condensed_matter

The general principle underlying all of these is that when a dimension becomes comparable to
the electron wavelength, the continuum of states in that direction collapses into discrete
levels, and the density of states changes shape. That change in the density of states alters
optical absorption, electrical transport, and heat capacity together, which is why
low-dimensional systems are not merely small versions of bulk materials but genuinely
different ones — the same collective-behaviour lesson that opened the subject, appearing again
at a different scale.


electromagnetism

## Electrostatics

Coulomb's law gives the force between two point charges as k q1 q2 / r^2 along the line
joining them, with k = 1/(4 pi eps_0) approximately 8.99e9 N m^2/C^2. Its inverse-square
form is not an approximation but has been verified to extraordinary precision, and that
precision matters because the exponent being exactly two is equivalent to the photon
being exactly massless. The law's structural similarity to Newtonian gravitation is
useful, but the differences dominate practice: charge comes in two signs so
electrostatic forces can cancel, and the electric force between two protons is roughly
10^36 times their gravitational attraction, which is why gravity is irrelevant at
atomic scales and why bulk matter is electrically neutral to extraordinary accuracy.


electromagnetism

The electric field E is force per unit charge, a vector field defined at every point
whether or not a test charge is present. Treating the field as physically real rather
than as a bookkeeping device is not merely a preference: fields carry energy and
momentum, and they propagate at finite speed, so the interaction between two charges is
mediated locally rather than acting at a distance. Field lines visualize the field —
originating on positive charge, terminating on negative, never crossing, with density
indicating magnitude.


electromagnetism

Gauss's law states that the electric flux through any closed surface equals the enclosed
charge divided by eps_0. It is equivalent to Coulomb's law for static fields but is far
more powerful when symmetry is available, because a well-chosen surface makes the flux
integral trivial. Three canonical results follow in a line each: a spherically symmetric
charge distribution produces an external field identical to a point charge at its
centre; an infinite line of charge gives a field falling as 1/r; an infinite plane gives
a uniform field independent of distance. That last result is the basis of the
parallel-plate capacitor and is worth internalizing because the distance-independence
surprises people every time.


electromagnetism

Electrostatics reference:
  Coulomb force          F = k q1 q2 / r^2,  k = 1/(4 pi eps_0) ≈ 8.99e9
  electric field         E = F/q;  point charge E = k q / r^2
  Gauss's law            flux of E through closed surface = Q_enc / eps_0
  potential              V = k q / r;  E = -grad V
  potential energy       U = q V;  for two charges U = k q1 q2 / r
  capacitance            C = Q/V;  parallel plate C = eps_0 eps_r A / d
  stored energy          U = (1/2) C V^2 = (1/2) Q V = Q^2/(2C)
  energy density         u = (1/2) eps_0 E^2
  dipole moment          p = q d;  torque in a field = p cross E


electromagnetism

Electric potential is the potential energy per unit charge and is a scalar, which makes
it far easier to work with than the field. Superposition of potentials is a simple sum
where superposition of fields requires vector addition, so the standard technique for a
complicated charge distribution is to compute the potential and then take the negative
gradient. Only potential differences are physical; the zero point is a convention,
usually taken at infinity for localized distributions and at "ground" in circuits.


electromagnetism

Q: A hollow conducting sphere carries charge Q. The electric field inside the cavity,
with no charge in the cavity, is:
A. kQ/r^2 pointing outward   B. Zero   C. Uniform and nonzero   D. Infinite at the
centre
Reasoning: Apply Gauss's law to a surface inside the conductor's material: the field
there is zero in electrostatic equilibrium, so the enclosed charge is zero, so all
charge resides on the outer surface. A Gaussian surface inside the cavity encloses no
charge and by symmetry the field vanishes. This is electrostatic shielding — the
principle behind the Faraday cage and behind shielded cable. Answer: B.


electromagnetism

Q: Two capacitors, 2 uF and 4 uF, are connected in series across 12 V. The charge on
the 2 uF capacitor is:
A. 24 uC   B. 16 uC   C. 48 uC   D. 8 uC
Reasoning: Series capacitors carry equal charge. The equivalent capacitance is
(2*4)/(2+4) = 4/3 uF, so Q = C V = (4/3)(12) = 16 uC on each. Note this is the reverse
of resistors, where series elements share current and parallel ones share voltage; for
capacitors the series combination adds reciprocals. Answer: B.


electromagnetism

## Conductors, Dielectrics, and Circuits

In electrostatic equilibrium a conductor has zero internal field, since any field would
drive its free charges until it is cancelled. Three consequences follow: all excess
charge resides on the surface, the surface is an equipotential, and the field just
outside is perpendicular to the surface with magnitude sigma/eps_0. Charge concentrates
where curvature is greatest, so the field is strongest at points and edges — which is
why lightning rods are pointed, why high-voltage hardware uses large-radius corona
rings, and why a sharp burr on a PCB trace is where dielectric breakdown starts.


electromagnetism

A dielectric placed in a field polarizes: its molecules develop or align dipole moments,
producing a bound surface charge that partially cancels the applied field. The internal
field is reduced by the relative permittivity eps_r, so a capacitor with a dielectric
holds eps_r times more charge at the same voltage. Real dielectrics also have a
breakdown strength — air about 3 kV/mm, common polymers 15-25 kV/mm — beyond which they
ionize and conduct, which together with eps_r sets the fundamental size-versus-voltage
tradeoff for every capacitor.


electromagnetism

Circuit analysis reduces field theory to lumped elements under the quasi-static
approximation, valid when the circuit is small compared with the wavelength at the
operating frequency. Ohm's law V = IR is a material property rather than a law of
nature, holding for ohmic conductors over a limited range and failing for
semiconductors, gas discharges, and anything heated by its own current. Kirchhoff's laws
are the real foundation: current into a node equals current out (charge conservation),
and voltages around a loop sum to zero (energy conservation, or equivalently the
electrostatic field being conservative).


electromagnetism

Circuit relations:
  Ohm's law            V = I R
  power                P = V I = I^2 R = V^2 / R
  resistance           R = rho L / A
  series resistors     R = R1 + R2 + ...
  parallel resistors   1/R = 1/R1 + 1/R2 + ...
  series capacitors    1/C = 1/C1 + 1/C2 + ...
  parallel capacitors  C = C1 + C2 + ...
  RC time constant     tau = RC; 63% in one tau, 99% in five
  RL time constant     tau = L/R
  inductor voltage     V = L dI/dt
  capacitor current    I = C dV/dt
  impedance            Z_R = R, Z_C = 1/(j omega C), Z_L = j omega L
  resonance            omega_0 = 1/sqrt(LC)


electromagnetism

Impedance extends resistance to sinusoidal steady state by using complex numbers, which
turns differential equations into algebra. A capacitor's impedance falls with frequency,
so it blocks DC and passes high frequencies; an inductor's rises, so it passes DC and
blocks high frequencies. This immediately gives filters: a series resistor with a shunt
capacitor is a low-pass with corner frequency 1/(2 pi RC), and swapping them gives a
high-pass. Series RLC resonance occurs where the capacitive and inductive reactances
cancel, leaving only resistance, so the impedance is minimized and current peaks.


electromagnetism

The quasi-static assumption fails as frequency rises and is the boundary between circuit
theory and transmission-line theory. The rule of thumb is that a conductor must be
treated as a transmission line when its length exceeds roughly a tenth of the signal
wavelength — or, for digital signals, when the propagation delay exceeds about a sixth
of the edge rise time, since a fast edge contains high-frequency content regardless of
the clock rate. This is why a 10 MHz clock on a short board is a simple net while a 1 ns
edge on the same trace requires controlled impedance and termination.


electromagnetism

## Magnetostatics and Induction

Magnetic fields are produced by moving charge. The Biot-Savart law gives the field from
a current element, and Ampere's law — the circulation of B around a closed loop equals
mu_0 times the enclosed current — plays the role Gauss's law plays in electrostatics,
making symmetric problems easy. A long straight wire gives B = mu_0 I/(2 pi r); a
solenoid gives a nearly uniform interior field B = mu_0 n I depending only on turns per
unit length; a toroid confines the field entirely within itself, which is why toroidal
inductors radiate so little and are preferred where interference matters.


electromagnetism

The magnetic force on a moving charge is q v cross B, perpendicular to both velocity and
field. Because it is always perpendicular to velocity, the magnetic force does no work —
it changes direction but never speed. A charge moving perpendicular to a uniform field
therefore travels in a circle of radius r = mv/(qB), the cyclotron radius, at a
frequency independent of speed, which is what makes the cyclotron work and what
determines confinement in magnetic traps and mass spectrometers.


electromagnetism

There are no magnetic monopoles, so far as any experiment has shown, which is the
content of Gauss's law for magnetism: the flux of B through any closed surface is zero.
Field lines therefore always close on themselves. The simplest magnetic source is the
dipole, produced by a current loop, and its field falls as 1/r^3 rather than the 1/r^2
of an electric monopole — which is why magnetic interactions fall off faster with
distance and why magnetic shielding works differently from electrostatic shielding,
requiring high-permeability material to redirect flux rather than a conductor to
terminate field lines.


electromagnetism

Faraday's law of induction states that a changing magnetic flux through a circuit
induces an electromotive force equal to the negative rate of change of flux. Lenz's law
is the meaning of that minus sign: the induced current opposes the change producing it,
which is required by energy conservation — an induced current that reinforced its cause
would produce runaway energy from nothing. Flux can change by changing the field, the
area, or the orientation, which correspond respectively to transformers, to sliding-bar
generators, and to rotating-coil alternators.


