Download Electrical oscillations and tuned circuits

Electrical oscillations
The oscillations of the current within an electrical
circuit are of fundamental importance in the
generation of waveforms of a variety of shapes
for radio, oscilloscopes, signal generators and so
forth. One of the simplest circuits for producing
these oscillations is a capacitor and an inductor
connected as shown in Figure 1. In
understanding these oscillations it is helpful to
compare them with the mechanical oscillations in
a spring.
Figure 1
The mass on the spring oscillates, transferring
stored potential energy in the spring to kinetic
energy of the mass and back again. The charge
in the electrical circuit also oscillates, transferring
stored energy in the electric field of the capacitor
to energy in the magnetic field in the inductor.
(a) This diagram represents the initial zero
energy situation for both Systems. The mass is
at rest and the springs are in equilibrium and no
energy exists as stored charge in the capacitor
or current in the inductor.
(b) The mass is displaced from its rest position,
and therefore potential energy is stored in the
spring. The capacitor is charged, thus
possessing potential energy.
(c) As soon as the mass is released it moves to
the right, gaining kinetic energy. The capacitor
begins to discharge through the inductor,
producing a current in it.
(d) The mass is now at rest; all the energy is
stored as potential energy in the spring. The
charge has now stopped moving and all energy
is stored in the capacitor, which is charged in the
opposite sense from its original state.
(e) This process now repeats itself, the mass
oscillating backwards and forwards and the
charge continually charging and discharging the
capacitor.
A continuous exchange of energy occurs from
potential to kinetic energy in the springs
½ Fe going to ½ mv2
and in the electrical circuit ½ QV going to ½ Li2
The amplitudes of the oscillations decrease with time, since energy is lost as other forms: (a) in
the spring as heating in the coils and air resistance, (b) in the inductor and connecting wires as
heat due to the flow of current within them.
1
Power in a.c. circuits
The power consumed in any circuit is given by equation
P = vi
Now for a capacitor
V = v0 sin (ωt) and i = io cos (ωt).
The power dissipated is therefore
P = i0v0 sin (ωt).cos (ωt) = i0v0 sin (2ωt)
But the average value of sin(2ωt) is zero, and therefore the power dissipated in a purely
capacitative circuit is also zero.
The same argument shows that the power dissipated in a purely inductive circuit is also zero.
For this reason capacitors and inductors are often used in a.c. circuits to limit the voltage, since
they do not waste any energy.
Filters and tuned circuits
The resonance effects in an L-C circuit maybe used to filter out selected regions of the
frequency spectrum.
Figure 2(a) shows an acceptor filter where frequencies of 1/[2√(LC)] will be passed by the filter.
Figure 2(b) shows a rejector filter where all frequencies except those of frequency 1/2π√(LC)
will be passed by the filter.
Figure 2(b)
f = 1/[2√LC]
all frequencies
C
All except
f = 1/[2√LC]
C
L
all frequencies
Figure 2(a)
f = 1/[2√LC]
L
If the capacitor (or inductor) is variable, then the circuit may be tuned to resonate at a particular
frequency. This is used in the tuning of a radio set. The aerial receives a broad hand of
frequencies and the capacitor is varied so that the circuit resonates at the frequency of the
required station. A simple circuit for the tuner section of a radio receiver is shown in Figure 3(a).
The response of the circuit with frequency is shown in Figure 3(b), in which R is the total series
resistance of the tuned circuit.
Intensity
small R
Figure 3(a)
Figure 3(b)
large R
Frequency
2