Decade (log scale)

Decade (log scale)

One decade is a factor of 10 difference between two numbers (an order of magnitude difference) measured on a logarithmic scale. It is especially useful when referring to frequencies and when describing frequency response of electronic systems, such as audio amplifiers and filters.

Calculations

The factor-of-ten in a decade can be in either direction: so one decade up from 100 Hz is 1000 Hz, and one decade down is 10 Hz. The factor-of-ten is what is important, not the unit used, so 3.14 rad/s is one decade down from 31.4 rad/s.

To determine the number of decades between two frequencies, use the logarithm of the ratio of the two values:

How many decades is it from 15 rad/s to 150,000 rad/s?
log 10(150000 / 15) = 4 decades
How many decades is it from 3.2 GHz to 4.7 MHz?
\log_{10} (4.7\times10^6 / 3.2\times10^9 ) = -2.83 decades
How many decades is one octave?
One octave is a factor of 2, so log 10(2) = 0.301 decades per octave

To find out what frequency is a certain number of decades from the original frequency, multiply by appropriate powers of 10:

What is 3 decades down from 220 Hz?
220 \times 10^{-3} = 0.22 Hz
What is 1.5 decades up from 10?
10 \times 10^{1.5} = 316.23

To find out the size of a step for a certain number of frequencies per decade, raise 10 to the power of the inverse of the number of steps:

What is the step size for 30 steps per decade?
101 / 30 = 1.079775 - or each step is 7.9775% larger than the last.
Bode plot showing the concept of a decade: each major division on the horizontal axis is one decade

Graphical representation and analysis

Decades on a logarithmic scale, rather than unit steps (steps of 1) or other linear scale, are commonly used on the horizontal axis when representing the frequency response of electronic circuits in graphical form, such as in Bode plots, since depicting large frequency ranges on a linear scale is often not practical. For example, an audio amplifier will usually have a frequency band ranging from 20 Hz to 20 kHz and representing the entire band using a decade log scale is very convenient. Typically the graph for such a representation would begin at 1 Hz (100) and go up to perhaps 100 kHz (105), to comfortably include the full audio band in a standard-sized graph paper, as shown below. Whereas in the same distance on a linear scale, with 10 as the major step-size, you might only get from 0 to 50.

1,10,100,1k,10k,100k using decades vs. 0,10,20,30,40,50 using linear scale

Electronic frequency responses are often described in terms of "per decade". The example Bode plot shows a slope of -20 dB/decade in the stopband, which means that for every factor-of-ten increase in frequency (going from 10 rad/s to 100 rad/s in the figure), the gain decreases by 20 dB.

See also


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Decade (disambiguation) — A decade is a period of ten years. Decade or Decades may also refer to: Music Decade (Neil Young album), 1977 Decade: Greatest Hits, an album by Duran Duran A Decade, an album by Our Lady Peace A Decade , a single by DJ Marcel Woods Decades , a… …   Wikipedia

  • Décade (physique) — Une décade est un facteur de 10 entre deux nombres. C est un concept important dans les représentations graphiques de type logarithmiques, en particulier pour les fréquences, par exemple lorsque nous décrivons la réponse en fréquence d un système …   Wikipédia en Français

  • Logarithmic scale — A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. Presentation of data on a logarithmic scale can be helpful when the data covers a large range of values ndash; the… …   Wikipedia

  • Cosmological decade — A cosmological decade ( CÐ ) is a division of the lifetime of the cosmos. The divisions are logarithmic in size, on base 10. Each successive cosmological decade represents a ten fold increase in the total age of the universe.[1] As expressed in… …   Wikipedia

  • List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia

  • Slide rule — For other uses, see Slide rule (disambiguation). A typical ten inch student slide rule (Pickett N902 T simplex trig). The slide rule, also known colloquially as a slipstick,[1] is a mechanical analog computer. The slide rule is used primarily for …   Wikipedia

  • Moore's law — Plot of CPU transistor counts against dates of introduction. Note the logarithmic vertical scale; the line corresponds to exponential growth with transistor count doubling every two years …   Wikipedia

  • Step response — The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the… …   Wikipedia

  • Image scanner — Desktop scanner, with the lid raised. An object has been laid on the glass, ready for scanning …   Wikipedia

  • Otis King — s patent calculator, model K; Photo by Richard Lyon. Otis Carter Formby King (1876–??) was a grocer and engineer in London who invented and produced a cylindrical slide rule with helical scales, primarily for business uses initially. The product… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”