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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. Must be >= 0 dB: a negative loss would be a passive component creating energy (same rule as the radar L).

Antenna Gain Limit (aperture and Harrington bounds)

G     = eta * (pi * D / lambda)^2          (circular aperture of diameter D)
D_max = (ka)^2 + 2ka,  k = 2*pi/lambda,  a = D/2    (Harrington bound)

Derivation: the gain of any aperture antenna is G = 4*pi*A_e / lambda^2 (Balanis, Antenna Theory, Sec. 2.16; Pozar, Microwave Engineering, Ch. 14), where the effective aperture A_e = eta * A_phys and A_phys = pi * D^2 / 4 for a circular aperture. Substituting gives G = eta * (pi*D/lambda)^2.

  • eta: aperture efficiency, 0 < eta <= 1 (illumination taper, spillover, blockage, surface error)
  • D: antenna diameter in meters
  • lambda: wavelength = c / f

Because A_e cannot exceed the physical area, eta <= 1 and G_ap = (pi*D/lambda)^2 is the largest gain a planar aperture of diameter D can have. It is not a rigorous bound on an arbitrary antenna that fits in a D-metre footprint: a half-wave dipole (2.15 dBi) in a 0.1 m footprint at 900 MHz exceeds G_ap = -0.5 dBi. The rigorous bound on the directivity of any antenna enclosed in a sphere of radius a = D/2 is Harrington's D_max = (ka)^2 + 2ka (Harrington, "Effect of antenna size on gain, bandwidth, and efficiency", J. Res. NBS 64D, 1960; Balanis, Antenna Theory, small-antenna limits). Since ka = pi*D/lambda, this is the aperture value (ka)^2 plus the 2ka term, so PhysBound's hard limit is

G_phys = max(G_ap, D_max) = D_max          (for every D, because 2ka > 0)

The two coincide to within 10*log10(1 + 2*lambda/(pi*D)) dB: 0.08 dB for a 1 m dish at 10 GHz, 0.03 dB for a 30 m dish at 1 GHz (large dishes are effectively unchanged), but

= 2.1 dB whenever D < lambda, where the aperture formula would falsely reject real electrically small antennas.

PhysBound uses three values:

Value Formula Behaviour
Physical limit G_phys = max((pi*D/lambda)^2, (ka)^2 + 2ka) Claimed gain above this is a PhysicalViolationError
Aperture value (eta = 1) G_ap = (pi * D / lambda)^2 Claimed gain between G_ap and G_phys is accepted with a warning (implied eta > 1: only a non-planar / electrically small radiator can do it)
Typical value (eta = 0.55, parabolic dish; Skolnik, Radar Handbook, Ch. 9) G_typ = 0.55 * (pi * D / lambda)^2 Claimed gain between G_typ and G_ap is accepted with a warning (it implies eta > 0.55, unusually efficient)

The gap between G_ap and G_typ is a constant -10*log10(0.55) = 2.6 dB. The aperture_efficiency parameter moves the warning threshold only; the hard limit is always max(G_ap, D_max).

All values are returned: tx_physical_limit_dbi / rx_physical_limit_dbi (hard limit), tx_aperture_limit_dbi / rx_aperture_limit_dbi (eta = 1), tx_typical_aperture_gain_dbi / rx_typical_aperture_gain_dbi (typical efficiency) and tx_limiting_bound / rx_limiting_bound ("harrington" when D < lambda, electrically small; "aperture" when D >= lambda), together with the implied planar-aperture efficiency eta_claim = G_claim / G_ap in the messages.

Worked examples: - README row 3: 0.3 m dish at 1 GHz, lambda = 0.2998 m, ka = pi*D/lambda = 3.144, so G_ap = 9.88 -> 9.95 dBi, D_max = 9.88 + 6.29 = 16.17 -> 12.09 dBi and G_typ = 5.44 -> 7.35 dBi. A 45 dBi claim would need eta = 3200. - README row 10: 0.1 m at 900 MHz, lambda = 0.333 m, ka = 0.943: G_ap = -0.51 dBi, D_max = 0.889 + 1.886 = 2.775 -> 4.43 dBi, G_typ = -3.10 dBi. A 20 dBi claim is rejected, but a 2.15 dBi half-wave dipole in the same footprint is valid (warned: implied eta = 1.84, non-planar radiator).

Caveats reported as warnings: - The Friis equation assumes far-field propagation, d > 2*D^2/lambda (Fraunhofer distance). Links closer than this are flagged. - Above 300 GHz the free-space model ignores atmospheric absorption.

