8/14/2019 Transient Response Counts When Choosing Phase Margin

http://slidepdf.com/reader/full/transient-response-counts-when-choosing-phase-margin 1/4Power Electronics Technology November 2008   www.powerelectronics.com18

Transient Response Counts

When Choosing Phase MarginBy Christophe Basso, Applications Manager,ON Semiconductor, Toulouse, France

An analytical derivation of the optimum con-verter phase margin for critically dampedresponse shows it is close to 76 degrees, wellabove the traditional recommendation of45 degrees.

 The design o a closed-loop switch-mode powersupply creates a path between the variable adesigner wants to monitor and the control pino the designer’s converter. Tis control pin canbe the peak current setpoint in a current-mode

power supply or the duty-cycle input o a voltage-mode con-troller. I the monitored variable deviates rom its imposedtarget, the controller reacts by either increasing or decreas-ing the delivered power to the load via an amplified error

signal ed to its control pin. However, requency-dependentgain and phase (H(s)) affect the power stage.o ensure that the power supply behaves as specified, the

designer must shape the return path (G(s)) to compensateor the power-stage response at certain requency points.Among the important parameters are:

l DC gain or the smallest static error and the lowestoutput impedance

l  Crossover requency or the required responsespeed.

At the crossover point, where the loop-gain module((s)) equals 1, the phase rotation affects the returningsignal. I the signal returns in phase with the control signal,these are the conditions that create an oscillator, which issomething one wants to avoid. o make sure the signal doesnot return in phase (i.e., with a 360-degree phase rotation),a designer must plan a certain amount o margin betweenthe phase rotation o (s) at the crossover requency andthe 360-degree limit, which is the phase margin. How much

phase margin should one ask or to provide perormanceand stability? extbooks ofen suggest 45 degrees. Shoulddesigners try to get more than that? Let us analyze howmuch.

Second-Order SystemFig. 1 shows a LC low-pass filter where the resistor (R)

represents the network losses. Tis architecture could beseen as a simplified lossy output filter o an unloaded buckconverter. In that case, the input voltage (V

IN) is the average

level o the square-wave signal present at the power switch/reewheel diode cathode junction. For the purpose o this

analysis, this average voltage will be ac modulated, and weare looking or the expression o the output voltage acrossthe output capacitor. Te transer unction, H(s) = V

OU (s)/

VIN

 (s), o this structure will then be calculated.Using Laplace notation, Eq. 1 describes the transer

unction o this RLC network:

  (Eq. 1)H sLCs RCs

( ) .=+ +

1

12

By rearranging the expression, one can identiy the qual-ity coefficient and the resonant requency:

(Eq. 2)H ss s s s

QR R R R  

( ) ,=+ +

=+ +

1

2 1

1

12

2

2

2wz

w w w 

Fig. 1. The buck converter here is represented by a simple low-pass

filter. The script at the bottom calculates values of R (network

losses) in response to changing quality coefficient values (Q).

+

 V IN

R1{R} L1{L}

C1{C}

 V OUT

Parameters:

fO = 235 kHzL = 10 µH

2 2

1C

4 L=

π

 1o

LCω =  

1R 

{C}2{Q}

4{L}

=

×

Q = 10

fO

8/14/2019 Transient Response Counts When Choosing Phase Margin

http://slidepdf.com/reader/full/transient-response-counts-when-choosing-phase-margin 2/4 www.powerelectronics.com  Power Electronics Technology November 200819

where ωR  is the resonant requency:

  (Eq. 3)wR LC

=1

, 

ζ is the damping actor:

  (Eq. 4)z = R C

L4  and

Q is the quality coefficient:  (Eq. 5)Q =

1

2z.

Te idea now is to evaluate the response to a 1-V inputstep and change the quality coefficient values by tweakingresistor R1. Tis resistor is representative o the losses inthe network such as the equivalent series resistance (ESR)o the inductor. In Fig. 1, the calculation is automatedo R, whose value is evaluated according to the selectedquality coefficient. One also could multiply Eq. 1 by 1/sec

and calculate the inverse Laplace transorm to obtain thetemporal response. In this case, a SPICE simulation is aster.Te results appear in Fig. 2.

