠⎞

⎜⎝⎛ −⋅⋅⋅− 2

SLi )D(KDDLTR5.0 ′⋅−′⋅⋅⋅⋅−

EPCM1 D

LTR5.0 i ⋅⋅⋅− ⎟

⎠⎞

⎜⎝⎛ ⋅⋅⋅−⋅− D

LTR5.0D i DD

LTR5.0 i ′⋅⋅⋅⋅−

EPCM2 D

LTR5.0K iSL ⋅⎟⎠⎞

⎜⎝⎛ ⋅⋅− ⎟

⎠⎞

⎜⎝⎛ ⋅⋅⋅−⋅− D

LTR5.0D i DKDD

LTR5.0 SLi ⋅+′⋅⋅⋅⋅−

EVCM1 0 ⎟⎠⎞

⎜⎝⎛ ′⋅⋅⋅−⋅− D

LTR5.0D i DD

LTR5.0 i ′⋅⋅⋅⋅

EVCM2 DKSL ′⋅− ⎟⎠⎞

⎜⎝⎛ ′⋅⋅⋅−⋅− D

LTR5.0D i DKDD

LTR5.0 SLi ′⋅−′⋅⋅⋅⋅

VCM3 DKSL ′⋅− ⎟⎠⎞

⎜⎝⎛ ′⋅⋅⋅⋅− D

LTR5.0D i DKDD

LTR5.0 SLi ′⋅−′⋅⋅⋅⋅−

EPCM3 D


⎜⎝⎛ ⋅⋅− D

LTR5.0KD iSL ⋅⎟⎠⎞

⎜⎝⎛ ⋅⋅−⋅− DDK

LTR5.0 SLi ′⋅⋅⎟

⎠⎞

⎜⎝⎛ +⋅⋅−

EPCM4 D


⎜⎝⎛ ⋅⋅− ⎟

⎠⎞

⎜⎝⎛ ⋅⋅⋅−⋅− D

LTR5.0D i DKDD

LTR5.0 SLi ⋅+′⋅⋅⋅⋅−

The low frequency linear model of figure A-4 can be used for all transfer functions including the input impedance and output impedance. The line to output and control to output transfer functions can be reduced to the simplified transfer functions of the averaged model represented by equations 39 and 40. The only difference is the inductor pole frequency which becomes:

L

RKω im

L⋅

= (A.67)

This is considered to be the “true” inductor pole frequency without the sampling gain term.

A-10


Derivation of HP(s) from H(s) Let the control to inductor current transfer function of the continuous-time model from equation A.32 equal that of the unified model from equation A.40.

)s(HRKZZ)s(HK

)s(HRKZZK

vi

PimLO

Pm

imLO

m

C

L

⋅⋅++⋅

=⋅⋅++

= (A.68)

Solve for HP(s).

)1)s(H(RKZZ

ZZ)s(H

imLO

LOP −⋅⋅++

+= (A.69)

Evaluation of H(s)-1 yields:

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⋅+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

′⋅⋅⋅++

+=

2n

2

znmmpi

LimLO

LOP

ωs

Qωs

K1

K1

RZ

RKZZ

ZZ)s(H (A.70)

Utilization of equation A.70 in the unified model does not result in accurate plots of the current loop gain for all operating modes. It does yield accurate plots of the control to output gain. To simplify the analysis, let ZL/Ri=s·L/Ri. Evaluation of ωn and QZ results in:

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛−

′⋅⋅⋅

⋅++

+=

2n

2

mmpiL

im

L

O

L

O

P

ωs

2T

K1

K1

RLs

ZRK

ZZ

1

ZZ

1)s(H (A.71)

Let: 2T

K1

K1

RLK

mmpie −⎟

⎟⎠

⎞⎜⎜⎝

⎛−

′⋅= (A.72)

This allows HP(s) to be expressed as:

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅⋅

⋅++

+=

2n

2

eL

im

L

O

L

O

P

ωsKs

ZRK

ZZ

1

ZZ

1)s(H (A.73)

Ke is evaluated for each operating mode in table A-2.