Antenna Gain Tool (antenna_gain)

Standalone version of the aperture check above, for a single antenna given either its circular diameter D or its physical area A_phys.

lambda   = c / f
D        = sqrt(4 * A_phys / pi)                 (equivalent circular diameter, if area given)
A_phys   = pi * D^2 / 4
A_e      = eta * A_phys                          (effective aperture)
D_0      = 4*pi*A_phys / lambda^2 = (pi*D/lambda)^2   (planar-aperture directivity, eta = 1)
D_max    = (ka)^2 + 2ka,  k = 2*pi/lambda,  a = D/2    (Harrington bound; = D_0 + 2ka)
G_phys   = max(D_0, D_max) = D_max                (hard physical gain limit)
G        = eta * D_0 = 4*pi*A_e / lambda^2       (gain at aperture efficiency eta)
HPBW     ~ 70   * lambda / D  degrees            (tapered parabolic reflector, rule of thumb)
HPBW     ~ 58.4 * lambda / D  degrees            (uniformly illuminated circular aperture)
R_ff     = 2 * D^2 / lambda                      (far-field / Fraunhofer distance)
eta_claim = G_claim / D_0                        (efficiency implied by a gain claim)

Sources: - G = 4*pi*A_e/lambda^2 and G = e_ap * D_0: Balanis, Antenna Theory, Sec. 2.16 (aperture efficiency and directivity of aperture antennas); Pozar, Microwave Engineering, Ch. 14. - HPBW of a uniformly illuminated circular aperture 29.2 deg * lambda/a, a = D/2, i.e. 58.4 deg * lambda/D: Balanis, Sec. 12.5, Table 12.2. Reflectors with typical edge taper broaden this to about 70 deg * lambda/D (Balanis, Sec. 15.4; Stutzman & Thiele, Antenna Theory and Design, Sec. 9.4). Both values are returned and flagged as approximations. - Far-field region begins at R = 2 D^2 / lambda: Balanis, Sec. 2.2.4. - Typical parabolic-dish aperture efficiency eta = 0.55: Skolnik, Radar Handbook, Ch. 9. - Harrington bound D_max = (ka)^2 + 2ka for an antenna enclosed in a sphere of radius a = D/2: Harrington, J. Res. NBS 64D (1960); Balanis. Governs for D < lambda.

Outputs: wavelength_m, diameter_m, physical_aperture_m2, effective_aperture_m2, physical_limit_dbi (max(D_0, D_max)), aperture_limit_dbi (D_0, eta = 1), harrington_limit_dbi (D_max), limiting_bound ("harrington" for D < lambda, "aperture" for D >= lambda), typical_gain_dbi (eta = aperture_efficiency), directivity_linear (D_0), half_power_beamwidth_deg (70 lambda/D), half_power_beamwidth_uniform_deg (58.4 lambda/D), far_field_distance_m, and when claimed_gain_dbi is supplied implied_efficiency and claim_is_valid.

Validation uses the same thresholds as the link budget tool: a claim above physical_limit_dbi raises PhysicalViolationError (law: "Antenna Aperture Limit"); a claim between typical_gain_dbi and physical_limit_dbi is a warning (with a note that only a non-planar / electrically small radiator can exceed aperture_limit_dbi).

Worked example: 1 m dish at 10 GHz, lambda = 0.029979 m, ka = pi*D/lambda = 104.8, D_0 = 10983 -> 40.41 dBi, D_max = 10983 + 209.6 = 11193 -> 40.49 dBi, G(eta=0.55) = 37.81 dBi, A_e = 0.432 m^2, HPBW ~ 2.10 deg (uniform: 1.75 deg), R_ff = 66.7 m. A 45 dBi claim would need eta = 2.9. Small-antenna example: a 2.15 dBi half-wave dipole in a 0.1 m footprint at 900 MHz (ka = 0.943, D_0 = -0.51 dBi, D_max = 4.43 dBi) is valid with limiting_bound = "harrington"; the planar-aperture formula alone would have rejected it.

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.

Effective Input Noise Temperature

T_e = T_0 * (F_total - 1),    T_0 = 290 K

Where F_total is the cascaded noise factor (linear). Noise figure is defined (IEEE Std; Pozar, Microwave Engineering, Sec. 10.1) as the SNR degradation when the source is at the reference temperature T_0 = 290 K:

F = (S_i/N_i) / (S_o/N_o) = (T_0 + T_e) / T_0    =>    T_e = T_0 (F - 1)

T_e characterises the receiver hardware and therefore does not depend on the temperature_k (antenna/source temperature) supplied to the noise_floor tool. Example: NF = 1.66 dB gives T_e = 290 * (1.466 - 1) = 135.0 K whether the antenna looks at a 290 K or a 77 K source. (Earlier versions computed temperature_k * (F - 1), which understated T_e by T_0 / T_A for cold sources; this was wrong and has been fixed.)

Receiver Sensitivity

S_min = k_B * (T_A + T_e) * B * SNR_req,      T_e = T_0 * (F - 1)

The total input-referred noise is the source (antenna) noise k_B * T_A * B plus the receiver's own noise k_B * T_e * B. In dB:

S_min(dBm) = 10*log10(k_B * (T_A + T_e) * B / 1e-3) + SNR_req
  • T_A: source/antenna noise temperature (temperature_k, default 290 K)
  • T_e: receiver effective input noise temperature, referenced to T_0 = 290 K
  • NF: system noise figure in dB, F = 10^(NF/10)
  • SNR_req: required SNR at the detector in dB

When T_A = T_0 = 290 K, k_B (T_0 + T_0(F-1)) B = k_B T_0 B F and this reduces to the familiar S_min = N_floor + NF + SNR_req. For other source temperatures the familiar form is incorrect because NF is defined at 290 K: with T_A = 77 K and NF = 3 dB, T_e = 288.6 K and the true noise is k_B * 365.6 K * B, 3.8 dB higher than N_floor(77 K) + 3 dB would suggest.

