Another example of a first-order system is an “RC” circuit, consisting of a resistor R and a capacitor C as shown in the next figure. The same equations apply to this circuit, with time constant = RC. Functionally, a capacitor can be regarded as an integrator because the voltage across it equals the time integral of the current (I(t)) that passes through it divided by the capacitance. The same current passes through the resistor and produces a voltage across it equalling I(t)R, so equating the current passing through the resistor and through the capacitor gives dVout / dt = ( Vin - Vout ) / (RC). This equation has the same form as the equation for the derivative of temperature, dT(t) / dt in the preceding example, if the time constant 𝜏 is equal to RC.
Sensors exhibiting first-order dynamic response can be more complicated, but the simplest (like the preceding examples) have exponential response and are characterized by
where H*(x) represents the static transfer function or the transfer curve, giving the output value that would be produced by a steady measurand with the value x(t).
The above diagram, based on the equation, is a functional depiction of a first-order dynamic system having time constant 𝜏. It shows how the output M from a sensor having first-order dynamic response depends on the measurand x . The red box is the complete instrument transfer function H. However, x and M both generally vary with time. The triangular symbols represent amplifiers producing an output that is the difference between the two inputs multiplied by the indicated gain (1 or 1 / 𝜏 in these two examples). The box with an integration symbol integrates the input (in this case, the time derivative dM(t) / dt of the measured quantity as given by the preceding equation). Integrating the input produces the measurement, M. However, that measurement is also fed back to the left amplifier, which produces an output determined by the difference between H*(x) and M. The diagram is a representation equivalent to the differential equation shown above, the RC circuit diagram, or the diagram with the generic transfer function H. The transfer function depicted can be considered a low-pass filter because only slowly varying input fluctuations will be passed through to the instrument output. Rapidly varying input fluctuations will be dampened (i.e., attenuated) and will not be passed to the output. In general, input variations of frequency less than 1 / 𝜏 (where 𝜏 is the time constant) will be passed through to the instrument output, whereas input variations at frequencies higher than the instrument’s temporal resolution (1 / 𝜏 ) will be attenuated.