electromagnetism

Magnetic quantities:
  force on charge        F = q v cross B
  force on wire          F = I L cross B
  Biot-Savart            dB = (mu_0/4 pi) I dl cross r_hat / r^2
  Ampere's law           circulation of B = mu_0 I_enc
  long wire              B = mu_0 I / (2 pi r)
  solenoid               B = mu_0 n I  (interior, long solenoid)
  Faraday's law          emf = -d(flux)/dt
  motional emf           emf = B L v
  inductance             L = flux linkage / I;  emf = -L dI/dt
  stored energy          U = (1/2) L I^2
  energy density         u = B^2 / (2 mu_0)
  mutual inductance      transformer: V_s/V_p = N_s/N_p


electromagnetism

Eddy currents are induced in any bulk conductor experiencing changing flux, and they
dissipate energy resistively. This is a loss mechanism in transformer cores, addressed
by laminating the core so induced current paths are interrupted, or by using ferrite
whose high resistivity suppresses them. It is also the operating principle of induction
heating, induction cooktops, metal detectors, eddy-current brakes, and the damping in
analog meter movements — a good example of the same physics being a defect in one
context and the product in another.


electromagnetism

[user] I'm getting 40 mV of noise on a sensor line running parallel to a motor cable
for about 300 mm. What's coupling in and how do I fix it?
[assistant] tool_call em.coupling_analysis({"parallel_length_mm": 300,
"separation_mm": 8, "aggressor": {"type": "pwm_motor", "current_a": 3.2,
"di_dt_a_us": 45, "freq_khz": 20}, "victim": {"impedance_ohms": 10000,
"observed_noise_mv": 40}})
[tool] {"mutual_inductance_nh": 62, "inductive_coupling_mv": 2.8,
"capacitive_coupling_pf": 4.1, "capacitive_noise_mv": 37, "dominant":
"capacitive", "victim_impedance_note": "high_z_amplifies_capacitive_coupling"}
[assistant] It is capacitive coupling, not magnetic, and your 10 kilohm sensor
impedance is what makes it severe.


electromagnetism

The breakdown: mutual inductance between the two runs is about 62 nH, and at your 45
A/us switching rate that induces only 2.8 mV. Capacitive coupling through roughly 4 pF
of mutual capacitance accounts for 37 mV of the 40 you measured. The reason capacitive
coupling dominates is your victim impedance — displacement current through the mutual
capacitance develops a voltage across whatever impedance it finds, and 10 kilohms
develops a thousand times more voltage than 10 ohms would for the same current.


electromagnetism

Fixes, in order of effectiveness for capacitive coupling specifically:


electromagnetism

Lower the victim impedance. This is the biggest lever and often the easiest. Buffer the
sensor with an op-amp near the sensor itself so the long run is driven from a low
impedance. Dropping from 10 k to 100 ohms cuts the coupled voltage by a factor of a
hundred.


electromagnetism

Add an electrostatic shield. A grounded shield around the sensor line intercepts the
displacement current and returns it to ground instead of to your signal. Ground the
shield at one end only for low-frequency signals to avoid a ground loop. Note this is
the fix for capacitive coupling; it does almost nothing for magnetic coupling, which
needs twisted pair or a high-permeability shield instead.


electromagnetism

Increase separation and reduce parallel run length. Capacitance falls with separation,
so routing the sensor line away from the motor cable — ideally crossing at right angles
rather than running parallel — helps directly.


electromagnetism

Add a low-pass filter at the receiver. Your signal is presumably far below 20 kHz, so an
RC filter with a corner around 1 kHz removes the switching noise entirely. Do this even
after the other fixes; it costs two components.


electromagnetism

If it had been inductive coupling the prescription would be different — twisted pair,
minimized loop area, and slower switching edges — which is why identifying which
mechanism dominates is the first step rather than applying all the remedies at once.


electromagnetism

## Maxwell's Equations and Electromagnetic Waves

Maxwell's four equations describe all classical electromagnetism. Gauss's law relates
electric flux to charge. Gauss's law for magnetism states that magnetic flux through a
closed surface is zero, encoding the absence of monopoles. Faraday's law relates the
circulation of E to changing magnetic flux. The Ampere-Maxwell law relates the
circulation of B to current plus the rate of change of electric flux.


electromagnetism

That last term, the displacement current, was Maxwell's addition and it is the one that
changed physics. Ampere's law as originally stated is inconsistent for time-varying
currents — apply it to a surface passing between the plates of a charging capacitor and
it gives a different answer than a surface cutting the wire. Adding a term proportional
to the rate of change of electric flux restores consistency, and it makes the equations
symmetric: a changing magnetic field produces an electric field, and a changing electric
field produces a magnetic field. Each sustaining the other permits a self-propagating
disturbance.


electromagnetism

Maxwell's equations, differential form in vacuum:
  div E   = rho / eps_0                      Gauss
  div B   = 0                                no monopoles
  curl E  = -dB/dt                           Faraday
  curl B  = mu_0 J + mu_0 eps_0 dE/dt        Ampere-Maxwell
  Lorentz force  F = q(E + v cross B)
  wave speed     c = 1/sqrt(mu_0 eps_0) ≈ 2.998e8 m/s
  in a medium    v = c/n,  n = sqrt(eps_r mu_r)
  impedance of free space  Z_0 = sqrt(mu_0/eps_0) ≈ 377 ohms


electromagnetism

Taking the curl of Faraday's law and substituting Ampere-Maxwell yields the wave
equation, with propagation speed 1/sqrt(mu_0 eps_0). That this computed number matched
the measured speed of light was the moment optics became a branch of electromagnetism —
one of the great unifications in physics, and one obtained by manipulating equations
rather than by experiment.


electromagnetism

Electromagnetic waves are transverse, with E and B perpendicular to each other and to
the direction of propagation, in phase, with magnitudes related by E = cB. They carry
energy at a rate given by the Poynting vector S = (1/mu_0) E cross B, and they carry
momentum, which is why radiation pressure exists and why solar sails work. The full
spectrum from radio through gamma rays differs only in frequency; the same equations
govern all of it, and the apparent differences in behaviour come from how each frequency
range interacts with matter of a given structure size.


electromagnetism

Polarization describes the orientation of the E field. Unpolarized light has random
orientation; a linear polarizer transmits one component, giving Malus's law where the
transmitted intensity is proportional to cos^2 of the angle between the polarizer and
the incident polarization. Circular polarization results from two perpendicular
components of equal amplitude and 90 degrees relative phase. Polarization is why
sunglasses reduce glare (light reflected at shallow angles is partially polarized
horizontally), how LCD displays work, and how radio links reuse frequency by
transmitting orthogonal polarizations on the same channel.


electromagnetism

## Radiation and Antennas

Accelerating charge radiates. A static charge has a static field, a uniformly moving
charge carries its field along with it, but an accelerating charge produces a
disturbance that detaches and propagates away carrying energy. The Larmor formula gives
the radiated power as proportional to the square of the acceleration, which is why
radiation is negligible for slowly varying currents and dominant at high frequency.
This single fact is why classical physics fails for atoms — an orbiting electron
accelerates and should radiate away its energy in about 10^-11 seconds — and it was one
of the principal motivations for quantum mechanics.


electromagnetism

An antenna converts guided waves on a transmission line into radiated waves in space,
and reciprocity guarantees that its transmitting and receiving properties are identical.
The half-wave dipole is the reference element: a conductor about half a wavelength long,
fed at the centre, with a radiation resistance near 73 ohms — conveniently close to
common coaxial cable impedances — and a doughnut-shaped pattern with nulls along its
axis. A quarter-wave monopole over a ground plane behaves like half a dipole, with about
36 ohms radiation resistance, and its performance depends critically on the ground plane
being adequate, which is why a handheld radio's antenna works differently depending on
how it is held.


electromagnetism

Antenna gain is not amplification; it is a redistribution of the same radiated power
into a narrower beam. An antenna with 10 dBi gain concentrates power ten times more
intensely in its main lobe than an isotropic radiator would, necessarily at the expense
of other directions. Directivity and beamwidth are inversely related — higher gain means
a narrower beam and tighter pointing requirements — which is the central tradeoff in
link design, and why a satellite dish must be aimed carefully while an omnidirectional
whip need not.


electromagnetism

Antenna and link quantities:
  wavelength             lambda = c / f;  300/f_MHz gives lambda in metres
  half-wave dipole       L ≈ 0.48 lambda, R_rad ≈ 73 ohms, gain 2.15 dBi
  quarter-wave monopole  L ≈ 0.24 lambda over a ground plane, R_rad ≈ 36 ohms
  gain vs aperture       G = 4 pi A_eff / lambda^2
  free-space path loss   FSPL_dB = 20 log10(d) + 20 log10(f) + 32.45  (km, MHz)
  Friis equation         P_r = P_t G_t G_r (lambda / (4 pi d))^2
  VSWR                   ratio of max to min standing-wave voltage; 1:1 is a
                         perfect match, 2:1 reflects about 11% of power
  return loss            dB below incident power reflected; higher is better
  skin depth             delta = sqrt(2 rho / (omega mu)); current flows in a thin
                         surface layer at high frequency


electromagnetism

Impedance matching is what determines whether power reaches the antenna or reflects
back down the feedline. A mismatch creates standing waves, so some power returns to the
transmitter — potentially damaging it — and the mismatch loss compounds with cable loss.
Matching networks, baluns, and antenna tuners all exist to transform the antenna's
complex impedance to the system impedance, conventionally 50 ohms for radio and 75 ohms
for video. The 50 ohm convention is itself a compromise: minimum coaxial cable
attenuation occurs near 77 ohms and maximum power handling near 30 ohms, and 50 is
roughly the geometric compromise between them.


electromagnetism

Skin effect concentrates high-frequency current in a thin surface layer, so the
effective resistance rises with the square root of frequency. At 1 MHz in copper the
skin depth is about 65 micrometres and at 1 GHz about 2 micrometres, which means the
interior of a conductor is entirely unused at radio frequencies. The practical
consequences: hollow tubing performs as well as solid rod for RF, silver plating on
a base conductor is effective because only the surface carries current, litz wire with
many individually insulated strands reduces loss at intermediate frequencies, and PCB
trace surface roughness becomes a significant loss mechanism in multi-gigahertz designs
because the current is forced to follow the longer rough path.


