Page 271 - Mechatronics with Experiments

Printer: Yet to Come
October 28, 2014 11:15 254mm×178mm
JWST499-Cetinkunt
JWST499-c05
ELECTRONIC COMPONENTS FOR MECHATRONIC SYSTEMS 257
Again, the above two different states of the RC circuit can be represented with the first
equation with a pulse input voltage representation,
( t )
1
V (t) = R ⋅ i(t) + i( )d ; t ≤ t ≤ t f (5.70)
s
1
C ∫
t 1
where we used the fact that Q(t ) = 0.0 since initially the capacitor is assumed to be
1
uncharged and
V (t) = 24 ⋅ (1(t − t ) − 1(t − t )) (5.71)
s
1
2
The above differential equation can be solved using Laplace transforms method for i(t), then
V (t) and V (t) can be obtained from the current–voltage relationship across the capacitor
c
R
®
and resistor components. Using Simulink for numerical solution is also an easier option
(Figure 5.7).
V (t) = 0.0; t ≤ t ≤ t (5.72)
c 0 1
V (t) = 24 ⋅ (1 − e −(t−t 1 )∕(RC) ) V = 24 ⋅ (1 − e (t−0.0001)∕0.0001 )V; t ≤ t ≤ t (5.73)
c 1 2
V (t) = V (t ) ⋅ (e −(t−t 2 )∕(RC) )V = 23.56 ⋅ e −(t−0.0005)∕0.0001 )V; t ≤ t ≤ t (5.74)
c c 2 2 f
It is important to recognize that both circuits are first-order filters and that the response speed
is dominated by the time constant of the circuit, = L∕R = 100 μs for the RL circuit and
= RC = 100 μs for the RC circuit. Similarly, we can confirm our physical expectations by
comparing these results. Initially, the capacitor is uncharged and there is no voltage across
it. Hence the current will be developed as if there is only a resistor on the circuit. Then, as a
result of the current, the capacitor will start to charge and develop voltage potential across
it. As a result, the available voltage across the resistor will decrease. Eventually, when the
capacitor charge is so large that the voltage across t is equal to the supply voltage, the
voltage across the resistor will be zero, hence the current will be zero. When the circuit is
switched to zero supply voltage, the capacitor voltage will drive the current in the opposite
direction until the charge it holds is fully drained. The charge and discharge of the capacitor
is an exponential function, whose time constant is determined by R ⋅ C.

5.4.2 Amplifier: Gain, Input Impedance,
and Output Impedance
Electronic circuits consist of connections between components where the output of one com-
ponent is connected to the input of another component. The input and output impedances
of connected components are important and can significantly affect the transmitted signal.
For instance, op-amps are used to amplify and filter its input signal before passing it to the
next component.
An ideal amplifier, in its simplest form, amplifies its input signal (Figure 5.8) and
presents the result as its output signal. It does not change the original signal shape (frequency
content) due to the fact that the signal is connected to the amplifier. Consider an operational
amplifier (op-amp) connected to an input source (i.e., a sensor signal) and an output load.
Ideally, the op-amp has a gain (K amp ), input impedance (Z or R ), and output impedance
in
in
(Z out or R out ). An ideal op-amp has infinite input impedance and zero output impedance. In
reality, it has a very large input impedance and very small output impedance. The Z is the
in
generalized version of R , and Z out is the generalized version of R out .
in