IC Design of Power Management Circuits (III)
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Transcript of IC Design of Power Management Circuits (III)
IC Design ofPower Management Circuits (III)
Wing-Hung KiIntegrated Power Electronics Laboratory
ECE Dept., HKUSTClear Water Bay, Hong Kong
www.ee.ust.hk/~eeki
International Symposium on Integrated CircuitsSingapore, Dec. 14, 2009
Ki 2
Part III
Switching Converters:Stability and Compensation
Ki 3
Stability and Compensation
Nyquist criteriaSystem loop gainPhase margin vs transient responseType I, II, III compensatorsCompensation for voltage mode controlCompensation for current mode control
Content
Ki 4
Consider the feedback system:
Feedback Systems
F(s)in
G(s)
out
Note that F(s) and G(s) are ratios of polynomials in s, that is,
= F
F
n (s)F(s)
d (s)= G
G
n (s)G(s)
d (s)
The closed loop transfer function is
= = =+ +
out F(s) F(s)H(s)
in 1 F(s)G(s) 1 T(s)
and the loop gain is
= =n(s)
T(s) F(s)G(s)d(s)
Ki 5
Stability Criteria
Local stability: all poles of T(s) (= all roots of d(s)) are in LHP
System stability:∗
all poles of H(s) are in LHP⇒
all zeros of (1+T(s)) are in LHP⇒
all roots of (n(s)+d(s)) are in LHP
If all functional blocks satisfy local stability, the Nyquist criterion for system stability is:
∗
Nyquist plot of 1+T(s) does not encircle (0,0)⇒
Nyquist plot of T(s) does not encircle (-1,0)
If all functional blocks satisfies local stability, the Bode plot criteria for system stability is:
phase margin φm >0o and gain margin GM>0dB
Ki 6
1st Order Loop Gain Function
=+
o
1
TT(s)
s1p
+ω = 0ω = +∞
1−1
oTRe
Im
unit circle
oT / 2
o-45
−∞
0
−0
ωUGF
oT
ωUGF
| T |
/ A
ω
ω
− o90
1p
− o45
1p
Nyquist Plot
Bode Plots
stable
Ki 7
2nd Order Loop Gain Function
=⎛ ⎞⎛ ⎞
+ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
o
1 2
TT(s)
s s1 1p p
−1 oTRe
Im
0
ωUGF
oT
ωUGF
| T |
/ A
ω
ω
− o90
2p
− o180φm
1p
φm
Nyquist Plot
Bode Plots
stable
Ki 8
3rd Order Loop Gain Function
=⎛ ⎞⎛ ⎞ ⎛ ⎞
+ + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠
o
1 2 3
TT(s)
s s s1 1 1p p p
−1 oTRe
Im
0
ωUGF
oT
ωUGF
| T |
/ A
ω
ω
− o90
2p
− o180φ <m 0
1p
φm
Nyquist Plot
Bode Plots
unstable(encirclement
of -1)
3p
− o270
Ki 9
Observations on Loop Gain Function
• 1st order systems are unconditional stable.
• 2nd order systems are stable, but a high damping factor would cause large overshoot and excessive ringing before settling to the steady state.
• For 3rd order systems, if the 3rd pole p3 is less than 10X of the unity gain frequency ωUGF , the system is unstable.
Hence, for a stable system, the loop gain function could be approximated by a 2nd order loop gain function with the 2nd pole p2 usually larger than ωUGF to achieve small overshoot.
Ki 10
Loop Gain Function and Transient Response
The transient response of a feedback system is given by
where L-1(⋅) is the inverse Laplace transform of (⋅).
