For automatic control, it is necessary to do dynamic analysis of any process. Because the automatic controller depends on the dynamic action of the process. Process analysis elements of process dynamics can be easily known. These elements are mainly of four types.

1. Proportional Element.

2. Capacitance Element.

3. Time constant element.

4. Oscillatory Element.

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## 1. Proportional Element

If the output of the system is proportional to the input then it is called proportional element. For this let us consider the capillary example. The flow rate of liquid through the capillary is changeable variable. It is considered as input variable. It is assumed that the capillaries create a laminar resistance. Hence this flow head equation is written as follows.

**C=RM**

**For example**: Fig. Let us consider the capillary displayed in. The flow rate m from the proportional element capillary is the variable whose value is being changed. Thus, the head C of the input variable is the result of the changed flow rate. Hence it can be considered as output variable. Capillaries create a laminar resistance. Hence, the flow head equation can be determined as follows.

C = RM

Where

C = Output variable

M = Input variable

We can write the above equation as follows.

Output = System function * input

**R = C/M**

For unit function C = R

R is system resistance. The response of the proportional element to step change in input is shown in the following graph. It is clear from this that this step repeats with the output magnitude.

**Types of Process Variables and Degree of Freedom in Process **

## 2. Capacitance Element

If the output system is based on capacitance then it is called capacitance element. If the rate of change in the output of the system is proportional to the input, then such an element is called capacitance element. To study capacitance element fig. Let us consider the physical diagram shown in.

The flow M outside or inside the tank is called the input variable and the head C of the liquid in the tank is called the output variable.

m = dc/dt

C x dc/dt = m

m = C dc/dt

C = m/(dc/dt)

Where

C = Capacitance

c = output variable

t = time

m = input variable

In order to obtain the system function of operation notation of different equations must be used.

S = d/dt

Where

S = Differential opration

Then the equation may be written

(Cs)c = m or c = (1/C.S)m

**c/m = 1/CS**

1/CS is called the system function which is also called capacitance element. Other capacitance elements are as follows.

Electrical capacitance, gas thermal capacitance, mechanical capacitance.

## 3. Time Constant Element

The time constant element is shown in fig. and it simplified by the liquid tank an resistance. The input variable is the in flow is 'm' and the output variable is the tank head 'c' for the tank capacitance C.

Any system made up of a series arrangement of capacitance and resistance, in which there is a negative increase in the rate of change of capacitance per potential due to increase in the output caused by resistance, is called a time constant element. Time constant is displayed by a liquid tank and resistance as in the following figure.

If its input variable is inflow rate m and output variable is head c, then for tank capacitance C.

C.dc/dt = q-m ............(1)

Where q is out flow for the fluid resistance.

q = C/R ..............(2)

Combining equation 1 & 2

C.dc/dt = m - C/R

Rc dc/dt + C = mR .............(3)

यहां R*c का गुणफल समय होता है इस R*c को T से represent करते हैं।

T.dc/dt +C = mR

Laplace transform

T(SC) + C = mR

C (Ts+1) = mR

C = m(R/(Ts+1)

**C/m = R/(Ts+1) ...........(4)**

Equation (4) is the characteristics of the system function of the time constant element.

**Note**: Time constant elements are produced by the combination of electrical, liquid, gas and thermal resistance and capacitance.

## 4. Oscillatory Element

Oscillatory element is shown in fig. Although it is not encountered in ordinary liquid gas and thermal process. It is typical of many measuring instruments such as Bourdon Tube pressure Gauge.

Consider the mass spring and damping system of fig. Newton's second law of motion gives.

M d²c/dt² = - B dc/dt - KC + m ..........(1)

Where

M = Mass of system

C = Displacement

B = Viscose damping Co-efficient

K = Spring Co-efficient or Hooks constant

However, this liquid is not used for ordinary purposes like gas and heat. Therefore, to study the oscillatory element, let us consider mass spring and damping system.

M d²c/dt² = - B dc/dt - KC + m ..........(1)

This equation taking Laplace transform

MS²C + BSC + KC = m

C (MS² + BS + K) = m

C = m * 1/(MS² + BS + K)

For conversation the following definition are often employed.

T = √ m/K …......(a) time characteristics

S = √ B/4Km ..........(b) damping ratio

By (a) and (b) equation

**C = 1/K/(TS + 2fTS +1)**

This C system function is the characteristics of Oscillatory Element. When damping ratio is less than 1 it means under damping.