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C
Rc
RL
PWMCCM
c
pa
c'Control
Compensator
Rp
+
_
Vin
Rs
ISOLATION
0
Figure 1. PWM switch model in flyback circuit.Simple linear circuits replace the switching
action of the real circuit.
These models have their strengths and uses, andhelp designers in their work. And, in almost all ofthe papers that derive models, measurements areused to confirm the validity of the work [1-4].This leads naturally to a conclusion for the
reader modeling is accurate and sufficient.
What the modeling papers dont tell you is theamount of effort that goes into the test circuits tomake sure that measurements and predictionsagree closely. For example, the converter isalways operated in the center of its range, andnoise is completely eliminated from the system.This often requires converter grounding setupsand instrumentation that may be completelyunrealistic in practical production power
supplies. When you measure a real productionsupply, it is often extremely difficult to get it toconform closely to the published theory.
III. MODEL BREAKDOWN
There are two main reasons why modelingfrequently fails for a circuit:
1. The model is not detailed enough to accountfor the circuit operation.
2. The circuit elements of the model are notaccurately known or modeled.
Examples of circuit operation that can cause themodel to fail are:
1. Discontinuous conduction mode operation(and associated ringing waveforms afterdiode shuts off).
2. Snubber operation, especially losslesssnubbers.3. Light load operation, especially for current-
mode control.
4. High-ripple circuits such as flybackconverters.
5. Control system noise and ripple.6. Propagation delays.7. Semiconductor nonlinearities diode offsets,
junction drops, etc.
8. Multiple output converters.9. High and low temperature operation.If you look through this list, and your circuit hasnone of these issues, the models can be veryaccurate, assuming you meet the second criteria
of using the correct component parasitics.Unfortunately, almost every real converter builtfor production has at least one of thesecharacteristics. Which gets back to the main pointof this paper measure your power supplycontrol loop stability!
IV. FREQUENCY RESPONSE
ANALYZER FUNCTIONS
So what is required to measure the control loop
of a power system? Basically you need anoscillator to inject into the circuit, and two testchannels to measure the response. There areseveral functions needed to do the job properly:
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A. A sinusoidal oscillator output signal ofadjustable size that is swept over severaldecades automatically.
B. An accurate measurement algorithm thatcompares the signals for gain and phaseinformation accurate to 0.1dB for gain, and
2 degrees for phase.
C.A narrowband noise measurement systemthat will reject all the switching, linefrequency, and other noise in the test signals.
D.Dynamic range of at least 80dB toaccommodate the range of signal sizes andgains that will be encountered.
A. Swept-Sine Output
A switching converter needs to be measured fromlow frequencies of interest up to high enoughfrequencies to show the major system dynamics.
This is typically below 50Hz at the low end, (lessthan this for motor control circuits and PFCcircuits) up to the switching frequency. (Half theswitching frequency is theoretically more thansufficient, but there are often practical benefits inseeing the higher frequency data.)
A typical switching power supply operating at100kHz will be measured from 10Hz to 100kHz,and will have a crossover frequency of 2kHz -10kHz. Across this range, you want a piece ofequipment that can output a test sine wave, andautomatically step it in logarithmic increments.For each test frequency, the instrument needs tomeasure two signals at the test frequency, andcompare their magnitudes and phase.
B. Noise rejectionFig. 2 shows a typical power supply loop gainmeasured with a frequency response analyzer.
-60
-40
-20
0
20
40
60
80
10 100 1 k 10 k 100 k
Frequency (Hz)
Gain (dB)
Figure 2. Typical loop gain showing high gain atlow frequency, and 5kHz crossover frequency.
At the low frequency end of a power supply loopmeasurement, the gain of the system will be veryhigh in excess of 60dB, and sometimes as highas 80dB. Consider the size of the signals neededto make this measurement. You must ensure thatyou are measuring small-signal with a small acperturbation on top of the dc operating point. Forthis, the output signal will typically be 100mV orless.
The input signal to the loop, for a 100mV output,will be 0.1mV for a gain of 60dB, and 10V for again of 80dB. It is common, however, to have upto a few hundred mV of noise in the signals. It isimpossible to measure the test signal in so muchnoise without a specialized instrument. That isthe second main function of the frequency
response analyzer to extract the test frequencyonly, with a very narrow bandwidth, so systemnoise does not interfere with the measurements.
100mV/div
Figure 3. Typical test signal to be measured, withswitching spikes and other noise. Less than
0.1mV test signal must be accurately measured inthe presence of over 200mV noise.