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
Radar T_s > 0 K S_min = k T_s B -> 0 gives unbounded range
Aperture efficiency 0 < eta <= 1 Effective area cannot exceed physical area
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 (radar L, link L_tx, L_rx) Passive system cannot create energy
Num pulses >= 1 At least one pulse required
PRF > 0 Hz Pulsed radar must emit pulses
Pulse width tau > 0 s A pulse must carry finite energy
Duty cycle tau * PRF < 1 Receiver must be un-blanked for part of every PRI (else CW, not pulsed)

Pulse-Doppler Radar Ambiguity (radar_ambiguity)

Sources: Skolnik, Introduction to Radar Systems, 3rd ed., Ch. 2 (range ambiguity, duty cycle) and Ch. 3 (MTI, blind speeds, Doppler); Richards, Fundamentals of Radar Signal Processing, 2nd ed., Ch. 1 (Doppler shift, range resolution), Ch. 3 (Doppler sampling and aliasing) and Ch. 5 (Doppler processing, range-Doppler dilemma).

lambda      = c / f
PRI         = 1 / PRF
R_ua        = c / (2 * PRF)             maximum unambiguous range
v_blind     = lambda * PRF / 2          first blind speed (f_d = PRF)
v_ua        = lambda * PRF / 4          unambiguous radial velocity (|f_d| <= PRF/2)
f_d         = 2 * v_r / lambda          Doppler shift, closing velocity positive
R_ua * v_ua = c * lambda / 8            range-Doppler dilemma invariant
dR          = c * tau / 2               range resolution, unmodulated pulse of width tau
dR          = c / (2 * B)               range resolution with pulse compression, bandwidth B
R_min       = c * tau / 2               eclipsing: receiver blanked while transmitting
duty        = tau * PRF
  • R_ua: an echo from beyond R_ua arrives after the next pulse is transmitted and is folded to the apparent range R mod R_ua (second-time-around echo).
  • v_ua / v_blind: the pulse train samples the Doppler phase at the PRF, so by the sampling theorem only |f_d| <= PRF/2 is unambiguous; f_d = n * PRF is indistinguishable from stationary clutter (MTI blind speeds). A target with |f_d| > PRF/2 is reported as aliased, with the apparent velocity obtained by folding f_d into (-PRF/2, PRF/2].
  • Range-Doppler dilemma: because R_ua ~ 1/PRF and v_ua ~ PRF, their product is fixed at c * lambda / 8 for a given carrier. Raising the PRF buys velocity coverage at the expense of range and vice versa; only a change of wavelength (or multiple-PRF staggering / Chinese-remainder unfolding, outside the scope of this tool) can move the product.
  • Range resolution: for a simple pulse the effective bandwidth is B ~ 1/tau, so dR = c*tau/2. Pulse compression (chirp, phase codes) decouples resolution from pulse width: pass bandwidth_hz and the tool uses c/(2B) instead, warning if B*tau < 1.

Outputs: wavelength_m, pulse_repetition_interval_s, max_unambiguous_range_m / _km, first_blind_speed_m_s, max_unambiguous_velocity_m_s, max_unambiguous_doppler_hz (PRF/2), range_velocity_product_m2_s (c*lambda/8), and when the corresponding inputs are supplied doppler_shift_hz, doppler_aliased, apparent_velocity_m_s, range_resolution_m, minimum_range_m, duty_cycle.

Violations (PhysicalViolationError):

Claim Law violated Limit
claimed_unambiguous_range_m > c/(2 PRF) Radar Range Ambiguity R_ua
claimed_unambiguous_velocity_m_s > lambda PRF/4 Radar Doppler Ambiguity v_ua
R_claim * v_claim > c lambda/8 (both given) Range-Doppler Dilemma c lambda/8
claimed_range_resolution_m < c tau/2 (or c/(2B)) Radar Range Resolution dR
PRF <= 0, tau <= 0, tau * PRF >= 1 Pulsed Radar Timing -

Warnings: target Doppler aliased (|f_d| > PRF/2), target at or beyond the first blind speed, duty cycle > 0.5, B*tau < 1, a single range or velocity claim that leaves the other quantity tightly constrained by the dilemma, and f > 300 GHz.

Worked example (README rows 15-16): X-band 10 GHz, lambda = 2.998 cm. PRF = 1 kHz: R_ua = 149.9 km, v_ua = +/-7.49 m/s, blind speed 14.99 m/s. PRF = 10 kHz: R_ua = 15.0 km, v_ua = +/-74.9 m/s; a claimed 500 m/s unambiguous velocity is 6.7x the limit. PRF = 100 kHz: R_ua = 1.5 km. In every case R_ua * v_ua = 1.123e6 m^2/s, so 150 km with +/-300 m/s (4.5e7 m^2/s) is impossible at any PRF. A 1 us unmodulated pulse gives dR = R_min = 149.9 m; a 10 m resolution requires B >= 15 MHz of pulse compression.