FREQUENCY RATIOS: OCTAVES vs. DECADES
Mastering the Logarithmic Language of Bode Plots
f: 100.0 Hz
10 Hz
100 Hz
1 kHz
10 kHz
LIVE CONVERSION:
+1 Octave: --
+1 Decade: --
LOG-LOG SCALE: MAGNITUDE RESPONSE
━━ Current Response
- - Octave Span
- - Decade Span
Octaves & Decades: The Logarithmic Ratios
In linear math, $100 - 50 = 50$. In engineering, we don't care about the difference; we care about the ratio. A jump from 10Hz to 20Hz is "the same size" as a jump from 10kHz to 20kHz. Both are exactly one octave.
The Octave (Ratio 2:1)
An octave is a doubling or halving of frequency.
f2 = 2 × f1
In a standard Type-1 system ($1/s$), the gain drops by 6 dB per octave.
The Decade (Ratio 10:1)
A decade is a ten-fold increase or decrease.
f2 = 10 × f1
In a standard Type-1 system ($1/s$), the gain drops by 20 dB per decade.
The Critical Math: dB/oct ↔ dB/dec
Why is -20 dB/dec the same as -6 dB/oct? We use the $20 \log_{10}$ rule. Let $N$ be the number of integrators ($1/s^N$):
Decade Calculation:
Change in dB = $20 \log_{10}(1/10^N) = -20N \text{ dB/decade}$
Octave Calculation:
Change in dB = $20 \log_{10}(1/2^N) = -20N \times 0.301 \approx -6N \text{ dB/octave}$
Change in dB = $20 \log_{10}(1/10^N) = -20N \text{ dB/decade}$
Octave Calculation:
Change in dB = $20 \log_{10}(1/2^N) = -20N \times 0.301 \approx -6N \text{ dB/octave}$
Simulator Logic:
- X-Axis (Log): Notice how the space between 10 and 100 is the same as 100 and 1000. This is because $\log_{10}(100) - \log_{10}(10) = 1$.
- The Markers: Move the slider. Observe that the Red Dot (Octave) always stays a fixed horizontal distance from the White Dot. The Blue Dot (Decade) is also at a fixed, larger distance. This proves that ratios are constant distances on a log scale.
- The Slopes: Toggle between -20 and -40 dB/dec. Note how the "steepness" doubles. -40 dB/dec is exactly equal to -12 dB/oct.