start error budgeting section and use widely uVrms

1 files changed, 290 insertions, 17 deletions

diff --git a/module-design.typ b/module-design.typ
index cc4ae9d..3e389d3 100644
--- a/module-design.typ
+++ b/module-design.typ
@@ -1,4 +1,4 @@
-#import "@preview/unify:0.8.1": numrange, qty, qtyrange, unit
+#import "@preview/unify:0.8.1": num, numrange, qty, qtyrange, unit
 #import "@preview/diverential:0.3.0": *
 
 = Module design <module-design>
@@ -12,7 +12,7 @@ The challenge is substantial. The Metrologic tier demands $qty(10, "ppm")$
 precision, $qty(90, "dB")$ signal-to-noise ratio, and
 $qty(0.014, "ppm per hour")$ drift over a DC to $qty(20, "kHz")$ bandwidth.
 These specifications, taken together, require that a $qty(20, "V")$ signal range
-be resolved to $qty(200, "uV")$, that noise remain below $qty(632, "uV") upright("RMS")$,
+be resolved to $qty(200, "uV")$, that noise remain below $qty(632.455, "uVrms")$,
 and that accumulated drift not exceed $qty(10, "ppm")$ over thirty days of
 continuous operation. Achieving this with commodity through-hole components —
 resistors, capacitors, and op-amps available from any electronics distributor —
@@ -127,8 +127,8 @@ operators.
     [$upright(C)$], [Capacitance], [$unit(F)$],
     [$upright(L)$], [Inductance], [$unit(H)$],
     [$upright(P)$], [Power], [$unit(W)$],
-    [$upright(f)$], [Frequency], [$unit("Hz")$],
-    [$omega$], [Angular frequency ($omega = 2 pi upright(f)$)], [$unit("radian per second")$],
+    [$f$], [Frequency], [$unit("Hz")$],
+    [$omega$], [Angular frequency ($omega = 2 pi f$)], [$unit("radian per second")$],
 
     [$tau$], [Time constant ($tau = upright("RC") "or" upright("L/R")$)], [$unit(s)$],
 
@@ -205,7 +205,7 @@ operators.
     [$epsilon_"total"$], [Total combined error], [$unit("ppm")$ or $unit("uV")$],
 
     [$epsilon_"systematic"$], [Systematic error component], [$unit("ppm")$],
-    [$epsilon_"random"$], [Random error component], [$unit("uV") upright("RMS")$],
+    [$epsilon_"random"$], [Random error component], [$unit("uVrms")$],
 
     [$epsilon_"offset"$], [Offset error], [$unit("uV")$],
     [$epsilon_"gain"$], [Gain error], [$unit("ppm")$],
@@ -235,11 +235,9 @@ operators.
 
     [$upright(e)_(n,"white")$], [White noise component], [$unit("nV")/sqrt(unit("Hz"))$],
 
-    [$upright(e)_(n,1/upright(f))$],
-    [$1/upright(f)$ noise component],
-    [$unit("nV")/sqrt(unit("Hz"))$ at $qty(1, "Hz")$],
+    [$upright(e)_(n,1/f)$], [$1/f$ noise component], [$unit("nV")/sqrt(unit("Hz"))$ at $qty(1, "Hz")$],
 
-    [$upright(f)_c$], [Noise corner frequency ($1/upright(f)$ to white)], [$unit("Hz")$],
+    [$f_c$], [Noise corner frequency ($1/f$ to white)], [$unit("Hz")$],
 
     [$upright(V)_n$], [Total noise voltage], [$unit("V") upright("RMS")$],
     [$upright("CNI")$], [Current noise index (potentiometers)], [$unit("dB")$],
@@ -255,8 +253,8 @@ operators.
     columns: 3,
     table.header([Symbol], [Definition], [Typical unit]),
     [$upright(D)$], [Duty cycle], [$unit("percent")$ or dimensionless],
-    [$upright(f)_"carrier"$], [Carrier frequency], [$unit("Hz")$],
-    [$upright(f)_0$], [Resonant or center frequency], [$unit("Hz")$],
+    [$f_"carrier"$], [Carrier frequency], [$unit("Hz")$],
+    [$f_0$], [Resonant or center frequency], [$unit("Hz")$],
     [$phi$], [Phase], [$unit("radian")$ or $unit("degree")$],
     [$Delta phi$], [Phase error or shift], [$unit("radian")$ or $unit("degree")$],
 
