Formula for Right Half Plane Zero in a Boost Converter
Formula for Right Half Plane Zero in a Boost Converter
(OP)
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
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





RE: Formula for Right Half Plane Zero in a Boost Converter
Switching Power Supplies By Keith Billings (Earlier Edition) had a good description and equation that worked for me but my copy grew legs and walked off. The book also has good sections on magnetics especially powered iron cores and litz wire. I'd buy another copy but I don't have a need these days.
http://www
RE: Formula for Right Half Plane Zero in a Boost Converter
I was thinking of some practical method of geting the RHPZ corner frequency.
I was thinking that it must be measurable from an actual boost converter (simulation or real).
Can we say If it wasn't measurable then it wouldn't matter?
-But i wonder how to isolate this measurement from others in the converter.
I reckon to make a boost SMPS and use no compensation...ie just the feedback divider.
Switch from no load to full load and see how long it takes to stabilise out at the full load. -either that or put a watch on the boost diode current and time how long before the current goes from no load current to full load current.
.....in that timing i thought would be an indication of the corner freq of the RHPZ.
then you can check this with the formulae.
RE: Formula for Right Half Plane Zero in a Boost Converter
DH
RE: Formula for Right Half Plane Zero in a Boost Converter
1) Does the RHP Zero exist for both continuous and discontinuous switching?
2)The RHP Zero shifts frequency [f] with changes in the
Vin Level and the question is; f decreases as Vin decreases? And this results in conditional stability conditions?
RE: Formula for Right Half Plane Zero in a Boost Converter
Thus, while you can predict the location of the RHP zero, unless the converter operates with a fixed or only slightly varying load, you don't ever want to encounter it in the field - the converter will get stuck in the above described metastable state.
RE: Formula for Right Half Plane Zero in a Boost Converter
Yes, it's true that designers new to SMPS's need to educate themselves of these issues with CCM, some of which you've pointed out. There are many applications for which CCM is used in flybacks and boost converters. I don't believe anyone is asking for trouble by doing so, unless they don't know what they're doing.
DH
RE: Formula for Right Half Plane Zero in a Boost Converter
That said, as you have a phase/gain analyzer at your disposal *you* probably know what you are doing... Most people don't bother buying one of those things just for the heck of it :)
RE: Formula for Right Half Plane Zero in a Boost Converter
The flyback which I'm working on works fine (28V input with typical O/V, U/V, and other input spikes) but my customer now changed their parameters and they need it to work if the input droops to 6V. That's where the CCM is occuring and thus, the lively RHPzero came to life. I had to dust off an old textbook to analyze it.
RE: Formula for Right Half Plane Zero in a Boost Converter
Hey, I found yet another equation to determine the RHPZ frequency:
F = (Rl * Vin^2)/(Lsec * Vout * (Vin + Vout))
Where Rl is the load resistance and Lsec is the secondary inductance of the flyback transformer (if L is in uH then F is in MHz)).
This is from Marty Brown's "Practical Switching Power Supply Design" which, to be frank, over-simplifies a lot of the math but actually is a decent reference for some of the practical aspects. Getting a bit long in the tooth, though, just like the old standards for SMPS design by Billings and the late Pressman.
RE: Formula for Right Half Plane Zero in a Boost Converter
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RE: Formula for Right Half Plane Zero in a Boost Converter
Renovator1 i also use the Marty Brown book, in fact its my favourite reference....-with his flyback examples he always uses opto feedback with TL431.
I dont know how he gets his compensation done though because the following article:
http://powerelectronics.com/mag/50107.pdf
...seems to suggest that control loop compensation with TL431/opto is different to the equations he proposes.
RE: Formula for Right Half Plane Zero in a Boost Converter
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RE: Formula for Right Half Plane Zero in a Boost Converter
I used several times this formula with good results.
It also results from a mathematical small signal process description.
RE: Formula for Right Half Plane Zero in a Boost Converter
so i suspect you can do this by putting a small cap on the lower feedback divider resistor and gradually upping it until the No_load to Full_load switching works OK.
eventually the duty cycle will be prevented from changing that quickly that the "channel" through the boost diode is not "strangled" off too quickly.
RE: Formula for Right Half Plane Zero in a Boost Converter
I always use CCM when the load current is large enough. If one understands the intricate details of SMPS operation, CCM is no problem at all. In DCM, the stresses on the parts due to high peak currents and voltages are substantial. CCM offers lower noise and higher efficiency.
The downside to CCM is the RHPZ which forces the servo loop to roll off to unity gain at a lower freq than if DCM were used. In DCM there is no RHPZ. Thus CCM offers lower bandwidth than DCM.
Overall, I use DCM when the output current is small. Noise and stress are not an issue in that case, and DCM offers higher bandwidth. For large output currents, I always use CCM. Otherwise noise and peak stresses are excessive as well as losses.
