Hybrid model for bar and transducer
A mechanical-electrical hybrid model of detector before the SQUID amplifier is drawn in the picture below. It is made of two mechanical oscillators and an electrical oscillator
We use the following notation:
x differential longitudinal strain of the bar
y relative distance of the capacitor plates
q the electric charge flowing through the output inductance
Y0 full distance of the condenser plates
Q0 total electric charge stored in the capacitor
mi equivalent masses of the oscillators
ki their equivalent elastic constants
bi their equivalent damping factors
C1 capacitance of the decoupling pick-up capacitor
L1 inductance of the superconducting impedance-matching transformer
RL resistance of the dissipative elements of the superconducting transformer
fG force acting on the bar due to the passage of a gravitational wave
hybrid model of the sub-system made of the bar, the resonant transducer
and the resonant impedance matching transformer.
A) x, y and q are very small compared to Y0 and Q0 so linear approximation is always possible.
B) we shall not report the static components of y and q, we can always remove them from the equations redefining Y0 and Q0.
C) the absolute value of the charge Q0 (or of the electric field E0) enter the dynamic equations as a sensitivity tuning coefficient.
Let us call
· the condenser charge at time t,
· the effective electric field between the plates,
· C(t)=C0(1-y/Y0) the capacitance of the transducer ( ),
· Fel the mechanical reaction of the field E(t) on m2:
The dynamics equations of the hybrid model in the variables x, y e q are: