EMC quantities and their uncertainty

Transcription

1 EMC quantities and their uncertainty Carlo Carobbi Dipartimento di Elettronica e Telecomunicazioni Università degli Studi di Firenze Politecnico di Torino, I3P - 17 Giugno

2 EMC quantities Frequency domain (1 of 2) Absolute Voltage V db(µv) Current I db(µa) Impedance Z db(ω) or Ω Power P dbm or mw Specific Absorption Rate SAR W/kg Electric field strength E db(µv/m) or V/m Magnetic field strength H db(µa/m) or A/m Magnetic flux density B db(µt) or µt Power density S db(w/m 2 ) or W/m 2 Antenna calibration factor ACF db(1/m) Normalized site attenuation NSA db(m 2 ) Bandwidth(resol., noise, impulse) B Hz Voltage spectral density (coherent) V imp db(µv/hz) Voltage spectral density (incoherent) V n db(µv/ Hz) Politecnico di Torino, I3P - 17 Giugno

3 EMC quantities Frequency domain (2 of 2) Relative Attenuation A db Gain G db Insertion loss IL db Return loss RL db Voltage standing wave ratio VSWR Site voltage standing wave ratio SVSWR db Signal to noise ratio SNR db Spurious signal (spectral purity) dbc Overload factor (margin of linearity) db Noise figure NF db Phase 10kHz offset dbc Flatness (frequency response) db Linearity (display scale fidelity) db/db Politecnico di Torino, I3P - 17 Giugno

4 EMC quantities Time domain Absolute Peak amplitude V pk V Amplitude at 30 ns, 60 ns V 30,60 V Rise time (10 to 90 %, 20 to 80 % ) t r ns, μs Fall time (90 to 10 %, 80 to 20 % ) t f ns, μs Time constant (0 to 63 %, 100 to 37 %) T C, T D ms Duration t d ns, μs Relative Overshoot % Undershoot % Modulation depth % Politecnico di Torino, I3P - 17 Giugno

5 Absolute vs. relative Absolute uncertainty includes relative uncertainty Absolute uncertainty = Reference source uncertainty + Relative uncertainty Examples of absolute and relative measurements follow Politecnico di Torino, I3P - 17 Giugno

6 Example of absolute measurement TG SA CABLE TG = Tracking Generator SA = Spectrum Analyzer CABLE = RG 58 C/U, 50 cm TG output power setting = 20 dbm ( MHz) SA reading =?? ±!! Politecnico di Torino, I3P - 17 Giugno

7 Absolute measurement displayed result -18 TG setting = 20 dbm -20 Δ MAX = 0,9 db -22 P [dbm] SA reading 2 db MAX = maximum deviation between SA reading and TG setting Δ = SA reading TG setting What is the expected Δ (or SA reading) and its uncertainty? f [MHz] Politecnico di Torino, I3P - 17 Giugno

8 Measurement model (absolute) = P SA,READ + Δ R P TG,SET = (P SA,CAL ) + SA P TG,SET + Δ R = P TG,CAL A + Δ MM + SA P TG,SET + Δ R = SA + P TG,CAL P TG,SET + Δ MM + Δ R A = SA + TG + Δ MM + Δ R A Basic equations: P SA,READ = P SA,CAL + SA P SA,CAL = P TG,CAL A + Δ MM P TG,CAL P TG,SET = TG Meaning of the symbols (db units): P SA,READ = SA reading Δ R = Correction for non-repeatability P TG,SET = TG output power setting P SA,CAL = SA calibrated input power SA = SA residual error A = Cable attenuation = 20log(1/ S 21 ) Δ MM = Correction for mismatch = 20log( 1 (S 21 ) 2 Γ TG Γ SA ) P TG,CAL = TG calibrated output power TG = TG residual error Politecnico di Torino, I3P - 17 Giugno

9 Mismatch correction Necessary because: RF generators are calibrated by using nearly matched RF measuring instruments RF measuring instruments are calibrated by using nearly matched RF generators Politecnico di Torino, I3P - 17 Giugno

