Sensing principles formulas

Delta y_max is the largest deviation of the actual output from the reference straight line.

R_p is the total track resistance. Reduces to x for R_L infinite. The error is zero at both ends. For R_L much larger than R_p: as a fraction of the ideal reading the error is about (R_p/R_L)x(1-x), largest at x = 0.5; as a fraction of V_s (full scale) it is about (R_p/R_L)x^2(1-x), largest at x = 2/3, where it is about 0.148 R_p/R_L.

nu is Poisson's ratio of the gauge material. 1 + 2nu is the dimensional effect; the last term is the piezoresistive effect. For metal foil gauges the resistivity term is small and G is about 2. Semiconductor gauges have G of order 100 or more, mainly from the piezoresistive term, but are more temperature-sensitive and less linear.

Valid for all arms initially R and ΔR much less than R; sign of V_o depends on which terminals are taken as output.

Equal-arm bridge with one active arm; the fractional error is about (ΔR/R)/2. Half and full bridges with equal and opposite changes, each pair in series across the supply, have no such error; axial-plus-transverse arrangements keep a small one.

Input-stage resistors R_1 (two equal), gain resistor R_g, difference stage with matched R_2 at the inputs and R_3 as feedback and ground resistors. Gain is set by R_g alone when R_3/R_2 is fixed.

Fringing neglected. Sensing by varying plate gap d (output C is non-linear in d), overlap area A (linear in displacement), or dielectric (varying the effective ε_r).

Exactly linear in displacement x of the centre plate; needs a ratio-measuring circuit such as a bridge or ratiometric amplifier.

C_1 and C_2 in series across the AC excitation V_s, the other half of the bridge two equal impedances, V_o measured across the detector diagonal by a high-impedance detector. The phase of V_o (0 or 180 degrees) shows the direction of x.

N turns, l_g the total air-gap length, A_g the gap area; valid when the core's reluctance is negligible beside the gap's. L is then proportional to 1/l_g, so a single-gap sensor is non-linear; a differential (push-pull) pair in a bridge improves linearity, as for the capacitor.

t is the crystal thickness, p = F/A the applied stress; g in V·m/N. Follows from q = dF and C_p = ε_0 ε_r A/t, for an open-circuit crystal with no cable or amplifier load.

Mid-band value, leakage neglected. Output changes if the cable is changed; this is the reason a charge amplifier is preferred.

Op-amp with feedback capacitor C_f; independent of cable capacitance when the op-amp gain is high. Sign is inverting.

A_ol is the op-amp's open-loop gain. When (1 + A_ol)C_f is much larger than C_t, V_o tends to −q/C_f, which is why the cable capacitance hardly matters.

R_f in parallel with C_f gives a high-pass response; at f_c the gain magnitude is 3 dB below its mid-band value 1/C_f. A steady (DC) force gives a charge that leaks away, so true static force cannot be measured.

Faraday's law for a coil of N turns linking flux Φ; Blv for a conductor of length l moving at speed v, with B, l and v mutually perpendicular. Electromagnetic velocity pickups and flowmeters use this principle.

A ferromagnetic wheel with T teeth turning at n rev/min past a magnetic pickup gives pulses at frequency f in Hz. The pulse amplitude also rises with speed, so the speed is read from the frequency.

B perpendicular to the plate and to the current; n is carrier density, q the carrier charge magnitude, so this gives the magnitude of V_H. The sign of R_H follows the carrier sign (negative for electrons), which identifies the carrier type. V_H reverses sign when B (or I) is reversed. Semiconductors give a larger V_H than metals because n is much smaller.