Zero Order
Instruments
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.
First Order
Instruments
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.
Second Order
Instruments
A second order linear instrument has an output which is
given by a non-homogeneous second order linear differential equation
d 2y(t)/dt 2 + 2. rho
.omega.dy(t)/dt +omega 2.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.