10 Basics of RF power measurement In linear units: P M = P G,CAL (1 Г M 2 )/ 1 Г G Г M 2 P M = power that a generator G delivers to a power meter M P G,CAL = power that G delivers to the perfectly matched M (Г M = 0) P G,CAL = P M P M,CAL = power delivered to M when fed by the perfectly matched G (Г G = 0) corrected for M mismatch P M,CAL = P M /(1 Г M 2 ) = P G,CAL When G and M are used in the field P M,CAL = P G,CAL / 1 Г G Г M 2 Note: SA is a voltmeter but the conclusion is similar: V M,CAL = (P G,CAL 50Ω) 1/2 / 1 Г G Г M. See HP AN 56, Microwave mismatch error analysis, Oct Politecnico di Torino, I3P - 17 Giugno

11 Calculation of mismatch correction Mismatch correction (in db) Δ MM = 20log( 1 Г G Г M ) depends both on the magnitude and phase of Г G and Г M It is impractical to determine magnitude and phase of Г G and Г M in the frequency range of interest Politecnico di Torino, I3P - 17 Giugno

12 Mismatch correction as a random variable Г G MAX and Г M MAX are usually available from manufacturer specifications or calibration (maximum over a specified frequency range) No information is available about the phase of the product Г G Г M apart from being somewhere between 0 and 2π The random variable (RV) corresponding to Δ MM is (Δ MM ) RV = 20log( 1 ke jψ ) where k = Г G MAX Г M MAX < 1 and ψ is a RV having a rectangular pdf within 0 and 2π It can be demonstrated that (Δ MM ) RV has a non-symmetric U-shaped pdf and δ MM = 0 u(δ MM ) = 20log(e)/ 2 k (approximation valid for any practical value of k) Note: A lower case letter represents the expected value, u is the standard uncertainty, U is the expanded uncertainty Politecnico di Torino, I3P - 17 Giugno

13 Uncertainty budget (absolute) Error source Max error (db) Pdf Divisor Standard deviation (db) TG level (1) 0,5 Uniform 3 0,29 TG flatness (2) 1,0 Uniform 3 0,58 TG level switch. (3) 1,0 Uniform 3 0,58 SA cal. signal 0,3 Uniform 3 0,17 SA IF gain 0,5 Uniform 3 0,29 SA flatness 0,5 Uniform 3 0,29 SA scale fidelity (4) R 0,2 Uniform 3 0,12 SA attenuation switch. (5) 0 Uniform 3 0 SA RBW switch. (5) 0 Uniform 3 0 TG-SA mismatch (6) R 0,97 U - shaped 2 0,68 TG-SA tracking (7) Repeatability R 0,01 Normal 1 0,01 Cable attenuation 0 0 Δ = TG + SA + Δ MM + Δ R A Δ TG = i Δ TG,i, δ TG = 0 db, u(δ TG ) = 0,87 db Δ SA = j Δ SA,j, δ SA = 0 db, u(δ SA ) = 0,46 db δ MM = 0 db, u(δ MM ) = 0,68 db δ R = 0 db, u(δ R ) = 0,01 db a(f) = 0,01 f db, u(a) = 0 db (f in MHz) Cable attenuation leads to a correction for a frequency dependent systematic effect a(f) whose magnitude is comprised between 0 (DC) and 0,33 db (1000 MHz) with negligible uncertainty Notes (100 khz 3 GHz): (30 MHz & 10 dbm) = REF (2) With respect to REF (3) With respect to 10 dbm level setting (4) 0,2 db/1 db (5) Ref. settings are used (ATT = 10 db, RBW = 300 khz) (6) TG output VSWR = SA input VSWR = 2:1 (max) (7) Not specified by the manufacturer Combined std. dev. u(δ) 1,20 Politecnico di Torino, I3P - 17 Giugno

14 Best estimate and expanded uncertainty δ = 0,01 f db (f in MHz) U(δ) = 2,4 db (2 std. dev., % confidence level) Politecnico di Torino, I3P - 17 Giugno