@@ -329,7 +327,7 @@ Standard SI prefixes are used throughout:
     [micro], [$upright(mu)$], [$10^(-6)$],
     [nano], [$upright(n)$], [$10^(-9)$],
     [pico], [$upright(p)$], [$10^(-12)$],
-    [femto], [$upright(f)$], [$10^(-15)$],
+    [femto], [$f$], [$10^(-15)$],
   ),
   caption: [Unit prefixes],
 ) <table-unit-prefixes>
@@ -398,7 +396,7 @@ Standard SI prefixes are used throughout:
 / Signal-to-Noise Ratio ($upright("SNR")$): The ratio of signal power to noise
   power, typically expressed in decibels. The SAME Metrologic tier specifies
   $qty(90, "dB") upright("SNR")$, corresponding to approximately
-  $qty(632, "uV") upright("RMS")$ noise referred to the $qty(20, "Vpp")$ signal
+  $qty(632.455, "uVrms")$ noise referred to the $qty(20, "Vpp")$ signal
   range.
 / Referred-to-Input ($upright("RTI")$): An error specification expressed as an
   equivalent error at the circuit input. $upright("RTI")$ allows comparison of
@@ -408,9 +406,9 @@ Standard SI prefixes are used throughout:
   must be met at the output.
 / Noise Gain: The gain seen by error sources at the amplifier input, which
   differs from signal gain in inverting configurations. For an inverting
-  amplifier with feedback resistor $upright(R)_upright(f)$ and input resistor
+  amplifier with feedback resistor $upright(R)_f$ and input resistor
   $upright(R)_"in"$, noise gain equals
-  $1 + upright(R)_upright(f)/upright(R)_"in"$.
+  $1 + upright(R)_f/upright(R)_"in"$.
 / Error Budget: A systematic accounting of all error sources and their
   contributions to total system error. Errors are typically combined by
   root-sum-square ($upright("RSS")$) for independent sources or algebraic sum
@@ -593,7 +591,7 @@ Standard SI prefixes are used throughout:
 / Thermal Noise (Johnson-Nyquist Noise): Voltage fluctuations arising from the
   thermal agitation of charge carriers in any resistive element. Thermal noise
   power is proportional to temperature, resistance, and bandwidth.
-/ Flicker Noise ($1/upright(f)$ Noise): Noise whose power spectral density is
+/ Flicker Noise ($1/f$ Noise): Noise whose power spectral density is
   inversely proportional to frequency. Flicker noise dominates at low
   frequencies and is particularly significant in DC-coupled precision circuits.
 / Shot Noise: Noise arising from the discrete nature of electric charge,
@@ -723,7 +721,7 @@ When digital logic is present, it must be completely isolated from the analog
 signal path. The module must ensure:
 + No measurable ripple from digital switching reaches the analog outputs.
 Digital supply noise must be attenuated to below the system noise floor
-($qty(632, "uV") upright("RMS")$ for $qty(90, "dB") upright("SNR")$).
+($qty(632.455, "uVrms")$ for $qty(90, "dB") upright("SNR")$).
 + No RF emissions escape the cassette. The module must meet the EMI shielding
 requirements specified in @cassette-grounding-requirements, with
 particular attention to the higher-frequency harmonics generated by digital
@@ -963,6 +961,281 @@ content.
 == Error budgeting and system-level compensation
 <error-budgeting-and-system-level-compensation>
 
