Page 62 - Mechatronics with Experiments

48 MECHATRONICS
A digital computer is a discrete-event device. It can work with finite samples of signals. The
sampling rate can be programmed based on the clock frequency. Every sampling period, the
sensor signals are converted to digital form by the A/D converter (the sampling operation).
If the command signals are generated from an external analog device, this also must be
sampled. During the same sampling period, control calculations must be performed, and
the results must be sent out through the D/A converter. The A/D and D/A conversions are
finite precision operations. Therefore there is always a quantization error.

2.2 THE SAMPLING OPERATION AND SIGNAL
RECONSTRUCTION

Due to the fact that the controller is a digital computer, the following problems are introduced
in a closed loop control system: time delay associated with signal conversion and processing,
sampling, quantization error due to finite precision, and reconstruction of signals.

2.2.1 Sampling: A/D Operation

In this section, we will focus on the sampling only and its implications. We will consider
the sampling operation in the following order:

1. physical circuit of the sampler,
2. mathematical model of sampling,
3. implications of sampling.

2.2.2 Sampling Circuit

Consider the sample and hold circuit shown in Figure 2.3. When the switch is turned ON,
the output will track the input signal. This is the sampling operation. When the switch is
turned OFF, the output will stay constant at the last value. This is the hold operation.
While the switch is ON, the output voltage is
t
1
y(t) = i( )d (2.1)
C ∫ 0
where
y(t) − y(t)
i(t) = (2.2)
R
Taking the Laplace transforms of the differential equations, and substituting the value of i
from the second equation gives the input–output transfer function of the sample and hold
circuit.
1
y(s) = y(s) (2.3)
(RCs + 1)
While the switch is OFF i(t) = 0; y remains constant (hold operation).
Let T be sampling period, T is the portion of T for which the switch stays ON,
0
T is the remaining portion of the sampling period during which the switch stays OFF
1
(Figure 2.4). If the input signal y(t) changes as a step function, the output signal will track
it according to the solution of the transfer function in response to the step input. Figure 2.4
shows the typical response of a realistic sample and hold circuit of an A/D converter.