The exact transient response is affected by F(s), however, if only T(s) is considered, we may consider the modified feedback system:
− − ⎛ ⎞⎛ ⎞= × ×⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠1 1
o1 1 F(s)v (s) H(s)s s 1 T(s)
L = L
F(s)in
G(s)
out T(s)in '
1
out '
=+F(s)
H(s)1 T(s)
=+T(s)
H'(s)1 T(s)
Ki 11
For a loop gain function approximated by a 2-pole function:
= ≈⎛ ⎞⎛ ⎞ ⎛ ⎞
+ + +⎜ ⎟⎜ ⎟ ⎜ ⎟ω⎝ ⎠⎝ ⎠ ⎝ ⎠
o
1 2 UGF 2
T 1T(s)
s s s s1 1 1p p p
ωUGF
The closed loop function (with unity gain feedback) is
2p
− o90
− o180φm
= =+
+ +ω ω
2
UGF UGF 2
T(s) 1H'(s)
1 T(s) s s1p
| T |
/ T
=+ +
ω ω
2
2o o
1H'(s)
1 s s1Q
Write H'(s) in standard 2nd order form:
ω = ωo UGF 2p
ω= UGF
2
Qp
Compensator Considerations
1p
Ki 12
Relationship between p2 /ωUGF and φm
ω2
UGF
p ω=UGF
2
1p k
π−
−24Q 1e−φω
1 2m
UGF
p=tan
k Q overshoot phase margin
1
≈2.73 3
=3 1.73
1 0.163 o45
o70
o60
o30=1 / 3 0.577
≈0.605 0.6 0.01
0.76 0.064
1.316 0.275
Note that φm =60o gives an overshoot of 6.4%, and the 1% settling time (tset ) would be very long. By setting p2 =3ωUGF , then φm =70o, and the overshoot is only 1%.
Ki 13
Type I Compensator (0Z1P)
1R1C
inV
= − = −out
in 1 1
V 1A(s)
V sC R
outVrefV
ω =UGF1 1
1C R
− o180
− o90
| A |
/ A
The simplest Type I compensator is an integrator with ωUGF = 1/C1 R1.
Assume Aop (s) is first order with ωt >> 1/C1 R1 :
= ≈+ ω ω
opop
1 t
A 1A (s)1 s / s /
⇒ ≈+ ω1 1 t
1A(s)sC R (1 s / )
ωt
ω
ω
Ki 14
Type I compensator can be implemented using transconductance amplifier (OTA). OTA has a very high output resistance ro and cannot drive resistive loads.
mg
or oC
inVoutV
refV
= − = −+
out m o
in o o
V g rA(s)
V 1 sC r
Type I Compensator (0Z1P)
m og r o o1 / C r
ω = mUGF
o
gC
| A |
/ A
Using OTA may save one IC pin.
ω
ω
− o90
Ki 15
1R1C
inVoutV
refV
+= −
+ +out 2 2
in 1 2 1 1 2 2
V 1 sC RV s(C C )R [1 s(C || C )R ]
2R2C
+≈ <<
+2 2
1 22 1 1 2
(1 sC R )A(s) (C C )
sC R (1 sC R )
Type II Compensator (1Z2P)
1 2
1C R2 2
1C R
2 11 /C R
ω =UGF
1 11 / C R
| A |
/ A
o90 phaseboosting
Type II compensator consists of a pole-zero pair with ωz <ωp , and a maximum phase boosting of 90o is possible.
ω
ω
− o90
Ki 16
mg
or
2C
inVoutV
refV2R
+= − = −
+ +out m o 2 2
in 2 o 2
V g r (1 sC R )A(s)
V 1 sC (r R )
+≈
+m o 2 2
2 o
g r (1 sC R )A(s)
(1 sC r )
Type II Compensator (1Z1P)
m og r
2 2
1C R
2 o
1C r
Type II compensator can also be implemented using OTA.
ω = mUGF
2
gC
| A |
/ A
ω
ω
− o90
Ki 17
1R
1C
inVoutV
refV
+ + += −
+ + +out 2 2 3 1 3
in 1 2 1 1 2 2 3 3
V (1 sC R )[1 sC (R R )]V s(C C )R (1 sC || C R )(1 sC R )
2R2C
3R3C
Type III Compensator (2Z3P)
+ +≈
+ +2 2 3 1
2 1 1 2 3 3
1 2 1 3
(1 sC R )(1 sC R )A(s)
sC R (1 sC R )(1 sC R )
(C <<C , R >>R )
1 2
1C R
2 2
1C R
2 1
1C R
3 3
1C R
o180boosting
− o90
| A |
/ A
ω
=UGF
1 3
1C R
3 1
1C R
Type III compensator consists of two pole-zero pairs, and phase boosting of 180o is possible to compensate for complex poles.