+=== Introduction <error-budgeting-introduction>
+
+The goal of this section is to demonstrate, with mathematical rigor, that the
+SAME Metrologic tier specifications are achievable:
+
+#figure(
+  table(
+    columns: 2,
+    table.header([Parameter], [Requirement]),
+    [Precision], [$qty(10, "ppm")$ ($qty(0.001, "percent")$)],
+    [Signal-to-noise ratio], [$qty(90, "dB")$],
+    [Drift rate], [$lt.eq qty(0.014, "ppm per hour")$],
+    [Bandwidth], [DC to $qty(20, "kHz")$],
+    [Drift-free operation], [$qty(30, "day")$],
+  ),
+  caption: [Summary of metrologic precision tier requirements],
+) <table-error-budgeting-introduction>
+
+We will proceed as follows:
++ Establish the relationship between these specifications,
++ Enumerate all error sources in a precision analog signal path,
++ Quantify each error source for commodity components,
++ Show that the naive error budget exceeds our specifications,
++ Introduce compensation topologies that reduce specific error terms,
++ Construct the compensated error budget,
++ Demonstrate that the compensated budget meets specifications.
+
+This section provides the theoretical framework. @error-compensation-strategies,
+@advanced-compensation-topologies and @implicit-computation provide detailed
+analysis and proof for each compensation topology. @reference-implementation
+provides concrete implementations.
+
+=== Fundamental relationships <error-budgeting-fundamental-realtionships>
+
+Before enumerating errors, we must establish how our specifications relate to
+each other.
+
+==== Precision and signal range
+<fundamental-relationships-precision-and-signal-range>
+
+The SAME signal range is $plus.minus qty(10, "V")$, a span of $qty(20, "V")$.
+Precision of $qty(10, "ppm")$ means the maximum acceptable error is:
+
+$ epsilon_"max" = qty(20, "V") times (10 times 10^(-6)) = qty(200, "uV") $
+
+This is our total error budget. Every error source - noise, drift, offset,
+nonlinearity - must sum to less than $qty(200, "uV")$ referred to output
+($upright("RTO")$), or equivalently, $qty(10, "ppm")$ referred to full scale
+($upright("FS")$)
+
+
+==== $upright("SNR")$ and noise floor <snr-and-noise-floor>
+
+A $qty(90, "dB") upright("SNR")$ with a $qty(20, "Vpp")$ signal range implies an
+$upright("RMS")$ noise floor of:
+
+$
+  upright(V)_("noise","rms") = qty(20, "V")/10^(90/20) = qty(20, "V")/31623 approx qty(632.455, "uVrms")
+$
+
+For Gaussian noise, the peak-to-peak value is approximately $6 times$ the
+$upright("RMS")$ value ($2 Phi (3) - 1 approx qty(99.7, "percent")$ containment):
+
+$
+  upright(V)_("noise","pp") approx 6 times qty(632.455, "uV") = qty(3.79473, "mVpp")
+$
+
+This appears to conflict with our $qty(200, "uV")$ precision requirement. The
+resolution is that _precision_ and _noise_ are different specifications:
+/ Precision: Applies to the _mean_ of the signal - systematic errors, offset,
+  gain error, linearity.
+/ Noise: Applies to the _variance_ - random fluctuations around the mean.
+
+A measurement averaged over sufficient time can achieve precision better than
+the instantaneous noise floor. However, for real-time computation (DC to
+$qty(20, "kHz")$ bandwidth), we cannot average indefinitely.
+
+The relationship between noise and achievable precision depends on the
+observation time $tau$:
+
+$
+  epsilon_"noise" = upright(V)_("noise","rms") / sqrt(2 times upright("BW") times tau)
+$
+
+For a single-sample measurement at $qty(20, "kHz")$ bandwidth,
+$tau = qty(25, "us")$: it represents one Nyquist interval
+($f_"sample" = 2 upright("BW") = qty(40, "kHz")$, giving
+$T_"sample" = qty(25, "us")$).
+
+$