energy_systems

## Thermodynamic Limits and Energy Accounting

Energy analysis begins with quantities and units, and confusion between them is the source of most
errors in public discussion of the subject. Energy is measured in joules or kilowatt-hours; power
is energy per unit time, measured in watts. A battery stores energy; a generator delivers power. A
statement that a plant produces some number of megawatts describes its rate, and multiplying by
hours of operation gives the energy delivered — a distinction that matters enormously when
comparing intermittent sources with continuous ones.


energy_systems

The second law imposes limits that no engineering can circumvent. A heat engine converting thermal
energy to work is bounded by the Carnot efficiency, which depends only on the ratio of the cold
and hot reservoir temperatures. Real engines fall well below it because of irreversibilities, but
the bound explains the structure of every thermal plant: raising the hot temperature is the primary
route to efficiency, which is why materials science limits determine power plant performance.


energy_systems

Exergy is the more useful accounting concept for systems analysis because it measures the fraction
of energy that can actually be converted to work. A joule of high-temperature heat and a joule of
low-temperature heat are equal in energy and very unequal in exergy, which is why heating a
building with electricity generated from heat is thermodynamically wasteful — high-grade energy is
degraded to a task requiring only low-grade energy. A heat pump avoids this by moving heat rather
than producing it, which is why it delivers several units of heat per unit of electricity without
violating anything.


energy_systems

Fundamentals and limits:
  energy               joules or kilowatt-hours
  power                energy per time; watts
  capacity             the maximum power rating
  capacity factor      actual energy delivered divided by rated power times time
  the common confusion  comparing capacity between intermittent and continuous
                       sources without capacity factors is meaningless
  Carnot efficiency    one minus the ratio of cold to hot absolute temperature
  the practical consequence  raise the hot temperature; materials limit it
  combined cycle       a second engine on the first one's waste heat; the highest
                       thermal efficiencies achieved
  cogeneration         use the waste heat directly; total utilization far above
                       electrical efficiency alone
  exergy               the work-capable fraction; the correct currency for
                       systems analysis
  energy grade         high-grade electricity degraded to low-grade heat is a
                       thermodynamic waste even when it is convenient
  heat pump            moves heat rather than generating it; coefficient of
                       performance above one is not an anomaly
  energy return on investment  energy delivered over energy invested to build and
                       operate; a system-level metric that constrains what is
                       viable


energy_systems

Q: A heat pump delivers 3.5 kWh of heat per kWh of electricity consumed. This:
A. Violates conservation of energy
B. Is normal; the device moves existing heat from a colder reservoir rather than converting
   electricity into heat, so the delivered heat exceeds the electrical input without creating
   energy
C. Indicates a measurement error
D. Only works in warm climates
Reasoning: A heat pump is a refrigerator run in reverse. The electrical work drives a thermodynamic
cycle that transfers heat up a temperature gradient; the heat delivered is the heat extracted plus
the work input. The coefficient of performance falls as the temperature difference grows, which is
why cold-climate performance is a real design concern, but values well above one are routine.
Answer: B.


energy_systems

## Generation Technologies

Thermal generation dominates historically and burns a fuel to raise steam or drive a gas turbine.
Its characteristics are dispatchable output, substantial thermal inertia limiting how quickly it can
change, and efficiency determined by the top cycle temperature. Combined cycle gas plants reach the
highest thermal efficiencies of any deployed technology by running a gas turbine and then using its
exhaust heat to raise steam for a second turbine.


energy_systems

Nuclear fission generates heat without combustion, with very high energy density, very low
operating fuel cost, and high capital cost. Its economics are therefore dominated by construction
cost and financing, and its operational profile favours continuous full-power operation. Its safety
engineering is defence in depth: multiple independent barriers, passive cooling where achievable, and
containment sized for the worst credible failure. Waste management is a long-timescale materials and
institutional problem rather than an unsolved technical one.


energy_systems

Renewable generation is characterized by zero fuel cost, low marginal cost, and — for wind and solar
— variability that is weather-driven and only partly predictable. This inverts the traditional
system economics: with near-zero marginal cost the merit order places them first, and the challenge
moves from fuel supply to matching a variable supply with demand. Hydroelectricity is the exception
among renewables in being dispatchable and storable, which makes it disproportionately valuable to a
system with high variable penetration.


energy_systems

Generation:
  thermal              fuel to heat to steam or gas turbine; dispatchable, slow
                       to ramp
  combined cycle       highest deployed thermal efficiency
  nuclear fission      high energy density, high capital cost, low fuel cost
  the nuclear economics  dominated by construction and financing
  defence in depth     independent barriers, passive safety, containment
  hydroelectric        dispatchable, storable, geographically constrained
  pumped hydro         the dominant grid storage technology by installed energy
  wind                 variable, partly predictable, zero marginal cost
  solar photovoltaic   variable with a predictable daily shape and unpredictable
                       cloud modulation
  geothermal           continuous, geographically constrained
  the merit order      dispatch by marginal cost; zero-marginal-cost sources run
                       first
  the price consequence  high renewable penetration suppresses wholesale prices
                       during high output, which affects the economics of every
                       generator including the renewables themselves
  curtailment          generation deliberately reduced because it cannot be used
                       or transmitted; a signal that storage or transmission is
                       the binding constraint


energy_systems

## The Grid and System Operation

An alternating current grid is a synchronized machine. Every rotating generator connected to it
turns in lockstep at the system frequency, and that frequency is a direct real-time indicator of
the balance between generation and load: excess generation raises it, excess load lowers it.
Maintaining it within a narrow band is the fundamental control task of system operation, performed
continuously and on timescales from seconds to hours.


energy_systems

The control hierarchy has distinct layers. Inertia — the kinetic energy stored in rotating masses —
resists frequency change instantaneously with no control action at all. Primary control, the
governor response of generators, acts within seconds to arrest a deviation. Secondary control
restores the frequency to nominal over minutes. Tertiary control and economic dispatch reposition
the fleet over longer periods. Each layer buys time for the next, and the loss of inertia as
synchronous generators are replaced by inverter-connected sources removes the instantaneous layer,
which is why synthetic inertia from fast-responding inverters and storage has become an active
engineering concern.


energy_systems

Reactive power is the aspect of grid operation least familiar outside the field and it is essential.
Alternating current systems with inductive loads and lines require reactive power to sustain
voltage, it does not travel far, and it must be supplied locally. Voltage collapse — a runaway in
which falling voltage increases current, increasing losses, further lowering voltage — is a real
failure mode responsible for major blackouts, and reactive support is what prevents it.


energy_systems

Grid operation:
  synchronism          all rotating generators locked to system frequency
  frequency            the real-time indicator of supply-demand balance
  inertia              stored kinetic energy resisting change instantaneously
  the inertia concern  inverter-connected sources supply none inherently
  synthetic inertia    fast inverter response emulating it
  primary control      governor response in seconds
  secondary control    frequency restoration over minutes
  economic dispatch    least-cost allocation subject to constraints
  unit commitment      which plants to start, given start-up costs and minimum
                       run times
  reserves             spinning and non-spinning capacity held for contingency
  the N-1 criterion    the system must survive the loss of any single element
  reactive power       sustains voltage; local, and it cannot be transmitted far
  voltage collapse     a runaway instability; a genuine blackout mechanism
  protection           relays isolating faults in cycles; coordination failures
                       cause cascading outages
  cascading failure    an initial loss overloading neighbours in sequence; the
                       mechanism behind every large blackout
  black start          restoring a collapsed grid from generators that can start
                       without external power; a capability that must be
                       deliberately maintained


energy_systems

## Storage, Flexibility, and Transmission

Storage decouples generation from consumption in time, and the correct way to compare technologies
is by the duration they serve rather than by capacity alone. Short-duration storage handles
frequency regulation and smoothing over seconds to minutes. Medium-duration storage shifts solar
output from midday to evening over hours. Long-duration storage covers multi-day weather events and
seasonal mismatch, and it is the category with no economical solution at scale, which is the central
unresolved problem in high-renewable system design.


energy_systems

Round-trip efficiency and cost per unit of energy stored are the two figures that determine where a
technology fits. Batteries have high round-trip efficiency and high energy cost, which suits short
durations with many cycles. Pumped hydro has good efficiency and low energy cost with severe
geographic constraints. Hydrogen and thermal storage have poor round-trip efficiency and low energy
cost, which is the profile that suits long duration and few cycles — efficiency matters less when
the alternative is not storing at all.


energy_systems

Demand flexibility is storage's underrated complement, because shifting when energy is consumed
costs less than storing it. Water heating, space conditioning within a comfort band, vehicle
charging, and many industrial processes tolerate substantial time shifting, and the aggregate
resource is large. Realizing it requires price signals reaching the consuming device and automation
acting on them, since manual response does not scale.


energy_systems

Storage and flexibility:
  duration is the axis  compare technologies by the duration they serve
  short duration       frequency regulation and smoothing; batteries
  medium duration      intraday shifting; batteries and pumped hydro
  long duration        multi-day and seasonal; the unresolved problem
  round-trip efficiency  energy out over energy in
  energy cost          cost per stored kilowatt-hour; determines viable duration
  power cost           cost per kilowatt of charge or discharge rate
  the profile match    poor efficiency and low energy cost suits long duration
  demand flexibility   shifting consumption is cheaper than storing energy
  the flexible loads   water heating, conditioning, vehicle charging, industrial
                       processes
  the requirement      automated response to price signals; manual does not scale
  transmission         moves energy in space as storage moves it in time
  geographic smoothing  aggregating variable sources over a wide area reduces
                       aggregate variability substantially
  interconnection      the cheapest form of flexibility where it can be built
  the constraint       transmission is slow to permit and build; it is frequently
                       the binding constraint on renewable deployment rather than
                       generation cost
  losses               resistive losses scale with current squared, which is why
                       transmission voltage is high; high voltage direct current
                       is preferred for long distances and undersea links


energy_systems

## Efficiency, Losses, and Systems Analysis

Efficiency improvements compete with supply expansion and usually win on cost, which makes the
demand side the first place to look. The relevant comparison is the cost of a saved unit of energy
against the cost of a delivered one, and for building envelopes, lighting, motors, and industrial
process heat the saved unit is frequently cheaper by a wide margin. This is well established and
persistently under-exploited, because the costs and benefits often fall on different parties.


