App Note Noninvasive Stability V1 3
Contents
1. 102 10 104 10 10 10 f Hz mum R1 Mag Gain 200 ffHz TR2 r Cursor 1 8 315k 21 757 200 2 150 N xX F 100 50 102 10 104 10 10 107 f Hz mum R2 Unwrapped Phase Gain The loop gain Bode plot with a 100 uF tantalum capacitor shows a phase margin of m 21 8 at the crossover frequency of f 8 3 kHz In the higher frequency range additional crossover frequencies can exist The Bode 100 performs measurements up to 40 MHz allowing investigation of these high frequency effects which are often related to capacitor PCB or connection parasitics OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 14 of 24 Measurements using the 100 uF electrolytic capacitor Oop f Hz TR1 dB Cursor 1 SIN78k 28 422f 40 i i O 20 X L 0 20 102 102. OMICRON as LAB Bode 100 Application Note Traditional and Non Invasive Stability Measurements Using the Bode 100 and the Picotest J2111A Current Injector By Florian Hammerle amp Steve Sandler 2015 by OMICRON Lab V1 3 Visit www omicron lab com for more information Contact support omicron lab com for technical support Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 2 of 24 Table of Contents 1 EXECUTIVE SUMMAR Waseca esac sce ee acces ee ec cece ceca wc eae ese cee eset cee essen eee 3 IN TAS K oonan saves secs cestcuueecunccuavectcbeeusecauec eevee E causecunsedeseateaucsasusaucenas 4 3 MEASUREMENT SETUP amp RESULTS 00 0 cio cc eccccssccsscccscccscccccceccsccssccsscccsccesscessseccsccnsccesccesscessaccsecces 5 3 1 STABILITY MEASUREMENT OF THE CONTROL LOOP csssccccccsesecccccscccccccsssecccccaucecccccauccececeaueecccuauceeceecausescecaaunsess 5 3I ere icl0 felegic gies 10 eretneerer te wert Tent E er rere renee terre rece terre Tere rn were ere tee 5 OT TC VCO I S EENE E N A E ATE EE OE EATE A A E LTO A T T 7 CARL Oe 2116 g OREA AEE T T AAAA ET TATAAS ee ANETA IAAT AE AT 8 e E EEE A Ao N eee E EEE EE E E E EA E EE E E EEE A EA 9 TV SAE MOSU errete rrene ARRE TE E cian asec ae OEEO AEREE EEEE 13 3 2 OUTPUT IMPEDANCE MEASUREMENT errcrnernei sienne R 15 Zt Meas WreEmMeEn o SUD arrai aaea E E AAA E A AE ONO EEE OE 15
3. Bode 100 The Bode 100 is set up as follows Measurement Mode Frequency Sweep Mode Fis Start Frequency 100 Hz Stop Frequency 10 MHz Sweep Mode Logarithmic Number of Points 201 Receiver Bandwidth 3 Hz Attenuator 1 amp 2 0 dB Level 0 dBm OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 17 of 24 To switch on the external reference start the device configuration window and click on the external reference switch symbol In addition the input impedance of channel 1 has to be set to 50 Q while channel 2 stays in high impedance mode Configuration Device Configuration Connection Setup Measurement Gain Phase Impedance Reflection SOURCE i o Sop je RECEIVER 1 Bandwidth RECEIVER 2 ATTN 1 0dB_ v Trace 1 amp 2 settings Format Q Ta j Ymar 3 00 min 1 00m Y Scale Lin C Log TR Log TRA Data gt Memory 1 Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 18 of 24 3 2 3 Phase Margin Calculation According to reference Erickson amp Maksimovic 2004 the phase margin m is related to the quality factor Q by COS Pm g sin m The quality factor at the crossover frequency can be calculated from the measured group delay by Q T 1 f Ty Hence the phase margin at crossov
4. 104 10 106 10 f Hz mums R1 Mag Gain e307 ffHz TR2 Cursor 1 59 178k 91 409 200 2 150 N xX 100 50 0 102 10 104 10 106 10 f Hz mum R2 Unwrapped Phase Gain The loop gain Bode plot with a 100 uF electrolytic capacitor shows a phase margin of m 91 4 at the crossover frequency of f 59 2 kHz OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 15 of 24 3 2 Output Impedance Measurement Together with the Picotest J2111A Current Injector the Bode 100 offers a simple and non invasive method to measure the output impedance of a regulating system The output impedance data provides a measurement of the phase margin without the need to inject a signal into the control loop This is the only way to measure the phase margin of a fixed voltage regulator where the control loop is not available for a traditional Bode measurement OUTPUT NPUT A 4 Wy HI oe OMICRON Ley J Bode 100 G cI LAB i Network Analyzer Input Regulator or Switcher Out DUT Current Monitor Signal lt Filter amp Load Modulated Injector Current Output Simple Voltage Connection Analyzer Modulation Oscillator High PSRR Power J2111A Solid State Adapter Current Injector Output impedance measurement using the J2111A Source Picotest Signal Injector Documentation 2010 3 2 1 Measur
