0195136047.pdf

104 TIME-DEPENDENT CIRCUIT ANALYSIS

The variablescan assume real, imaginary, or complex values. The time-invariant dc source is

represented by settings=0. The use ofs=jωwould imply sinusoidal excitation.

Note thatAestis the only function for which a linear combination of

K 1 Aest+K 2

d

dt

(

Aest

)

+K 3

∫

Aest

in whichK 1 ,K 2 , andK 3 are constants has the same shape or waveform as the original signal.

Therefore, if the excitation to a linear system isAest, then the response will have the same

waveform.

Recall the volt–ampere relationships (for ideal elements) with time-varying excitation.

For the resistorR:

vR=RiR (3.1.5)

iR=GvR (3.1.6)

For the inductorL:

vL=L

diL

dt

(3.1.7)

iL=

1

L

∫

vLdt (3.1.8)

For the capacitorC:

vC=

1

C

∫

iCdt (3.1.9)

iC=C

dvC

dt

(3.1.10)

With exponential excitation in whichv(t)=Vestandi(t)=Iest, it can be seen that the following

holds good because exponential excitations produce exponential responses with the same expo-

nents. (Notationwise, note thatv(t) andi(t) represent thereal-valuedsignals, whereasv(t) and

i(t) representcomplex-valuedsignals.)

ForR:

VR=RIR (3.1.11)

IR=GVR (3.1.12)

ForL:

VL=(sL)IL (3.1.13)

IL=( 1 /sL)VL (3.1.14)

ForC:

VC=( 1 /sC)IC (3.1.15)

IC=(sC)VC (3.1.16)

The preceding equations resemble the Ohm’s law relation. The quantitiesR,sL, and 1/sChave the

dimension of ohms, whereasG,1/sL, andsChave the dimension of siemens, or 1/ohm. The ratio

of voltage to current in the frequency domain at a pair of terminals is known as theimpedance,

designated byZ(s), whereas that of current to voltage is called theadmittance,designated by