 Ratiometric cancellation with matched signal and reference filters

Ratiometric cancellation with matched signal and reference filters preserves a resistance ratio despite changing excitation when both paths apply the same dynamics to the shared electric current. Matching only the steady gain is insufficient during a transient.

Let constant sensor and reference resistances be $R_T,R_{\mathrm{ref}}$, and let both paths be linear time-invariant systems with impulse responses $h_T,h_{\mathrm{ref}}$. With consistent initial conditions, their filtered electric potential differences arewhere $\ast$ is convolution. If $h_T=h_{\mathrm{ref}}$, simultaneous sampling gives $y_T/y_{\mathrm{ref}}=R_T/R_{\mathrm{ref}}$ whenever the denominator is nonzero. Arbitrary unequal stored initial states need not cancel. Nor can a time-varying $R_T$ generally be pulled outside the convolution.

For unit steady gains, write $I(t)=I_0(1+\eta (t))$ after settling at $I_0$. Define $q_T=h_T\ast \eta$ and $q_{\mathrm{ref}}=h_{\mathrm{ref}}\ast \eta$. ThenThis displays exactly which excitation component survives. For small fluctuations the fractional residual is $q_T-q_{\mathrm{ref}}$, so both amplitude and timing differences between the transfer functions matter.

As an original transient example, step the excitation up by ten percent from a settled state. Two first-order low-pass filters with time constants ${\tau }_T=1\mathrm{ms}$ and ${\tau }_{\mathrm{ref}}=2\mathrm{ms}$ give, for $t\ge 0$,At $t=1\mathrm{ms}$ the reading is high by approximately $2.2962\%$, although both filters have the same final gain and the error tends to zero. Setting both time constants to $1\mathrm{ms}$ cancels this particular excitation step at every time. Sampling skew after matched ratiometric filters can still break simultaneous cancellation even with identical hardware.

The primary ADS1248 experiment used a resistance decade box, a $1\mathrm{mA}$ excitation and a $2\mathrm{k\Omega }$ reference. Reference-only filtering produced signal-dependent noise; matching the signal and reference paths substantially reduced that dependence. The report provides the circuit and noise-versus-input plots, rather than merely recommending a common nominal cutoff.^[ref]