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A Frequency Response Analyzer, also referredto as a Network Analyzer, is specificallydesigned to provide these functions. Untilrecently, it was necessary to spend $30k or moreto get an instrument suitable for the task ofmeasuring switching power supplies and theircomponents. Today, this type of instrument can
be purchased for under $10k, making it bothaffordable and indispensable for all powerengineering companies [5].
V. LOOP GAIN MEASUREMENTS
Power supplies are extremely high dc-gainsystems. They use integrators to maximize the dcgain, and ensure that the output voltage dcregulation is tight. The power supply and controlcircuits cannot, therefore, be run with the loop
open. Its simply not possible to hold the dcoperating point steady with an open loop system.
Fortunately, there are established anddocumented techniques for measuring the openloop gain of a system while the loop is keptphysically closed. The only invasion into thecircuit is through the insertion of a test resistor.The technique is very accurate as it does indeedmeasure the true open loop gain of the system,not a gain modified by the injection technique
itself.
Channel BChannel A Out
NETWORK ANALYZER SYSTEM
20 ohm
Load
Power Stage
+
_
E/A_
+ V ref
PWM
1 2
3
4
Figure 4. Feedback loop system showingfrequency response analyzer connection and
injection resistor.
This is shown in Fig. 4. The only complicationwith this technique is that the signal injected intothe circuit must be injected differentially acrossthe resistor, not with respect to ground. This istypically done with a signal injectiontransformer. The output signal from the networkanalyzer is coupled through an isolation
transformer.
Channel BChannel A Out
20 ohm
Low
Frequency
High Gain
Power System
Injected
Signal
LoopOutputLoop
Input
20 ohm
Injected
Signal
Loop
OutputLoop
Input
20 ohm
Injected
Signal
Loop
OutputLoop
Input
Crossover
Frequency
Unity Gain
High
Frequency
Low Gain
Frequency
Response
Analyzer
Figure 5. Signal distributions at different loop
gains high gain, crossover, and low gain. Theinjected signal size stays constant across thewhole frequency range, and gets distributed
between the input and output of the loop,depending on the loop gain.
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Its a little unusual how this works the signalsize across the resistor is constant. The vectorsum of the injected signal and return signal areexactly equal to this injected signal. The powersupply feedback system will adjust the signalsizes according to the gain needed. For example,if the gain of the system is 60dB, and the injected
signal is 100mV, almost the entire injected signalappears across the output and only 0.1mV acrossthe input.
At the crossover frequency, the injected signal isdistributed equally between input and outputsignals. And, when the gain of the system is verylow, beyond the crossover frequency, most of thesignal is applied to the loop input, and only asmall fraction to the loop output.
An animated example of this process can be seenat the Ridley Engineering web site [5], showinggraphically how the injected signal is dividedbetween loop input and output.
In order not to disturb the operating point of thesystem, the injection resistor is kept smallrelative to other components in the circuit.
Typically, a 20or 50resistor is used in series
with several kalready in the circuit.
During measurement, it is usually advisable tomonitor critical waveforms of the control systemto make sure that they remain in the small-signalregion. An oscilloscope probe on the output ofthe error amplifier, and output of the powersupply is usually sufficient. As the gain of theloop changes, it is customary to adjust the size ofthe signal injection to keep the signals largeenough to be measured, but small enough keepthe system linear.
Other concerns and techniques for measuring
loop gains properly are given in [5].
VI. CASE STUDY NO. 1
A printer power supply company was ready torelease a design to manufacturing. A substantialamount of modeling and prediction ensuredsystem stability. The power supply wasconsidered ready for production with quantities
exceeding 10,000 units per month.
The company had a frequency response analyzerin house, ready for evaluation. But, like most ofus, the engineer was too wrapped up in theproduction details to take loop gainmeasurements.
Finally he was persuaded to do a quick test withthe analyzer on this latest product. Reluctantly,he did, taking a couple of hours out of his
schedule.
-180
-160
-140
-120
-100
-80
Phase (deg)
Measured Loop Phase
Predicted Loop Phase
-40
-20
0
20
40
60
100 1 k 10 k 100 kFrequency (Hz)
Gain (dB)
Predicted Loop Gain
Measured Loop Gain
100 1 k 10 k 100 kFrequency (Hz)
Figure 6. Predicted and measured power supplyloop gains using manufacturers data for
capacitor esr. Over 50 degrees phase error!