ω
ω
+ o90
Ki 18
PWM Voltage Mode Control
S
RQ
Q
refVA(s)
EACMP
gV
oV
ckramp
L
CLR
1R
2R
obVav
av
PM
NM
A regulated switching converter consists of the power stage and the feedback circuit.
For a buck converter, if an on-chip charge pump is not available, then the NMOS power switch is replaced by a PMOS power switch.
avramp
ck
Q
Q
Ki 19
Loop Gains of Voltage Mode CCM Converters
The system loop gain is T(s) = A(s)×H(s), where A(s) is the frequency response of the EA (compensator). Loop gains of voltage mode PWM CCM converters with trailing-edge modulation are compiled. Parasitic resistances except ESR are excluded [Ki 98].
+= ×
+ +
o esr
2m
L
bV 1 sCRT(s) A(s) .
sLDV 1 s LCR
Buck:
Boost:
Buck-boost:
−= ×
+ +
2o L
2m
2 2L
bV [1 sL / (D ' R )]T(s) A(s) .
D ' V sL s LC1D' R D'
−= ×
+ +
2o L
2m
2 2L
b | V | [1 sDL / (D ' R )]T(s) A(s) .
DD' V sL s LC1D' R D '
Ki 20
Voltage Mode Compensation (1)
Example: Consider a buck converter with the following parameters:
Vdd =4.2V, Vo =1.8V (D=0.429), Vm =0.5V, b=0.667L=2μH, C=3.3μF, RL =1.8Ω
(Io =1A), Resr =100mΩ, fs =1MHz
The system loop gain is given by
× + × × += ⋅ =
+ + + +ω ω
2 2
2 2o o
5.6 [1 s /(3M)] A(s) 5.6 [1 s /(3M)]T(s) A(s)
1 s s 1 s s1 12.3 390k Q(390k)
The system loop gain consists of a pair of complex poles, and one strategy is to use dominant pole compensation.
For a buck converter, the complex pole frequency ωo /2π
is 10 to 30 times lower than the switching frequency fs .
Ki 21
Voltage Mode Compensation (2)
60
40
20
0dB
=+1330
A(s)1 s /10
⇒5.6 15dB390k
3M1M 10M100k10k1k10010
80
0.1 1
− o90
− o180
=1330 62.5dB
− o270
/H(s)/ A(s)
o0 ω
ω
× +=
+ +2
2
5.6 (1 s /3M)H(s)
1 s s12.3 390k (390k)
Dominant pole compensation
Ki 22
Voltage Mode Compensation (3)
60
40
20
0dB
× +=
⎛ ⎞+ + +⎜ ⎟
⎝ ⎠
2
2
7500 (1 s /3M)T(s)
1 s s(1 s /10) 12.3 390k 390k
390k
3M
1M 10M100k10k1k10010
80
0.1 1
− o90
− o180
=7500 77.5dB
− o270
/H(s)
/ T(s)
φ =mo90
ω=
UGF75k
o0 ω
ω
Ki 23
Stability inferred from Line and Load Transients
Measuring loop gain could be difficult, and for some circuits, and especially integrated circuits, due to loading effect and that loop- breaking points may not be accessible, stability is inferred by simulating or measuring the line transient and/or load transient.
If the circuit is stable and has adequate phase margin, line and load transients will show first order responses.
If the circuit is stable but has a phase margin less than 70o, line and load transients will show minor ringing.
If the circuit is unstable, line and load transient will show serious ringing/oscillation.
Ki 24
Current Mode PWM with Compensation Ramp
In practice, the output of EA (Va ) should not be tempered, and a compensation ramp of +mc is added to m1 instead.
S
RQ
Q
refVA(s)
EACMP
ddV
oV
ck
L
CLR
1R
2R
obVav
i
i /N
fNR
PM
NM
V2I
ramp from OSCav
DT
1 c f(m m )R+ 2 c f(m m )R− −
bvbv
compensationramp
ddV
Ki 25
Loop Gain of Current Mode Buck Converter (1)
The loop gain of a current-mode CCM buck converter with trailing- edge modulation is shown below. Others can be found in [Ki 98].
