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PhysBound Formula Reference

Every formula PhysBound uses to validate physics claims. All constants are CODATA 2018 exact values sourced from SciPy.

Physical Constants

Symbol Value Unit Source
c 299,792,458 m/s Speed of light (SI exact)
k_B 1.380649 x 10^-23 J/K Boltzmann constant (SI exact)
h 6.62607015 x 10^-34 J*s Planck constant (SI exact)
T_ref 290 K IEEE standard reference temperature
N_0 -174.0 dBm/Hz Thermal noise floor at T_ref

Free-Space Path Loss (FSPL)

FSPL(dB) = 20*log10(d) + 20*log10(f) + 20*log10(4*pi/c)
  • d: distance in meters
  • f: frequency in Hz
  • c: speed of light in m/s

Equivalent compact form: FSPL(dB) = 32.45 + 20*log10(f_MHz) + 20*log10(d_km)

Applicability: free-space (line-of-sight, no multipath). PhysBound warns above 300 GHz where atmospheric absorption invalidates the model.

Friis Transmission Equation

P_rx = P_tx + G_tx + G_rx - FSPL - L_tx - L_rx

All values in dB/dBm/dBi:

  • P_tx: transmit power (dBm)
  • G_tx, G_rx: antenna gains (dBi)
  • FSPL: free-space path loss (dB)
  • L_tx, L_rx: miscellaneous losses (dB), e.g., cable, connector, mismatch

Antenna Aperture Gain Limit

G_max = eta * (pi * D / lambda)^2
  • eta: aperture efficiency (default: 0.55 for parabolic dishes)
  • D: antenna diameter in meters
  • lambda: wavelength = c / f

Returns gain in linear scale; converted to dBi via 10*log10(G_max).

Any claimed gain exceeding G_max for a given antenna size and frequency is a physics violation.

Shannon-Hartley Channel Capacity

C = B * log2(1 + SNR)
  • C: maximum channel capacity in bits per second
  • B: channel bandwidth in Hz
  • SNR: signal-to-noise ratio (linear, not dB)

Spectral Efficiency

eta = C / B = log2(1 + SNR)    [bps/Hz]

SNR Conversion

SNR_linear = 10^(SNR_dB / 10)
SNR_dB = 10 * log10(SNR_linear)

Any throughput claim exceeding C for a given bandwidth and SNR is a physics violation. PhysBound flags the exact excess percentage.

Thermal Noise Power

N = k_B * T * B
  • k_B: Boltzmann constant
  • T: system temperature in Kelvin
  • B: bandwidth in Hz

In dBm: N(dBm) = 10 * log10(k_B * T * B / 1e-3)

At the IEEE reference (290K, 1 Hz): N = -174.0 dBm/Hz. This is the fundamental lower bound on receiver noise.

Friis Noise Cascade

F_total = F_1 + (F_2 - 1)/G_1 + (F_3 - 1)/(G_1 * G_2) + ...
  • F_n: noise factor of stage n (linear, = 10^(NF_dB/10))
  • G_n: gain of stage n (linear)

All values are in linear scale internally; inputs and outputs use dB.

Key insight: the first stage dominates the system noise figure. A low-noise first stage (LNA) with high gain suppresses the noise contribution of subsequent stages.

System Noise Temperature

T_sys = T_ref * (F_total - 1)

Where F_total is the cascaded noise factor (linear).

Receiver Sensitivity

S_min = N_floor + NF + SNR_req

All in dB:

  • N_floor: thermal noise floor in dBm (= 10*log10(k_B * T * B / 1e-3))
  • NF: system noise figure in dB
  • SNR_req: required SNR at the detector in dB

S_min is the minimum signal power (in dBm) the receiver can detect.

Monostatic Radar Range Equation

Maximum Detection Range

R_max = [P_t * G^2 * lambda^2 * sigma / ((4*pi)^3 * S_min * L)]^(1/4)
  • P_t: peak transmit power in watts
  • G: antenna gain (linear, monostatic: same antenna TX/RX)
  • lambda: wavelength = c / f (meters)
  • sigma: radar cross section (RCS) in m^2
  • S_min: minimum detectable signal power in watts
  • L: total system losses (linear)

Signal-to-Noise Ratio (SNR Form)

SNR = P_t * G^2 * lambda^2 * sigma / ((4*pi)^3 * k_B * T_s * B_n * R^4 * L)
  • k_B: Boltzmann constant
  • T_s: system noise temperature in Kelvin
  • B_n: noise bandwidth in Hz
  • R: range in meters

Minimum Detectable Signal

S_min = k_B * T_s * B_n * SNR_min / N_pulses

Where N_pulses provides coherent integration gain.

Key Physical Insight: The Fourth-Root Law

Range scales as the fourth root of power, gain squared, RCS, and wavelength squared:

  • Doubling P_t increases R_max by factor of 2^(1/4) = 1.189 (NOT 2x)
  • Doubling antenna gain (linear) increases R_max by factor of 2^(1/2) = 1.414
  • 10x larger RCS increases R_max by factor of 10^(1/4) = 1.778

Any claimed detection range exceeding R_max for the given parameters is a physics violation.

Input Validation Guards

PhysBound enforces these constraints on all inputs before computation:

Constraint Physical Basis
Frequency > 0 Hz Causality; EM wave must propagate
Distance > 0 m Causality; non-degenerate link
Bandwidth > 0 Hz Information-theoretic requirement
Temperature >= 0 K Third Law of Thermodynamics
SNR > 0 (linear) Signal must carry energy
Noise Figure >= 0 dB Quantum noise limit
Antenna diameter > 0 m Physical aperture must exist
Power > 0 W Conservation of Energy
RCS > 0 m^2 Physical target must scatter energy
Losses >= 0 dB Passive system cannot create energy
Num pulses >= 1 At least one pulse required