Formula for Right Half Plane Zero in a Boost Converter 4

[DHambley]

Lloyd Dixon gave us this equation long ago for the corner frequency of the RHP zero for a Boost converter:

F = (1/2pi)* R/L * (1-D)^2/D
= (1/2pi)* R/L * (Vi/Vo)^2/D

Other app notes from TI give this equation:

F = (1/2pi)* R/L * (1-D)^2
= (1/2pi)* R/L * (Vi/Vo)^2
There's no D (duty cyle) in this denominator!

National Semi gives this formula:

F = (1/2pi)* 1/L * (Vin*D)/Iin

Well, these three just don't match! Any thoughts on who's right? I've always used the first formula above.

DH

 

[Fluorescence]

here is someting else on RHPZ...

http://pearlx.snu.ac.kr/Publication/00066413.pdf

 

[DHambley]

Nobody is debating the DC tansfer function of a boost vs flyback. (well actually some people are)

The issue is the frequency of the RHPZ, not the simple DC transfer function of a boost converter.

d(Vout)/D(duty) as a funcion of frequency will show you the location of the RHPZ.

The input voltage is a DC offset. This dissapears in the first derivative.
Thus,
For the Boost, Let Vout* = (Vout-Vin)
For the Flyback Let Vout* = (Vout)
Now calculate the frequency responce of Vout* vs pulse width. You will find that the transfer functions are identical. (That is after factoring the turns ratio to 1:1 and ignoring the diode drop).

 

[cabraham]

This question is easily answered by measuring frequency response of a boost converter. Using a low duty factor like 0.20 easily shows the diference between the boost and buck-boost. If the D term is omitted from the denominator, and D = 0.20, then the RHPZ freq is 5 times higher than if the D is included in the denominator. You can measure and verify.

TI appnotes SLVA057, SLVA059, & SLVA061 completely characterize the buck, boost, and buck-boost. The RHPZ freq values differ. Every university website has a D in the denominator for the buck-boost, but not the boost.

Are you saying that the whole world, academic, and industry is wrong, and you are solely right? Later this week I'll hand write a derivation for both converters affirming the values of RHPZ. BR.

Claude

 

[cabraham]

Greetings all,

Here is the boost converter rhpz derivation. The next post will be for the buck-boost rhpz.

Claude

 

[cabraham]

As promised, here is the buck-boost rhpz derivation. BR.

Claude

 

[cabraham]

Sorry, but I omitted the "L" in the denominator of the boost computation. At the bottom the left hand equation is missing the "L" in the denominator, but the L was re-inserted into the final answer. The final answer is correct, but one step along the way had an error. I updated the mistake and attach it. Thanks.

Claude

 

[Fluorescence]

Hello,

Also of interest here is the effect that putting slope compensation in has. -Especially since slope compensation is so often needed in Continuous mode boosts and flybacks.

In other words, i wonder what would be the RHPZ frequency when slope compensation is used ?

Also, i wonder if there if there is a point where the slope compensation is so much, that the RHPZ problem disappears?

 

[cabraham]

The rhpz freq is not changed by slope comp. The slope comp, SC, is mixed with the sensed current signal, Isns, and the PWM is terminated when the sum of SC + Isns equals the error signal.

The rhpz freq is determined by the time required for the inductor current to ratchet to its new level. SC adds a little "voltage mode" behavior to the system, and reduces the speed of the inner (current) control loop. The SC neither speeds up nor slows down the inductor's ramp-up or ramp-down time when slewing current in the presence of a step load change. Also, if we set the Isns to zero by shorting the sensing resistor, and increase the slope comp so that the PWM is entirely controlled by the SC ramp, the system operates in pure VMC (voltage mode control). The rhpz for VMC is at the same freq as with CMC. It doesn't change. It is an artifact of R, L, & D.

The actual load current, which is determined by R, the inductance L, and the duty factor, D, all determine the rhpz freq. During the ramp time due to a load step change, SC & Isns are not active since the servo loop is pinned against its rail and the error amp is saturated.

