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Allan Deviation

Allan Deviation. J. Mauricio López R. Time and Frequency Division Centro Nacional de Metrología, CENAM jlopez@cenam.mx. Real Clocks.

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Allan Deviation

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  1. Allan Deviation J. Mauricio López R. Time and Frequency Division Centro Nacional de Metrología, CENAM jlopez@cenam.mx

  2. Real Clocks There not exist the perfect clock, all the real clocks are unestable. The output frequency of a clock changes with time. A correct mathematical tool is needed to characterise the frequency instability of oscillators.

  3. +A -A A: amplitude : frequency t: time Mathematical model for ideal frequency signal

  4. Quartz oscillators are a very common equipment to produce frequency signals but their frequency output could be more or less instable as function of many parameters

  5. V(t) =[V0 + (t)] sen[20t + (t)] tiempo V(t) = V0 sen(20t) tiempo Mathematical model for real (instable) frequency signals

  6. The Time – Frequency relation Frequency Period

  7. Frecuencia estable ( oscilador ideal) F (t) 1 V - 1 T T T 3 1 2 Tiempo pn F pn V(t) = V sin(2 t) (t) = 2 t 0 0 0 Frecuencia inestable ( Oscilador real) F (t) 1 V - 1 T T T 1 2 3 Tiempo e pn f V(t) =[V + (t)] sin[2 t + (t)] F pn f (t) = 2 t + (t) 0 0 0 1 1 d ( t ) d ( t ) Φ f n n + ( t ) = = d t d t 0 2 2 π π V(t) = salida del oscilador , V = Amplitud nominal pico - a - pico 0 e n (t) = amplitud de ruido , = frecuencia nominal 0 F f (t) = Fase , (t) = ruido de fase The time – frequency relation

  8. V(t) =[V0 + (t)] sen[20t + (t)]

  9. Exactitud y Estabilidad Ni exactitud ni precisión Precisión sin exactitud Exactitud sin precisión Exacto y preciso f f f f 0 Tiempo Tiempo Tiempo Tiempo Alta exactitud a largo tiempo e inestable a corto tiempo Estable de baja exactitud Inestable de baja exactitud Alta estabilidad y alta exactitud

  10. Short term instability 30 25 20 f/f (ppm) 15 10 Time / Days 10 15 20 25 5 Aging and short term stability

  11. An example of noise in measurements

  12. 3 X 10-11 0.1 s average time 0 -3 X 10-11 100 s 3 X 10-11 1.0 s average time 0 100 s -3 X 10-11 Frequency noise 4-23

  13. Frequency noise Sz(f) = hf  = 0  = -1  = -2  = -3 name White Flicker Random walk Time dependence Las graficas muestran las fluctuaciones de la variable z(t), la cual puede ser, por ejemplo, la salida de un contador (f vs. t), o la medición de fase ([t] vs. t). Los gráficos muestran tanto la dependencia temporal como la dependencia en frecuencia; h es el coeficiente de amplitud.

  14. Concepto de la Varianza de Allan Fase

  15. - N 1 1 ( ) ( ) å 2 2 t = - σ y y ( ) + y i 1 i - 2 N 1 = i 1 Allan variance donde: Allan variance Number of measurements Frequencymeasurement

  16. Allan variance donde: Allan variance Phase measurement Númber of measurements Time window = mt0

  17. Distribución c2 Para df < 100 donde: Estimado de la Varianza de Allan Número de grados de libertad Varianza de Allan verdadera Barras de Incertidumbre

  18. Barras de Incertidumbre Distribución X2

  19. ( ) ( ) 2 2 s df s df 2 y < s < y ( ) ( ) y c c 2 2 0 . 975 0 , 025 Barras de Incertidumbre Barra Inferior Barra Superior Tablas X2

  20. Tabla X2

  21. Para df > 100 Barras de incertidumbre

  22. Para df > 100 Barra Superior 1 ( ) ( ) 2 c = - 2 0 , 025 h 1 , 96 2 Barra Inferior 1 ( ) ( ) 2 c = + 2 0 , 975 h 1 , 96 donde: 2 = - h 2 df 1 Barras de incertidumbre

  23. ( ) ( ) + - N 1 N 2 m = df ( ) White Phase Modulation - 2 N m ( ) ( ) é ù - + - æ ö æ ö N 1 2 m 1 N 1 Flicker Phase Modulation = ç ÷ ç ÷ df exp ln ln ê ú 2 n 4 è ø è ø ë û ( ) ( ) - - 2 é ù 3 N 1 2 N 2 4 m White Frquency Modulation = - df ê ú + 2 2 m N 4 m 5 ë û NBS Technical note 679 Número de Grados de Libertad

  24. Flicker Frequency Modulation Random-Walk Frequency Modulation NBS Technical note 679 Número de Grados de Libertad

  25. Time window dependence of the y() -1 y() -1 12 -12 0 Tipo de ruido: White phase Flicker phase White freq. Flicker freq. Random walk freq. Por debajo del ruido “fliker”, los cristales de cuarzo tipicamente tienen una dependencia -1 (white phase noise). Los patrones atómicos de frecuencia muestran una dependencia del tipo -1/2 (white frequency noise) para tiempos de promediación cercanos al tiempo de ataque del lazo de amarre, y -1 para tiempos menores del tiempo de ataque. Tipicamente los ’s para el ruido flicker son: 1 s para osciladores de cuarzo, 103s para relojes de rubidio y 105s para Cesio.

  26. Example of Allan deviationcalculation

  27. Example of Allan deviationcalculation

  28. Example of Allan deviationcalculation

  29. Allan Deviation J. Mauricio López R. Time and Frequency Division Centro Nacional de Metrología, CENAM jlopez@cenam.mx

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