To his amazement, the power supply was almostunstable, with only 35 degrees of phase margin atthe selected nominal operating point 50 degreeslower than expected. The production release wasdelayed for a day to figure out the problem, andfix it quickly.
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The model the engineer was using was fine, andthat all the values corresponded to themanufacturers data for the passive components.And thats where the problem lay the wrongvalue of ESR for the output capacitor was beingused. It was correct according to the data sheets,
which called out a maximum value of 7.5for a
tantalum capacitor. The real value of the ESRwas only 0.25 a factor of 30 lower than thepublished maximum! This corresponded to achange in gain of 30dB at higher frequency. And,since we often cross a control loop over in theregion where the output capacitors are resistive,the whole loop gain was depending on this value.
-40
-20
0
20
40
60
100 1 k 10 k 100 kFrequency (Hz)
Gain (dB)
Predicted LoopGain
Measured LoopGain
-180
-160
-140
-120
-100
-80
Phase (deg)
Measured LoopPhase
Predicted LoopPhase
100 1 k 10 k 100 kFrequency (Hz)
Figure. 7. Same measured loop gain as Fig. 6,but predicted loop gain using measured values of
capacitor esr (30 times lower!).
A refinement to the model values confirmed themeasured loop gain, and the need to add a coupleof additional compensation components to theboard.
Cost to the program was only a few hoursengineering time, and the changes wereincorporated in the final production board run thenext day. The savings in potential product recalland re-engineering were substantial.
VII. CASE STUDY NO. 2
Researchers in the academic community whospend all their time on theory and modeling oftenadvise engineers that they should be veryconcerned about stability. The fact is that mostdesigners have so many other issues on their
plate during development, this is just one moreitem on the list, and its frequently overlooked
Its often inconvenient to stop and think aboutcontrol systems. When deeply involved in thedevelopment of power supplies including all thedetails of design, layout, parts selection, testing,etc. its easy to neglect control. At the end of theproject, when final testing is almost complete, weall find it tempting to assume the fact theconverter works assures us that everything is OKin the loop without needing to measure itdirectly. Even the most experienced of us makewrong assumptions, and forget the importance ofthis step.
This example is of an off-line flyback converterwith dual outputs. The simplified schematic isshown in Fig. 8. One of the outputs is used topower an electronic load, and has a large amountof storage capacitance. This output is isolatedfrom the mains input. The second output isreferenced to the primary side, and used to power
the control IC. This second output is also used forfeedback regulation, since it does not needisolation in the feedback path.
Like the previous example, the power supply wasbuilt, tested, and ready for prototypemanufacturing. And, like many engineers, it wasassumed that a simple DCM flyback circuit likethis would behave as predicted by small-signalcircuit models.
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The small-signal model initially used for thedesign is shown in Fig. 9. This makes theassumption that the outputs of the converter trackeach other well, and all the output capacitancescan be reflected to a single output. The convertermodel then reduces to just a single-outputflyback (i.e. the model format that appears in
almost all theoretical papers there is littlepublished on how to handle multiple outputsproperly.)
+
10uF 383
4.32K
10K
0.1 4.7nF
0.1
2.7 OHM
+
47uF
3N60E
1nF
2.00K+
10nF
COMP1
VFB
IS
RT/CTGND
OUT
VCC
VREF
U501UC2843
13 T
50 T
MUR120
20 1
2
3
MUR120
+
4700uF
+
4700uF 2.5 W
Main Output14T
Auxiliary
Regulated Output
Figure 8. Simplified schematic of flyback powersupply with auxiliary regulation. Main output has
a large storage capacitance. Auxiliary output isused for feedback regulation, and to power the
controller. The 20resistor in between points(1) and (2) is used for experimental signal
injection.
C
Rc
RL
PWMCCM
c
pa
c'Control
Compensator
Rp
+
_
Vin
Rs
+_Err
0
(-13/14)
ISOLATION
0
Figure 9. Small-signal circuit model. (Simplestassumption the two outputs track each other
according to the turns ratios).
-160
-120
-80
-40
0
1 10 100 1 k 10 k 100 kFrequency (Hz)
Measured PowerStage
Predicted Power Stage
Phase (deg)
-60
-40
-20
0
10
1 10 100 1 k 10 k 100 kFrequency (Hz)
Measured Power Stage
Predicted Power Stage
Gain (dB)
Figure 10. Measured and predicted loop gains.
