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That is, we can immediately compute the modulus and the argument of the
phasor of the current if we know the values of the circuit components, the
source voltage phasor, and the frequency of the source.

In-Class Exercises

Using the expression for the circuit resonance frequency ω previously intro-
0
duced in Pb. 6.32, for the RLC circuit:
Pb. 6.51 Show that the system’s total impedance can be written as:


 1
Z = R + jω Lν − , where ν = ω = ω LC
0   ν ω
0
Pb. 6.52 Show that Z()ν = Z( / );ν1 and from this result, deduce the value
of ν at which the impedance is entirely real.
Pb. 6.53 Find the magnitude and the phase of the total impedance.

Pb. 6.54 Selecting for the values of the circuit elements LC = 1, RC = 3, and
ω = 1, compare the results that you obtain through the phasor analytical
method with the numerical results for the voltage across the capacitor in an
RLC circuit that you found while solving Eq. (4.36).

The Transfer Function
As you would have discovered solving Pb. 6.54, the ratio of the phasor of the
potential difference across the capacitor with that of the ac source can be
directly calculated once the value of the current phasor is known. This ratio
is called the Transfer Function for this circuit if the voltage across the capaci-
tor is taken as the output of this circuit. It is obtained by combining Eqs. (6.85)
and (6.86) and is given by:

V ˜ 1
c = = H ω (6.87)
()
V ˜ (jRC −ω ω 2 LC + ) 1
s
The Transfer Function concept can be generalized to any ac circuit. It refers
to the ratio of the output voltage phasor to the input voltage phasor. It incor-
porates all the relevant information on the details of the circuit. It is the stan-
dard form for representing the response of a circuit to a single sinusoidal
function input.

© 2001 by CRC Press LLC