Q-Multiplier
Q-Multiplier
A block diagram for
a Q-multiplier is reported in figure 1.
Feedback
When a portion of the output signal is
feedback toward the input and recombined in phase with the input signal we have
a positive or regenerative feedback.
The system may oscillate at the frequencies
for which the phase shift seen from the signal through the loop is an integer
multiple of 2p if
the open loop gain is greater than one. If the open loop gain is less than one,
the system is stable.
The following analysis
is performed by means of scattering parameters. The transmission and reflection
coefficients for the whole system are
where the upper scripts i,
r and o relate to the input directional coupler,
the resonator and output directional coupler respectively. The amplifier gain
is Ga and q accounts for the phase shift.
The resonator
transmission coefficient, close to resonance frequency 
is
where Qc
is the unloaded quality factor of the resonator, Ql is the
loaded cavity factor and 
.
Replacing it in the
we obtain
We set now the open loop phase shift at 
equal to 2p.
The Q-multiplier transmission coefficient 
becomes
Where
. We lose the 
term because of the
previous hypothesis on the open loop phase shift (at 
). Additionally, close to 
, we neglect the delay line phase variations with the
frequency, with respect to the resonator phase shift contribution.
We can simplify this equation as follows:
We have rewritten the system transmission
coefficient with a form similar to that of a single resonator, where
has the meaning of a multiplication factor for
the resonator quality. These equations show how the gain and the selectivity of
the Q-multiplier increase rapidly when the open loop gain 
at 
, approaches the unity (see figure 2).
At the resonance, 
, the forward gain of the Q-multiplier 
, with respect to the resonator insertion loss 
, is
Except
for the couplers insertion losses 
and 
we have a gain equal
to 
. In the same way the selectivity of the Q-multiplier
increases as 
.
A simplified model
for the noise analysis is reported in figure 3. Here we assume only two noise
sources: the input termination resistance at the conventional temperature T0
= 290 K and the amplifier noise represented by means of the amplifier
equivalent temperature Ta.
The output noise
power is:
The equivalent noise temperature 
of the Q-multiplier
system is easily obtained from the following equivalence
This expression simply states that the system
equivalent noise temperature 
is the amplifier
equivalent noise temperature 
reported back to the
system input.
I wish you remember that for a non-dissipative
directional coupler 
, so that we can rewrite the 
expression in this
way
where 
is the coupling
coefficient.
Figure 4 reports the behavior of 
versus 
coupling. The
equivalent input noise temperature increases quite rapidly increasing the input
coupling.