Figure 10. shows an extreme when things goawry. The gain is dramatically off in the criticalcrossover frequency region, and the two phase
curves have almost no resemblance to each
other!
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This is a more subtle problem than simply usingthe wrong component values. First, a morecomplex model is needed to predict the multipleoutput converter response. Fig. 11 shows theequivalent circuit model that separates the outputcapacitors, and allows for winding resistances ofthe individual outputs.
Inserting the winding resistances still produceslarge errors. The gain is much too high inmeasurement versus prediction. More detailedanalysis shows the problem to be the equivalentseries resistance of the diode! This is a parasiticthat is often overlooked in modeling. Very few ofthe papers on modeling highlight the diode esr asbeing an issue its always assumed to be verylow, and therefore somewhat irrelevant.
L
PWMCCM
c
pa
c'ControlRp
+_
Vin
C1
Rc1
RL1
Rs1
+_
0
(-13/14)
0
ISOLATION
C2
Rc2
RL2
Rs2
Compensator
Figure 11. Two-output flyback converter withmore accurate modeling of output circuits. Diode
esr, included in Rs2, is the critical parasiticcausing errors.
Whats different here? The second diode, on theauxiliary output, is used at very low current. And,
as semiconductor experts are well aware, theincremental series resistance of a diode at low
currents is very high in this case, about 1. Thisresistance is high enough to decouple the largestorage capacitor on the main output from thefeedback output, substantially increasing theconverter gain.
100 kFrequency (Hz)
-60
-40
-20
0
10
1 10 100 1 k 10 k
Measured Power Stage
Predicted Power Stage
Gain (dB)
-160
-120
-80
-40
0
Measured Power
Stage
Predicted Power Stage
Phase (deg)
Frequency (Hz)
1 10 100 1 k 10 k 100 k
Figure 12. Loop gain measurement andpredictions with proper diode esr value in the
model.
For both of these design examples, furtherdeviations from predicted results can be expectedas the power supplies are operated at the cornersof the electrical specifications, and at temperatureextremes. You should always measure yoursystems across the full range of operation tominimize risk of subsequent problems andfailure.
VIII. COMPONENT
MEASUREMENTS
Both case studies uncovered an important aspectof switching power supply design. Parasitics ofcomponents in a power converter can havedramatic effects on the overall system operation,especially on stability. It is, therefore, crucial tounderstand which elements are important in thedesign (thats part of the role of good modeling)and to know how to get their values.
Many manufacturers of power components donot give adequate data for their parts for properconverter design. For example, capacitormanufacturers may specify an esr at 120Hz, arelic of the days when all power was processed atline frequency. The esr at 100kHz will typicallybe substantially different and often lower. It is upto the design engineer to determine which valueto use in modeling and control design.
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Similarly, the parasitics of magnetics parts arecritical to good design. Changing leakageinductance, for example, can make or break(literally!) a power supply design. Prototypes andmanufacturing units of transformers must beproperly characterized, specified, and controlledin volume production to ensure system reliability.
If you have a frequency response analyzer onyour bench for loop gain measurements, you alsohave a very powerful tool for characterizing theimpedances of components, and finding thecritical parasitic values.
A frequency response analyzer is designed tomeasure transfer functions of a circuit, in thepresence of substantial noise. However, it canalso be used as a very effective engineering toolfor impedance characterization of components.
This is a very important function of such aninstrument, since a dedicated RLC meter, capableof operating over the range of interest of powerconversion components, can be very costly.
While a frequency response analyzer does nothave the same resolution capabilities of a trueRLC meter, its ability to measure impedancecontinuously over a wide frequency range makesit very applicable to power components.Parasitics of magnetics, capacitors, and other
components are often a function of frequency.Most RLC meters do not provide this continuousdata. And it can be very misleading if thecomponent manufacturers do not have the propertest frequency for measurements.
-100
-50
0
50
100
150
10 100 1 k 10 k 100 k 1 M 10 M
100 kOhm
1 mOhm
1 p F
4 nH
Impedance (dB Ohm)
Frequency (Hz)
Figure 13. Range of impedance measurementsfor the frequency response analyzer using simple
measurement setup circuits.
It is not necessary to have sophisticated setupsfor measuring impedance to the degree ofaccuracy needed for power components. Somevery simple circuits can provide a wide dynamicrange of measurement, as shown in Fig. 13.Careful selection of test circuits allows the
measurement of high impedances up to 100k,
and of capacitors as small as 1 pF. This allowscharacterization of almost any power transformerat its resonant frequency.
