## 2 System Description
### Sensitivity Analysis  
#### 2.3.1 Magnetic Field Sensor Measurement Sensitivity Analysis  
The sensitivity of a magnetic field sensor is defined as the power spectral density of its intrinsic noise in the frequency domain. As a tool for measuring magnetic flux density, one of the most critical parameters of an inductive magnetic field sensor is its sensitivity, also referred to as equivalent input magnetic field noise. This type of noise originates internally within the sensor, including contributions from the coils and the amplification circuitry. Consequently, it cannot be mitigated through shielding layers and represents the inherent noise of the sensor, which can only be reduced but not entirely eliminated. Unfortunately, in practical applications, this noise often determines the minimum measurable magnetic flux density of the sensor, as well as the overall performance and suitability of the sensor for specific applications. Figure 6: Schematic of parallel-plate capacitor and low-noise amplifier circuit. Therefore, the equivalent input magnetic field noise is a critical consideration in both the analysis and design of the sensor. Overall, inductive magnetic field sensors primarily comprise three major noise sources: coil resistor thermal noise (TR noise), circuit equivalent input voltage noise (CV noise), and circuit equivalent input current noise (CI noise). In practice, noise sources are influenced by changes in sensor parameters, temperature variations, and frequency fluctuations. Therefore, the sensitivity of the sensor should be evaluated based on the actual design model and operating environment of the sensor, and ultimately verified through laboratory testing. The following sections analyze these three types of noise in detail. 1. Coil Noise  
For inductive magnetic field sensors, the coil comprises thousands of turns and can be modeled as a series combination of an ideal inductor (\(L_{pc}\)) and resistor (\(R_{sc}\)), which is then placed in parallel with an ideal capacitor (\(C_{sc}\)). Therefore, the noise introduced by the coil can be equivalently represented by the noise introduced by the resistor \(R_{sc}\). The thermal noise (TR noise) in the resistor arises from internal electronic motion. This type of noise was first experimentally verified by [PERSON] in 1928 and later theoretically derived by [PERSON]. Therefore, the thermal noise in the resistor is also referred to as Nyquist noise. Let \(k_{B}\) denote [PERSON]'s constant (\(1.380649\times 10^{-23}J/K\)), and \(T_{c}\) represent the absolute temperature. The equivalent input voltage noise (\(e_{TR}\)) at the sensor input due to TR noise is expressed as follows:  
\[e_{TR}(f)=\sqrt{4k_{B}T_{c}R_{sc}}\quad(V/\sqrt{\mathrm{Hz}}). \tag{6}\]  
2. Circuit-Induced Noise  
Noise introduced by the amplification circuitry is one of the significant sources of noise in inductive magnetic field sensors and can generally be categorized into two types: equivalent input voltage noise and equivalent input current noise. In reality, all components on the circuit board, including both active and passive devices, contribute to the total noise, making it difficult to isolate and measure each component's contribution individually. In this work, the overall noise from the circuit is equivalently modeled as a voltage source, namely CV noise (\(e_{w1}\)), and a current source, namely CI noise ( \(i_{w}\) ). These two noise sources are modeled and analyzed using direct measurement methods. Similar to resistors, the noise introduced by the amplification circuit can also be decomposed into a white noise component and a \(1/f\) noise component. The primary characteristic is that at high frequencies, white noise dominates, whereas at low frequencies, \(1/f\) noise prevails. The frequency at which white noise equals \(1/f\) noise is referred to as the noise corner frequency of the circuit's equivalent input noise. For the sensors in this project, operating in the high-frequency range, the noise is predominantly dominated by the white noise from the first-order amplifier.