Buck:( )× +
= ×⎛ ⎞ ⎛ ⎞−
+ + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
esrf 1
2 1
L 1 1 L
1 1b 1 sCRCR n D' T
T(s) A(s)(n D ' D)T1 1 1 1 1s s
CR n D' T n D' T C R L
The two poles are in general real.
−= + > ⇒ >c 2
1 c 11
m m 2 Dwith n 1 , m nm 2 2D'
<L2Land R
D ' T
Ki 26
Loop Gain of Current Mode Buck Converter (2)
If the poles are real and far apart, the denominator could be simplified.
Buck: + ω= ×
⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟ω ω⎝ ⎠⎝ ⎠
L a z
f
a t1
R ||R 1 s /T(s) A(s) b. .
R s s1 1
= = +−
ca 1
1 1
mLR n 1
(n D ' D)T m
For two real poles that are farther apart, pole-zero compensation could be used to extend the bandwidth.
ω = ω = ω =z a t1c L a 1
1 1 1 CR C(R ||R ) n D ' T
Ki 27
Example: Consider a current mode buck converter with the same parameters as those of the voltage mode converter for comparison.
Vdd = 4.2V, Vo = 1.8V (D = 0.429), b = 0.667, fs = 1MHz, Rf = 1ΩL = 2μH, C = 3.3μF, RL = 1.8Ω
(Io =1A), Resr = 100mΩ, mc = m2
+= ×
⎛ ⎞⎛ ⎞+ +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
0.8(1 s /3M)T(s) A(s)s s1 1
290k 880k
Current Mode Compensation (1)
⇒
n1 =1.75, 1/n1 D’T ≈
1/1μ
The system loop gain is given by
Ki 28
We may assume the poles are far apart and use the simplified equation, and we have
+= ×
⎛ ⎞⎛ ⎞+ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
0.8(1 s /3M)T(s) A(s)s s1 1
250k 1M
Current Mode Compensation (2)
n1 =1.75, Ra =3.5Ω, ωz =3M rad/s, ωa =250k rad/s, ωt1 =1M rad/s
The system loop gain is then given by
Instead of a pair of complex poles as in voltage mode control, two separate poles are obtained, and both dominant-pole compensation and pole-zero compensation could be employed.
Ki 29
Current Mode Compensation (3)
60
40
20
0dB
=+10000
A(s)(1 s /10)
⇒ −0.8 2dB250k
1M 10M100k
10k1k10010
H(s)
1
− o90
− o180
/H(s)
o0 ω
ω
−20
1M
3M
80Dominant pole compensation
Ki 30
Current Mode Compensation (4)
60
40
20
0dB250k
1M 10M100k10k1k100101
− o90
− o180/ T(s)
o0 ω
ω
−20
1M
80
ω=
UGF80k
⇒8000 78dB
φ=
mo70
× +=
+ + +8000 (1 s /3M)
T(s)(1 s /10)(1 s /250k)(1 s /1M)
Ki 31
Pole-zero cancellation
− o90/H(s)
o0 ω
/ A(s)
1M 10M100k10k1k100101ω
60
40
20
0dB
250k 3M
H(s)−20
+=
+(1 s /250k)
A(s)(s /375k)(1 s /3M)
375k
Current Mode Compensation (5)
− o180
Ki 32
Bandwidth increased by 4 times to 300k rad/s
1M 10M100k10k1k100101
− o90
− o180
o0 ω
ω
60
=+
1T(s)
(s / 300k)(1 s /1M)40
20
0dBω =UGF 300k
φ=
mo70
/ T(s)
Current Mode Compensation (6)
Ki 33
References: Switching Converter Compensation
[Brown 01] M. Brown, Power Supply Cookbook, EDN, 2001.
[Ki 98] W. H. Ki, "Signal flow graph in loop gain analysis of DC-DC PWM CCM switching converters," IEEE Trans. on Circ. and Syst. 1, pp.644-655, June 1998.
[Ma 03a] D. Ma, W. H. Ki, C. Y. Tsui and P. Mok, "Single-inductor multiple-output switching converters with time-multiplexing control in discontinuous conduction mode", IEEE J. of Solid-State Circ., pp.89-100, Jan. 2003.