5. be used to measure switching regulators as well The measurements are performed on the Picotest Voltage Regulator Test Standard VRTS testing board using the OMICRON Lab B WIT injection transformer and the Picotest J2111A Current Injector The Current Injector together with the Bode 100 allows direct measurement of the output impedance group delay and Q of the system Using this information the phase margin of the system can be calculated without breaking the feedback loop of the controller This method is therefore non invasive In this application note the results of the non invasive measurement are compared to the classical Bode plot loop gain measurements Additionally the influence of the output capacitor ESR on the phase margin is investigated Two different output capacitors are used for the phase margin measurements and the results are compared Additional information on stability measurement with the Bode 100 can be found in OMICRON Lab 2009 Measurement of DC DC Converters with Bode 100 1 See http www picotest com products_injectors html Equivalent Series Resistance OMICRON E LAB Smart Measurement Solutions _ Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 4 of 24 2 Measurement Task The phase margin of the LM317 linear voltage regulator is evaluated using two different methods 1 Traditional stability measurement via the Loop Gain Phase Bode plot 2
6. can switch off the calibration and check the influence of the calibration If the calibration influence on the measurement results is high even if two similar voltage probes are used the measurement setup may be inaccurate pe Probe Calibration The calibration can be switched ON and OFF by clicking on the calibration indicator TA Probe Calibration GAIN OFF Wu f OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 9 of 24 3 1 4 Measurement We will first measure the Bode plot with the tantalum capacitor Starting a single sweep leads to the following Bode plot p Pis F 80 60 7 40 20 07 TR1 dB 20 40 604 102 108 104 105 108 107 f Hz mum 1 R1 Mag Gain 250 502 103 104 105 108 107 f Hz mum R2 Unwrapped Phase Gain The marked ranges indicate that the measurement result is not correct The distortions are due to the excessive measurement level which causes nonlinearities of the system to be measured This is not a result of the analyzer but is due to large signal effects within the regulator Steven M Sandler 2011 OMICRON Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 10 of 24 The injection signal level need
7. Non Invasive output impedance measurement The two measurements are then compared The Picotest VRTS kit is used as the basis for the testing The VRTS can be used to perform most of the common voltage regulator measurements using the Bode 100 in conjunction with the Picotest Signal Injectors The kit includes the regulators and capacitors used for the measurements in this application note Voltage Regulator Test Standard board Source Picotest Voltage Regulator Test Standard 2010 To highlight the influence of the output capacitance on the phase margin of the regulator two different Capacitors are used for the measurements The two capacitors are the 100 uF tantalum capacitor capacitor no 1 and the 100 uF aluminum electrolytic capacitor capacitor no 3 OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 5 of 24 3 Measurement Setup amp Results 3 1 Stability Measurement of the Control Loop We can measure the loop gain T s of the LM317 feedback system by breaking the control loop and injecting a small signal voltage into the feedback pin This can be done with the B WIT wideband injection transformer and two 1 1 voltage probes A constant load current of 25 mA is achieved by switching on the positive bias current of the J2111A Picotest Current Injector The injector can provide positive negative or zero bias so that the J2111A ca
