Electrical impedance
Electrical impedance, or simply impedance, describes a measure of opposition
to alternating current (AC). Electrical impedance extends the concept of
resistance to AC circuits, describing not only the relative amplitudes of the
voltage and current, but also the relative phases. When the circuit is driven with
direct current (DC) there is no distinction between impedance and resistance;
the latter can be thought of as impedance with zero phase angle.
The symbol for impedance is usually Z and it may be represented by its
magnitude and phase. However, complex number representation is more
powerful for circuit analysis purposes. The term impedance was coined by
Oliver Heaviside in July 1886. Arthur Kennelly was the first to represent
impedance with complex numbers in 1893.
•The magnitude of the complex impedance is the ratio of the voltage amplitude
to the current amplitude.
•The phase of the complex impedance is the phase shift by which the current is
ahead of the voltage.
The meaning of electrical impedance can
be understood by substituting it into
Ohm's law:
An AC supply applying a voltage V,
across a load Z, driving a current I.
The magnitude of the impedance acts just like resistance, giving the drop in
voltage amplitude across an impedance for a given current . The phase factor
tells us that the current lags the voltage by a phase of (i.e. in the time domain, the
current signal is shifted to the right with respect to the voltage signal).
Just as impedance extends Ohm's law to cover AC circuits, other results from DC
circuit analysis can also be extended to AC circuits by replacing resistance with
impedance. In order to simplify calculations, sinusoidal voltage and current waves
are commonly represented as complex-valued functions of time:
This representation using complex exponentials may be justified by noting that (by
Euler's formula):
i.e. a real-valued sinusoidal function (which may represent our voltage or current
waveform) may be broken into two complex-valued functions. By the principle of
superposition, we may analyze the behavior of the sinusoid on the left-hand side
by analyzing the behavior of the two complex terms on the right-hand side. Given
the symmetry, we only need to perform the analysis for one right-hand term; the
results will be identical for the other. At the end of any calculation, we may return
to real-valued sinusoids by further noting that
In other words, we simply take the real part of the result.
The impedance of an ideal resistor is purely real and is referred to as a
resistive impedance:
Ideal inductors and capacitors have a purely imaginary reactive
impedance:
Capacitive
reactance
Inductive
reactance
Combining impedances
Series combination
Parallel combination
LC circuit
An LC circuit is a resonant circuit or tuned circuit
that consists of an inductor L and a capacitor C.
When connected together, an electric current
can alternate between them at the circuit's
resonant frequency. LC circuits are used either
for generating signals at a particular frequency,
or picking out a signal at a particular frequency
from a more complex signal. They are key
components in many applications such as
oscillators, filters, tuners and frequency mixers.
An LC circuit is an idealized model since it
assumes there is no dissipation of energy due to
resistance.
The resonant frequency
of the LC circuit.
Series LC
circuit
What happens when
?
Parallel LC
circuit
What happens when
?
RLC circuit
Tuned circuits, such as this one, have many
applications particularly for oscillating circuits and
in radio and communication engineering. They
can be used to select a certain narrow range of
frequencies from the total spectrum of ambient
radio waves. For example, AM/FM radios
typically use an RLC circuit to tune a radio
frequency. Most commonly a variable capacitor
allows you to change the value of C in the circuit
and tune to stations on different frequencies.
Other practical designs vary the inductance L to
adjust tuning.
An example of the application of
resonant circuits is the selection of
AM radio stations by the radio
receiver. The selectivity of the tuning
must be high enough to discriminate
strongly against stations above and
below in carrier frequency, but not so
high as to discriminate against the
"sidebands" created by the imposition
of the signal by amplitude modulation.
What if there is no emf?
attenuation
a)
damped oscillations
b)
overdamping, no oscillations