15 Monte Carlo verification pdf (1/dB) GUM and GUM Supplement 1 confidence intervals (95.45% confidence level) are practically the same: GUM: ( 2.40, +2,40) db GUM Supplement 1: ( 2.35, + 2,37) db Note: GUM-S1 confidence interval is such that the probabilities in the tails of the pdf are equal Δ + - A a (db) Politecnico di Torino, I3P - 17 Giugno

16 A plot to summarize Δ (SA reading) δ + U(δ) 2 db It is evident that instrument imperfection is over-estimated The displayed trace consists of 1001 samples On a statistical basis 45 to 46 samples are expected to exceed the limits δ±u(δ) Δ [db] 0 2U(δ)=4,8 db -2-4 δ U(δ) f [MHz] Politecnico di Torino, I3P - 17 Giugno

17 We add a long section of cable 50 cm 10 m TG SA CABLE TG = Tracking Generator SA = Spectrum Analyzer CABLE = RG 58 C/U, 50 cm + 10 m 10 m cable attenuation =?? ±!! Politecnico di Torino, I3P - 17 Giugno

18 Displayed result and comparison P [dbm] SA reading #1 (50 cm) SA reading #2 (10 m + 50 cm) 2 db Systematic effects can be clearly identified The uncertainty budget in the second case is the same as in the first one except for SA scale fidelity The manufacturer specifies 0,2 db/1 db and 1 db/10 db Since the range of the difference is about 8 db the maximum linearity error can be 1 db We obtain, for the second case U (δ) = 2,7 db f [MHz] Politecnico di Torino, I3P - 17 Giugno

19 What about the difference of (long + short) short? Gain [db] SA reading #2 SA reading #1 A LONG + Δ A 2 db f [MHz] The difference is very smooth It is obviously dominated by the attenuation of the long cable We can conclude that the difference IS the long-cable attenuation (changed of sign, A LONG ) + small correction (Δ A ) Due to CORRELATION the uncertainty of the difference is NOT [U(δ) 2 + U (δ) 2 ] 1/2 = 3,6 db!! Politecnico di Torino, I3P - 17 Giugno

20 Measurement model (relative) P SA,READ #2 P SA,READ #1 = A LONG + Δ SCALE FIDELITY + Δ MM #2 Δ MM #1 + Δ R #2 Δ R #1 or A READ = A LONG + Δ A Only the R error sources contribute to the relative measurement (see the uncertainty budget for absolute measurement) Scale fidelity (1,0 db) Mismatch (0,97 db) contributes two times in quadrature Repeatability (0,01 db) contributes two times in quadrature Politecnico di Torino, I3P - 17 Giugno

21 Uncertainty budget (relative - I) Error source Max error (db) Pdf Divisor Standard deviation (db) SA scale fidelity 1,0 Uniform 3 0,58 TG-SA mismatch (S) 0,97 U - shaped 2 0,68 TG-SA mismatch (S+L) 0,97 U - shaped 2 0,68 Repeatability (S) 0,01 Normal 1 0,01 Repeatability (S+L) 0,01 Normal 1 0,01 Combined std. dev. u(δ A ) 1,12 I am not satisfied! 2,2 db relative vs. 2,4 db absolute (in terms of 2 std. dev.) I hope for a much lower instrument imperfection It is evident that manufacturer specifications are too pessimistic (both on scale fidelity and TG and SA mismatch) Politecnico di Torino, I3P - 17 Giugno

22 Monte Carlo verification GUM: ( 2.24, +2,24) db GUM Supplement 1: ( 2.17, + 2,25) db pdf (1/dB) ΔA Δ A (db) Politecnico di Torino, I3P - 17 Giugno

23 I am not satisfied! 2 0 A READ + U(δ A ) -2-4 Gain [db] A READ A READ U(δ A ) 2U(δ A ) = 4,5 db db f [MHz] Politecnico di Torino, I3P - 17 Giugno

