Signals and Systems - Electrical Engineering

2.3 LTI Continuous-Time Systems 123

When the relation is a straight line through the origin with a constant slopeC, using the current–
charge relationi(t)=dq(t)/dt, we get the differential equation

i(t)=Cdvc(t)/dt

characterizing the capacitor. Lettingi(t)be the input, solving this differential equation gives as
output the voltage

vc(t)=

1

C

∫t

0

i(τ)dτ+vc( 0 ) (2.5)

which explains the way the capacitor works. For timet>0, the capacitor accumulates charge on
its plates beyond the original charge due to an initial voltagevc( 0 ). The capacitor is seen to be a
linear system ifvc( 0 )=0; otherwise it is not. In fact, whenvc( 0 )=0, the outputs corresponding
toi 1 (t)andi 2 (t)are

vc 1 (t)=

1

C

∫t

0

i 1 (τ)dτ

vc 2 (t)=

1

C

∫t

0

i 2 (τ)dτ

respectively, and the output due to a combinationai 1 (t)+bi 2 (t)is

1

C

∫t

0

[ai 1 (τ)+bi 2 (τ)]dτ=avc 1 (t)+bvc 2 (t)

Thus, a linear capacitor is a linear system if it is not initially charged. When the initial condition is
not zero, the capacitor is affected by the current inputi(t)as well as by the initial conditionvc( 0 ),
and as such it is not possible to satisfy linearity, as only the current input can be changed. The
capacitor is thus an incrementally linear system.

The inductorLis the dual of the capacitor (replacing currents by voltages andCbyLin the above
equations, we obtain the equations for the inductor). Alinear inductoris characterized by the
magnetic flux–current relation

φ(t)=LiL(t) (2.6)

being a straight line of slopeL>0. If the plot of the magnetic fluxφ(t)and the currentiL(t)is not
a line, the inductor is nonlinear. The voltage across the inductor is

v(t)=

dφ(t)

dt

=L

diL(t)

dt