First order and second order systems

 

In instrumentation and control, systems are often represented by mathematical models that describe their behavior. These models can be classified based on their order, which is determined by the highest derivative of the output variable that appears in the system's differential equation. First-order and second-order systems are commonly used in instrumentation and control because they provide a simple and effective way to model and analyze the behavior of complex systems.

A first-order system is a system whose behavior can be described by a first-order ordinary differential equation of the form:

τ (dy/dt) + y = K*u

where y is the output variable, u is the input variable, K is the steady-state gain, and τ is the time constant. The time constant represents the time it takes for the system to reach 63.2% of its steady-state value after a step input is applied.

The transfer function of a first-order system is given by:

G(s) = K/(τs + 1)

where s is the Laplace variable. The transfer function represents the relationship between the input and output variables in the frequency domain. The steady-state gain of a first-order system is the ratio of the steady-state output to the steady-state input, and it is equal to the transfer function evaluated at s=0.

First-order systems are widely used in instrumentation and control because they provide a simple and effective way to model the behavior of many physical systems. 

For example, a first-order system can be used to model the thermal behavior of a building or a room, where the temperature response of the system to a sudden change in heating or cooling is characterized by a time constant and a steady-state gain.

A second-order system is a system whose behavior can be described by a second-order ordinary differential equation of the form:

md^2y/dt^2 + cdy/dt + ky = F*u

where y is the output variable, u is the input variable, F is the external force or disturbance, m is the mass, c is the damping coefficient, and k is the stiffness coefficient. The response of a second-order system to a step input can exhibit overshoot, ringing, and oscillation.

The transfer function of a second-order system is given by:

G(s) = K/(ms^2 + cs + k)

where K is the steady-state gain, m is the mass, c is the damping coefficient, and k is the stiffness coefficient. The transfer function represents the relationship between the input and output variables in the frequency domain.

The natural frequency of a second-order system is given by:

ωn = √(k/m)

where ωn is the frequency at which the system oscillates without damping. 

The damping ratio of a second-order system is given by:

ζ = c/(2√(mk))

where ζ is a measure of the system's ability to dissipate energy. The steady-state gain of a second-order system is the ratio of the steady-state output to the steady-state input, and it is equal to the transfer function evaluated at s=0.

Second-order systems are widely used in instrumentation and control because they provide a more accurate and detailed representation of the behavior of many physical systems. For example, a second-order system can be used to model the behavior of a suspension system in a vehicle, where the response of the system to bumps and vibrations in the road is characterized by a natural frequency, damping ratio, and steady-state gain.

In control systems, the choice of whether to use a first-order or second-order system depends on the specific application and the desired level of accuracy and complexity. First-order systems are typically used for applications where the response of the system is slow and smooth, and where a simple and easy-to-implement control strategy is sufficient.