energy_systems

Losses accumulate along the chain from primary energy to delivered service, and analysing them at
the wrong point produces the wrong conclusion. A high-efficiency appliance supplied by a
low-efficiency generation and transmission chain delivers less benefit than its rating suggests, and
the correct accounting runs from primary energy to the service actually wanted — a lit room, a
warm building, a moved tonne — rather than to the intermediate energy carrier.


energy_systems

Systems analysis is where the counterintuitive results appear. Rebound effects return part of the
savings from an efficiency gain as increased use, and estimates of the magnitude vary widely by
sector. Marginal versus average emissions matter for evaluating any load change: adding load
displaces the marginal generator, not the average one, so the emissions consequence can be
substantially different from the grid average. And the system value of a generator depends on when
it produces relative to demand, so two sources with identical cost per unit of energy can have very
different value to the system.


energy_systems

Efficiency and systems analysis:
  the demand side first  saved energy is usually cheaper than delivered energy
  the split incentive  costs and benefits falling on different parties; the main
                       reason cheap efficiency goes unrealized
  the full chain       primary energy to delivered service, not to the carrier
  the service framing  a lit room, a warm building, a moved tonne
  rebound effect       part of the saving returned as increased use
  marginal emissions   what a load change actually displaces; frequently very
                       different from the grid average
  time-resolved analysis  hourly or better; annual averages hide the entire
                       problem in a variable system
  value factor         the market value of a source relative to the average
                       price; falls as penetration rises for correlated sources
  the correlation problem  all solar produces at the same time, so each addition
                       is worth less than the last
  levelized cost       cost per unit of energy over the lifetime; useful and
                       incomplete because it ignores when the energy arrives
  system cost          the honest metric; includes the storage, transmission, and
                       flexibility a source requires
  the modelling caution  results from an energy system model follow from its
                       assumptions about cost trajectories, demand growth, and
                       what is allowed to be built, and the sensitivity analysis
                       is where the information is


fluid_dynamics

## Fluid Statics and Kinematics

A fluid is a substance that deforms continuously under shear stress, which distinguishes it
from a solid that reaches a static deformation. Both liquids and gases qualify, differing in
compressibility — liquids are nearly incompressible under ordinary conditions while gases are
not — and that distinction determines which equations apply.


fluid_dynamics

Hydrostatic pressure increases with depth because the fluid above has weight, giving the
familiar linear relation for constant density. Its consequences include buoyancy, formalized as
Archimedes' principle: the upward force equals the weight of displaced fluid, which follows
directly from integrating the pressure over a submerged surface. Pascal's principle, that
pressure applied to an enclosed fluid transmits undiminished, is the basis of hydraulic force
multiplication — a small force on a small piston produces a large force on a large one, with
the work conserved because the large piston moves proportionally less.


fluid_dynamics

The kinematic description of flow admits two viewpoints that recur throughout the subject.
The Lagrangian view follows individual fluid parcels, which is natural for deriving the
equations from Newton's laws applied to a mass. The Eulerian view describes the field at fixed
points in space, which is what instruments measure and what computation uses. Converting
between them introduces the material derivative, which accounts for both the local change at a
fixed point and the change from the parcel moving into a region with different conditions —
the convective term, and the source of the nonlinearity that makes fluid dynamics hard.


fluid_dynamics

Statics and kinematics:
  hydrostatic pressure   P = P_0 + rho g h
  Archimedes             buoyant force equals displaced fluid weight
  Pascal's principle     pressure transmits undiminished; hydraulic multiplication
  streamline             tangent to the velocity field at an instant
  pathline               the trajectory of one parcel over time
  streakline             the locus of parcels that passed through a point; what dye
                         injection visualizes
  steady flow            all three coincide; in unsteady flow they differ
  material derivative    D/Dt = d/dt + (v · grad); local plus convective change
  convective term        the nonlinearity; the source of turbulence and of the
                         difficulty of the subject
  vorticity              curl of velocity; twice the local rotation rate
  irrotational flow      zero vorticity; admits a velocity potential and enormous
                         simplification
  circulation            the line integral of velocity around a closed curve;
                         directly related to lift


fluid_dynamics

Q: A block floats with 80% of its volume submerged in water. Its density is closest to:
A. 200 kg/m^3   B. 800 kg/m^3   C. 1250 kg/m^3   D. 1000 kg/m^3
Reasoning: Floating equilibrium requires the buoyant force to equal the weight, so the
displaced fluid mass equals the object's mass. Displaced volume is 0.8 of the object's volume,
so 0.8 times water density equals the object's density, giving 800 kg/m^3. Answer: B.


fluid_dynamics

## Conservation Laws and Bernoulli

The governing equations express conservation of mass, momentum, and energy for a fluid.
Continuity states that mass cannot accumulate in a fixed volume without a net inflow, which for
incompressible flow reduces to the velocity field being divergence-free. In a pipe of varying
area this gives the familiar result that velocity is inversely proportional to cross-sectional
area, so a constriction speeds the flow.


fluid_dynamics

Bernoulli's equation relates pressure, velocity, and elevation along a streamline for steady,
incompressible, inviscid flow. Its content is energy conservation: the sum of pressure energy,
kinetic energy, and potential energy per unit volume is constant along a streamline. The
counterintuitive consequence — that faster flow has lower pressure — explains the lift on a
curved surface, the operation of a venturi meter, and why a shower curtain is pulled inward.


fluid_dynamics

The equation's assumptions are restrictive and routinely violated in casual application.
Steady flow excludes anything time-varying. Incompressibility excludes gas flow above roughly
thirty percent of the speed of sound. Inviscid excludes anything where friction matters, which
includes flow near a wall and any flow that has separated. And it applies along a streamline,
so comparing points on different streamlines requires additionally that the flow be
irrotational. The standard incorrect explanation of aircraft lift — that air travelling over
the longer upper surface must arrive simultaneously and therefore goes faster — fails on the
premise itself, since there is no reason for simultaneous arrival, and the parcels above in
fact arrive earlier.


fluid_dynamics

Conservation relations:
  continuity, general    d(rho)/dt + div(rho v) = 0
  continuity, incompress. div v = 0
  pipe flow              A1 v1 = A2 v2
  Bernoulli              P + (1/2) rho v^2 + rho g h = constant along a streamline
  stagnation pressure    P + (1/2) rho v^2; what a pitot tube measures
  dynamic pressure       (1/2) rho v^2
  Torricelli             v = sqrt(2 g h) for efflux from a tank; Bernoulli applied
  venturi                pressure drop at a constriction measures flow rate
  momentum equation      the integral form gives forces on ducts, vanes, and jets
  head loss              the departure from Bernoulli caused by friction; what
                         real pipe flow analysis is about
  Bernoulli assumptions  steady, incompressible, inviscid, along a streamline;
                         each is violated routinely in casual application


fluid_dynamics

Lift is worth stating correctly because the popular explanation is wrong and a better one is
available without mathematics. A wing deflects air downward, and by Newton's third law the air
pushes the wing upward — this is complete and correct as far as it goes. The circulation
picture is the more quantitative version: the flow around a lifting wing has net circulation,
and the Kutta-Joukowski theorem relates lift to that circulation directly. The angle of attack
and the trailing-edge shape together determine the circulation, which is why a symmetric
aerofoil generates lift at positive angle of attack and none at zero, contradicting any
explanation resting on the upper surface being longer.


fluid_dynamics

## Navier-Stokes and Dimensionless Numbers

The Navier-Stokes equations are Newton's second law applied to a fluid element, with terms
for pressure gradient, viscous stress, and body forces balanced against the material
acceleration. They are believed to describe all classical fluid behaviour, and their general
solution is unknown — whether smooth solutions always exist in three dimensions is one of the
Millennium Prize problems, and the difficulty comes from the convective nonlinearity.


fluid_dynamics

Because analytical solutions exist only for a handful of simplified cases, dimensional
analysis carries much of the practical load. The Buckingham pi theorem states that a
relationship among n variables involving k independent dimensions can be expressed among n-k
dimensionless groups, which reduces the parameter space enormously and identifies which
combinations govern behaviour.


fluid_dynamics

The Reynolds number is the most important of these. It is the ratio of inertial to viscous
forces, and it determines the character of the flow rather than merely its magnitude. Below a
critical value the flow is laminar, with smooth layers and viscous forces dominating; above it
the flow is turbulent, with chaotic eddies and inertia dominating. Two geometrically similar
flows at the same Reynolds number behave identically regardless of their physical scale, which
is dynamic similarity and is what makes scale-model testing meaningful.


fluid_dynamics

Dimensionless groups:
  Reynolds       rho v L / mu;  inertial over viscous; laminar versus turbulent
  Mach           v / c;  compressibility; below 0.3 gas flow is effectively
                 incompressible
  Froude         v / sqrt(g L);  inertia over gravity; free-surface and ship waves
  Weber          rho v^2 L / sigma;  inertia over surface tension; droplet breakup
  Prandtl        momentum over thermal diffusivity; boundary layer thickness ratio
  Nusselt        convective over conductive heat transfer
  Rayleigh       buoyancy-driven convection onset
  Strouhal       vortex shedding frequency; f L / v, about 0.2 for a cylinder over
                 a wide Reynolds range
  Peclet         advection over diffusion
  drag coefficient  drag / (0.5 rho v^2 A); a function of Reynolds number
  pipe transition   laminar below Re ~2300, turbulent above ~4000
  dynamic similarity  matched dimensionless groups give matched behaviour


fluid_dynamics

Matching dimensionless numbers is what makes model testing valid and is also its central
difficulty. A ship model at one-fiftieth scale cannot simultaneously match Reynolds and Froude
numbers — matching Froude requires a velocity scaled by the square root of length, and matching
Reynolds requires velocity scaled inversely with length, which are incompatible. The standard
resolution is to match Froude, since wave-making resistance dominates, and to compute the
viscous contribution separately using empirical correlations. Recognizing which group governs
the phenomenon of interest is therefore a prerequisite to designing a valid experiment.