As one can see, low coefficient values lead to a completelyoscillation-ree response, whereas values above 0.5 givebirth to overshoots. As the quality coefficient increases,meaning ewer losses, the overshoot gets larger. I thequality coefficient would go to infinity, it would imply anundamped LC network, keeping oscillations going urtherto an excitation.

Looking for RootsA study o Eq. 2’s denominator reveals the roots orwhich H(s) goes to infinity. Mathematically, it correspondsto:  (Eq. 6)s s

QR R 

2

21 0

w w+ + = .

Te roots come easily as ollows:  (Eq. 7)s s

QQR 

1 22

21 1 4, ( ).= - ± -

w

In Eq. 7, the term under the square root can either bepositive or negative, depending on the quality coefficient value. For values below 0.5, the so-called overdamped case,

the term under the square root remains positive and bothroots s

1 and s

2 are separated real roots.

Te step response is sluggish, as shown in Fig. 2. Whenthe quality coefficient reaches 0.5, called the criticallydamped case, the roots are still real but are now coincident.Te step response is much aster, but still does not exhibitovershoot.

Now, i the quality coefficient grows urther, this is anunderdamped case and the roots welcome an imaginaryportion that increases as the quality coefficient goes up.Tis results in a ast-step response now eaturing overshootand oscillations.

I the quality coefficient reaches infinity, the real portiono roots s

1 and s

2 ades away and the system reely oscillates.

Tis means there is no more damping (losses) brought by

the real terms. Analyzing the trajectory o these roots iscalled root locus analysis. Such an analysis shows how theroots are positioned in the s-plane and give an indicationo how they move relative to some parameters.

Keep in mind that it is Q in this example here, but it

could be the gain k o a system where, at some point whenk increases, the roots migrate in the right-hal plane andcause instability. Fig. 3 describes the path taken by s

1 and

s2 as the quality coefficient changes.

Approximation of an Open-Loop ResponseBased on what has already been disclosed, it would be

interesting to model the closed-loop dc-dc converter withan equation where a quality coefficient term would appear.Tat way, a designer could select the parameter that affectsthis quality coefficient to shape the output response he orshe is looking or: a response that is slow but without anyovershoot, or vice versa, a response that is aster but ac-cepts a little overshoot. Let us start the derivation processby looking at Fig. 4.

Time (µs)

    V   o   u   t   #   6 ,

    V   o   u   t   #   5 ,

    V   o   u   t   #

   4 ,

    V   o   u   t   #   3 ,

    V   o   u   t    (    V    )

1.80

1.40

1.00

0.600

0.200

5.00 15.0 25.0 35.0 45.0

Q = 0.5

Q < 0.5 over dampingQ = 0.5 critical dampingQ > 0.5 under damping

Q < 0.5 overdampingQ = 0.5 criticaldampingQ > 0.5 underdamping

Fast response and no overshoot!Fast response and no overshoot

Overshoot = 65%Overshoot = 65%

 Asymptotically stable Asymptotically stable

Q = 5

Q = 1Q = 1

Q = 0.707Q = 0.707

Q = 0.1Q = 0.1

Fig. 2. When Q is swept from 0.1 to 5, the response to a step is slow

(Q = 0.1) and without overshoot, whereas Q values above 0.5

 produce overshoots but are fast. As Q increases, meaning lower

losses, the overshoots get larger.

 

Low Q Low Q

Q = 0.5

Q = ∞

σ

High Q

High Q

LHP RHP

Fig. 3. Root locus analysis helps to understand how the roots move

relative to a selected parameter, such as the quality coefficient (Q)

of the LC network.

8/14/2019 Transient Response Counts When Choosing Phase Margin

http://slidepdf.com/reader/full/transient-response-counts-when-choosing-phase-margin 3/4Power Electronics Technology November 2008   www.powerelectronics.com20

  TRANSIENT RESPONSE

Fig. 4 shows the complete loop gain (s) made o theconverter power-stage transer unction, H(s), urthershaped by the compensator transer unction, G(s). Teexample here is dealing with a continuous-conductionmode (CCM) buck converter operated in voltage-modecontrol. In this figure, concentrate on the area around thecrossover requency, which represents one important designparameter o the dc-dc converter trying to be stabilized.

