0195136047.pdf

3.2 TRANSIENTS IN CIRCUITS 133

In our example,

vC(t)=

(

−

40

9

+ 10

)

e−t/^30 − 10 =

(

50

9

e−t/^30 − 10

)

V, fort> 0

In the circuit of Figure E3.2.4(d) fort= 0 −, notice thatvx( 0 −)= 50 /9V.Fort>0,vx(t)

=− 5 iC(t).

The capacitor current is found by

iC(t)=C

dvC(t)

dt

=( 5 )

(

−

50

9

·

1

30

)

e−t/^30 =−

25

27

e−t/^30 A

Thenvx(t)=− 5 iC(t)=( 125 / 27 )e−t/^30 A, fort>0. Note thatvx

(

0 +

)

= 125 /27 V, and

vx,ss=0. The resistor voltage has changed value instantaneously att=0.

One should recognize that, in the absence of an infinite current, the voltage across a capacitor
cannot change instantaneously: i.e.,vC( 0 −)=vC( 0 +). In the absence of an infinite voltage,
the current in a inductor cannot change instantaneously: i.e.,iL( 0 −)=iL( 0 +). Note that an
instantaneous change in inductor current or capacitor voltage must be accompanied by a change
of stored energy in zero time, requiring an infinite power source. Also observe that the voltages
across a resistor,vR( 0 −)andvR( 0 +), are in general not equal to each other, unless the equality
condition is forced byiL( 0 −)orvC( 0 −). The voltage across a resistor can change instantaneously
in its value.
So far we have considered circuits with only one energy-storage element, which are known
asfirst-order circuits,characterized by first-order differential equations, regardless of how many
resistors the circuit may contain. Now let us take up seriesLCand parallelLCcircuits, both
involving the two storage elements, as shown in Figures 3.2.2 and 3.2.3, which are known as
second-order circuitscharacterized by second-order differential equations.
Referring to Figure 3.2.2, the KVL equation around the loop is
vL(t)+vC(t)+RThi(t)=vTh(t) (3.2.18)

vTh(t) iL(t) = iC(t) = i(t)

b

iL(t) = iC(t) a

C

L

Ideal

sources

and

linear

resistors

Thévenin equivalent

b

(t > 0)

a

+

− +

−

+

−

vC(t)

vL(t)

C

L

RTh

Figure 3.2.2SeriesLCcase with ideal sources and linear resistors.

iEQ(t) RThv(t) = vL(t) = vC(t)L vL(t)

b

a

C

L C

Ideal iL(t) iC(t)

sources

and

linear

resistors b

(t > 0) Norton equivalent

a

+

−

vC(t)

+

−

iL(t) iC(t)

Figure 3.2.3ParallelLCcase with ideal sources and linear resistors.