A-11


TABLE A-2 EVALUATION OF FOR EACH OPERATING MODE eK

Mode

mpK1′

mK

1 eK

PCM1

IN

SLi V

V)D1(

LTR +−⋅⋅

IN

SLi V

VLTR)D5.0( +⋅⋅−

0

PCM2 DK)D1(

LTR SLi ⋅+−⋅⋅ DK2

LTR)D5.0( SLi ⋅⋅+⋅⋅−

iSL R

LDK ⋅⋅−

VCM1

IN

SLi V

VD

LTR +⋅⋅

IN

SLi V

VLTR)5.0D( +⋅⋅−

0

VCM2 )D1(KD

LTR SLi −⋅+⋅⋅ )D1(K2

LTR)5.0D( SLi −⋅⋅+⋅⋅−

iSL R

LDK ⋅′⋅−

EPCM1

IN

SL

VV

IN

SLi V

VLTR)5.0D( +⋅⋅−

TD ⋅−

EPCM2 SLK SLi K

LTR)5.0D( +⋅⋅−

TD ⋅−

EVCM1

IN

SL

VV

IN

SLi V

VLTR)D5.0( +⋅⋅−

TD ⋅′−

EVCM2 SLK SLi K

LTR)D5.0( +⋅⋅− TD ⋅′−

VCM3 SLi KD

LTR +⋅⋅ SLi K

LTR)5.0D( +⋅⋅−

0

EPCM3

IN

SLSL V

V)D1(K +−⋅

IN

SLSLi V

VK)D21(

LTR)5.0D( +⋅⋅−+⋅⋅−

iSL R

LDKTD ⋅⋅+⋅−

EPCM4

IN

SLSL V

VK +

IN

SLSLi V

VK

LTR)5.0D( ++⋅⋅−

TD ⋅−

Ke=0 for PCM1, VCM1 and VCM3.

Evaluation of the high frequency asymptote is done by recognizing that 1ZZ

1L

O ≈+ .

With ZL=s·L, HP(s) reduces to:

n

P

ωQs1

1)s(H⋅+

= where

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅

⋅

=

TRKLπ

1Q

im

(A.74)

For PCM1, VCM1 and VCM3, this value of Q is equivalent to that defined by equation A.59.

A-12


Unified Linear Model Using General Gain Parameters Low Frequency Model of Figure A-4

Line Rejection Starting from equation A.37 with HP(s)=1:

LO

OIN

O

iOCmINO ZZ

ZKv

ZR

vvKDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅−⋅+⋅= (A.75)

Since : VOC Gvv ⋅−=

LO

OIN

O

iOVOmINO ZZ

ZKv

ZR

vGvKDvv+

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−⋅−⋅−⋅+⋅= (A.76)

This allows the closed loop low frequency line rejection to be expressed as:

imOVmLO

Om

IN

O

RKZGKZZZ)KKD(

vv

⋅+⋅⋅++⋅⋅−

= (A.77)

By setting GV=0, this reduces to the open loop low frequency audio susceptibility equation. Note: Introduction of either HP(s) or H(s) will still result in a deviation from the actual response at higher frequency.


Output Impedance Starting with , write the transfer functions from the model of figure A-4. There is no perturbation from the input, so .

SWv0v IN =

(A.78) )RiGv(Kv iLVOmSW ⋅−⋅−⋅= The output voltage is perturbed by an external current source such that: (A.79) OOLO Z)ii(v ⋅+= The inductor current can be expressed as:

L

OSWL Z

)vv(i

−= (A.80)

Combining equations, the closed loop output impedance is found to be:

imL

VmO

O

O

O

RKZ)GK1(Z

1

Ziv

⋅+⋅+⋅

+= (A.81)

By setting GV=0, this reduces to the open loop output impedance. To incorporate the sampling gain term, multiply Ri by H(s).

A-13



Input Impedance From A.60, write the equation for : SWv (A.82) )ZZ(i)KvRiv(KDvv LOLINiLCmINSW +⋅=⋅−⋅−⋅+⋅= Write the equation for from A.61 and A.65: INi

)KvRiv(R

DKDii INiLC

O

mLIN ⋅−⋅−⋅

⋅+⋅= (A.83)

From the model of figure A-4: (A.84) OVLVOC ZGiGvv ⋅⋅−=⋅−= Combining equations A.82 and A.84:

imOVmLO

mINL RKZGKZZ

)KKD(vi

⋅+⋅⋅++⋅−⋅

= (A.85)

Combining equations A.83 and A.84:

O

im

O

OVm

O

mININ

L

RDRK

RDZGK

D

RDKK

vii

⋅⋅−

⋅⋅⋅−

⋅⋅⋅+

= (A.86)

Set equation A.85 equal to A.86. After much algebraic manipulation, the closed loop input impedance is found to be:

imOVm

OLOOm

imOVm

LO

2O

IN

IN

RKZGK

R)ZZR(DKK

1

RKZGKZZ

1

DR

iv

⋅+⋅⋅

−++⋅⋅

+

⋅+⋅⋅+

+⋅−= (A.87)

By setting GV=0, the open loop input impedance is found. Note: Introduction of either HP(s) or H(s) will still result in a deviation from the actual response at higher frequency.

