Damping

Damping

          When torque is applied in a system in a direction opposite to its motion, it is known as Damping. In case of coulomb damping, the opposition is constant and, thus there will be a constant difference between the input and the output under steady conditions. In the viscous damping provided by dashpot, the opposition is proportional to the relative velocity. As the relative velocity is zero in the steady state, the damping is also zero.

Transfer Function

Transfer Function

          The transfer function is a mathematical expression showing the relation between output and the input to  each unit or block of a control system. If the system has a single input and a single output, it can be represented by block diagram as shown in Fig., 

Transfer function is defined as the ratio of output over the input with all initial conditions equal to zero. Mathematically, 

Transfer function =  θ0 / θ_i

              Where      θ0 = Output signal of the block of a system, and 

                               θ_i  = Input signal to the block of a system.

Thus, the output from an element be obtained by multiplying the input signal with the transfer function.

Note : From the transfer function of the individual block, the equation of motion of system can be formulated.

Overall Transfer Function

Overall Transfer Function

          In the previous topic, we have discussed the transfer function of a block. In this we are going to know the Overall transfer function of a control system. A control system actually consists of several such blocks which are connected in series. The overall transfer function of the series is the product of the individual transfer functions. Consider a block diagram of any control system represented by the three blocks as shown in Fig., 

Fig. Overall transfer function.

          Thus, if  are individual transfer functions of three blocks in series, then th overall transfer function of the system is given as 

Where       K = Constant representing the overall amplification or gain, and 

                  G(D) = Some function of the operator D.

Note: The above equation only true if there is no interaction between the blocks, that is the output from one block is not affected by its connection to the subsequent blocks.

Open-Loop Transfer Function

Open-Loop Transfer Function

          The open loop transfer function is defined as the overall transfer function of the forward path elements. Consider an open loop control system consisting of several elements having individual transfer function such F₁(D), F₂(D), F₃ (D)  as shown on in Fig.,  

Fig. Open loop control system.

 Thus

                                    = F1 (D ) X F2 (D ) X F3 (D ) = KG (D)

The simplified block diagram of the open loop transfer function is shown in Fig.,

Fig. Simplified open loop control system.

Closed - Loop Transfer Function

Closed - Loop Transfer Function

          The closed - loop transfer function is defined as the overall transfer function of the entire control system. It is a mathematical expression (algorithm) describing the net result of the effects of a closed (feedback) loop. Consider a closed loop transfer function or feedback loop consisting of several elements as shown on Fig.,

Fig Closed-loop transfer function.

Now, for the forward path element, we know that

where      K G (D ) = F1 (D ) X F2 (D ) X F3 (D)

On rearranging, we get

               θo = K G ( D )θi − K G ( D)θo

or             [1 + K G ( D )] θo = K G ( D)θi

          The above expression shows the transfer function for the closed-loop control system. Thus the block diagram may be further simplified as shown in Fig., where the entire system is represented by a single block.

Fig. Simplified closed-loop system.