Lecture 11: Introduction to electronic analog circuits 361-1-3661
1
7.3.
Oscillators for high frequencies: LC oscillators:
Our aim is to develop oscillators with high frequency
stability at high frequencies. We have to find a different
approach because, as will be shown in the next lecture, the
small-signal voltage gain decreases with frequency, and,
therefore, the frequency stability of the Wien-bridge oscillator
would be low at high frequencies.
Our approach will be based on employing parallel
RLC
resonant circuits as feedback networks. (At high frequencies
inductances are small and inexpensive.) The phase response of
a resonant circuit is real at the frequency of resonance, and the
phase condition of the Barkhausen criterion will hold true;
hence, we will set the oscillator frequency at the resonant
frequency of its feedback network:
ω
1
=
0
.
We saw in the previous lecture that to obtain high
frequency stability the slope of the phase response of the
feedback network, or its equivalent quality factor, should be
high. By definition, equivalent quality factor,
Q
equiv
, of a
parallel resonant circuit equals its intrinsic quality factor:
Q
equiv
=
Q
RLC
=R
/(
0
L
). Hence, to keep
Q
as high as possible for
a given
L
and
0
we have to keep
R
as high as possible.
Before starting the development of an oscillator, let us first
see what the physical meaning of
R
is in a parallel resonant
circuit. Fig. 1 shows that for a given
, for example,
0
,
R
is
inversely proportional to the resistance
R'
of the inductance
wire: the lower
R'
: the higher
R
, the higher
Q
RLC
.
7.3.1.
Hartley and Colpitts oscillators
According to the positive-feedback approach, we have to
connect an amplifier to the
RLC
feedback network and to
close the feedback loop to obtain an
LC
oscillator. In order not
to reduce the quality factor of the
RLC
network, we have to
choose only amplifiers with high output impedance.
Otherwise, the output impedance of the amplifiers will shunt
R
in Fig. 1 and reduce
Q
RLC
. Hence, we choose either CE or
CB amplifiers.
In a general case, the small-signal voltage gains of these
amplifiers are greater than one. Therefore the transmission of
the
RLC
feedback network at the resonant frequency should
be less than unity to satisfy the amplitude condition of the
Barkhausen criterion. This transmission should also be
negative to satisfy the phase condition of the Barkhausen
criterion for a CE amplifier and positive for a CB amplifier.
These requirements can be met if we divide the voltage at
the output of the
RLC
network as shown in Fig. 2. Without the
reactive voltage divider in Fig. 2(b), the feedback network
transmission at
1
=
0
would be equal to unity. This is so
R'
C
L'
R
C
L
Re[
Ζ
RLC
]
Im[
RLC
]
j
0
L'
R'
j
0
L
R
L
f
Q
R
Q
R
L
f
L
f
R
0
0
0
2
2
2
π
=
=
′
′
=
Fig. 1. A practical
R'L'C
circuit (above), a transformation of
L'
and
R'
into
L
and
R
(in the middle), and an electrical equivalent of the physical
R'L'C
circuit: a parallel
RLC
resonant circuit (below).