A-14

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS APPENDIX B

LM3495 Unified Linear Model using General Gain Parameters

Low Frequency Model without Sampling Gain Term

Vp

Vsw Vo

=OUT/IN

IN OUT

10nC1

AC 0 0V2

Km/Ro

80.589

P1 S1D

[1] [100m] 200mRo

3.4m

Rs

1u

L1

10kR1AC 1 0

V1

3m

Rl

1mRc

200uCo

10kR2

Km16.1178

K

5.0257m

Gi

4

Gv

750u1.5kR3

72MegR4 100p

C2

K=D*(1/Km-Kn)

Vin

Vc

Vop

Figure B-1: SIMETRIX LM3495 Linear SPICE Model

Frequency / Hertz

100200400 1k 2k 4k 10k 20k 40k 100k 400k 1M

Impe

danc

e @

V1

/ Ohm

Y3

192021222324252627

Gai

n / d

B

Y2

-120-110-100-90-80-70-60-50-40

Pha

se /

degr

ees

Y1

-200-150-100-50

050

100150

Line Rejection and Input Impedance

(a)

(b)

(c)

(a) Vo/Vin Phase (Y1)(b) Vo/Vin Gain (Y2)(c) Impedance @ V1 (Y3)

Figure B-2: Line Rejection and Negative Input Impedance

B-1

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS – APPENDIX B by Robert Sheehan

Frequency / Hertz

100 200 400 1k 2k 4k 10k 20k 40k 100k 200k 400k 1M

Gai

n / d

BY2

-120-110-100-90-80-70-60-50-40

Pha

se /

degr

ees

Y1

-140-120-100-80-60-40-20

0

Audio Susceptibility - Vc=0

(a)

(b)

(a) Vo/Vin Phase (Y1)(b) Vo/Vin Gain (Y2)

Figure B-3: Audio Susceptibility

Frequency / Hertz

100 200 400 1k 2k 4k 10k 20k 40k 100k 200k 400k 1M

Gai

n / d

B

Y2

-60-50-40-30-20-10

01020

Pha

se /

degr

ees

Y1

-140-120-100-80-60-40-20

0

Control To Output

(a)

(b)

(a) Vo/Vc Phase (Y1)(b) Vo/Vc Gain (Y2)

Figure B-4: Control to Output

B-2


Frequency / Hertz

100 200 400 1k 2k 4k 10k 20k 40k 100k 200k 400k 1M

Gai

n / d

B

Y2

-20

-10

0

10

20

30

40

Pha

se /

degr

ees

Y1

100

120

140

160

180

200

Error Amplifier

(a)

(b)

(a) Vc/Vop Phase (Y1)(b) Vc/Vop Gain (Y2)

Figure B-5: Error Amplifier

Frequency / Hertz

100 200 400 1k 2k 4k 10k 20k 40k 100k 200k 400k 1M

Gai

n / d

B

Y2

-60

-40

-20

0

20

40

60

Pha

se /

degr

ees

Y1

20

40

60

80

100

120

Voltage Loop

(a)

(b)

(a) Vo/Vop Phase (Y1)(b) Vo/Vop Gain (Y2)

Figure B-6: Voltage Loop

B-3


AC 1 0I1

Vop

Vc

Vin

K=D*(1/Km-Kn)

100pC2

72MegR4 1.5k

R3

Gv

750u

Gi

4

K

5.0257m

Km16.1178

10kR2

200uCo

1mRc

3m

Rl

AC 0 0V1

10kR1

1u

L1

3.4m

Rs

200mRo

P1 S1D

[1] [100m]

Km/Ro

80.589

AC 0 0V2

10nC1

=OUT/IN

INOUT

VoVsw

Vp

H1

1

Figure B-7: SIMETRIX LM3495 Linear Model for Output Impedance

Frequency / Hertz

100 200 400 1k 2k 4k 10k 20k 40k 100k 200k 400k 1M

Gai

n / d

B

Y2

-100

-80

-60

-40

-20

0

Pha

se /

degr

ees

Y1

-50

0

50

100

150

Output Impedance - Zout

(a)

(b)

(c)

(d)

(a) Closed Loop Phase (Y1)(b) Closed Loop Gain (Y2)

(c) Open Loop Phase (Y1)(d) Open Loop Gain (Y2)

Figure B-8: Output Impedance

B-4


H1

1

Vp

Vsw Vo

=OUT/ININ OUT

10nC1

AC 0 0V2

Km/Ro

80.589

P1 S1

D

[1] [100m] 200mRo

3.4m

Rs

1u

L1

10kR1

AC 1 0V1

3m

Rl

1mRc

200uCo

10kR2

Km16.1178

K

5.0257m

Gi

4

Gv

750u1.5kR3

72MegR4 100p

C2

K=D*(1/Km-Kn)

Vin

Vc

Vop

Figure B-9: SIMETRIX LM3495 Linear Model for Input Impedance

Frequency / Hertz

100 200 400 1k 2k 4k 10k 20k 40k 100k 200k 400k 1M

Gai

n / d

B

Y2

2525.5

2626.5

2727.5

2828.5

29

Pha

se /

degr

ees

Y1

-150-100-50

050

100150200

Input Impedance - Zin

(a)

