Bode magnitude & phase plots
Derive Bode representations from logarithms and the pole/zero factors of chapter 03 rather than memorizing slope tables. Define amplitude-ratio decibels carefully, keep dimensional transfer functions referenced, derive exact and asymptotic first-order pole/zero contributions, explain decades and octaves, add factor contributions in magnitude and phase, quantify corner error, and work a multi-break example before using Bode as a design shortcut.
- Define decibels from a dimensionless power ratio and derive the 20log10 amplitude-ratio form under the appropriate same-quantity conditions.
- Explain why logarithms require dimensionless ratios and how dimensional transfer magnitudes such as transimpedance need an explicit reference.
- Explain why a logarithmic frequency axis turns equal frequency ratios into equal horizontal distances.
- Derive the exact Bode magnitude and phase of one real first-order pole factor.
- Derive the low- and high-frequency asymptotes and the −20 dB/decade pole slope from logarithmic limits.
- Derive the corresponding +20 dB/decade and positive phase contribution of one real first-order zero.
- Add pole/zero contributions using logarithm and argument identities rather than visual heuristics.
- Build and check a multi-break Bode response from a factored H(s), including the exact −3.010 dB first-order corner offset from the magnitude asymptote.
Chapter 03 exposed H(s) as a product of pole and zero factors. Bode plots are useful because they turn those products into sums and compress many decades of frequency onto one page. But every familiar slope rule follows from logarithms and asymptotic limits; none needs to be memorized as an isolated table.
04.1Start from the decibel definition
For a dimensionless power ratio, the decibel is defined by
If two amplitudes A₂ and A₁ are related to power through the same proportionality so that P∝A², then
For a dimensionless voltage transfer H=Vout/Vin, Bode magnitude is conventionally
This is a logarithmic representation of a voltage ratio. It is not automatically a power gain unless the impedance conditions justify that interpretation.
04.2Never take a logarithm of a dimensional quantity without a reference
A logarithm requires a dimensionless argument. Therefore an expression such as 20 log10(1000 Ω) is not dimensionally meaningful by itself.
If a transimpedance magnitude is reported relative to 1 Ω, one may write
Likewise, absolute voltage levels such as dBV use an explicit 1 V reference. AC-02 uses ordinary dB primarily for dimensionless transfer ratios unless another reference is stated.
04.3Why the frequency axis is logarithmic
Define a horizontal coordinate proportional to log10ω. Then
so every multiplication of frequency by 10 occupies the same width. That interval is one decade. A factor of 2 is one octave.
Figure 4.1 — Why is the Bode frequency axis logarithmic?
The one question is: how do equal frequency ratios become equal horizontal distances?
On a log10 axis, multiplying frequency by 10 always adds the same amount to log10(f). The intervals 10→100, 100→1000 and 1 k→10 k therefore occupy equal widths.
A logarithmic frequency axis is therefore ideal for systems whose characteristic frequencies span orders of magnitude.
04.4Derive the Bode contribution of one first-order pole
For the normalized pole factor
the exact magnitude in dB is
and the exact phase is
Now take limits rather than memorize slopes.
At high frequency, increasing Ω by one decade adds 1 to log₁₀Ω, so the asymptote changes by −20 dB per decade.
At the corner Ω=1, the exact magnitude is
while the two straight asymptotes meet at 0 dB. The familiar “−3 dB corner” is an exact property of this normalized first-order pole factor, not the definition of every possible bandwidth in every system.
04.5Derive the corresponding zero contribution
For one real first-order zero factor
the exact contributions are
Thus the high-frequency asymptotic magnitude rises at +20 dB/decade and the phase tends toward +90°.
A pole and zero at the same characteristic frequency have opposite external Bode contributions in the ideal reduced transfer.
04.6Products become sums — the real reason Bode is convenient
If
then magnitude multiplication becomes dB addition:
and complex arguments add:
Bode construction is therefore factor bookkeeping backed by exact logarithm and phase identities.
04.7Phase straight lines are approximations, not laws
The exact first-order pole phase is −atan(ω/ωp). A common hand-sketch approximation is
At 0.1ωₚ the exact phase is −5.711°, and at 10ωₚ it is −84.289°. The approximation is useful for reasoning, but exact calculations should use atan2/atan when numerical accuracy matters.
A first-order zero contributes the opposite sign.
04.8Worked multi-break example
Consider the dimensionless transfer
The low-frequency gain is
The asymptotic magnitude slopes follow directly from factor activation:
The exact magnitude expression is
and the exact phase is
Selected exact checks are:
Because the break frequencies are only one decade apart, their exact transitions overlap. The asymptotic sketch remains useful, but the exact function is the authority.
04.9A Bode plot is a model visualization, not the measurement itself
A measured network analyzer or swept-sine experiment returns sampled estimates of magnitude and phase. A Bode plot is a chosen representation of those data or of a model prediction.
Before comparing model and measurement, verify:
- frequency units and angular-frequency conversion;
- the same input/output references and terminations;
- whether plotted magnitude is linear, dB, dBV, dBΩ or another referenced quantity;
- phase branch/unwrapping choices;
- instrument bandwidth, source/load impedance and calibration plane.
A visually straight slope is not by itself proof of one ideal pole; several nearby effects can imitate an asymptote over a limited range.
04.10Mastery exercises
- Convert magnitude ratios 1, 1/√2, 0.1 and 10 to dB.
- Why is 20log10(100 Ω) incomplete? Give a dimensionally valid dBΩ expression.
- Derive the −20 dB/decade asymptote of one first-order pole from its exact expression.
- Derive the +20 dB/decade asymptote of one first-order zero.
- At one pole corner, compare exact magnitude with the straight-line asymptote.
- For H(s)=10/[(1+s/100)(1+s/10000)], give the asymptotic slopes in each frequency region.
- For the same H, state the low-frequency magnitude in dB and the high-frequency phase limit.
- Explain why exact phase should be preferred over the straight-line phase approximation when fitting measured data.
Solutions
1. 0 dB, −3.0103 dB, −20 dB and +20 dB respectively.
2. The logarithm needs a dimensionless ratio. For example 20log₁₀(|Z|/1 Ω) gives a value in dBΩ by convention.
3. At ω≫ωₚ, −10log₁₀[1+(ω/ωₚ)²]≈−20log₁₀(ω/ωₚ), hence −20 dB per factor-10 increase.
4. The zero has the opposite sign, giving +20log₁₀(ω/ω_z) at high frequency.
5. The exact value is −3.0103 dB while the low- and high-frequency magnitude asymptotes intersect at 0 dB.
6. 0 dB/dec below 100 rad/s, −20 dB/dec from 100 to 10000 rad/s, then −40 dB/dec above 10000 rad/s.
7. 20log₁₀10=20 dB; two real poles contribute a high-frequency phase limit of −180°.
8. The straight-line phase is only a sketch approximation and can bias extracted corner frequencies or delay interpretations.
Chapter 05 will close AC-02 by turning magnitude/phase curves into bandwidth and first filter interpretations, while carefully separating the common −3 dB convention from more general bandwidth definitions.