# Mechanisms of Noise Transmission in Velocity Broad-Band Seismometers: Modeling and Analysis
## 3 Methodology
### Noise Transfer Model in Feedback Path
#### 3.2.2 Spectral Characteristics Analysis  
From Equation (33), it can be inferred that the noise transfer function of the integrator is determined by the transfer function of the integrator \(H_{I}(s)\), the impedance ratio  
Figure 4: Equivalent circuit model for feedback path analysis.  
\(\frac{Z_{1}(s)}{(Z_{1}(s)+R_{2})R_{coil}}\), the gain of the force transducer \(E\), and the ratio of the feedback mechanical constant to the input mechanical constant \(\frac{K_{f}}{K_{m}}\).  
Before the characteristics of the integrator noise transfer function is analyzed, it is essential to further examine one of the influencing factors: the impedance ratio. We can approximate the impedance ratio \(\frac{Z_{1}(s)}{(Z_{1}(s)+R_{2})R_{coil}}\approx\frac{1}{R_{2}}\). The detailed approximation process can be found in Appendix B, specifically in Equation (A7).  
In summary, the integrator noise transfer function can be rewritten, and the approximate noise transfer function of the integrator can be expressed as follows:  
\[|H_{int}^{a_{i}}(s)|=20\log\left(\left|\frac{-H_{I}(s)K_{i}^{a}K_{f}}{R_{2}K_{ m}}\right|\right)\,\text{dB}. \tag{38}\]  
Reducing the integrator gain, \(K_{i}^{a}\), and \(K_{f}\), or increasing \(R_{2}\) and \(K_{m}\), can result in a reduction in the transfer function gain. Its frequency spectrum characteristics are primarily determined by \(H_{I}(s)\), exhibiting a monotonic decrease with a slope of \(-20\) dB/dec. The frequency \(\omega_{int,z}\) passing through the \(0\) dB point can be expressed as follows:  
\[\omega_{int,z}=\frac{K_{i}^{a}K_{f}}{\tau R_{2}K_{m}}. \tag{39}\]  
Similarly, from Equations (35) and (37), it can be deduced that the impedance ratio behavior of \(\frac{1}{Z_{3}(s)+R_{coil}}\) plays a critical role in shaping the amplitude-frequency characteristics of the force transducer's resistor thermal noise transfer function. At low frequencies, where \(Z_{3}(s)\) is predominantly resistive, the term \(\frac{1}{Z_{3}(s)+R_{coil}}\) results in a low gain, while at higher frequencies, \(Z_{3}(s)\) decreases due to the capacitive effect, the gain increases. A detailed analysis can be found in Appendix A, specifically in Equation (A8). In addition to the impedance behavior, the overall amplitude of the transfer function is modulated by the terms \(E\), \(K_{f}\), and \(K_{m}\), which determine the magnitude of the transfer function. Thus, the frequency response of the transfer function is governed primarily by the impedance ratio characteristics, while the terms \(E\), \(K_{f}\), and \(K_{m}\) affect the overall gain.  
In summary, the force transducer noise transfer function demonstrates lower gain at low frequencies, effectively enhancing noise suppression in this range. Conversely, at higher frequencies, the gain increases, leading to greater noise transmission. Decreasing \(E\) and \(K_{f}\), or increasing \(K_{m}\), can contribute to a reduction in the noise transfer function gain. The angular frequency \(\omega_{ft,z}\), at which the amplitude of the noise transfer function reaches \(0\) dB, can be expressed as follows:  
\[\omega_{ft,z}=\sqrt{\frac{-a^{2}R_{coil}^{2}+a^{2}b^{2}-2 aR_{coil}-1}{C_{1}{} ^{2}(R_{coil}^{2}-b^{2})}} \tag{40}\]  
where \(a=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) and \(b=\frac{E.K_{f}}{K_{m}}\). \(\omega_{ft,z}\) exhibit a clear inverse relationship with \(C_{1}\), while its relationship with other influencing factors requires specific analysis based on particular conditions.