At the low impedance range, 4-terminal Kelvinmeasurements allow a frequency response
analyzer to effectively measure down to 1mand 4nH of lead inductance. Impedances belowthis range require expensive calibrated setupsused on high-end RLC meters to measureanything other than dc resistance. (DC resistance
can also easily be measured in the
range withvery simple lab instrumentation an accuratevoltmeter and current source.)
IX. TRANSFORMER IMPEDANCE
MEASUREMENTS
One powerful use of a frequency responseanalyzer is the measurement of impedances oftransformers and inductors. Extended impedanceversus frequency plots provide significant design
and performance data.
As in classical line-frequency transformer design,the proper way to characterize a powertransformer for high-frequency applications iswith priamry-side impedance measurements withthe secondaries (i) open-circuited and (ii) short-circuited. This provides a wealth of designinformation, and should be measured andarchived for every design that is done, and foreach step of the transfer of the design intomanufacturing.
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0
20
40
60
80
100
120
1 2
3
4
5
6
10 100 1 k 10 k 100 k 1 M 10 M
Frequency (Hz)
Open CircuitImpedance
Short CircuitImpedance
Impedance (dB Ohm)
Figure 14. Transformer impedancemeasurements. Open circuit and short circuit
measurements provide a wealth of designinformation.
Fig. 14 shows some example plots of powertransformer impedance, measured from theprimary, with the secondaries first open, thenshort-circuited. Several asymptotes on the curvesof Fig. 14 are numbered, corresponding to theregions used for measurement of criticaltransformer parameters:
1. Primary winding resistance2. Primary + reflected secondary winding
resistance
3. Magnetizing inductance4. Leakage inductance5. Open-circuit resonance
(and equivalent winding capacitance)
6. Short circuit resonance(and its equivalent capacitance)
Dont underestimate the value to the design andmanufacturing process of measuring these curves
early in the product cycle. Magnetizinginductance is seen as the dominant component inthe open circuit measurement from 1kHz to100kHz. In manufacturing tests, it should bemeasured in this range. (Typical testing is done at1kHz).
The leakage inductance impedance is onlydominant in the short circuit measurements from50kHz to 3MHz, and it should be measured inthis range by the manufacturer. Specifying and
measuring the leakage at 1kHz will lead to
unacceptable accuracy, and poor quality
control.
In some designs, it will become apparent that theleakage inductance asymptote is not a straight20dB/decade slope. Instead, the leakage willdecrease with frequency, due to proximity effectsin the windings. If this effect is pronounced, greatcare must be taken in choosing the test frequencyto avoid errors.
Fig. 15 shows an example of open-circuitimpedance measurements on a production powersupply. Curve number 2 is the impedance of a
transformer built in the engineering lab. Curvenumber 1 was a pre-production prototype. Theopen circuit resonance of the engineering partwas at 550kHz, and of the pre-productionprototype, 340kHz. Both frequencies were highenough not to cause initial concern, with aswitching frequency of 70kHz. The engineeringprototype worked, however, and themanufacturing part did not. The power supplyfailed to start.
0
20
40
60
80
100
120
10 k 100 k 1 M 10 M
Frequency (Hz)
Open Circuit
Impedance
1 2
Impedance (dB Ohm)
Figure 15. Transformer production problem allparameters of the transformer are the same
except the resonant frequency the manufactureromitted a single layer of tape, and the power
supply no longer worked!
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The change in resonant frequency actuallycorresponded to a change of winding capacitancefrom 16pF to 52pF. This was due to themanufacturer omitting a layer of tape in thewindings! There were no safety issues in thiscase and the output was line-side referenced. Theincreased capacitance caused a current spike in
the control circuit, which prematurely tripped thePWM controller, and prevented proper circuitoperation.
Sometimes, even in the best designs, (scarythough this is to management and purchasingdepartments!) a layer of tape is all that standsbetween success and failure of a product.Frequency response measurements during thedevelopment process are a tremendous help indetecting understanding, and solving subtle butpotentially expensive problems like this.
There are many more details, of course, tofrequency response testing and assessment ofmagnetics. These are covered in considerablepractical detail in Ridley Engineerings designcourses [6].
