Solution of a second order system using LabVIEW

 

The second-order system model is a set of mathematical equations used to describe the behavior of nonlinear systems. It consists of two nonlinear differential equations and a set of constants that define how those variables change over time. The system is defined by its state variables and the values for their respective operators at each instant in time.

Standard mathematical form of a second-order differential equation is as below:

Where A and B are constants.

For control system, the standard mathematical form of second order system with Unit step input is as below:

Where ξ is the damping factor and ωn is the natural frequency.

The system will be underdamped when 0 < ξ < 1. The particular solution,
yp(t) = 1 determines the steady state solution of the system. The homogeneous solution determines the transient solution which is

 

1.     Methods

This project report will be executed using LabVIEW. The first part of the report will be presenting Numeric block based code for analyzing the response of the system based on the following solution:

Where ξ is the damping factor and ωn is the natural frequency. If T is the total time for which response is analyzed, N is the number of points then time step is ∆t = T/N. At i^th iteration, t= i*∆t = i*T/N.

We will be using this equation for different parameters of ζ , ωn, T and N to analyze its response on graph.

In next part of the lab we will be using another method which is call formula node. In formula node we write the formula in script and then give input and output arguments to it.

Below code is written using Numeric blocks in LabvIEW.

We can simulate the same model using Formula script node also.

Current Sinking and Current sourcing configuration of a device

Current sinking refers to a configuration in electrical circuits where a device, such as an LED or transistor, is connected to a power source and "sinks" current into the ground. This means that current flows from the power source, through the device, and into the ground.

Current sourcing, on the other hand, is when a device "sources" current out of a power source and into a load. In this configuration, current flows from the power source, through the device, and into a load, such as a motor or a resistor.

In both cases, the device acts as a switch that controls the flow of current in the circuit. The choice of whether to use current sinking or sourcing depends on the specific requirements of the circuit and the characteristics of the device being used. sinking and sourcing terminology applies only to DC input and output circuits. Input and output points that are sinking or sourcing can conduct current in one direction only. 

The figure below depicts a sinking input. To properly connect the external supply, it must be connected so the input provides a path to supply common(-). So, start at the PLC input terminal, follow through the input sensing circuit, exit at the common terminal, and connect the supply (-) to the common terminal. By adding the switch between the supply (+) and the input, the circuit is completed. Current flows in the direction of the arrow when the switch is closed.

The four possible combinations of input/output sinking/sourcing circuits are shown below. The common terminal is the terminal that serves as the common return path for all I/O points in the bank.

Why 4-20 mA signal is used for transmission?

4 to 20 mA Current Transmission

In many process control applications, signals are transmitted in the form of current in the range of 4 to 20 mA. Current loops are used not only for receiving information from sensors and field instrumentation, they are also used for transmitting control signals to actuators or other devices to regulate a controlled action. For long distance signal transmission, current signal is preferred because of

(1) Compatibility: The 4-20 mA signal is widely used in industrial process control because it is a widely accepted standard. This means that a wide range of instrumentation and control equipment is available that can use this signal, making it easy to integrate into many different types of control systems.

(2) Noise immunity from EMI:  Industrial environments can be noisy places, with electrical interference from other equipment and power sources. The 4-20 mA signal is designed to be immuned to this type of noise, which helps to ensure that the signal remains accurate and reliable even in noisy environments.

(3) Unaffected by voltage drop along the line.

(4) No stray Thermocouples at joints, hence no error due to thermoelectric emf.

(5) Current signal can be transmitted over long distance till the compliance voltage requirement is met.

(6) Self monitoring ability. currents less than 4 mA and higher than 20 mA can indicate a fault in the circuit.

(7) Power Transmission: The 4-20 mA signal is self-powered, meaning that it can be transmitted over long distances without the need for an external power source.

4 mA lower limit known as "Live zero" provides ability to detect cable or connection fault. The current upto 3.6-3.8mA is used to power the loop instruments in loop-powered mode.

How current transmission provide noise immunity?

(1) Loop wiring: The 4-20 mA signal is typically transmitted in a loop configuration, where the signal is transmitted from the field device back to the control system. This creates a closed loop circuit that helps to reduce the impact of noise and interference on the signal.

(2) Low frequency: The 4-20 mA signal operates at a relatively low frequency, typically in the range of a few kilohertz. This low frequency reduces the impact of high-frequency noise and interference that can be present in industrial environments.

(3) Signal strength: The 4-20 mA signal is transmitted at a relatively high current level, typically between 4 and 20 milliamperes. This high current level helps to ensure that the signal remains strong and robust, even in the presence of noise and interference.

(4) Common mode rejection: Many industrial control systems are designed to reject common mode noise, which is a type of noise that affects both the positive and negative parts of the signal in a similar way. The 4-20 mA signal is typically transmitted in a differential configuration, which helps to reduce the impact of common mode noise.