Capacitive and
inductive loads oppose the flow of alternating currents. This
opposition is expressed as impedance at a given frequency. The effect
of a real-world impedance load is observed as an attenuation of the
signal and a phase shift. Because of the nature of the impedance, it
is denoted as a vector whose angle is the same as the phase angle
between voltage and current, and the magnitude of the impedance is the
same as the quotient between the voltage and current magnitudes, as
follows:
Note: Bold
values denote vector quantities or complex numbers.
Z
= V/I
Numerically, the impedance vector
is represented as a complex number either in polar form (magnitude and
phase) or rectangular form (real and imaginary). The following
equation expresses impedance in rectangular form:
Z
= R
+ jX
where
R
and X
are resistance and reactance, respectively. When
X
= 0, the load is purely resistive; when
R
= 0, the load is purely reactive. For capacitors, the reactance can be
expressed as follows:
Xc
= –1/(2pfCs)
For inductors, the reactance can
be expressed as follows:
XL
= 2pfLs
In real-world applications, loads
are neither purely reactive nor purely resistive. However, they can be
easily modeled either as a series or parallel combination of a
resistive and a reactive load using the formulas above.
To simplify mathematical
manipulation, calculation, and analysis, it is sometimes convenient to
express the impedance as its reciprocal quantity, or admittance.
Admittance is defined as
Y
= 1/Z
= I/V
and can be written as
Y
= G
+ jB
where
G
and B
are the rectangular (real and imaginary) components, known as
conductance and susceptance respectively. The conductance
G
is the reciprocal of the parallel resistance, as follows:
G
= 1/RP
The susceptance for capacitors is
expressed as follows:
BC
= 2fCP
= 1/XC
The susceptance for inductors is
expressed as follows:
BL
= 1/2fLP
= 1/XL
In general, is mathematically
easier to manipulate parallel loads as admittances and series loads as
impedances.
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