(b)

(c)

(d)

(a) Closed Loop Phase (Y1)(b) Closed Loop Gain (Y2)

(c) Open Loop Phase (Y1)(d) Open Loop Gain (Y2)

Figure B-10: Input Impedance

B-5


Vcs

AC 1 0

V3

Vop

Vc

Vin

K=D*(1/Km-Kn)

100pC2

72MegR4 1.5k

R3

Gv

750u

Gi

4

K

5.0257m

Km16.1178

10kR2

200uCo

1mRc

3m

Rl

AC 0 0V1

10kR1

1u

L1

3.4m

Rs

200mRo

P1 S1

D

[1] [100m]

Km/Ro

80.589

AC 0 0V2

10nC1

=OUT/ININ OUT

VoVsw

Vp

Vcsp

Figure B-11: SIMETRIX LM3495 Linear Model for Current Loop

Frequency / Hertz

100 200 400 1k 2k 4k 10k 20k 40k 100k 200k 400k 1M

Gai

n / d

B

Y2

-40

-30

-20

-10

0

10

20

Pha

se /

degr

ees

Y1

-250

-200

-150

-100

-50

0

Current Loop

(a)

(b)

(a) Vcs/Vcsp Phase (Y1)(b) Vcs/Vcsp Gain (Y2)

Figure B-12: Current Loop

B-6

PES02 Presentation October 24, 2006



by

Robert Sheehan


Santa Clara, CA

ERRATA

For

PES02 Tuesday, October 24, 2006

8:30am – 9:30am

Power Electronics Technology Exhibition and Conference October 24-26, 2006

Long Beach Convention Center Long Beach, CA

Corrections to the Conference Proceedings CD version, which are included in the print version: 1. Figure 4: Delete last sentence, “By setting H(s)=1, the averaged model is obtained.” 2. Table 2A: VCM1, change Sn=(VIN-VO)·Ri/L to Sn=VO·Ri/L. 3. Table 5: Last section for PCM2 and VCM2, change KSL from “0.05” to “0.10” (two places). 4. Figure 10: Graph legends, change last entries for “Gain Q=0.318” to “Phase Q=0.318” (three places). 5. Equation 71: Change from “iL·Ri - 0.5…” to “iL·Ri + 0.5…” 6. Section 4. General Slope Compensation Requirements, second paragraph: Change from “when the sum of the inductor current’s slope…” to “when the sum of the sensed inductor current’s slope…” Additional sections which are not on the Conference Proceedings CD version: Appendix A Appendix B

PES02 Presentation October 24, 2006

EMULATED CURRENT MODE CONTROL FOR BUCK REGULATORS by Robert Sheehan Notes:

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Products Applications

Audio www.ti.com/audio Communications and Telecom www.ti.com/communications

Amplifiers amplifier.ti.com Computers and Peripherals www.ti.com/computers

Data Converters dataconverter.ti.com Consumer Electronics www.ti.com/consumer-apps

DLP® Products www.dlp.com Energy and Lighting www.ti.com/energy

DSP dsp.ti.com Industrial www.ti.com/industrial

Clocks and Timers www.ti.com/clocks Medical www.ti.com/medical

Interface interface.ti.com Security www.ti.com/security

Logic logic.ti.com Space, Avionics and Defense www.ti.com/space-avionics-defense

Power Mgmt power.ti.com Transportation and Automotive www.ti.com/automotive

Microcontrollers microcontroller.ti.com Video and Imaging www.ti.com/video

RFID www.ti-rfid.com

OMAP Mobile Processors www.ti.com/omap

Wireless Connectivity www.ti.com/wirelessconnectivity

TI E2E Community Home Page e2e.ti.com

Mailing Address: Texas Instruments, Post Office Box 655303, Dallas, Texas 75265Copyright © 2011, Texas Instruments Incorporated

http://www.ti.com/audio

http://www.ti.com/communications

http://amplifier.ti.com

http://www.ti.com/computers

http://dataconverter.ti.com

http://www.ti.com/consumer-apps

http://www.dlp.com

http://www.ti.com/energy

http://dsp.ti.com

http://www.ti.com/industrial

http://www.ti.com/clocks

http://www.ti.com/medical

http://interface.ti.com

http://www.ti.com/security

http://logic.ti.com

http://www.ti.com/space-avionics-defense

http://power.ti.com

http://www.ti.com/automotive

http://microcontroller.ti.com

http://www.ti.com/video

http://www.ti-rfid.com

http://www.ti.com/omap

http://www.ti.com/wirelessconnectivity

http://e2e.ti.com

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