# Graphical Conventions Across Disciplines

*Every field has its own favourite plot style for a complex response
function of ω. They are all the same information — a 2-D curve traced as
ω sweeps — displayed through disciplinary habits of the eye.*

## The object being plotted

In every case the underlying datum is a complex-valued function χ(ω) (or
H, Z, ε, n, S_ij, …) of real frequency ω. That is four real numbers of
interest at each ω: Re, Im, |·|, and arg. K–K says the four are **not
independent** — knowing two of them on the real axis pins down the other
two. Different plotting traditions emphasise different projections of
this common object.

## Bode plot — electrical engineering and controls

Two stacked panels, both with `log ω` on the x-axis:

- Top: `20 log₁₀ |H(ω)|` in dB.
- Bottom: phase `∠H(ω)` in degrees.

Features the eye is trained on: straight-line asymptotes at ±20 dB/decade
per pole/zero, corner frequencies at pole/zero locations, characteristic
−90°/pole phase drops. A first-order low-pass shows a textbook −3 dB
knee with a 45° phase at the corner; a resonant second-order system
shows a peak whose height is set by Q.

The Bode **gain–phase theorem** is K–K in this representation: for
minimum-phase systems, `∠H(ω)` is determined by `log |H(ω)|` (and vice
versa) via a Hilbert transform in log-frequency. Non-minimum-phase
zeros — all-pass filters, RHP zeros, transport delays — add phase lag
that is *not* implied by the gain, and are the practical exception.

## Nyquist / polar plot — controls and impedance spectroscopy

A single 2-D plot, Im vs Re of the complex response, parametrised by ω.
In controls the plot is of `H(jω)` and the eye looks for **encirclements
of −1** (Nyquist stability criterion). In electrochemical impedance
spectroscopy the plot is of Z(ω) with the convention of plotting `−Im Z`
upward, because capacitive impedances have `Im Z < 0` and experimenters
prefer arcs that go *up*.

An RC circuit traces a semicircle from (R, 0) at high ω down to the
origin at ω → 0; a Warburg diffusion element traces a 45° line. An
electrochemical cell with charge-transfer plus diffusion gives a
semicircle smoothly morphing into a 45° tail — instantly recognisable to
a corrosion engineer.

## Cole–Cole plot — dielectric spectroscopy

The complex relative permittivity `ε_r(ω) = ε'_r − jε''_r` (engineering
convention common in this community) is plotted as **−ε'' on the
vertical axis vs ε' on the horizontal axis**, with ω implicit along the
curve. A single Debye relaxation

```
ε_r(ω) = ε_∞ + (ε_s − ε_∞) / (1 + jωτ)
```

produces a **perfect semicircle** of diameter `ε_s − ε_∞` sitting on the
real axis between ε_∞ (ω → ∞) and ε_s (ω → 0). The top of the arc is at
ωτ = 1 — the relaxation frequency is read directly off the plot.

Real materials rarely give a clean semicircle. The community has a zoo
of empirical extensions, each with a standard plot shape:

- **Cole–Cole** (1941): `ε_r = ε_∞ + (ε_s − ε_∞) / [1 + (jωτ)^{1−α}]`.
  The semicircle is *depressed*: its centre sits below the real axis,
  and the full shape is still an arc but flatter. α ∈ [0, 1) measures
  the broadening from a distribution of relaxation times.
- **Cole–Davidson** (1950): `ε_r = ε_∞ + (ε_s − ε_∞) / (1 + jωτ)^β`.
  Asymmetric, skewed arc — high-frequency side is elongated.
- **Havriliak–Negami** (1967): combines both, `… / [1 + (jωτ)^{1−α}]^β`.
  The most general of the four; fits polymer relaxations particularly
  well.

An experienced dielectric spectroscopist reads α and β off the shape of
the arc at a glance. K–K is satisfied automatically because each of
these functional forms is the frequency-domain response of a causal
linear system.

## Dispersion and extinction curves — photonics and solid-state

Optical constants are traditionally plotted as two linear panels, x-axis
either ω, or photon energy ℏω in eV, or wavelength λ in nm:

- **n(ω)** — real part of the refractive index, "dispersion curve".
- **κ(ω)** — imaginary part of the refractive index, "extinction
  coefficient". Sometimes `α(ω) = 2ωκ/c` is plotted instead, the
  absorption coefficient in cm⁻¹.

Around a resonance at ω₀ the eye expects the textbook pairing: κ(ω) is a
Lorentzian-like peak; n(ω) climbs below the resonance, dips sharply
through ω₀ (anomalous dispersion), then settles to a new value above.
Databases like Palik and Adachi present optical constants in exactly
this two-panel linear form for every material of interest. K–K is the
tool that checks consistency between the two panels and reconstructs one
from the other when only reflectance data is available.

## Susceptibility spectrum — spectroscopy

Lineshape plots of `χ″(ω)` vs ω appear all over spectroscopy: IR
absorption, NMR, Raman, EPR, optical spectra. The eye looks for peaks,
their positions (level energies), widths (decay rates, Q factors), and
integrated areas (transition matrix elements and, via sum rules, total
oscillator strength). χ′(ω) is usually shown alongside or implied; the
pair together is sometimes called a dispersion–absorption plot.

## S-parameters and the Smith chart — microwave engineering

At microwave frequencies the fundamental measured quantity is the
scattering matrix S_ij(ω). Two display traditions dominate:

- **Magnitude + phase** plots: `|S_ij|` in dB and `∠S_ij` in degrees,
  both vs frequency on a linear or log axis. This is the microwave
  Bode plot.
- **Smith chart**: a conformal map of the complex Γ-plane (reflection
  coefficient) onto the unit disk, with grid lines of constant
  normalised resistance and reactance. An impedance-matching engineer
  reads off series/parallel L/C transformations as motions along these
  grid lines.

A passive microwave network has `|S_ii| ≤ 1`, so on the Smith chart
every physical input reflection sits inside the unit circle. A
narrowband resonance traces a loop; a broadband termination traces a
short arc. K–K applied to S-parameters (Chapter on microwave networks)
says the magnitude and phase of each `S_ij(ω)` are linked by a Hilbert
transform plus whatever non-minimum-phase zeros the network contains.

## They are all the same function

Figure these out and the K–K literature stops looking like five
unrelated subjects. A Lorentzian dielectric resonance shows up as:

- Bode: a peak in gain, a ±90° phase swing through resonance.
- Nyquist: an arc hooking below the real axis.
- Cole–Cole: a perfect semicircle from ε_∞ to ε_s.
- n/κ: a peak in κ, an anomalous-dispersion wiggle in n.
- Smith chart: a loop on the chart.

Every one of these pictures encodes the same complex function of ω, and
every one is constrained by Kramers–Kronig in the same way. Switching
freely between them is one of the most practically useful skills the
K–K framework confers.

## See also

- [16_sign_conventions.md](16_sign_conventions.md)
- [18_named_quantities.md](18_named_quantities.md)