# Mechanisms of Noise Transmission in Velocity Broad-Band Seismometers: Modeling and Analysis
## 2 Theoretical Model of VBB Seismometer  
### Seismometer System Model  
To effectively study the VBB seismometer, it is crucial to develop a comprehensive system model. Figure 3 shows the structure of a typical VBB seismometer, including its components and noise sources. The seismometer's (referred to as the VBB seismometer in this paper unless otherwise specified) closed-loop system is typically divided into three sections: the forward path, feedback path, and differential amplifier. The forward path consists of four subsystems: the suspension pendulum, capacitive half-bridge, preamplifier-demodulator, and filter. The feedback path includes a PID controller and force transducer, which, together with the forward path, form the feedback loop. The differential amplifier buffers the output and generates differential signals. The input \(a_{i}(t)\), representing ground motion acceleration, is measured in \(\text{m}/\text{s}^{2}\), where \(t\) is time in seconds. The Laplace transform of \(a_{i}(t)\), denoted as \(A_{i}(s)\), is expressed in terms of the Laplace variable \(s\). This ground motion is transformed by the mechanical constant \(K_{m}\) into the normal acceleration \(a_{A}(t)\) at the sensing point of the mechanical pendulum. Figure 3: VBB seismometer system block diagram and noise sources. Figure 2: Two approaches to optimizing sensor noise: noise source or transmission path. The input mechanical constant is defined as follows:  
\[K_{m}=\frac{d_{1}d_{2}}{d_{3}}^{2}. \tag{1}\]  
Here, \(d_{1}\) is the distance between the pendulum's rotation axis and its center of mass, \(d_{2}\) is the distance from the pendulum rotation axis to the center axis of the displacement transducer, and \(d_{3}\) is the rotational radius, given by \(d_{3}=\sqrt{\frac{b}{M}}\), where \(J_{0}\) is the moment of inertia of the mechanical pendulum relative to the pendulum rotation axis, and \(M\) is the mass of the pendulum. The suspension pendulum, modeled as a second-order spring-mass system, transforms the acceleration input \(a_{A}(t)\) into a displacement output. Its transfer function, denoted as \(H_{a}^{d}(s)\), is given by [24]:  
\[H_{a}^{d}(s)=\frac{1}{s^{2}+2\zeta\omega_{0}s+{\omega_{0}}^{2}}, \tag{2}\]  
where \(\zeta\) represents the damping ratio, and \(\omega_{0}\) is the undamped natural angular frequency. The relationship between \(\omega_{0}\) and the natural frequency \(f_{0}\) of the spring-mass system is given by  
\[\omega_{0}=2\pi f_{0}. \tag{3}\]  
The displacement output from the suspension pendulum is directed to the capacitive displacement transducer. This transducer consists of a capacitive half-bridge with three plates, where either the central or outer plate moves along with the pendulum's mass [25; 26], converting the mechanical displacement into an electrical signal. The sensitivity of the capacitive displacement transducer is denoted as \(K_{d}^{c}\). For small displacements, the differential capacitance \(\Delta C\) produced via the capacitive half-bridge can be approximated as follows:  
\[\Delta C=\frac{\varepsilon_{0}\varepsilon_{r}S}{d-\Delta d}-\frac{\varepsilon _{0}\varepsilon_{r}S}{d+\Delta d}\approx 2C_{0}\frac{\Delta d}{d}. \tag{4}\]  
Here, \(\varepsilon_{0}\) is the vacuum permittivity, \(\varepsilon_{r}\) the relative permittivity, \(S\) the area of the plates, \(d\) the distance between the plates, \(\Delta d\) the displacement of the pendulum, and \(C_{0}\) the initial capacitance. Thus, for small displacements, the capacitive bridge's transfer function can be approximated using the linear scaling coefficient \(K_{d}^{c}\):  
\[K_{d}^{c}=\frac{2C_{0}}{d}. \tag{5}\]  
The signal from the capacitive bridge is amplified by a preamplifier and then demodulated, transforming the differential capacitance into a voltage signal. The transfer function of the preamplifier-demodulator subsystem, \(K_{c}^{\circ}\), is expressed as follows:  
\[K_{c}^{\circ}=A_{1}, \tag{6}\]  
where \(A_{1}\) is a dimensionless constant.