8. asy way of measuring the ESR References Erickson R W amp Maksimovic D 2004 Fundamentals of Power Electronics Springer OMICRON Lab 2009 www omicron lab com application notes Retrieved 12 2010 from Measurement of DC DC converters with Bode 100 OMICRON Lab 2010 Capacitor ESR Measurement Application Note www omicron lab com application notes Retrieved 12 2010 Picotest 2010 Signal Injector Documentation Version 1 0c Picotest 2010 Voltage Regulator Test Standard Version 1 0d Steven M Sandler T B 2011 Network Analyzer Signal Levels Affect Measurement Results 37 1 OMICRON E LAB Smart Measurement Solutions _ Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 24 of 24 OMICRON Lab is a division of OMICRON electronics specialized in providing Smart Measurement Solutions to professionals such as scientists engineers and teachers engaged in the field of electronics It simplifies measurement tasks and provides its customers with more time to focus on their real business OMICRON Lab was established in 2006 and is meanwhile serving customers in more than 40 countries Offices in America Europe East Asia and an international network of distributors enable a fast and extraordinary customer support OMICRON Lab products stand for high quality offered at the best price value ratio on the market The products reliability and ease of use guarante
9. ation point Ch2 Freq 2 600kKH2 Ch2 Ampl 10 00mMmA Ch2 5 00mAQ2 Step load response with tantalum output capacitor Ch2 Freq 2 600KH2 Ch2 Ampl 10 00mA Ch2 5 00mAQ Step load response with electrolytic aluminum capacitor The step load response shows that the electrolytic capacitor suppresses ringing The measurement with the tantalum capacitor shows ringing at a frequency of about 1 1 we e AN OMICRON N LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 23 of 24 4 Conclusion The Bode 100 can be used to measure a traditional Bode response as well as a non invasive output impedance measurement when combined with the Picotest J2111A Current Injector The non invasive measurement has been shown to be in excellent agreement with the traditional measurement offering a simple and reliable method to evaluate the stability of voltage regulators without breaking the feedback loop The non invasive method therefore allows the stability of regulators to be assessed when the feedback loop is not accessible as in the case of a fixed voltage regulator In addition it can be seen that the equivalent series resistance has a very high influence on the Stability of the voltage regulator As the ESR is not always specified in the high frequency range it can be useful to measure the ESR The Bode 100 with the impedance adapters offers an e
10. dea DOV T a E EAEE E TE EEA AE A EAE EE E E N E 16 22 Phase Mri C ACUO mirre ea ne TEE N OIE OTENE OEE E TTE OAT 18 EPR EU PIM AEE AAE AE E A EE O EEE ET 19 3 3 EQUIVALENT SERIES RESISTANCE eee me E AE 21 3 4 STEP LOAD RESPONSE ice tesscccusccenacntocassavteceseuvaccetdubecune ie tananseatanens tievenscdutecensdubensnaueteavedtnthedesdutancusaubeaweauedecesdavsaadeceasnos 22 4 CONCLUSION nE EE 23 Note Basic procedures such as setting up adjusting and calibrating the Bode 100 are described in the Bode 100 user manual You can download the Bode 100 user manual at http www omicron lab com bode 100 downloads html 3 The J2111A does not require calibration The J2111A comes with and uses the J2170 High PSRR power supply Note All measurements in this application note have been performed with the Bode Analyzer Suite V2 43 SR1 Use this version or a higher version to perform the measurements shown in this document You can download the latest version at www omicron lab com bode 100 downloads You can download the latest Picotest Injector manual at http www picotest com products_injectors html OMICRON E LAB nua Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 3 of 24 1 Executive Summary This application note shows how the phase margin of a linear voltage regulator LM317 can be measured using the Bode 100 and additional accessories The same techniques can
11. e trouble free operation Close customer relationship and more than 25 years in house experience enable the development of innovative products close to the field Europe Middle East Africa Asia Pacific Americas OMICRON electronics GmbH OMICRON electronics Asia Limited OMICRON electronics Corp USA Phone 43 59495 Phone 852 3767 5500 Phone 1 713 830 4660 Fax 43 59495 9999 Fax 852 3767 5400 Fax 1 713 830 4661 info omicron lab com www omicron lab com
