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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
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