X. CAPACITOR MEASUREMENTS
In the loop gain design example earlier in thispaper, a wrong capacitor esr value almost caused
a very expensive problem in a power supplydesign. This was despite the fact thatconventional wisdom would define the capacitoras better than its specification. The loop gainmeasurement was far from prediction, and thecapacitor esr was found to be 30 times lower thanthe specified maximum from the data sheets.
Surprisingly, this is not an uncommonexperience. Manufacturers often do not have thetime, equipment, or incentive to properly
characterize and document their components forthe design engineer. It is good engineeringpractice to measure every capacitor destined foryour power supply, and make sure the acceptablerange of the parasitic components are welldefined. And you should do this across the entiretemperature range the components will operatein.
Fig. 16 shows a set of capacitor measurementsuseful for power supply design. The tantalumcapacitor is the type used in the example of loopgain earlier in the paper. The measured esr is
about -20dB (0.1), compared with the datasheet which specifies a maximum impedance at
100kHz of 7.5.
Also plotted in the figure is the impedance of a22F electrolytic capacitor. The tantalum andelectrolytic have the same impedances up to afew kHz. The esr of this particular electrolytic
was 0dB, or 1. Clearly, the tantalum capacitorwill do a better job of filtering higher frequencynoise.
-40
-20
0
20
40
60
80
10 100 1 k 10 k 100 k
Tantalum 22 uF
Electrolytic 22 uF
MLC 1 uF
1 M
Frequency (Hz)
Impedance (dB Ohm)
Figure 16. Impedances of different types of
capacitors.
The final curve is the impedance of a multilayerceramic capacitor. Notice that above 200kHz, theMLC, with a value of only 1F, has a lowerimpedance than the 22F electrolytic capacitor.For equivalent values, 22 of the 1F MLC capswould give vastly superior performance. Onereason we dont use this type of capacitorextensively in power supplies is the cost.
XI. SUMMARYIn todays marketplace, you cannot risk productfailure. Modeling alone does not guaranteecontrol loop stability. Measurement of controlloops and components with a frequency responseanalyzer is an essential design step.
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The critical factors to remember in all yourdesigns are:
You mustmeasure power supply control loopstability for reliable design. This should bedone at every stage of development andprototyping. In some instances, it makes
sense to incorporate loop measurement aspart of manufacturing, especially for lowvolume production, and high reliabilitysupplies.
Modeling is almost always inaccurate thefirst time. Simplified models often fail toaccount for the parasitics and circuit eventsthat you may see in your design. The firstcontrol loop stability measurement willusually surprise you.
Frequency response analyzers are nowaffordable. Theres no reason for anycompany developing power supplies to bewithout one you simply cannot afford therisk of shipping a product with an unknownstability margin.
Component impedance measurement is avaluable capability of frequency responseanalyzers. Measurement of powercomponents gives you critical design data
that is often unavailable from manufacturers.
Ray Ridley has specialized in the modeling,design, analysis, and measurement of switchingpower supplies for over 20 years. He hasdesigned many power converters that have beenplaced in successful commercial production. Inaddition he has consulted both on the design ofpower converters and on the engineeringprocesses required for successful powerconverter designs.
Ridley Engineering, Inc. is a recognized industryleader in switching power supply design, and isthe only company today offering a combinationof the most advanced application theory, designsoftware, design hardware, training courses, andin-depth modeling of power systems.
XII. REFERENCES
R.D. Middlebrook, S. Cuk, A General UnifiedApproach to Modeling Switching ConverterPower Stages, IEEE Power ElectronicsSpecialists Conference, 1984 Record, pp. 18-34.
V. Vorprian, "Simplified Analysis of PWMConverters Using the Model of the PWM Switch:Parts I and II",IEEE Transactions on Aerospaceand Electronic Systems, March 1990, Vol. 26,No. 2, pp. 490-505.
V. Vorprian, R. Tymerski, K.H. Liu, F.C. Lee,Generalized Resonant Switches Part 2: Analysisand Circuit Models, VPEC Power ElectronicsSeminar Proceedings, Virginia PolytechnicInstitute and State University, Blacksburg, VA,
pp. 124-131, 1986.
R.B. Ridley, "A New Small-Signal Model forCurrent-Mode Control", PhD Dissertation,Virginia Polytechnic Institute and StateUniversity, November, 1990. See web site toorder.
Switching power supply design information,design tips, frequency response analyzers, andeducational material for power supplies can befound at the web site located at:
http://www.ridleyengineering.com
Ridley Engineering, Inc. High-FrequencyMagnetics Design professional engineeringseminar taught semi-annually. See [5].
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