12. ement Setup The figure above shows the basic measurement setup to measure the output impedance of a regulator system with the Bode 100 and the Picotest J2111A Current Injector The output of the Bode 100 is connected to the modulation input of the J2111A MOD A signal at the MOD input of the injector leads to a change in load current according to the input signal at a gain of 10 mA V The monitor output of the injector then delivers a voltage signal that is proportional to the current flowing through the injector output 1 A 1 V when terminated with 50 Q This signal is measured at channel 1 of the Bode 100 The output voltage is measured using a 1 1 probe with channel 2 Performing a gain measurement with an external reference leads to the output impedance Vch2 A Vout out Veni lout OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 16 of 24 Output impedance measurement example 3 2 2 Device Setup Current Injector J2111A The positive bias of the current injector has to be switched on bias as the Bode output voltage does not have an offset and the LM317 is a positive voltage regulator The positive bias will provide a 25 mA offset current allowing the current injector to operate in class A mode For best performance the output wires from the J2111A should be twisted or a coax They are shown here untwisted for clarity
13. er frequency can be calculated from an output impedance measurement using the above relationships The Bode Analyzer Suite supports the direct phase margin calculation from the output impedance measurement There are two different ways to measure the phase margin 1 Basic PM Calculation single cursor 2 Advanced PM Calculation two cursors The basic PM calculation uses one cursor value to determine the phase margin value from the Q T peak This method is very accurate for low phase margin values below approximately 40 For higher phase margin systems the output impedance peak will not exactly occur at the same frequency as the Q T peak The advanced PM calculation accounts for this difference in frequency Therefore it needs two cursor values to calculate the phase margin One cursor must be placed at the peak of the output impedance magnitude and the other cursor must be placed at the peak of Q T7 In the following we use the Basic PM Calculation single cursor As you will see later this method is sufficient in this case since the peak in impedance and the peak in Q T occur at the same frequency The basic PM calculation is activated by right clicking in the cursor area of the Bode Analyzer Suite as shown in the figure below Cursor 4 341 kHz 5513 de 2120 25 005 Place cursor at the OTg peak E Curzor 2 delta C2 01 Jump To Mext Cut Copy A Basic PM Calculation single cursor alculation hwo curso
14. hase margin which are displayed in the cursor table OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 20 of 24 The output impedance measurement with a 100 uF tantalum capacitor shows a phase margin of Qm 22 7 at the crossover frequency of f 8 0 kHz These results are in agreement with the results from the loop gain measurement 21 8 and f 8 3 kHz Next we connect the electrolytic capacitor to the output and restart the measurement 20 0 6 0 5 L J 0 4 CS lt a EA A AE E S E a E S E A e A S E Le e a T D 03 4 pe a Cee ee oe ee ne oe Ey esae eel le ee dea bee i a JV 10 i E 0 2 ba fitz TAB TR2 TR2 PM T00 Cursor 1 50 878k 294 401m 255 658m gt 71 30 eee asses G 4 102 10 104 10 10 10 f HZ memes TR1 Mag Gain TR2 QTg Gain The aluminum capacitor has a very high ESR which results in high damping As the phase margin is gt 1 the damping is very high and no resonance peak appears in the output impedance This output impedance curve therefore shows a very stable system with high damping and the display indicates a phase margin of om gt 71 OMICRON Smart Measurement So
15. lutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 21 of 24 3 3 Equivalent Series Resistance The great difference in stability of the system depending on the output capacitor is caused by the different ESR of the capacitors The ESR of the two capacitors is shown in the following figures The measurements were performed using the Bode 100 with the B WIC impedance adapter see also OMICRON Lab 2010 The tantalum capacitor has a very low resistance of about 50 mQ in the vicinity of the crossover frequency The electrolytic capacitor has a series resistance of about 1 5 Q ESR of capacitor 1 tantalum capacitor 1 0 2 00 T T T gt D TR1 Ohm 102 108 104 108 108 107 f Hz mum R1 Rs Impedance ESR of capacitor 3 electrolytic capacitor NO O oh on T FTE TR1 Ohm O1 TT T sd 108 104 105 106 107 f Hz pal el el O N mum R1 Rs Impedance OMICRON E LAB Smart Measurement Solutions _ Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 22 of 24 The same measurement setup used for the output impedance measurement can also be used to measure the step load response The Bode 100 output has to be replaced with a function generator and the inputs with an oscilloscope The chosen step size is 10 mA around the 25 mA oper
16. measurement and without shifting the crossover frequency To reduce the measurement noise the shaped level function of the Bode 100 can also be used The Bode 100 also allows averaging and selectable Receiver Bandwidth for noise reduction OMICRON E LAB Smart Measurement Solutions _ Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 12 of 24 Activate the Shaped Level feature as shown in the following picture Configuration a Shaped Layel Level 0 00 dBm Next the shaped level function has to be entered SS In the Shaped Level window frequency and the associated level can be entered This enables the Bode 100 to reduce the level only at the points where a reduction is necessary and to increase the level in regions were the measurement shows too much noise Shaped Level File Tools Help OK MCancel G Print A Print Preview Output Level Reference Level 100 000 Hz Preview Output Level It is possible to use an optimal measurement level for every frequency range using a shaped level as shown in the picture above OMICRON ee LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 13 of 24 3 1 5 Measurement Result Measurements using the 100 uF tantalum capacitor
17. n operate in class A mode for use with a Network Analyzer The negative bias is for use with negative voltages while the positive bias is for positive voltages The Current Injector is normally in parallel with the normal circuit load current and impedance In this case the J2111A Current Injector is acting as a constant current load 3 1 1 Measurement Setup The VRTS board is powered using a universal wall adapter power supply which comes with interchangeable plugs for use in various countries The J2111A is powered using the J2170 High PSRR power supply The LM317 IC is plugged into the board as shown below Please make sure that the polarity is correct as shown in the picture below The LM317 provided with the kit is configured with a 410 Q to 249 voltage divider to deliver a 3 3V output voltage The injection resistor has a value of 5 Q It is recommended that you measure the output voltage to verify its 3 3 V before continuing Stability measurement of the LM317 board using VRTS Bode 100 B WIT and J2111A Current Injector OMICRON se LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 6 of 24 The B WIT injection transformer connects the Bode 100 to the test board BODE connectors as shown below Two oscilloscope probes are connected to the same connectors as the injection transformer The picture below shows the connection points on the test board I
18. r 1 amp 2 0 dB Level 0 dBm To switch on the external reference start the device configuration window and click on the external reference switch symbol Configuration RECEIVER 2 To directly measure the Bode plot we want to display the FF Trace 2 TR2 magnitude in dB and the phase of the loop gain T To do so the second trace in the Bode Analyzer Suite has to be activated By setting the correct Diagram Setup the phase can be displayed in a separate diagram Diagram Setup Auto Always Two Diagrams OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 8 of 24 Trace 1 amp 2 Settings ace RD A Unwrapped Fhase Unwrapped Phase 7 Begin 10 000khz M End 10 000MHz M Format Mag dB Ymin 5 00 00 dB Scale Lin Log TRI Log TRI Data gt Memory 1 Data gt Memory 1 Trace 1 settings Trace 2 settings 3 1 3 Calibration A calibration has to be performed if the two voltage probes are not identical As we are measuring a voltage gain we need a THRU calibration To do so both probes are connected to the same injection point as shown in the left picture below and the THRU calibration is started THRU calibration setup Measurement setup The calibration removes differences between the two probes It is recommended that you check the influence of the THRU calibration To do so you
