Zero-Order-Hold Transfer Function Derivation
The Zero-Order-Hold (ZOH) is a fundamental component in digital control systems. Its primary function is to reconstruct a continuous-time signal from a sequence of discrete samples, holding each sample's value constant over the sampling interval.
The impulse response, h(t), of an ideal ZOH circuit, assuming the sampling interval is T, is a rectangular pulse defined as:
$ h(t) = \begin{cases} 1 & \text{for } 0 \le t < T \\ 0 & \text{otherwise} \end{cases} $
The transfer function, H(s), in the Laplace domain is obtained by taking the Laplace transform of the impulse response h(t):
$ H(s) = \mathcal{L}\{h(t)\} = \int_0^\infty h(t) e^{-st} dt $
By substituting the definition of h(t) and the integration limits (from 0 to T):
$ H(s) = \int_0^T 1 \cdot e^{-st} dt $
Evaluating this definite integral:
$ H(s) = \left[ -\frac{1}{s} e^{-st} \right]_0^T $
Applying the limits of integration:
$ H(s) = \left( -\frac{1}{s} e^{-sT} \right) - \left( -\frac{1}{s} e^{-s \cdot 0} \right) $
$ H(s) = -\frac{1}{s} e^{-sT} + \frac{1}{s} $
Simplifying the expression by factoring out 1/s:
$ H(s) = \frac{1}{s} (1 - e^{-sT}) $
This result represents the transfer function of the Zero-Order-Hold system for a sampling interval T.
Important Questions from Basics of Control Systems
- Match the LIST-I with LIST-IIChoose the correct answer from the options given below:
LIST-I LIST-II A. Nyquist Criterion I. Polar Plots B. Bode Plot II. Rectangular Plots C. Phase Margin III. Amount of additional phase lag at the gain crossover frequency required to bring the system to the verge of instability D. Gain Margin IV. It is the reciprocal of the magnitude $|G(j\omega)|$ at the frequency at which the phase angle is $-180^\circ$. - Which of the following statement is not correct in respect of Transfer Function of linear Time Invariant System
A. In any open loop control system, the output is not compared with the reference input.
B. Traffic control by means of signals operated on a time basis is an example of closed loop control.
C. Closed loop control always implies the use of feedback control action in order to reduce system error.
D. Washing machine is an example of open loop system.
E. Stability is a major concern in open loop control system.
Choose the correct answer from the options given below:
Read the following statements about the transfer function of a system :
(a) The transfer function provides complete insight into the structure of the system.
(b) It offers a symbolic picture about the dynamic characteristics of the system.
(c) It does not give any insight into the structure of the system.
(d) If the transfer functions of individual components of the system are known, the overall characteristics of the system can be determined just by taking their product.
Which of the above statements are correct ?
The mass-spring-damper system shown in the following figure represents :
