Page 226 - The Mechatronics Handbook

0066_Frame_C11 Page 32 Wednesday, January 9, 2002 4:14 PM

or, equivalently, by observing that the current flowing in the series circuit is related to the capacitor
voltage by i(t) = Cdv C /dt, and that Eq. (11.55) can be rewritten as

d v C t()
2
RC-------- + LC------------------ + v C t() = v S t() (11.57)
dv C
dt dt 2
Note that although different variables appear in the preceding differential equations, both Eqs. (11.55)
and (11.57) can be rearranged to appear in the same general form as follows:

2
d yt() dy t()
a 2 -------------- + a 1 ------------ + a 0 yt() = Ft() (11.58)
2
dt dt
where the general variable y(t) represents either the series current of the circuit of Fig. 11.49 or the
capacitor voltage. By analogy with Eq. (11.54), we call Eq. (11.58) a second-order ordinary differential
equation with constant coefficients. As the number of energy-storage elements in a circuit increases, one
can therefore expect that higher-order differential equations will result.

Phasors and Impedance
In this section, we introduce an efficient notation to make it possible to represent sinusoidal signals as
complex numbers, and to eliminate the need for solving differential equations.
Phasors
Let us recall that it is possible to express a generalized sinusoid as the real part of a complex vector
whose argument, or angle, is given by (ωt + φ) and whose length, or magnitude, is equal to the peak
amplitude of the sinusoid. The complex phasor corresponding to the sinusoidal signal Acos(ωt + φ)
jφ
is therefore defined to be the complex number Ae :

jf
Ae = complex phasor notation for A cos ( wt + f) (11.59)
1. Any sinusoidal signal may be mathematically represented in one of two ways: a time-domain form

vt() = Acos ( wt + f)

and a frequency-domain (or phasor) form

(
V jw) = Ae jf

2. A phasor is a complex number, expressed in polar form, consisting of a magnitude equal to the
peak amplitude of the sinusoidal signal and a phase angle equal to the phase shift of the sinusoidal
signal referenced to a cosine signal.
3. When using phasor notation, it is important to make a note of the specific frequency, ω, of the
sinusoidal signal, since this is not explicitly apparent in the phasor expression.
Impedance
We now analyze the i-v relationship of the three ideal circuit elements in light of the new phasor notation.
The result will be a new formulation in which resistors, capacitors, and inductors will be described in
the same notation. A direct consequence of this result will be that the circuit theorems of section 11.3
will be extended to AC circuits. In the context of AC circuits, any one of the three ideal circuit elements

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