The SC+Isns does, however, influence the low freq dominant pole in the small signal transfer function. The slope comp increases said freq, but not drastically.

Does this help? BR (best regards).

Claude

 

[Fluorescence]

Thankyou very much, soryy if this was hijacking the post away from OP.

 

[cwptt]

Can anyone explain in mathematical terms how the RHPZ is always still lurking out there although the converter is in DCM, CCM. All authors I read just state that it is still there without showing any proof. Perhaps I should understand it by inference to what was stated in their articles. And what about the "D" in the denominator that "hgldr" eluded to in thread240-229945. Which equation is (most) correct? Also, any information (procedure)on the spread, placement of poles and zeroes for best compensation of the loop will be helpful. Please give reference(s)to any applicable application notes.

Tnks a 10E6,
Charles

 

[cabraham]

In dcm the rhpz for all practical purposes does not exist. It is located above the bandwidth of the error amp and has negligible influence. In ccm it falls in the range of the error amp bandwidth and it is generally what limits said bw.

Scroll up a few posts and download my hand written derivations for boost and buck-boost. It details the freq of said rhpz in ccm. The boost has no "D" in the denominator, whereas the buck-boost has a "D" in the denominator. Otherwise they look identical. I submitted the boost, then, the buck-boost next post, then the corrected boost 3rd post. Download the 2nd and 3rd files as they have no major typos.

As far as app notes go, I recommend Tex Instr notes SLVA057, SLVA059, & SLVA061. They cover the buck, boost, & buck-boost. Visit the TI site power management knowledge base and download app notes for freq compensation. One app note has solved computations for buck, flyback, etc. operating in dcm, ccm w/ voltage mode ctl, current mode, etc. The file, I believe has the title "discontinuous flyback - direct duty cycle control", or something like that. It is very helpful. I hope this helps. BR.

Claude

 

[Fluorescence]

Hello,

Sorry if i've missed something here, but Equation 13 (page 21) of the following "Control Loop Cookbook" by Lloyd Dixon

http://focus.ti.com/lit/ml/slup113a/slup113a.pdf

gives the Boost's and Flyback's RHPZ as having a D in the denominator.

Sorry to drag this up again, but can we now confirm that this is wrong? -its just that Lloyd Dixon's "cookbook" is taken as the scriptures for many of the senior engineers at places where i've worked.

 

[cabraham]

Lloyd's cookbook is indeed "scriptures" in my mind as well. Few individuals, if any, have contributed more to understanding the SMPS than Lloyd.

He starts out the paragraph (pg 21 per above post) stating that both boost and buck-boost converters have a RHPZ when the inductor current is in continuous mode, or CCM. He then states the classic equation giving the value of a RHPZ when using the buck-boost topology, which has the "D" in the denominator. He didn't say that the two topologies have the same RHPZ value. In other TI/Untirode publications, the boost RHPZ specifically excludes the "D" from the denominator. I've attached an app note from Aimtron showing the boost RHPZ on pg. 13, and the denominator has no "D". Also, just refer to TI/Unitrode app note SLVA061. At the top of pg. 30, the boost RHPZ is given with no denominator "D" factor.

My handwritten derivation gives the same result as TI/Unitrode and university web sites. Anyone still not sure should build a boost converter and set the input & output to obtain a 20% duty factor or less. With a D of 0.20, there will be a factor of 5 difference between having D in the denominator vs. not having it. A frequency response swept measurement of gain and phase will confirm the location of the RHPZ. A factor of 5 makes it too obvious.

It's ok to question anything. Typos and even conceptual errors are to be found even in the most credible sources. But this issue has been thoroughly examined by many and independently verified. Going through my math should help somewhat. If I've erred I not only accept correction, but I welcome and appreciate it. Best regards.

Claude

 

[treez]

interesting point (specially with today's PFC laws) would be the RHPZ frequency in a boost converter PFC.

Presumably, the voltage loop bandwidth is set so low in a PFC that the RHPZ is never a problem ?

 

[cabraham]

In a boost PFC there is no RHPZ. For a PFC the controlled quantity is not the output voltage, but rather the input current. As soon as the duty factor command is increased, the input current immediately increases. Thee is no "wrong way" phenomena in the PFC case.