fluid_dynamics

## Laminar Flow, Turbulence, and Boundary Layers

Laminar flow is orderly, with fluid moving in smooth layers that do not mix except by
molecular diffusion. It is analytically tractable, and pipe flow is the canonical case:
the velocity profile is parabolic, the flow rate depends on the fourth power of the radius by
the Hagen-Poiseuille relation, and pressure drop is linear in flow rate. That fourth-power
dependence has striking consequences — halving a pipe's radius reduces flow to one sixteenth at
the same pressure, which is why small changes in arterial diameter have large physiological
effects.


fluid_dynamics

Turbulent flow is chaotic, with velocity fluctuating irregularly and eddies spanning a wide
range of scales. It is dramatically better at mixing and at transferring heat and momentum, and
dramatically worse for pressure loss, with drop scaling closer to the square of flow rate.
The velocity profile is much flatter than the laminar parabola because turbulent mixing
transports momentum across the pipe, and the transition between the regimes occurs over a
range rather than at a sharp threshold, with the transition point sensitive to disturbances,
surface roughness, and inlet conditions.


fluid_dynamics

The energy cascade is the central conceptual picture of turbulence. Energy enters at large
scales set by the geometry, transfers to progressively smaller eddies through nonlinear
interaction without dissipation, and is finally dissipated by viscosity at the smallest scale.
Kolmogorov's theory predicts that the intermediate range has a universal statistical structure
independent of how the energy was injected, giving the observed spectrum with a characteristic
power law. The dissipation scale shrinks as Reynolds number rises, which is why direct
numerical simulation resolving all scales is computationally infeasible for practical Reynolds
numbers — the required grid points scale as Reynolds number to the power nine-fourths.


fluid_dynamics

Flow regimes and boundary layers:
  laminar pipe flow      parabolic profile; Q proportional to r^4; dP linear in Q
  turbulent pipe flow    flatter profile; dP roughly quadratic in Q
  transition             Re ~2300-4000; sensitive to disturbances
  energy cascade         large eddies to small; dissipation at the Kolmogorov scale
  Kolmogorov spectrum    universal statistics in the inertial range
  no-slip condition      fluid velocity equals the surface velocity at a wall; the
                         origin of all viscous drag
  boundary layer         the thin region where velocity rises from zero to free
                         stream; thickness grows along the surface
  laminar to turbulent   the boundary layer transitions downstream; a turbulent
                         layer has more skin friction and resists separation
  separation             the boundary layer detaches under an adverse pressure
                         gradient; produces a wake and large pressure drag
  stall                  separation on a wing at high angle of attack; lift
                         collapses
  drag crisis            drag coefficient drops sharply when the boundary layer
                         becomes turbulent; the reason golf balls have dimples
  vortex shedding        alternating vortices behind a bluff body; the Karman
                         street, and the source of resonant structural failures


fluid_dynamics

Boundary layer separation is the phenomenon that dominates practical aerodynamics and
hydrodynamics. Where the pressure rises in the flow direction — an adverse gradient — fluid
near the wall, already slowed by friction, may lack the momentum to continue and reverses,
detaching the layer from the surface. The resulting wake is a low-pressure region producing
pressure drag far exceeding skin friction, and on a wing it causes stall. Streamlining is
fundamentally about delaying separation by making the pressure recovery gradual, which is why a
teardrop shape has a small fraction of the drag of a sphere with the same frontal area.


fluid_dynamics

The dimpled golf ball is the counterintuitive case worth understanding. Dimples trip the
boundary layer into turbulence, which increases skin friction and, far more importantly, gives
the layer more momentum near the wall so it resists separation longer. The wake narrows
substantially, pressure drag falls, and the net drag is roughly halved — an increase in one
drag component producing a large decrease in the total.


fluid_dynamics

[user] I'm sizing a pump for a 200 m pipe run, 50 mm diameter, 3 L/s of water, with 8 m of
elevation gain.
[assistant] tool_call fluids.pipe_system({"length_m": 200, "diameter_mm": 50,
"flow_lps": 3.0, "elevation_m": 8, "fluid": "water_20c", "roughness_mm": 0.045,
"fittings": {"elbow_90": 6, "gate_valve": 2}})
[tool] {"velocity_m_s": 1.53, "reynolds": 76200, "regime": "turbulent",
"friction_factor": 0.0212, "major_loss_m": 10.1, "minor_loss_m": 0.9,
"static_head_m": 8, "total_head_m": 19.0, "hydraulic_power_w": 559,
"pump_power_at_65pct_w": 860}
[assistant] Total head is 19.0 metres, requiring about 560 watts of hydraulic power, so around
860 watts at the shaft assuming 65% pump efficiency.


fluid_dynamics

The breakdown is worth seeing because it shows where the energy goes. Static lift accounts for
8 metres — that is unavoidable work against gravity. Friction along the 200 metre run accounts
for 10.1 metres, more than the lift itself. Fittings contribute a further 0.9 metres, which is
minor here because you have few of them; in a compact system with many bends and valves, minor
losses frequently exceed the straight-run friction and the name becomes misleading.


fluid_dynamics

The flow is turbulent at Reynolds 76,200, so pressure drop scales close to the square of flow
rate. That scaling is the important design lever: increasing the pipe diameter from 50 mm to
65 mm drops the velocity to 0.90 m/s and cuts the friction loss to roughly 2.7 metres, taking
total head from 19 to about 11.6 metres and the pump power to roughly 520 watts. That is a 40%
reduction in operating energy for a modest increase in pipe cost, and over a system operating
continuously it pays back quickly.


fluid_dynamics

Two checks before finalizing. Confirm the net positive suction head available at the pump
inlet exceeds what the pump requires, since cavitation damages impellers rapidly and is a
function of the suction-side geometry rather than of the discharge calculation above. And
verify the velocity is within the conventional range — 1.53 m/s is comfortable, while above
roughly 3 m/s erosion and noise become concerns and below about 0.6 m/s sediment can settle in
horizontal runs.


fluid_dynamics

If the 3 L/s is a peak rather than a continuous requirement, size the pipe for the continuous
case and check that the peak is acceptable, rather than sizing everything for a rare
condition.


fluid_dynamics

## Compressible Flow and Applications

When flow speed becomes comparable to the speed of sound, density changes are no longer
negligible and the character of the flow changes qualitatively. The Mach number, the ratio of
flow speed to sound speed, is the governing parameter, with compressibility effects becoming
significant above roughly 0.3 and dominant near and above one.


fluid_dynamics

The most striking consequence is that the relationship between area and velocity reverses. In
subsonic flow, a constriction accelerates the flow, as continuity and Bernoulli require. In
supersonic flow, a constriction decelerates it and an expansion accelerates it, because the
density change outweighs the area change. This is why a converging-diverging nozzle is required
to produce supersonic exhaust: the flow accelerates to exactly sonic at the throat and
continues accelerating in the diverging section, which is the geometry of every rocket nozzle
and every supersonic wind tunnel.


fluid_dynamics

Shock waves are discontinuities that form when a supersonic flow must adjust to conditions it
cannot reach smoothly. Across a normal shock, the flow abruptly becomes subsonic with a rise in
pressure, density, and temperature and a loss of stagnation pressure — an irreversible process
that generates entropy. Oblique shocks turn the flow through an angle and are what forms on
supersonic aircraft, and the sonic boom is the pressure discontinuity of the shock system
sweeping across the ground.


fluid_dynamics

Compressible flow and applications:
  speed of sound        sqrt(gamma R T) in an ideal gas; ~343 m/s in air at 20 C
  Mach number           v / c;  M < 0.3 effectively incompressible
  subsonic              constriction accelerates
  supersonic            constriction decelerates; the area relation reverses
  choked flow           sonic at the throat; mass flow becomes independent of
                        further downstream pressure reduction
  converging-diverging  the nozzle geometry required for supersonic exhaust
  normal shock          abrupt transition to subsonic; stagnation pressure loss
  oblique shock         turns the flow; forms on supersonic bodies
  expansion fan         smooth isentropic turning in the opposite sense
  transonic             mixed subsonic and supersonic regions; the most difficult
                        regime and the origin of the historical drag rise
  wave drag             an additional drag component appearing near and above M = 1
  computational methods finite volume with shock-capturing schemes; turbulence
                        modelled rather than resolved at practical Reynolds numbers


fluid_dynamics

Computational fluid dynamics has become the primary analysis tool, and its central practical
issue is turbulence modelling. Resolving all turbulent scales directly is infeasible at
engineering Reynolds numbers, so the effect of the unresolved scales is modelled. Reynolds-
averaged approaches model all turbulence and are cheap, robust, and dependent on empirical
constants calibrated for particular flow classes — they are reliable for attached flows and
unreliable for separation, which is exactly what many engineering questions are about. Large
eddy simulation resolves the large scales and models only the small, at far greater cost and
with substantially better fidelity for separated and unsteady flows.


fluid_dynamics

The consequence for anyone using these tools is that a computational result requires
validation rather than acceptance. Grid convergence must be demonstrated by refining the mesh
and showing the answer stops changing. The turbulence model must be appropriate for the flow
regime. Boundary conditions must reflect the real situation, and the near-wall mesh resolution
must satisfy the requirements of the wall treatment being used. A converged solution on an
inadequate mesh with an inappropriate model produces a colourful and confident picture of
something that did not happen, and the discipline of validating against experimental data for
a similar configuration is what separates useful computation from decoration.


geophysics

## Seismology and the Structure of the Earth

Nearly everything known about the Earth's deep interior comes from seismic waves, because no direct
sample exists from below a few kilometres. An earthquake radiates elastic waves that travel through
the planet, refract and reflect at boundaries where properties change, and arrive at instruments
worldwide with travel times and amplitudes that encode the structure along their paths.


