# Design of Zero Order Hold and First Order Hold Circuit

**Introduction**

Sample & Hold Circuit is used to sample the given input signal and to hold the sampled value. Sample and hold circuit is used to sample an analog signal for a short interval of time in the range of 1 to 10µS and to hold on its last sampled value until the input signal is sampled again. The holding period may be from a few milliseconds to several seconds.

The following figure shows the block diagram of a typical sample and hold amplifier.

The Command terminal is in the form of a logic pulse. It controls whether to sample the input signal or hold the last sampled value of the input signal. When the pulse is high signal is sampled and when the pulse is low signal value is held. Thus the circuit has two modes of operation depending upon the logic level of S/H command signal.

#### Zero Order Hold

The **zero**–**order hold** (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval.

#### First Order Hold

The **first-order hold (FOH)** is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator. For the FOH, the signal is reconstructed as a piecewise linear approximation to the original signal that was sampled.

A mathematical model such as the FOH (or, more commonly, the zero-order hold) is necessary because, in the sampling and reconstruction theorem, a sequence of Dirac impulses, *x*_{s}(*t*), representing the discrete samples, *x*(*nT*), is low-pass filtered to recover the original signal that was sampled, *x*(*t*). However, outputting a sequence of Dirac impulses is impractical. Devices can be implemented, using a conventional DAC and some linear analog circuitry, to reconstruct the piecewise linear output for either the predictive or delayed FOH.

Follow the given steps to visualize zero order and first order hold circuits in MATLAB.

**Zero Order Hold Circuit**

1. Open MATLAB.

2. Define the continuous transfer function that you want to perform the hold operation on.

The transfer function formulated is shown below:

3. Convert the continuous transfer function into a discrete system and specify the sampling period and type of hold circuit as illustrated below.

4. Plot the continuous and discrete system on the same graph by using the command ‘step’ and ‘hold’.

The output of a zero order hold circuit is demonstrated as follows:

**First order Hold Circuit**

1. Open MATLAB

2. Define the continuous transfer function that you want to perform the hold operation on.

3. Convert the continuous transfer function into a discrete system and specify the sampling period and type of hold circuit as illustrated below.

4. Plot the continuous and discrete system on the same graph by using the command ‘step’ and hold.

The step response of a first order hold circuit for the given transfer function is as follows:

**Conclusion**

The step response to a zero order hold and first order hold circuit was observed in MATLAB.

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