[Ma 03b] D. Ma, W. H. Ki and C. Y. Tsui, "A pseudo-CCM / DCM SIMO switching converter with freewheel switching," IEEE J. of Solid-State Circ., pp.1007- 1014, June 2003.
Ki 34
SUPPLEMENTS
Ki 35
Voltage Mode Converters: Loop Gain Function
In discussing fast-transient converters, one important parameter is the loop bandwidth.
The loop gain function of the buck converter with voltage mode control operating in CCM ignoring ESR is given by [Ki 98]
T(s) o
2m
L
bV 1A(s) .sLDV 1 s LCR
= ×+ +
The resonance frequency ωo and the pole-Q are
oω1LC
= Q CRL
=
The converter enters DCM at
L(BCM)R 2LD'T
= ⇒ BCMQo
2 1D ' T
=ω
Ki 36
For voltage mode buck, the ripple voltage is given by
If ΔVo /Vo =0.01 and D=0.5, then the complex pole pair is at
Δ o
o
VV 2
s
D ' 18 LCf
=
To have adequate gain margin GM, say, 6dB, the unity gain bandwidth fUGF has to be reduced by 10×2=20 times:
oω s0.4f=
Voltage Mode Converters: Bandwidth Limitation
andBCMQ
o
2 1 10D ' T
= =ω
ofo sf
2 16ω
= ≈π
⇒
If fs =1MHz, then fUGF is at around fs /320 = 3.125kHz.
UGFf s s1 f f20 16 320
= × =
Ki 37
VM Buck: Loop Gain Function with Rδ
The unity gain frequency fUGF of fs /320 is too low. Fortunately (or unfortunately), the converter inevitably has parasitic resistors such as RESR , Rℓ
(inductor series resistor), Rs (switch resistance) and Rd (diode resistance), and the loop gain function is [Ki 98]
T(s)
δ
≈ ×⎛ ⎞
+ + +⎜ ⎟⎝ ⎠
o
2m
L
bV 1A(s) .DV L1 s CR s LC
Rwhere
Rδ ESR s dR R DR D'R≈ + + +
This Rδ
is at least 200mΩ, thus reducing QBCM to around 3. With GM to be 6dB, fUGF is reduced by 3×2=6 times, and
If fs =1MHz, then fUGF is at around fs /100 = 10kHz.
UGFf s s1 f f6 16 100
= × ≈
Ki 38
ωsωUGF
ω
ω
− o90
− o270
− o180
o0
|T|
/T
0dB
20dB
40dB
60dB
ωo
φm
− o45 / dec
o-180 ×Q/dec
=GM 6dB
oT
VM Buck: Dominant Pole Compensation
-20dB/dec
-60dB/dec
100X
ωp (dominant pole)
Ki 39
Current Mode Converters: Loop Gain Function
The loop gain function of the buck converter with current mode control operating in CCM ignoring ESR is given by [Ki 98]
T(s)
In general, the two poles are real, as discussed next.
−= + > ⇒ >c 2
1 c 11
m m 2 Dn 1 , m nm 2 2D'
f 1
2 1
L 1 1 L
1 1A(s)bCR n D' T
(n D ' D)T1 1 1 1 1s sCR n D' T n D' T C R L
×=
⎛ ⎞ ⎛ ⎞−+ + + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
L(BCM)R 2LD'T
=
with
and
Ki 40
To compute the upper limit of fUGF w.r.t. fs , we simplify the current mode case as follows. Let D=0.5 and choose n1 =2 such that sub-harmonic oscillation could be suppressed even for D=0.667. The loop gain function at heavy load is
Current Mode Converters: Bandwidth Limitation
T(s) ≈+ +
L
f L
R 1A(s)bR (1 sCR )(1 sT)
At RL(BCM) =2L/D’T,
BCMT (s) ≈+ +
L(BCM)
f
R 1A(s)bR (1 s8T)(1 sT)
Pole-zero cancellation at ω1 =1/CRL should be done at the highest load current Iomax (smallest load resistance). To achieve φm of 70o, fUGF should be 3 times lower than f2 , and fUGF ≈
fs /20. Hence, a current mode converter could have a unity gain frequency 5 times higher than its voltage mode counterpart.