geophysics

Two body wave types propagate through the volume. Compressional waves travel fastest, involve motion
parallel to propagation, and pass through liquids. Shear waves travel more slowly, involve
perpendicular motion, and cannot propagate in a liquid at all. That last fact is the single most
consequential observation in the history of the field: a shadow zone where shear waves do not arrive
on the far side of the Earth demonstrates a liquid outer core, and the shape of the compressional
wave shadow gives its size.


geophysics

Surface waves travel along the boundary, decay with depth, and carry most of the destructive energy
in a large earthquake. Their dispersion — the dependence of velocity on period, since longer periods
sample deeper — allows the shallow velocity structure to be inverted from a single record, which is
why surface wave analysis dominates crustal and upper mantle imaging. The layered picture of crust,
mantle, outer core, and inner core, with the discontinuities between them, was built entirely from
these observations before any other technique could address the question.


geophysics

Seismology and structure:
  body waves           compressional and shear, travelling through the volume
  compressional waves  fastest, parallel motion, propagate in liquids
  shear waves          slower, perpendicular motion, do not propagate in liquids
  the shear shadow     the demonstration of a liquid outer core
  surface waves        along the boundary, decaying with depth, most destructive
  dispersion           velocity varying with period; longer periods sample deeper
  crust-mantle boundary  a sharp velocity increase at shallow depth
  transition zone      phase changes producing discontinuities in the upper mantle
  core-mantle boundary  the largest property contrast in the planet
  inner core           solid, detected from waves that traverse it
  tomography           three-dimensional velocity structure from many crossing
                       paths; the same principle as medical imaging
  anisotropy           direction-dependent velocity from aligned minerals; a probe
                       of flow direction in the mantle
  attenuation          amplitude loss; sensitive to temperature and partial melt
  magnitude scales     logarithmic; moment magnitude is the physically meaningful
                       one and relates directly to the energy released
  the energy relation  one unit of magnitude corresponds to roughly thirty times
                       the energy, which is why the distinction between a large
                       and a very large event is far greater than the numbers
                       suggest


geophysics

Q: Shear waves are not detected on the side of the Earth opposite a large earthquake. This shows:
A. The waves were absorbed by rock
B. There is a liquid layer in the interior, because shear waves cannot propagate through a liquid
   and are therefore blocked entirely rather than merely attenuated
C. The instruments failed
D. The earthquake was too small
Reasoning: A liquid has no shear strength, so shear waves cannot exist in it. The complete absence
of shear arrivals across a well-defined shadow zone, rather than a reduction in amplitude, is
diagnostic of a liquid layer and was the discovery of the outer core. Answer: B.


geophysics

## Gravity, Geodesy, and the Shape of the Earth

The Earth's gravity field varies from place to place because mass is not uniformly distributed, and
measuring those variations maps subsurface density. A gravity survey measures the field with an
instrument sensitive to parts in a billion, and the interpretation requires removing the predictable
components — latitude, elevation, the mass of the intervening topography, and the tidal effect of
the Sun and Moon — before the residual anomaly reflects the geology.


geophysics

Isostasy is the principle that the crust floats on a denser mantle in approximate buoyant
equilibrium, so mountain ranges have deep low-density roots supporting them. It explains why the
observed gravity over a mountain range is far less than the topography alone would produce: the
excess mass above sea level is compensated by a deficit below. Departures from isostatic
equilibrium indicate that something is actively supporting or depressing the surface, which is how
dynamic support by mantle flow and glacial rebound are identified.


geophysics

Satellite geodesy transformed the field by making global measurement continuous. Satellite pairs
measuring their separation to micron precision detect mass redistribution, which has produced direct
measurements of ice sheet loss, groundwater depletion, and the seasonal cycling of water between
land and ocean. Precise satellite positioning measures crustal deformation at the millimetre level,
which is how plate motion, strain accumulation on faults, and volcanic inflation are now routinely
monitored.


geophysics

Gravity and geodesy:
  gravity anomaly      the residual after removing predictable components
  free-air correction  for elevation above the reference surface
  Bouguer correction   for the mass of the intervening topography
  terrain correction   for the surrounding relief
  isostasy             crustal blocks floating in buoyant equilibrium
  mountain roots       low-density material compensating the topography
  the compensation observation  gravity over mountains is far less than the
                       topography implies
  isostatic anomaly    what remains after compensation is accounted for
  dynamic topography   surface elevation supported by mantle flow
  glacial rebound      crust still rising from removed ice load; a direct probe of
                       mantle viscosity
  the geoid            the equipotential surface; the true reference for elevation
  satellite gravimetry  mass redistribution measured from orbit
  what it measures     ice loss, groundwater depletion, seasonal water cycling
  satellite positioning  millimetre crustal deformation
  interferometric radar  surface displacement mapped over wide areas; volcanic
                       inflation, subsidence, and coseismic slip


geophysics

## Geomagnetism and the Core

The Earth's magnetic field is generated by convection in the liquid iron outer core, a self-
sustaining dynamo in which fluid motion through a magnetic field generates currents that maintain
the field. The energy source is a combination of cooling and the buoyancy released as the inner core
solidifies, and the rotation of the planet organizes the flow into the pattern that produces a
predominantly dipolar field.


geophysics

The field is not static. It drifts, its intensity varies, and at irregular intervals averaging
hundreds of thousands of years it reverses polarity entirely, with a transition taking a few
thousand years during which the field is weak and complex. The record of these reversals is
preserved in rocks that cooled through their magnetic blocking temperature, locking in the field
direction at that time, and reading that record from the ocean floor provided the decisive evidence
for seafloor spreading.


geophysics

That evidence deserves emphasis because of how conclusive it was. Magnetic surveys of the ocean floor
revealed stripes of alternating polarity, symmetric about mid-ocean ridges, with the same sequence
on both sides. The only sensible explanation is that new crust forms at the ridge, records the field
direction as it cools, and spreads outward in both directions — which is seafloor spreading, and
which converted continental drift from a disputed hypothesis into the foundation of plate tectonics.


geophysics

Geomagnetism:
  the dynamo           convection in the liquid iron core sustaining the field
  the energy source    cooling plus buoyancy from inner core solidification
  dipole dominance     the field is mostly but not purely dipolar
  secular variation    drift and intensity change on decadal timescales
  reversals            irregular, averaging hundreds of thousands of years apart
  the transition       a few thousand years of weak complex field
  remanent magnetization  the field direction locked in as rock cools
  blocking temperature  below which magnetization is fixed
  magnetic stripes     alternating polarity, symmetric about ridges
  the decisive evidence  the same sequence on both sides of a ridge
  seafloor spreading   new crust forming and moving outward
  apparent polar wander  the pole's apparent path differing between continents;
                       resolved by the continents having moved, not the pole
  magnetostratigraphy  dating sequences by their reversal pattern
  the magnetosphere    the field's interaction with the solar wind; deflects
                       charged particles and is why the field matters for
                       habitability


geophysics

## Heat Flow, Mantle Dynamics, and Plate Tectonics

The Earth's internal heat drives everything geologically active, and it comes from two sources:
radioactive decay of long-lived isotopes in the mantle and crust, and heat left over from accretion
and core formation. The total output is modest compared with solar input at the surface — a fraction
of a watt per square metre — but it is the entire budget for mantle convection, plate motion,
volcanism, and the dynamo.


geophysics

Mantle convection is solid-state creep over geological time. The mantle is crystalline rock that
deforms by diffusion and dislocation mechanisms at rates of centimetres per year, which over
hundreds of millions of years is enough for large-scale overturn. Plates are the cold upper thermal
boundary layer of this convection, and the driving forces are dominated by the negative buoyancy of
subducting slabs pulling the plate behind them rather than by drag from below — which is why the
fastest plates are those with the largest subducting margins.


geophysics

The plate boundary types organize most surface geology. Divergent boundaries create new crust at
ridges. Convergent boundaries destroy it in subduction zones, producing the deepest earthquakes,
explosive volcanism, and mountain building. Transform boundaries slide past each other, producing
shallow earthquakes without volcanism. Recognizing which type is present tells you what hazards to
expect and what rocks to find.


geophysics

Heat and dynamics:
  heat sources         radioactive decay and primordial heat
  the surface flux     a fraction of a watt per square metre
  the budget           all internal geological activity comes from it
  mantle convection    solid-state creep at centimetres per year
  deformation mechanisms  diffusion and dislocation creep
  plates               the cold upper thermal boundary layer
  slab pull            the dominant driving force
  ridge push           secondary; gravitational sliding from elevated ridges
  basal drag           smaller than long assumed
  the speed correlation  plates with large subducting margins move fastest
  divergent boundaries  new crust at ridges
  convergent boundaries  subduction; deep earthquakes, explosive volcanism,
                       mountain building
  transform boundaries  lateral sliding; shallow earthquakes, no volcanism
  hotspots             volcanism away from boundaries; the plume hypothesis and
                       its ongoing debate
  the age-depth relation  ocean floor subsides predictably as it cools, one of
                       the cleanest quantitative relationships in the field


geophysics

## Exploration Methods, Hazards, and Inversion

Applied geophysics images the shallow subsurface for resources, engineering, groundwater, and
archaeology, using the same physics at smaller scale. Seismic reflection sends controlled energy
into the ground and records the echoes from layer boundaries; it is the highest-resolution method
and the most expensive. Electrical and electromagnetic methods measure conductivity, which responds
strongly to water content and to certain ores. Ground-penetrating radar images the top few metres in
resistive material and fails entirely in clay.


geophysics

Hazard assessment turns this understanding into estimates of what may happen and how often.
Earthquake hazard combines the locations and slip rates of faults, the recurrence statistics of
earthquake magnitudes, and models of how ground shaking attenuates with distance and amplifies in
soft sediment. Local site conditions frequently dominate — the same earthquake produces very
different shaking on rock and on soft basin fill, and the amplification factor can exceed a factor
of several.


