## 4 Sensitivity
### Dynamic Sensitivity  
The functional sensitivity combined with the electrical sensitivity already gives the sensor's total sensitivity with regard to static measurands, i.e., for a constant magnetic flux density. However, for dynamic measurements, i.e., if the signal to be measured is of the general form \(B_{\mathrm{x}}(t)=\hat{B}_{\mathrm{x}}\cos(2\pi f_{\mathrm{x}}t)\), the sensor's sensitivity decreases for higher frequencies \(f_{\mathrm{x}}\) as shown in measurements [12]. This effect is taken into account by the so-called dynamic sensitivity. In normalized form, the dynamic sensitivity, which typically exhibits a low-pass behavior, can be calculated according to  
\[S_{\mathrm{dyn}}(f_{\mathrm{x}})=\frac{1}{2}\cdot\left(\frac{\tilde{G}(f_{ \mathrm{ex}}+f_{\mathrm{x}})}{\tilde{G}(f_{\mathrm{ex}})}+\frac{\tilde{G}^{*}( f_{\mathrm{ex}}-f_{\mathrm{x}})}{\tilde{G}^{*}(f_{\mathrm{ex}})}\right) \tag{30}\]  
where \({}^{*}\) denotes a complex conjugate [41]. Because the optimum excitation frequencies differ in amplitude mode (Equation (26)) and in phase mode (Equation (26)), the AM dynamic sensitivity  
\[S_{\mathrm{dyn}}^{\mathrm{AM}}(f_{\mathrm{x}}) =\left.S_{\mathrm{dyn}}(f_{\mathrm{x}})\right|_{f_{\mathrm{ex}} =f_{\mathrm{ex}}^{\mathrm{AM}}}=\left|S_{\mathrm{dyn}}^{\mathrm{AM}}(f_{ \mathrm{x}})\right|\cdot e^{j\ \sigma_{\mathrm{dyn}}^{\mathrm{AM}}(f_{\mathrm{x}})} \tag{31}\] \[=\frac{f_{\mathrm{res}}^{2}\sqrt{81\ \frac{f_{\mathrm{res}}^{4}+144\ Q^{2}f_{ \mathrm{res}}^{2}f_{\mathrm{x}}^{2}+576\ Q^{4}f_{\mathrm{x}}^{4}}{64\ Q^{4}f_{ \mathrm{x}}^{4}+16\ Q^{2}f_{\mathrm{res}}^{2}f_{\mathrm{x}}^{2}+9\ f_{\mathrm{ res}}^{4}}}\cdot e^{j\ \left(\arctan\left(\frac{1}{\sqrt{8}}+\frac{9}{\sqrt{512}}\ \left(\frac{f_{\mathrm{ res}}}{\mathrm{0}f_{\mathrm{x}}}\right)^{2}\right)-\frac{\pi}{2}\right)}\]  
also differs from the PM dynamic sensitivity  
\[S_{\mathrm{dyn}}^{\mathrm{PM}}(f_{\mathrm{x}}) =S_{\mathrm{dyn}}(f_{\mathrm{x}})\Big{|}_{f_{\mathrm{ex}}=f_{ \mathrm{ex}}^{\mathrm{PM}}}=\left|S_{\mathrm{dyn}}^{\mathrm{PM}}(f_{\mathrm{ x}})\right|\cdot e^{j\ \vartheta_{\mathrm{dyn}}^{\mathrm{PM}}(f_{\mathrm{x}})} \tag{32}\] \[=\frac{1}{\sqrt{1+\left(\frac{2 Qf_{\mathrm{x}}}{f_{\mathrm{res}}} \right)^{2}}}\cdot e^{j\ \arctan\left(-\frac{2 Qf_{\mathrm{x}}}{f_{\mathrm{res}}}\right)}.\]  
For both operation modes, the amplitude frequency responses and the phase frequency responses are depicted in Figure 5. The \(-3\,\mathrm{dB}\) cutoff frequencies of  
\[f_{\mathrm{cs}}^{\mathrm{AM}}=\frac{f_{\mathrm{res}}\ \sqrt{2}\sqrt{\sqrt{10-1}}}{4Q} \approx\frac{f_{\mathrm{res}}}{2Q} \tag{33}\]  
and  
\[f_{\mathrm{cs}}^{\mathrm{PM}}=\frac{f_{\mathrm{res}}}{2Q} \tag{34}\]  
are almost identical. However, the amplitude response above the cutoff frequency decreases proportional to \(f_{\mathrm{x}}^{-2}\) (\(-40\,\mathrm{dB}/\mathrm{decade}\)) for AM, whereas the decrease for PM is only proportional to \(f_{\mathrm{x}}^{-1}\) (\(-20\,\mathrm{dB}/\mathrm{decade}\)).  
Figure 5: Illustrations of the dynamic sensitivities for amplitude mode and phase mode. The \(-3\,\mathrm{dB}\) cutoff frequencies are almost identical, whereas the slopes of the amplitude frequency responses differ.  
The plots are based on the sensor parameters specified in Table 1. (**a**) amplitude frequency responses, (**b**) phase frequency responses.