24 Why mismatch contributes two times (1 of 4)? P [dbm] db ~ 1,9 db ~ 200 MHz I deliberately mismatch TG output and SA input by using a 50 Ω feedthrough Г TG = Г SA = 1/3 20log[(1+1/9)/(1 1/9)] = 20log(10/8) = 1,94 db => peak to peak amplitude of the oscillation = 1,94 db Connection through the 50 cm cable => period of the oscillation = 200 MHz f [MHz] Politecnico di Torino, I3P - 17 Giugno

25 Why mismatch contributes two times (2 of 4)? db I add a section of cable of 1,5 m length P [dbm] ~ 1,8 db ~ 50 MHz f [MHz] Politecnico di Torino, I3P - 17 Giugno

26 Why mismatch contributes two times (3 of 4)? db Comparison short vs. (short + long) -26 P [dbm] f [MHz] Politecnico di Torino, I3P - 17 Giugno

27 Why mismatch contributes two times (4 of 4)? ~ 3,7 db Difference (short + long) short Gain [db] db f [MHz] Politecnico di Torino, I3P - 17 Giugno

28 VSWR verification VSWR measurement (10 to 1000 MHz) by using a second TG + SA combination and a directional bridge (> 40 db directivity) VSWR is much better than 2:1 (both at SA input and TG output) Internal attenuation (db) SA input RL (db) VSWR 0 > 10 < 1,92 Output power (dbm) TG output RL (db) VSWR 5 > 20 < 1,22 10 > 27 < 1,09 60 to 0 > 20 < 1,22 > 15 > 30 < 1,07 Politecnico di Torino, I3P - 17 Giugno

29 Linearity verification Linearity measurement by using a step attenuator (zero span, average over 100 sweeps) throughout the 20 db range (2 db/div) at 1 db steps from 0 to 10 db and at 2 db steps from 10 to 20 db (15 steps total) The RMS linearity error results 0,032 db (mean error = 0,01 db, max error = 0,065 db) Politecnico di Torino, I3P - 17 Giugno

30 Uncertainty budget (relative - II) Error source Max error (db) Pdf Divisor Standard deviation (db) SA scale fidelity 0,032 Normal 1 0,032 TG-SA mismatch (S) 0,039 U - shaped 2 0,027 TG-SA mismatch (S+L) 0,039 U - shaped 2 0,027 Repeatability (S) 0,01 Normal 1 0,01 Repeatability (S+L) 0,01 Normal 1 0,01 Combined std. dev. u(δ A ) 0,052 Now I am satisfied! Relative: Uncertainty passes from 2.2 db (manufacturer specifications) to 0,1 db (after VSWR and linearity verification, 2 std. dev.) Absolute: Introducing the actual VSWR and linearity error in the budget uncertainty passes from 2,4 db to 2,0 db (2 std. dev.) The absolute measurement can be further improved Politecnico di Torino, I3P - 17 Giugno

31 Monte Carlo verification pdf (1/dB) ΔA (db) Δ A (db) Politecnico di Torino, I3P - 17 Giugno

32 Now I am satisfied! A READ + U (δ A ) 1 db Gain [db] A READ U (δ A ) A READ 2U (δ A ) = 0,2 db f [MHz] Politecnico di Torino, I3P - 17 Giugno

33 Conclusion Many EMC quantities => many uncertainty budgets! EMC measurements are RF measurements We often work with a SUBSET of all the possible instrument settings and environmental parameters where instrument performance is much better than that stated by manufacturer specifications Few selected measurements and calibrations can dramatically reduce measurement uncertainty Direct link with instrument manufacturer is sometimes necessary Uncertainty evaluation is not a mathematical exercise Always start uncertainty evaluation by writing the MEASUREMENT MODEL The GUM uncertainty framework works well (also with db units!) Verification of the GUM confidence interval by using Monte Carlo method is useful in dubious cases (e.g. large uncertainty contribution from a U- shaped distributed quantity) Politecnico di Torino, I3P - 17 Giugno