geophysics

The unifying methodological problem across all of this is inversion: inferring subsurface properties
from surface measurements. It is inherently non-unique, because different structures can produce
identical observations, and every result therefore depends on the regularization or prior imposed to
select among the possibilities. A geophysical image is a model consistent with the data, not a
photograph, and its resolution varies with depth and position in ways that a rendered image does not
communicate. Reading the resolution analysis alongside the image is what separates competent
interpretation from over-reading.


geophysics

Exploration and inversion:
  seismic reflection   controlled energy and layer echoes; highest resolution
  seismic refraction   velocity structure from head waves; cheaper, coarser
  electrical resistivity  conductivity; responds to water and to ores
  electromagnetic methods  including airborne survey over large areas
  ground-penetrating radar  the top few metres; defeated by clay
  magnetics            buried magnetic bodies; fast and cheap reconnaissance
  gravity              density contrast; regional structure
  borehole logging     direct measurement along a well; the ground truth for
                       everything else
  seismic hazard       fault slip rates, recurrence statistics, attenuation
  site amplification   soft sediment amplifying shaking substantially
  liquefaction         saturated loose sediment losing strength during shaking
  tsunami generation   from seafloor displacement, not from shaking
  early warning        detecting the fast compressional wave to warn before the
                       slower destructive waves arrive; seconds to tens of
                       seconds, which is enough to stop trains and halt surgery
  the inversion problem  non-unique by nature
  regularization       the assumption selecting among consistent models
  resolution analysis  what the data can and cannot constrain
  the interpretive discipline  an image is a model consistent with the data, and
                       reading structure into poorly resolved regions is the most
                       common error in the field


metrology_and_measurement

## Units and the Redefined SI

Measurement requires a unit, and the question of what defines a unit has a history worth knowing
because the modern answer is structurally different from the historical one. A unit defined by an
artifact — a particular metal bar, a particular cylinder — is only as stable as the artifact, and
comparing against it requires physical access to it or to a copy whose relationship to it is
uncertain.


metrology_and_measurement

The 2019 redefinition completed a shift begun with the metre: every base unit is now defined by
fixing the numerical value of a fundamental constant. The second is defined by fixing a caesium
transition frequency, the metre by fixing the speed of light, the kilogram by fixing the Planck
constant, the ampere by fixing the elementary charge, the kelvin by fixing the Boltzmann constant,
the mole by fixing the Avogadro constant, and the candela by fixing a luminous efficacy.


metrology_and_measurement

The consequences are practical rather than philosophical. A definition based on a constant can be
realized anywhere by anyone with adequate apparatus rather than requiring comparison against a
unique object. It cannot drift, since the constant is fixed by definition rather than measured.
And the realization can improve without redefining the unit — a better experiment produces a more
accurate realization of the same definition, where previously an improved measurement of the
artifact changed what the unit meant.


metrology_and_measurement

Units and the SI:
  artifact definition  only as stable as the object; comparison requires access
  constant definition  realizable anywhere, cannot drift, and improves without
                       redefinition
  second               a fixed caesium hyperfine transition frequency
  metre                a fixed speed of light; defined via the second
  kilogram             a fixed Planck constant; realized by the Kibble balance and
                       by the silicon sphere route
  ampere               a fixed elementary charge
  kelvin               a fixed Boltzmann constant
  mole                 a fixed Avogadro constant; now independent of the kilogram
  candela              a fixed luminous efficacy; the only base unit incorporating
                       human perception
  derived units        combinations; newton, joule, watt, volt, ohm
  dimensional analysis  check that both sides of an equation have the same
                       dimensions; catches a large fraction of algebra errors
  the hierarchy        the second is the most accurately realized, and units
                       defined through it inherit that accuracy


metrology_and_measurement

Q: A unit defined by fixing a fundamental constant rather than by an artifact has the advantage
that:
A. It is easier to understand
B. It can be realized independently anywhere, cannot drift, and its realization can improve
   without changing the definition
C. It requires no apparatus
D. It is more precise by definition
Reasoning: An artifact definition makes the unit whatever that object is, so the unit drifts if the
object does and improving the measurement changes the unit. Fixing a constant separates the
definition from any realization, so better apparatus produces a better realization of an unchanged
unit and any laboratory with adequate equipment can realize it. Answer: B.


metrology_and_measurement

## Uncertainty

Every measurement has uncertainty, and a measurement reported without it is incomplete rather
than merely undocumented. The uncertainty states the range within which the true value is believed
to lie, together with the confidence attached, and it is what permits two measurements to be
compared or a measurement to be compared against a specification.


metrology_and_measurement

The distinction between error and uncertainty is worth keeping precise. Error is the difference
between the measured and the true value, which is unknown since the true value is unknown.
Uncertainty is the quantified doubt about the result, which is what can be stated. A known error is
corrected rather than reported; what remains after correction is uncertainty.


metrology_and_measurement

Uncertainty components are classified by how they are evaluated rather than by their nature. Type A
evaluation uses statistical analysis of repeated observations, giving the standard deviation of the
mean. Type B evaluation uses any other means — calibration certificates, manufacturer
specifications, physical reasoning, prior experience — converted to a standard uncertainty by
assuming a distribution. The classification is by method rather than by whether the component is
random or systematic, which is a common confusion.


metrology_and_measurement

Uncertainty:
  error                measured minus true; unknown, and corrected when known
  uncertainty          quantified doubt about the result; what is reported
  type A               evaluated statistically from repeated observations
  type B               evaluated by other means; converted assuming a distribution
  the classification   by method, not by random versus systematic
  standard uncertainty  expressed as a standard deviation
  combined uncertainty  components combined in quadrature when independent
  correlated components  do not combine in quadrature; the covariance term matters
                       and is frequently omitted
  expanded uncertainty  multiplied by a coverage factor; k = 2 for approximately
                       95% coverage
  coverage factor      state it, or the interval's meaning is undefined
  degrees of freedom   how well the uncertainty itself is known; few repeats give
                       a poorly known uncertainty
  sensitivity coefficient  how much the result changes per unit change in an input
  propagation          combine input uncertainties weighted by their sensitivity
                       coefficients
  significant figures  a crude uncertainty statement; report the uncertainty
                       explicitly instead
  the dominant term    identify it; reducing a component that is not dominant
                       accomplishes nothing


metrology_and_measurement

The quadrature combination deserves emphasis because it determines where effort is worth
spending. Independent uncertainty components combine as the square root of the sum of squares, so
a component half the size of the dominant one contributes only about ten percent to the total, and
one a third the size contributes about five percent. Effort spent reducing a minor component is
therefore nearly wasted, and identifying the dominant term before optimizing is what makes
measurement improvement efficient.


metrology_and_measurement

## Calibration and Traceability

Calibration establishes the relationship between an instrument's indication and the corresponding
value of the measurand, and it is what converts a reading into a measurement. An uncalibrated
instrument produces numbers whose relationship to reality is unknown, however precise and
repeatable those numbers are.


metrology_and_measurement

Traceability is the property that a measurement result can be related to a reference through a
documented unbroken chain of calibrations, each contributing to the uncertainty. This is what makes
measurements made in different places comparable — two laboratories whose instruments are traceable
to the same reference are measuring the same thing, and two whose instruments are not have no basis
for comparing their results.


metrology_and_measurement

The distinction between accuracy and precision is the one most often confused and it has direct
practical consequences. Precision is repeatability — how much the readings scatter. Accuracy is
closeness to the true value. An instrument can be precise and inaccurate, producing consistently
wrong readings, and this is the dangerous case because the consistency inspires confidence. It is
corrected by calibration; scatter is reduced by averaging, and no amount of averaging corrects an
offset.


metrology_and_measurement

Calibration and traceability:
  calibration          establishes the relationship between indication and value
  traceability         an unbroken documented chain to a reference
  the chain            each link adds uncertainty; the total accumulates
  primary standard     realizes the unit directly
  transfer standard    carries the value between laboratories
  working standard     used routinely; calibrated against a higher standard
  accuracy             closeness to the true value; corrected by calibration
  precision            repeatability; reduced by averaging
  the dangerous case   precise and inaccurate; consistency inspires confidence
  bias                 systematic offset; averaging does not remove it
  drift                change over time; sets the calibration interval
  calibration interval  from the observed drift, not from convention
  test uncertainty ratio  the standard's uncertainty relative to the tolerance
                       being verified; 4:1 is the conventional minimum
  guard banding        tightening the acceptance limit by the measurement
                       uncertainty; required when the uncertainty is not negligible
                       relative to the tolerance
  as-found and as-left  the readings before and after adjustment; the as-found
                       data is what tells you whether previous measurements were
                       valid


metrology_and_measurement

The as-found reading deserves emphasis because it is what makes calibration retrospectively
useful. An instrument found out of tolerance at calibration was presumably out of tolerance for
some period before, which calls into question every measurement made with it since the last
calibration. Recording the as-found value permits assessing how far out it was and therefore what
the consequences are, and discarding it in favour of only recording the adjusted state destroys
exactly the information needed to determine whether any previous result was affected.


metrology_and_measurement

## Precision Measurement Techniques

Precision measurement techniques share a set of strategies for defeating the noise and drift that
limit ordinary measurement, and understanding the strategies transfers across specific
applications.


metrology_and_measurement

Null methods compare the unknown against a reference and adjust until the difference is zero,
which means the detector's accuracy does not matter — only its ability to detect zero. A bridge
circuit measuring resistance by balancing against known resistances is accurate to the standards'
accuracy rather than to the detector's, which is why bridges dominated precision electrical
measurement.


metrology_and_measurement

Differential measurement measures the difference between two similar quantities rather than each
separately, which cancels whatever affects both. Two adjacent sensors in the same thermal and
electromagnetic environment share their disturbances, so the difference is far cleaner than either
signal.