Asymptotically looking at the curve within the rame revealsthe effects o an origin pole (ω0) and a high requency pole

(ω2). Mathematically, this approximation is:

  (Eq. 8)

ss s

( ) .Ⱦ

èçççç

ö

ø÷÷÷÷

  +æ

èçççç

ö

ø÷÷÷÷

1

10 2w w

In this approximated expression, extra poles and zerosare considered ar away rom the crossover requency,naturally limiting their impact on the transer unction.However, what is interesting is the response the dc-dcconverter delivers once its loop is closed. In other terms, let

us identiy the closed-loop transer unction derived romEq. 8. o obtain the closed-loop expression, evaluate:

(Eq. 9) s

s s s

( )

( ).

1

1

12

0 2 0

+=

+ +w w w

Eq. 9 is similar in orm to Eq. 2. Tereore, it can beput under the amiliar orm o a second-order system asdescribed in Eq. 10:

  (Eq. 10) s

s s s

QR R 

( )

( ).

1

1

12

2

+=

+ +w w

Te identification o the quality coefficient and theresonant requency is straightorward:

Q =   ww

0

2   (Eq. 11)

w w wR  = 0 2 .  (Eq. 12)

Tere is now an equation that describes the approximateclosed-loop response o the dc-dc converter and it includesa quality coefficient. Te next step is to establish a relation-ship between the closed-loop quality coefficient and the keydesign parameter, the open-loop phase margin. First, basedon Eq. 8, calculate the crossover requency determinedby the location o the origin pole and its associated high

requency pole. At the crossover point, it is known that the(s) module equals 1; thereore:

  (Eq. 13)

1

1

1

0

0

2

 j jCw

w

w

w

æ

èçççç

ö

ø

÷÷÷÷  +

æ

èçççç

ö

ø÷÷÷÷

= .

Extracting ωC and rearranging this equation gives:

  (Eq. 14)w

w  w

wC =

+æ

èçççç

ö

ø÷÷÷÷

  -2 0

2

2

1 4 1

2

.

I Eq. 12 is substituted into Eq. 14, a quality coefficient-dependent crossover requency can be obtained:

  (Eq. 15)w  w

C

Q=

  + -241 4 1

2

( ).

Eq. 15 shows how the closed-loop quality coefficientand the open-loop crossover requency are linked. It isimportant or this remark to be well understood: Q repre-sents the resulting closed-loop response quality coefficientbased on the open-loop pole/zero arrangement describingthe approximated open-loop compensated transer unc-tion in Eq. 8.

o continue urther with this analysis, evaluate the phaserotation o (s) at the crossover requency:

Fig. 4. The open-loop response of a compensated buck converter

can be approximated to a second-order system in the vicinity

of the crossover frequency, as shown by a plot of phase and

amplitude versus frequency.  

0.5

10.0

7.5

5.0

2.5

00 25 50 75 100

76 degrees

Q

φM

( )( )

( )

12 4

1 tan

tan

+   φ

φ

 

360

2φ×

π 

Fig. 5. The evolution of the closed-loop quality coefficient for

 phase margins from 0 degrees to 100 degrees. At Q = 0.5, the

 phase margin is 76 degrees.

 

   P   h   a   s   e   (   d   e   g   r   e   e   s   )

   M   o   d   u   l   e   (   d

   B   )

Frequency (Hz)

180

90.0

0

–90.0

–180

80.0

40.0

0

–40.0

–80.0

10 100 1 k 10 k 100 k  

0 degrees0 dB

|T(s)||T(s)|

argT(s )argT(s )argT(fC)argT(fC)

fCfC

fCM

ωοω0

ωοω2

-2-2

-1-1

8/14/2019 Transient Response Counts When Choosing Phase Margin

http://slidepdf.com/reader/full/transient-response-counts-when-choosing-phase-margin 4/4 www.powerelectronics.com  Power Electronics Technology November 200821

 TRANSIENT RESPONSE

 arg ( ) tan/

tan tan . CC C Cw

  w w w

w

w

w

p= - +

æ

èçççç

ö

ø

÷÷÷÷= - -- - -1 0 1

2

1

20 2

  (Eq. 16)Te phase margin represents the distance between the

total phase rotation at the crossover requency as given by

Eq. 16 and the –180-degree limit. In this case, the phasereversal brought by the operational amplifier is purposelyneglected. Hence:

j p wM C= + arg ( ),  (Eq. 17)

where ϕM

 is the phase margin.Substituting Eq. 16 into Eq. 17:

  (Eq. 18)j p  w

w

p p   w

wM

C C= - - = -- -tan tan .1

2

1

22 2

Recalling those “ar, ar away” trigonometric classes,this means:  (Eq. 19)tan tan .- -+ =1 1 1

2x

x

p

Tanks to Eq. 19, Eq. 16 can be updated as:

  (Eq. 20)j  w

wM

C

=   -tan .1 2

Te crossover requency versus the closed-loop qualitycoefficient were already defined in Eq. 15. o capitalize onthe definition in Eq. 20:

  (Eq. 21)jMQ

=+ -

æ

è

ççççç

ö

ø

÷÷÷÷÷-tan

( ).1

4

2

1 4 1

Te next step is to extract the closed-loop quality coe-ficient rom Eq. 21 and simpliy the result:

  (Eq. 22)Q M

M

M

M

=  +

=1 24 tan( )

tan( )

cos( )

sin( ).

j

j

j

j

Tis means there is now a relationship between the maindesign criterion, the open-loop phase margin and the qual-ity coefficient the loop will exhibit once closed. Te bestthing to do is to explore the various quality coefficients thatdifferent phase margin choices will bring (Fig. 5). 

I one wants to combine speed and a lack o overshoot,Fig. 2 suggests a quality coefficient o 0.5. Reading the cor-

responding phase margin in Fig. 5, it can be seen that adesign criterion o 76 degrees satisfies this request or sucha quality coefficient, ar away rom the 45 degrees recom-mended in the majority o textbooks.

What does it mean then? In the response to a load step,once the loop is closed, the open-loop phase margin mostlyaffects the recovery shape and a little o the undershootdepth. Tereore, it really depends on the kind o responsea designer is looking or or what the customer specificationsimpose on a design.

I a designer needs a ast recovery and a little overshootto be acceptable, then reducing the phase margin can bean option. On the contrary, i absolutely no overshoots aretolerated, the designer has no choice but to increase thephase margin to the detriment o the recovery speed.

Whatever solution designers selects, they have to makesure that — whatever the operating conditions, input/output, temperature and normal parametric variations(ESRs or instance) — the phase margin never goes below45 degrees. In other words, shooting or a typical valuearound 70 degrees should become a good design practice.

Transient Response and Phase MarginTe buck converter in this example uses one o the auto-

mated simulation platorms described in another paper.[1]

 Te technique allows designers to keep the same crossoverrequency while only working on the phase margin. Teoverall shape is the same as that presented in Fig. 4 witha 10-kHz crossover requency. Te output is subjected tostep ranging rom 1 A to 2 A in 1 µs. Te results appear inFig. 6. Te 76-degree phase margin gives a little overshooto 0.05%, whereas the 45-degree margin triples that over-shoot, still reasonable though given the vertical-axis scaleo 20 mV/division.

However, one can observe a aster recovery in the45-degree phase case (70 µs) versus the 76-degree case

(227 µs). Why do designers still have overshoot with the76 degrees when theory states there should be none? It isbecause Eq. 8 is a simplified view o the transer unctionin the vicinity o the crossover requency. I a designer hasthree or more poles installed near the crossover requency,the quality coefficient actor approximation done heredoes not work anymore and extra work will be required.[2] Nevertheless, as exemplified by Fig. 6, a small phase marginleads to a peaky closed-loop response. PETech

References1. Basso, C. Switch Mode Power Supplies: SPICE Simulationsand Practical Designs, McGraw-Hill, 2008.2. Erickson, R. and Maksimovic, D. Fundamentals of PowerElectronics, Kluwers Academic Press, 0-7923-7270-0.

 

5.08

5.04

5.00

4.96

4.92

0.800 1.10 1.40 1.70 2.00

t150µS/divt150 µs/div

 Vout(t)

20mV/div

 

 V OUT(t)

20 mV/div

 

Time (ms)

M=36°M=36 degrees

M=45°M=45 degrees

M=64°M=64 degrees

M=76°M=76 degrees

   V   o   l   t   a   g   e   (

   V   )

Fig. 6. Phase margins from 36 degrees to 76 degrees cause the

amplitude versus frequency plot to show a variation of the

transient response and recovery time around the 5-V target.