Remember that the boost converter has no short circuit protection. An additional FET will be needed to shut off power in the event of an output short.
Claude
RE: Formula for Right Half Plane Zero in a Boost Converter
The boost and flyback can both be modeled to be the same topology and thus, the same response. You can convert the flyback model to a boost by scaling the transformer to a 1:1 so that the inductance (as seen at the primary) is equal. Since the topologies can be scaled to be the same model, one response equation can't have the 1/D missing while the other equation keeps it.
Dixons equation has 1/D for a boost. The 1/D which you pointed out as being for a flyback, would also be true for a boost.
DH
RE: Formula for Right Half Plane Zero in a Boost Converter
Check any reference and you will find that what I posted is correct. If we take the xfmr-isolated flyback and scale the xfmr to 1:1, what we get is an *inverting buck-boost* topology, NOT a straight boost converter.
They are different. A boost is a member of the *general* flyback family, but the transfer function is quite different. The boost during the switch on time relies on the output cap to power the load. The inductor is energized via the input and to ground via the switch. When the switch is turned off, the inductor de-energizes into the output cap and load. But, the input source is conducting the inductor current. A portion of the per cycle energy is provided by the input power source, during the switch off time.
A buck-boost, however, does not have this direct input to output energy transfer. During the on time, the switch closes and the input source energizes the inductor while the output is disconnected from the input and inductor just like the boost. The cap holds the output. The big difference is that during the switch off time, the input is disconnected from the output. The inductor de-energizes through the diode and output load and cap.
With a buck-boost, all of the per cycle energy is stored in the inductor during the on time, and released to the output during the off time. The input never transfers energy directly as in the case with the boost converter.
The boost and buck-boost have differing transfer functions, because the "transfer" is different. The unitrode/TI archive of app notes, Natl Semi, Linear Tech, and every university says so. The transfer functions for various SMPS topologies has been studied, modeled, and tested to death.
Believe me, I wouldn't lie to you. Best regards.
Claude
RE: Formula for Right Half Plane Zero in a Boost Converter
However, the issue is the Frequency of the zero, not the 1/D in the DC responce euqation. This has nothing to do with the 1/D in Dixons equation.
RE: Formula for Right Half Plane Zero in a Boost Converter
Claude
RE: Formula for Right Half Plane Zero in a Boost Converter
http://pearlx.snu.ac.kr/Publication/00066413.pdf
RE: Formula for Right Half Plane Zero in a Boost Converter
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).
RE: Formula for Right Half Plane Zero in a Boost Converter
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
RE: Formula for Right Half Plane Zero in a Boost Converter
Here is the boost converter rhpz derivation. The next post will be for the buck-boost rhpz.
Claude
RE: Formula for Right Half Plane Zero in a Boost Converter
Claude
RE: Formula for Right Half Plane Zero in a Boost Converter
Claude
RE: Formula for Right Half Plane Zero in a Boost Converter
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?
RE: Formula for Right Half Plane Zero in a Boost Converter
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
RE: Formula for Right Half Plane Zero in a Boost Converter
RE: Formula for Right Half Plane Zero in a Boost Converter
Tnks a 10E6,
Charles
RE: Formula for Right Half Plane Zero in a Boost Converter
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
RE: Formula for Right Half Plane Zero in a Boost Converter
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.
RE: Formula for Right Half Plane Zero in a Boost Converter
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
RE: Formula for Right Half Plane Zero in a Boost Converter
Presumably, the voltage loop bandwidth is set so low in a PFC that the RHPZ is never a problem ?
RE: Formula for Right Half Plane Zero in a Boost Converter
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
RE: Formula for Right Half Plane Zero in a Boost Converter
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:-
[IMG]http://i41.tinypic.com/b7km82.jpg[/IMG]
**** 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:-
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RE: Formula for Right Half Plane Zero in a Boost Converter
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
RE: Formula for Right Half Plane Zero in a Boost Converter
Thanks again.
RE: Formula for Right Half Plane Zero in a Boost Converter
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.
RE: Formula for Right Half Plane Zero in a Boost Converter
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
RE: Formula for Right Half Plane Zero in a Boost Converter
By the way, the circuit has been in production for 2 months now. It works fine from 5V low-line to the 60V transient. I had found and verified the correct equations shortly after the first post. The closed loop tests show the RHPZ where the calculations say it should be.
RE: Formula for Right Half Plane Zero in a Boost Converter
This stuff is good to go over and review. None of us really know as much as we think we do. Twice a week in grad school class I am reminded of my limitations. Peer review and re-examining our own positions can only help us. If I'm wrong, I benefit by being corrected. Even when I'm right, reviewing my steps and thought process is a good sanity check. BR.
Claude