metrology_and_measurement

Modulation moves the measurement to a frequency where the noise is lower. Low-frequency noise
grows as frequency falls — the ubiquitous flicker noise — so measuring a slowly varying quantity
directly puts the signal exactly where the noise is worst. Chopping the signal at a chosen
frequency and detecting synchronously at that frequency, which is what a lock-in amplifier does,
recovers signals far below the broadband noise floor.


metrology_and_measurement

Precision techniques:
  null method          balance against a reference; the detector needs only to
                       detect zero
  bridge circuit       the electrical null method
  substitution         replace the unknown with a known that produces the same
                       indication; eliminates the instrument's nonlinearity
  differential         measure the difference; cancels common disturbances
  ratio measurement    ratios are frequently measurable far more accurately than
                       absolute values
  modulation           move the signal away from low-frequency noise
  lock-in detection    synchronous detection at the modulation frequency; extracts
                       signals far below the noise
  averaging            random noise falls as the square root of the count;
                       systematic error does not fall at all
  interferometry       measure displacement in wavelengths; the standard for
                       length at high precision
  frequency comparison  frequencies are measurable more accurately than any other
                       quantity, so converting a measurement into a frequency
                       measurement is a recurring strategy
  temperature control  the dominant systematic in most precision work
  vibration isolation  and acoustic isolation; the second dominant systematic
  Abbe error           an offset between the measurement axis and the axis of
                       interest turns an angular error into a linear one
  cosine error         a misaligned measurement axis reads short by the cosine of
                       the angle


metrology_and_measurement

The strategy of converting a measurement into a frequency measurement deserves emphasis because
it explains a great deal of precision instrument design. Frequency is measurable to parts in ten to
the eighteenth, better than any other physical quantity by many orders of magnitude, because
counting cycles over a long interval accumulates precision. A quantity converted into a frequency
shift — a resonator whose frequency depends on the measurand, an interferometer whose fringe rate
depends on velocity, an atomic transition whose frequency depends on a field — inherits that
precision. This is why quartz crystal microbalances, optical frequency combs, and atomic
magnetometers are built the way they are.


metrology_and_measurement

[user] My temperature measurement is repeatable to 0.01 C but disagrees with the reference
thermometer by 0.4 C. What does that tell me?
[assistant] tool_call metrology.diagnose({"repeatability_c": 0.01, "offset_vs_reference_c": 0.4,
"sensor": "platinum_rtd", "excitation_ma": 1.0, "wire_config": "two_wire",
"lead_resistance_ohm": 0.8, "calibrated": false})
[tool] {"issue": "systematic_offset_not_random_error",
"lead_resistance_contribution_c": 0.21, "self_heating_estimate_c": 0.06,
"uncalibrated_sensor_tolerance_c": 0.3, "conclusion": "precise_but_inaccurate",
"remedies": ["four_wire_connection", "reduce_excitation", "calibrate"]}
[assistant] You have excellent precision and a systematic error, which are independent properties
and require different remedies. Averaging more readings will not help at all.


metrology_and_measurement

Two identifiable contributions and one residual.


metrology_and_measurement

Your two-wire connection puts the lead resistance in series with the sensor, and the instrument
cannot distinguish it from the sensor's own resistance. At 0.8 ohms of lead resistance with a
platinum RTD, that is about 0.21 degrees of offset, always in the same direction. A four-wire
connection carries the excitation current on one pair and senses the voltage on another that
carries no current, so the lead resistance drops out entirely. This is the single largest and most
straightforward fix.


metrology_and_measurement

Self-heating from the excitation current contributes about 0.06 degrees. One milliamp through the
sensor dissipates power that raises its own temperature above the medium's, and the magnitude
depends on the thermal coupling — worse in still air, better in flowing liquid. Reducing the
excitation reduces the heating quadratically and reduces the signal linearly, so there is an
optimum. Measuring at two excitation levels and extrapolating to zero is the standard way to
quantify it.


metrology_and_measurement

The remainder is the sensor's own uncalibrated tolerance, which for a standard-grade element is a
few tenths of a degree. No wiring change addresses it; it requires calibration against a reference,
which gives you a correction to apply.


metrology_and_measurement

The general point worth taking from this: your repeatability of 0.01 degrees tells you what your
resolution and noise permit, and it says nothing whatever about accuracy. The two are independent,
and a highly repeatable instrument that is consistently wrong is the more dangerous situation
because the consistency reads as reliability. Comparing against a traceable reference is the only
thing that establishes accuracy, and it is what converts your readings from relative to absolute.


metrology_and_measurement

## Time, Frequency, and Statistical Treatment

Time and frequency are the most accurately realized quantities, and their accuracy propagates to
everything defined through them. A caesium standard realizes the second to parts in ten to the
sixteenth, and optical clocks based on transitions at far higher frequency reach parts in ten to
the eighteenth — accurate enough that the gravitational redshift over a centimetre of height
difference is measurable.


metrology_and_measurement

Frequency stability is characterized differently from other measurements because the noise is not
white. The Allan deviation, computed from the differences between successive averages, is used
instead of the standard deviation because the standard deviation of a signal with drift does not
converge as the averaging time grows. Plotting the Allan deviation against the averaging time
reveals the noise types present: white noise falls as the inverse square root of the averaging
time, flicker noise is flat, and drift rises — so the plot identifies the optimum averaging time
where the falling and rising contributions cross.


metrology_and_measurement

Statistical treatment of measurement data has requirements that general statistics does not
emphasize. The distinction between repeatability — the same operator, instrument, and conditions
over a short interval — and reproducibility — different operators, instruments, or laboratories —
matters because they differ substantially and quoting one when the other is meant misstates the
uncertainty by a large factor.


metrology_and_measurement

Time, frequency, and statistics:
  caesium standard     realizes the second; parts in 10^16
  optical clocks       parts in 10^18; sensitive to a centimetre of height
  frequency stability  characterized by Allan deviation, not standard deviation
  why not standard deviation  it does not converge for signals with drift
  Allan deviation plot  identifies the noise types and the optimum averaging time
  white noise          falls as the inverse square root of averaging time
  flicker noise        flat; averaging longer does not help
  drift                rises; averaging longer makes it worse
  the optimum          where the falling and rising contributions cross
  repeatability        same operator, instrument, and conditions, short interval
  reproducibility      different operators, instruments, or laboratories;
                       substantially larger
  outlier treatment    a documented criterion decided in advance; discarding
                       points that disagree is how systematic errors are
                       concealed
  regression uncertainty  the fit parameters have uncertainties and covariances;
                       propagating them requires the covariance rather than only
                       the variances
  round only at the end  intermediate rounding accumulates
  round the uncertainty first  then match the result's precision to it


metrology_and_measurement

The Allan deviation deserves the technical emphasis because it answers a question the standard
deviation cannot. For a stationary signal with white noise, averaging longer always improves the
estimate and the standard deviation is the right measure. For a real oscillator with flicker noise
and drift, averaging longer helps up to a point and then makes things worse, and the standard
deviation grows without bound as more data is included — which is the opposite of what a statistic
should do. The Allan deviation is constructed to converge for these signals and to identify which
noise process dominates at each timescale, which is exactly the information needed to choose an
averaging time.


metrology_and_measurement

## Instrumentation Practice

Practical measurement has a set of considerations that determine whether the theoretical accuracy
is achieved, and most measurement failures arise here rather than in the instrument.


metrology_and_measurement

Loading is the measurement disturbing what it measures. A voltmeter draws current, an ammeter drops
voltage, a thermometer conducts heat, a pressure sensor occupies volume, a flow sensor obstructs.
The disturbance is negligible when the instrument's impedance is far from the source's — high input
impedance for a voltage measurement, low for a current — and it is substantial when they are
comparable, which is exactly the case where the measurement seemed straightforward.


metrology_and_measurement

Grounding and shielding determine the noise floor of any sensitive electrical measurement. Ground
loops, where two points nominally at ground are connected by two paths, admit circulating current
that appears as noise. Single-point grounding at low frequency and a continuous reference plane at
high frequency are the standard responses, and the transition between them is a source of
persistent confusion.


metrology_and_measurement

Environmental effects are the systematic errors that appear when the measurement moves from the
laboratory. Temperature affects almost every sensor and most instruments. Humidity affects
capacitive sensors and insulation resistance. Vibration affects anything mechanical. Ambient light
affects optical measurements. Electromagnetic interference affects everything electrical. Each is
characterizable and each is more easily controlled than corrected.


metrology_and_measurement

Instrumentation practice:
  loading              the measurement disturbs the measurand
  input impedance      far from the source impedance, in the appropriate direction
  four-wire            separates excitation from sensing; eliminates lead
                       resistance
  guarding             drive a shield to the signal potential; eliminates leakage
                       currents in high-impedance measurement
  ground loop          two ground paths admitting circulating current
  single-point ground  at low frequency; a continuous plane at high frequency
  isolation            breaks the loop; optical, transformer, or capacitive
  shielding            electrostatic shields need a single ground connection;
                       magnetic shielding requires high-permeability material
  thermal EMF          junctions of dissimilar metals generate voltages at the
                       microvolt level; a dominant error in low-level DC
                       measurement, addressed by keeping junctions isothermal
  settling time        allow it; a reading taken before settling is a
                       reproducible wrong answer
  warm-up              instruments drift until thermally stable; specifications
                       assume warm-up
  environmental control  cheaper than correction
  document the conditions  a measurement without its conditions cannot be
                       reproduced or compared


metrology_and_measurement

Thermal electromotive force deserves the closing emphasis because it dominates precision
low-voltage measurement and is invisible to anyone not looking for it. Any junction between
dissimilar metals generates a voltage proportional to its temperature, typically microvolts per
degree, and a measurement circuit contains many such junctions — connectors, solder joints,
switch contacts, the transition from copper wire to a component lead. If the junctions are at
different temperatures the voltages do not cancel, and the resulting offset is comparable to or
larger than the signal in any measurement below the millivolt level. The remedies are keeping
junction pairs isothermal, using low-thermal-EMF connectors and solder, allowing thermal
equilibration, and reversing the measurement polarity and averaging — which cancels the offset
because it does not reverse with the signal.