19. rs OMICRON E LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 19 of 24 Activating the cursor calculation leads to an additional line in the cursor table showing the results of the calculations SF Frequency TRT MagDB TAZ OTs TA2PM Euler 8 042 kHz 4 521 dB 2 359 22 C40 Place cursor at the OTg peak Cursor 2 delta C2 L1 Q Tg PM Note The phase margin calculation is only available if one trace measurement format is set to Q 7 Note Q T is a function of group delay T T is calculated by numerical differentiation Therefore we recommend not to choose too many points in the sweep 201 points is the recommended choice Furthermore sometimes we recommend to use the logarithmic Y scale for a better visibility of the meaningful result areas 3 2 4 Measurement First we measure the phase margin with the tantalum output capacitor Starting a single sweep leads to the following measurement result 72 5 HE 42 0 HH 5 Ji a PE A 4 0 ATTN 10 5 EME TAN IN ETY ENAN TTA Aa A tt Meara Ma MA M e THe TR1 dB PA TR2 PM so ER L Cursor 1 8 042k 45521 2 359 22 748 O O 102 103 104 105 108 107 f Hz TRI Mag Gain TR2 QTg Gain Setting the cursor to the resonance peak in the output impedance leads to the crossover frequency and the calculated p
20. s to be decreased Reducing the measurement level to a value of 2 7dBm leads to the following Bode plot 80 _ TR1 dB 604 102 108 104 105 108 107 f Hz mum R Mag Gain 250 f Hz mum 1 R2 Unwrapped Phase Gain Now two unwanted effects appear OMICRON Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 11 of 24 Due to the low injection level the measurement shows more noise in the high gain magnitude range However in the more interesting zero gain area the measurement level is still too high The output level of the Bode 100 can further be reduced by connecting an external attenuator between the Bode output and the B WIT input In this example we are using the Picotest J2140A Attenuator Connecting a 20dB attenuator between the Bode 100 output and the B WIT and restarting the measurement leads to the following result 60 102 108 104 10 106 107 f Hz mum 1 R1 Mag Gain 2507 200 H 104 105 108 107 f Hz mum 1 R2 Unwrapped Phase Gain The nonlinearities disappear while the noise on the measurement increases To check if the output level is small enough it should be possible to increase the output level about 6 dB without the nonlinear effects reappearing on the
21. t should be noted that the probe ground connections are both connected to the VOUT connector to measure the voltage respect to the output voltage This is only true for floating voltage regulators such as the LM317 since the reference voltage is with respect to the output voltage and not to ground OUTPUT _ INPUT B WIT 100 Capacitor no 1 is a tantalum capacitor and capacitor no 3 a standard aluminum capacitor Both have a nominal capacitance value of 100 uF The figures below show the capacitors connected to the test board output wipe st k i GNU 62 e w Capacitor no 1 tantalum Capacitor no 3 aluminum With this setup we can measure the loop gain and determine the phase margin of the system For the stability measurement the Bode 100 needs to be configured correctly OMICRON N LAB Smart Measurement Solutions Bode 100 Application Note Traditional and Non Invasive Stability Measurements Page 7 of 24 3 1 2 Device Setup To measure the loop gain and phase two voltages at the injection point must be measured The Bode response is then calculated by V2 s V1 s T s This measurement can be performed directly with the Bode 100 using an external reference The Bode 100 is set up as follows Measurement Mode Frequency Sweep Mode Start Frequency 100 Hz Stop Frequency 10 MHz Sweep Mode Logarithmic Number of Points 201 or more Receiver Bandwidth 100 Hz Attenuato
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