Signals and Systems - Electrical Engineering

0.3 Analog or Discrete? 17

FIGURE 0.8
RC circuit.

+

−

vi(t) 1 Ω

1 F i(t)

FIGURE 0.9

Realization of first-order differential equation using

(a) a differentiator and (b) an integrator. (a)(b)

vi(t)

+

−

dvc(t)

dt

vc(t)

d(·)

dt

vi(t)

+

−

dvc(t)

dt vc(t)

∫(·)dt

constant-coefficient differential equations obtained from a simple RC circuit (Figure 0.8) with a con-

stant voltage sourcevi(t)as input and with resistorR= 1 ; and capacitorC=1 F (with huge plates!)

connected in series is given by

vi(t)=vc(t)+

dvc(t)

dt

(0.10)

with an initial voltagevc( 0 )across the capacitor.

Intuitively, in this circuit the capacitor starts with an initial charge ofvc( 0 ), and will continue charging

until it reaches saturation, at which point no more charge will flow (the current across the resistor and

the capacitor is zero). Therefore, the voltage across the capacitor is equal to the voltage source–that

is, the capacitor is acting as an open circuit given that the source is constant.

Suppose, ideally, that we have available devices that can perform differentiation. There is then the

tendency to propose that the differential equation (Eq. 0.10) be solved following the block diagram

shown in Figure (0.9). Although nothing is wrong analytically, the problem with this approach is that

in practice most signals are noisy (each device produces electronic noise) and the noise present in the

signal may cause large derivative values given its rapidly changing amplitudes. Thus, the realization

of the differential equation using differentiators is prone to being very noisy (i.e., not good). Instead

of, as proposed years ago by Lord Kelvin,^4 using differentiators we need to smooth out the process by

using integrators, so that the voltage across the capacitorvc(t)is obtained by integrating both sides of

Equation (0.10). Assuming that the source is switched on at timet=0 and that the capacitor has an

initial voltagevc( 0 ), using the inverse relation between derivatives and integrals gives

vc(t)=

∫t

0

[vi(τ)−vc(τ)]dτ+vc( 0 ) t≥ 0 (0.11)

(^4) William Thomson, Lord Kelvin, proposed in 1876 thedifferential analyzer, a type of analog computer capable of solving differential
equations of order 2 and higher. His brother James designed one of the first differential analyzers [78].