Fig. 1. The unit step function.
The dynamic response of a measuring instrument is the change in the output y caused by a change in the input x. Both x and y are functions of time t.
A zero order linear instrument has an output which is proportional to the input at all times in accordance with the equation
y(t) = Kx(t),
where K is a constant called the static gain of the instrument. The static gain is a measure of the sensitivity of the instrument.
An example of a zero order linear instrument is a wire strain gauge in which the change in the electrical resistance of the wire is proportional to the strain in the wire.
All instruments behave as zero order instruments when they give a static output in response to a static input.
A first order linear instrument has an output which is given by a non-homogeneous first order linear differential equation
tau.dy(t)/dt + y(t) = K.x(t),
where tau is a constant, called the time constant of the instrument.
In these instruments there is a time delay in their response to changes of input. The time constant tau is a measure of the time delay.
Thermometers for measuring temperature are first-order instruments. The time constant of a measurement of temperature is determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.
A cup anemometer for measuring wind speed is also a first order instrument. The time constant depends on the anemometer's moment of inertia.
A second order linear instrument has an output which is given by a non-homogeneous second order linear differential equation
d2y(t)/dt2 + 2.rho.omega.dy(t)/dt + omega2.y(t) = K.omega2.x(t),
where rho is a constant, called the damping factor of the instrument, and omega is a constant called the natural frequency of the instrument.
Under a static input a second order linear instrument tends to oscillate about its position of equilibrium. The natural frequency of the instrument is the frequency of these oscillations.
Friction in the instrument opposes these oscillations with a strength proportional to the rate of change of the output. The damping factor is a measure of this opposition to the oscillations.
An example of a second order linear instrument is a galvanometer which measures an electrical current by the torque on a coil carrying the current in a magnetic field. The rotation of the coil is opposed by a spring. The strength of the spring and the moment of inertia of the coil determine the natural frequency of the instrument. The damping of the oscillations is by mechanical friction and electrical eddy currents.
Another example of a second order linear instrument is a U-tube manometer for measuring pressure differences. The liquid in the U-tube tends to oscillate from side to side in the tube with a frequency determined by the weight of the liquid. The damping factor is determined by viscosity in the liquid and friction between the liquid and the sides of the tube.
Suppose that x(t) = 0 for t < 0, and x(t) = 1 for t > 0. This is called the unit step function (see Fig. 1).
Fig. 1. The unit step function.
Suppose also that y(t) = 0 for t < 0. We want to find y(t) for t > 0.
Since in a zero order instrument y(t) = K.x(t), we have y(t) = 0 for t < 0, and y = K for t > 0. Therefore the response to the unit step function is a step function with height K (see Fig. 2).
Fig. 2. The response of a zero order instrument to the unit step function with K = 1.5.
The response of a first order instrument to the unit step function for t > 0 is the solution of
tau.dy(t)/dt + y(t) = K
with the initial condition y(0) = 0. The solution is
y(t) = K[1 - exp(-t/tau)].
The initial rate of change of y(t) at t = 0 is K/tau (see Fig. 3).
Fig. 3. The response of a first order instrument to the unit step function with K = 1.5 and tau = 1.
After a time t = tau we have
y(tau) = K(1 - 1/e) = 0.632K.
After a long time y(t) approaches the value K. If tau is small the response of the instrument is fast. If tau is large the response of the instrument is slow.
The response of a second order instrument to the unit step function is the solution of
d2y(t)/dt2 + 2.rho.omega.dy(t)/dt + omega2.y(t) = K.omega2
with initial conditions
y(0) = 0, dy(0)/dt = 0.
The response depends on the damping factor rho.
If rho > 1, we have overdamping and y(t) slowly approaches the static value K. A graph of y(t) for rho = 3 with K = 1 and omega = 1 is shown in Fig. 4.
Fig. 4. The response of a second order instrument to the unit step function with overdamping (rho = 3) and critical damping (rho = 1).
If rho = 1, the response is
y(t) = K[1 - (1 + omega.t).exp(-omega.t)].
This condition is called critical damping. A graph of y(t) for rho = 1 with K = 1 and omega = 1 is also shown in Fig. 4.
If 0 < rho < 1, the condition is called underdamping. A graph for rho = 0.3 with K = 1 and omega = 1 is shown in Fig. 5. Here y(t) oscillates above and below the static value K with an amplitude that becomes smaller as y(t) slowly approaches K.
If rho = 0, there is no damping and y(t) oscillates above and below K with a constant amplitude equal to K and period 2.pi.omega. This response is also shown in Fig. 5.
Fig. 5. The response of a second order instrument to the unit step function with underdamping (rho = 0.3) and no damping (rho = 0).
It can be shown that the fastest approach to within 5% of the static value K is obtained when rho = 0.69. Second order instruments are often designed to have this optimal damping. In the response of an optimally damped instrument to a unit step function, y(t) slightly overshoots the static value K before reaching its final value (See Fig. 6).
Fig. 6. The response of an optimally damped second order instrument (rho = 0.69) to the unit step function.
When the input x(t) is changing with time, the output of a zero order instrument changes in exactly the same way as the input and is multiplied by the static gain K. But the outputs of first and second order instruments do not change in the same way. In these instruments the shape of the output function y(t) is different from the shape of the input function x(t). [For details consult: H. Schenck Theories of Engineering Experimentation (Third Ed.), McGraw Hill (1979), Cht. 5 by R. J. Hawks.]
By R. H. B. Exell, 2003. King Mongkut's University of Technology Thonburi. Home Page