Of course, the transfer function for input to output voltages does possess an RHPZ. But the input voltage to input current transfer function has no RHPZ at all.

Claude

 

[marginal]

Hello,

I am writing Regarding the Boost’s RHPZ.

I was wondering if the following Continuous mode boost LED driver would have an RHPZ problem……

****** Schematic of boost LED driver:-

**** UC3842 DATASHEET….

http://www.fairchildsemi.com/ds/UC/UC3842.pdf

This converter only has a current loop to control it.

The voltage loop is not normally operative……it only comes into play when a LED fails and the voltage must be stopped from running away.

Since the voltage loop is not operative, it would be pointless to put compensation components around the voltage error amplifier that exists inside the UC3842.

-However , this then means that I am defenceless against the RHPZ……since I will not be able to reduce the voltage loop bandwidth to get below the RHPZ frequency.

Do you believe this converter is therefore a bad idea ?

Thankyou for your time.

----------------------------------------------------
(Incidentally, the following LED driver catalog proposes many buck and boost circuits to power LEDs but there is no mention of RHPZ.

***LED driver catalog:-

http://www.linear.com/pc/downloadDocument.do?id=24935)

 

[cabraham]

Off the cuff here is my abbreviated answer. I don't have enough time to do an exhaustive analysis. With a boost cvtr in CCM driving LED lamps, the output is a constant current driver. The cap across the LEDs is sometimes not used. But during the on time, the inductor is disconnected from the LEDs and the cap provides the current. It will droop, then get recharged during the off time by the inductor.

Maybe the RHPZ is not a concern because of the following. With constant voltage output, a step change increase in the demanded load current requires a momentary increase in duty cycle D. The inductor energizes during this extended on time, and the cap is carrying the load for an extended time, resulting in droop. Since the off time is now decreased, there is less time for the cap to get recharged, hence the output is reduced further.

Eventually equilibrium is reached and the output turns around and increases. The time constant required for this to happen is related to the RHPZ. But with current drive output, the voltage is not regulated. There is no sudden change in output load current, so the duty factor does not get modulated, and the output does not droop. Off the cuff, it appears that the RHPZ is an artifact of the voltage control loop dynamics when the output current suddenly changes. But this application forces a steady output current. The only way I can foresee an RHPZ coming into play is if the output voltage demand was suddenly increased/decreased. Suppose there are 5 LEDs at the output, with a regulated 10 mA current. Suddenly a 6th LED in series is added (shunt switch opens). To maintain the same regulated current into 6 LEDs instead of 5 requires higher output voltage and higher duty factor. This is when the RHPZ shows up.

Do I make sense? BR.

Claude

 

[marginal]

Thankyou Cabraham, your answer looks great, i have copied it and will read it when i get home.
Thanks again.

 

[marginal]

Cabraham I am very grateful for your reply which makes a lot of sense to me.

Its strange that some Boost converter LED driver chips have datasheets that recommend using frequency compensation, some recommend adding slope compensation, -but some make no such recommendations….

Please see (if you wish) the ZXCS400 datasheet (Boost LED driver)

*****ZXCS400 DATASHEET

http://www.zetex.com/3.0/pdf/zxsc400.pdf

This chip (ZXCS400) does not have an oscillator and appears to use hysteretic mode.
Many LED driver chip datasheets say that hysteretic mode controllers do not need any frequency compensation or slope compensation .

If hysteretic mode controllers are devoid of the RHPZ problem, then I wonder why they are not far more common ?

The LM5022 Boost LED driver has facility for compensation components to get round the RHPZ problem (bottom of page 6, datasheet)

****LM5022 DATASHEET

http://cache.national.com/ds/LM/LM5022.pdf

The TPS61160 is a Boost LED driver that somehow manages to avoid the RHPZ by simply adding a 220nF compensation capacitor in all cases. (please see middle of page 15, TPS61160 datasheet)

*****TPS61160 DATASHEET

http://focus.ti.com/lit/ds/symlink/tps61161.pdf

Coming back to voltage mode boost converters, the TL497 is a constant on-time SMPS controller and none of the boost application circuits in the datasheet have any frequency compensation components. In fact, the TL497 has no error amplifier, but instead just has a comparator, so I am, wondering how TL497 boost converters get around the RHPZ?

