Instrument types and performance characteristics

 2.3.3 Second order instrument

If all coefficients a3 ...an other than a0, a1 and a2 in equation (2.2) are assumed zero, then we get:

                                           a2 *d² q0 /dt² + a1 * dq0/dt + a0q0 = b0qi                                            (2.7)

Applying the D operator again: a₂D²q₀ + a₁Dq₀ + a₀q₀ = b₀qi, and rearranging:

                                           q₀ = b₀qi/ a₀ + a₁D + a₂D²                                                                            (2.8)

It is convenient to re-express the variables a0, a1, a2 and b0 in equation (2.8) in terms of three parameters K (static sensitivity), ω (undamped natural frequency) and  (damping ratio), where:

                                                      K = b₀/a₀; ω = a₀/a₂;  = a₁/2a₀a₂

Re-expressing equation (2.8) in terms of K, ω and  we get:

                                                      q_0/ q_i = K /(D² /ω² + 2D/ω + 1)                                                     (2.9)

This is the standard equation for a second order system and any instrument whose response can be described by it is known as a second order instrument. If equation (2.9) is solved analytically, the shape of the step response obtained depends on the value of the damping ratio parameter . The output responses of a second order instrument for various values of  following a step change in the value of the measured quantity at time t are shown in Figure 2.12. For case (A) where  D 0, there is no damping and the instrument output exhibits constant amplitude oscillations when disturbed by any change in the physical quantity measured. For light damping of  D 0.2, repre[1]sented by case (B), the response to a step change in input is still oscillatory but the oscillations gradually die down. Further increase in the value of  reduces oscillations and overshoot still more, as shown by curves (C) and (D), and finally the response becomes very overdamped as shown by curve (E) where the output reading creeps up slowly towards the correct reading. Clearly, the extreme response curves (A) and (E) are grossly unsuitable for any measuring instrument. If an instrument were to be only ever subjected to step inputs, then the design strategy would be to aim towards a damping ratio of 0.707, which gives the critically damped response (C). Unfortunately, most of the physical quantities that instruments are required to measure do not change in the mathematically convenient form of steps, but rather in the form of ramps of varying slopes. As the form of the input variable changes, so the best value for  varies, and choice of  becomes one of compromise between those values that are best for each type of input variable behaviour anticipated. Commercial second order instruments, of which the accelerometer is a common example, are generally designed to have a damping ratio () somewhere in the range of 0.6–0.8.