+  epsilon_"noise" = qty(632.455, "uV") / sqrt(2 times 20000 times (25 times 10^(-6))) = qty(632.455, "uV") / 1 = qty(632.455, "uV")
+$
+
+This means noise alone consumes more than our entire precision budget for
+simgle-sample measurements.
+
+===== Resolution <snr-and-noise-floor-resolution>
+
+The $qty(10, "ppm")$ precision specification applies to _systematic_ errors
+only. The $qty(90, "dB") upright("SNR")$ specification applies to _random_
+errors. Both must be met, but they are separate budgets:
+
+#figure(
+  table(
+    columns: 3,
+    table.header([Budget], [Allocation], [Expressed as]),
+    [Systematic error budget], [$qty(200, "uV")$ ($qty(10, "ppm")$)], [Offset, drift, gain error, nonlinearity],
+
+    [Random error budget], [$qty(632.455, "uVrms")$ ($qty(90, "dB")$)], [Thermal noise, $1/f$ noise, interference],
+  ),
+  caption: [Error budgets allocation],
+) <table-snr-and-noise-floor-resolution>
+
+The $qty(90, "dB") upright("SNR")$ allows $qty(632.455, "uVrms")$ random
+noise, while the $qty(10, "ppm")$ precision requires $lt qty(200, "uV")$
+systematic error. These are independent budgets.
+
+In real-world analog computing (such as differential equation solving), feedback
+loops and integrators inherently ast as low-pass filters
+($tau gt.double qty(25, "uV")$). As a result, long-term or dynamic patch
+execution naturally attenuates the random noise variance toward zero, leaving
+the $qty(10, "ppm")$ systematic precision as the true controlling factor for
+continuous computation.
+
+==== Drift rate and long-term stability <drift-rate-and-long-term-stability>
+
+The drift rate specification of $qty(0.014, "ppm per hour")$ means:
+
+$
+  dv(epsilon, t) lt.eq 0.014 times 10^(-6) times qty(20, "volt per hour") = qty(280, "nano volt per hour")
+$
+
+Over 30 days (720 hours), the accumulated drift is:
+
+$ epsilon_("drift",qty(30, "day")) = 0.014 times 720 = qty(10.08, "ppm") $
+
+This is precisely calibrated so that 30 days of drift consumes the entire
+systematic error budget. After 30 days, recalibration is required.
+
+/ Important: The drift specification is a rate, not an absolute value. A module
+  may have an initial offset of $qty(5, "ppm")$ (within spec) and drift at
+  $qty(0.014, "ppm per hour")$. After 15 days, it reaches $qty(10, "ppm")$ total
+  and is now at the edge of specification. To guarantee a 30-day drift-free
+  computation period, active compensation topologies must trim or null
+  $epsilon_"initial"$ to near-zero ($lt qty(0.5, "ppm")$) at $t = 0$.
+
+==== Bandwidth and dynamic errors
+
+The DC to $qty(20, "kHz")$ bandwidth requirement introduces frequency-dependent
+errors:
+/ Gain error vs frequency: Amplifiers have finite gain-bandwidth product,
+/ Phase error vs frequency: Relevant for system-level computations (e.g.
+  solving differential equations),
+/ Slew rate limiting: Large signals at high frequency may exceed amplifier slew
+  rate,
+/ Settling time: Step inputs require the output to settle within
+  $qty(10, "ppm")$ of final value.
+
+For a sinusoidal signal at frequency $f$ to be reproduced within
+$qty(10, "ppm")$ amplitude error, the amplifier gain at the frequency must
+satisfy:
+
+$
+  A(f)/A(0) & gt.eq 1 - 10 times 10^(-6) \
+  A(f)/A(0) & gt.eq 0.99999
+$
+
+For a single-pole system with gain-bandwidth product $upright("GBW")$ and DC
+closed-loop gain $G$:
+
+$ A(f)/A(0) gt.eq 1/sqrt(1 + (f times G/upright("GBW"))^2) $
+
+Solving for $qty(10, "ppm")$ gain error at $qty(20, "kHz")$ with $G = 1$
+(unity gain buffer):
+
+$
+  upright("GBW") gt.eq 20000 / sqrt((1/0.99999)^2 - 1) approx 20000/sqrt(0.00002) approx qty(4.5, "MHz")
+$
+
+For $G = 10$:
+
+$ upright("GBW") gt.eq qty(45, "MHz") $
+
+This is achievable with modern op-amps, but it constrains component selection.