***TL497 DATASHEET

http://focus.ti.com/lit/ds/symlink/tl497a.pdf

Also, Quoting from page 10 (under “Theory of operation”) of the LM3404 Buck driver datasheet..........

“Hysteretic operation eliminates the need for small signal control loop compensation.”

*******LM3404 DATASHEET

http://www.national.com/ds/LM/LM3404.pdf

Is it true that hysteretic mode boost converters have no Right Half Plane Zero problem?
Also, is it possible to have a fixed frequency, hysteretic converter?

Thankyou for reading.

 

[cabraham]

Hysteretic mode does not use an error amp. It is a "bang-bang" type of controller. It does not have an oscillator, and it runs at variable frequency. The lack of error amp & servo loop means that no freq comp is needed. It is simple, and stability is never a concern because it is always unstable. Its natural tendency is oscillatory mode, bouncing between the upper and lower threshold values of the comparator.

I've only used hysteretic for buck topology. With a buck there is no RHPZ. A step input to increase the duty factor immediately results in an output that increases. Hence the hysteretic mode of control senses the output and turns off when the right value is attained.

I've heard of boost converters using HVMC (hysteretic voltage mode control), but I've never done it. With the boost, HVMC gets more involved. If the output drops below the threshold, the power switch must be turned on. As the inductor builds up flux/energy, the output cap is disconnected from the inductor and is carrying the load entirely on its own. If the output feedback info is such that the voltage keeps dropping, then the power switch will never shut off.

Some provision must be made for limiting the on time. Once the switch is shut off, the inductor will de-energize into the output cap & load, and the voltage will rise. Likewise for a buck-boost. A pure hysteretic approach cannot work with these topologies because the inductor gets energized while the cap is disconnected from the inductor and carrying the load alone. With a buck, as the inductor energizes during the on time, the output voltage is also rising. When Vout hits the upper threshold the switch can be shut off.

HVMC is very good at buck applications. To use HVMC with boost & buck-boost topologies, mods must be made.

As far as why HVMC isn't always used for buck converters, my preference is as follows.

HVMC controls output voltage by controlling cap ripple. HVMC is inherently noisy. With tantalum or aluminum electrolytic types, the esr determines the ripple voltage as well as the frequency. But esr is not a well controlled parameter, varying greatly with speciman, and temp (alum). One can use film or ceramics and add a low value resistor in series. This is my method. At minimum frequency the noise is hard to filter. At maximum frequency switching losses increase.

But, the output is then fed into circuitry having bypass caps, usually ceramic, which load the R-C output of the HVMC converter. This changes the frequency. The best way to limit this phenomena is to employ an L-C post filter. This requires an inductor, cap, and damper R-C network. The post filter also provides reduction in noise, a welcome benefit seeing how noisy HVMC is already.

HVMC is good for an amp or so, 2A tops. The noise becomes an issue. Fixed frequency is quieter but requires freq comp. HVMC is very fast, responding to transient load demands very well. Fixed freq readily lends itself to boost & buck-boost applications, whereas hysteretic is really a buck specific topology. To use hysteretic for boost/buck-boost applications requires limiting the on time and sensing voltage during the off time. This is necessary or it won't work. The transient speed advantage is comprimised and hysteretic becomes less attractive for these converters. It can and has been done, but for me, if I need a buck-boost or boost, fixed freq current mode control gives very good speed, low noise, and is easy to filter. Just my thoughts.

What control method, peak vs. avg current mode, input feedforward vs. direct duty cycle voltage mode, hysteretic voltage, hysteretic current, etc. is a never ending debate. All have their merits as well as limitations. Anyone experienced with SMPS could and has challenged the points I made above. After over a quarter century of SMPS experience, I am confident that what I just stated is valid. I welcome feedback.

Claude