Signals and Systems - Electrical Engineering

122 C H A P T E R 2: Continuous-Time Systems

Solution

Superposition is not satisfied for the first system. If the outputs forx 1 (t)andx 2 (t)arey 1 (t)=|x 1 (t)|

andy 2 (t)=|x 2 (t)|, respectively, the output forx 1 (t)+x 2 (t)is

y 12 (t)=|x 1 (t)+x 2 (t)|≤|x 1 (t)|+|x 2 (t)|=y 1 (t)+y 2 (t)

For the second system, if the response forx(t)isz(t)=cos(x(t)), the response for−x(t)is not−z(t)

because the cosine is an even function of its argument. Thus,

−x(t)→cos(−x(t))=cos(x(t))=z(t)

For the third system, ifx 1 (t)→v 1 (t)=(x 1 (t))^2 andx 2 (t)→v 2 (t)=(x 2 (t))^2 are corresponding

input–output pairs, then

x 1 (t)+x 2 (t)→(x 1 (t)+x 2 (t))^2 =(x 1 (t))^2 +(x 2 (t))^2 + 2 x 1 (t)x 2 (t)6=v 1 (t)+v 2 (t)

Thus, it is nonlinear. n

nExample 2.3

Consider each of the components of an RLC circuit and determine under what conditions they are

linear.

Solution

Because a resistor, a capacitor, and an inductor are one-port or two-terminal elements, input and

output variables are not obvious. However, from physics the cause and effect are well understood.

A resistorRhas a voltage–current relation

v(t)=Ri(t) (2.3)

If this relation is a straight line through the origin the resistor is linear; otherwise it is non-linear.

A diode is an example of a nonlinear resistor: and its voltage–current relation is nonlinear.

If the voltage-current relation is a straight line of constant slopeR, considering the current is the

input, superposition is satisfied. Indeed, if we apply to the resistor a currenti 1 (t)to getRi 1 (t)=

v 1 (t)and getRi 2 (t)=v 2 (t)when we apply a currenti 2 (t), then when a currentai 1 (t)+bi 2 (t),

for any constantsaandb, is applied, the voltage across the resistor isv(t)=R(ai 1 (t)+bi 2 (t))=

av 1 (t)+bv 2 (t)—that is, the resistor R is a linear system.

A capacitor is characterized by thecharge–voltage relation

q(t)=Cvc(t) (2.4)

If this relation is not a straight line, the capacitor is nonlinear. A varactor is a diode for which its

capacitance depends nonlinearly on the voltage applied to its terminals, and thus it is a nonlinear

capacitor.