+
+=== Complete enumeration of error sources
+<complete-enumeration-of-error-sources>
+
+To construct a rigorous error budget, we must enumerate every physical mechanism
+that can cause the output of an analog circuit to deviate from its ideal
+mathematical function. We categorize these into five classes:
+/ Noise: Random fluctuations (contributes to $upright("SNR")$ budget).
+/ Drift: Time-varying systematic errors (contributes to drift rate budget).
+/ Static errors: Fixed systematic errors at a given operating point (contributes
+  to precision budget).
+/ Dynamic errors: Frequency-dependent systematic errors (contributes to
+  precision budget at bandwidth edges).
+/ Nonlinearity: Signal-amplitude-dependent systematic errors (contributes to
+  precision budget).
+
+For each error source, we will provide:
+/ Physical mechanism: Why does this error exist?
+/ Mathematical model: How do we quantify it?
+/ Typical magnitude: What values do commodity components exhibit?
+/ Where it enters: Which circuit nodes are vulnerable?
+/ Compensation strategy: Reference to subsequent sections where mitigation is
+  proven.
+
+==== Noise sources <enumeration-noise-sources>
+
+Noise sources contribute to the random error budget. They determine the
+$upright("SNR")$ floor of the system.
+
+===== Resistor thermal noise (Johnson-Nyquist noise) <resistor-thermal-noise>
+
+====== Physical mechanism <resistor-thermal-noise-physical-mechanism>
+
+Thermal agitation of charge carriers in a resistor produces a random voltage
+across its terminals. This is a fundamental thermodynamic phenomenon; it cannot
+be eliminated, only minimized.
+
+====== Mathematical model <resistor-thermal-noise-mathematical-model>
+
+The $upright("RMS")$ noise voltage spectral density of a resistor $R$ at
+absolute temperature $T$ is:
+
+$ e_n = sqrt(4 k_B T R) $
+
+Where:
+- $k_B = qty("1.381e-23", "joule per kelvin")$ (Boltzmann constant),
+- $T =$ absolute temperature ($unit(K)$),
+- $R =$ resistance ($unit("ohm")$).
+
+The total $upright("RMS")$ noise over bandwidth $upright("BW")$ is:
+
+$ V_(n,"rms") = sqrt(4 k_B T R upright("BW")) $
+
+====== Typical magnitude <resistor-thermal-noise-typical-magnitude>
+
+At $T = qty(300, "K")$ ($qty(27, "dC")$), for a $qty(10, "kilo ohm")$ resistor
+over $qty(20, "kHz")$ bandwidth:
+
+$
+  V_(n,"rms") & = sqrt(4 times num("1.381e-23") times 300 times 10000 times 20000) \
+  V_(n,"rms") & approx qty(1.82, "uVrms")
+$
+
+For a $qty(1, "mega ohm")$ resistor (input impedance):
+
+$
+  V_(n,"rms") & = sqrt(4 times num("1.381e-23") times 300 times 1000000 times 20000) \
+  V_(n,"rms") & approx qty(18.2, "uVrms")
+$
+
+====== Where it enters <resistor-thermal-noise-where-it-enters>
+
+- Input resistors in summing amplifiers,
+- Feedback resistors (noise is gained at output),
+- High-impedance nodes (input buffers, integrators).
+
+====== Scaling law <resistor-thermal-noise-scaling-law>
+
+Thermal noise scales with $sqrt(R)$. Lower resistance values reduce noise but
+increase power consumption and loading effects. This creates a fundamental
+tradeoff.
+
+====== Compensation strategy <resistor-thermal-noise-compensation-strategy>
+
+Thermal noise is fundamental and cannot be compensated. Mitigation is through:
+- Minimizing resistance values where possible,
+- Bandwidth limiting (but conflicts with the $qty(20, "kHz")$ requirement),
+- Operating at lower temperatures (impractical for SAME).
+
+For SAME, thermal noise contribution must be budgeted, not eliminated.
+
 == Error compensation strategies <error-compensation-strategies>
 
 == Advanced compensation topologies <advanced-compensation-topologies>