## 2 Theory  
In previous research, it was found that the eddy currents in the SR can be modeled as a superposition of spatial patterns with amplitudes given by the elements of a vector \(j(t)\), we can describe their excitation and dynamics by linear differential equations [18; 19]  
\[\mathbf{M}\frac{\mathrm{d}j(t)}{\mathrm{d}t}+\mathbf{R}j(t)+\mathbf{m}\frac{ \mathrm{d}l(t)}{\mathrm{d}t}=0, \tag{1}\]  
The elements on the diagonal of the symmetric matrix \(\mathbf{M}\) are the self-inductances of the spatial eddy-current patterns and off-diagonal terms are mutual inductances of the spatial eddy-current patterns. The matrix \(\mathbf{R}\) contains the resistances of the current patterns on its diagonal and the off-diagonal \"mutual resistances\" describe the Ohmic coupling between patterns that share the same conductor. The third term is an electromotive force (EMF) induced by the current \(l(t)\) in a coil, which couples to the eddy-current patterns via mutual inductance in the vector \(\mathbf{m}\). The connection between magnetic field and eddy current can be expressed as  
\[B(t)=\mathbf{\beta}^{T}j(t). \tag{2}\]Here, \(\beta\) is a coupling vector in which elements are corresponding to the points and magnetic-field components of interests [20] (the superscript \(T\) denotes the transpose). Residual magnetic interference of pulse-induced transients has a significant influence on the magnetic signal to be measured and the residual magnetic is much smaller than the source magnetic that excites it. To observe it precisely, the residual magnetic shall be measured after the current through the \(B_{p}\) coil is completely switched off, because at that time it will dominate the magnetic field in the SR. The current in \(B_{p}\) coil can be expressed as  
\[\left\{ \begin{array}{l}I(t)=k(1-\theta(t))\\ \frac{\mathrm{d}I(t)}{\mathrm{d}t}=-k\frac{\mathrm{d}\theta(t)}{\mathrm{d}t}= -k\delta(t)\end{array} \right.. \tag{3}\]  
Here, \(\theta(t)\) is a unit step function, \(\delta(t)\) is the unit pulse function, \(k\) is the maximum value of current in \(B_{p}\) coil. Substituting \(\frac{\mathrm{d}I(t)}{\mathrm{d}t}=-k\delta(t)\) into (1) and applying the Fourier transform, one can get \(\mathrm{i}\omega\mathbf{\hat{M}}\hat{j}(\omega)+\mathbf{R}\hat{j}(\omega)-k \mathbf{m}=0\), then  
\[\hat{j}(\omega)=k(\mathrm{i}\omega\mathbf{M}+\mathbf{R})^{-1}\mathbf{m}=k \hat{j}_{ISR}(\omega) \tag{4}\]  
in which \(\hat{j}_{ISR}(\omega)=(\mathrm{i}\omega\mathbf{M}+\mathbf{R})^{-1}\mathbf{m}\). The _ISR_ means the inverse-step response. By applying Fourier transform to (1), we may obtain \(\mathrm{i}\omega\mathbf{M}\hat{j}(\omega)+\mathbf{R}\hat{j}(\omega)+\mathrm{i }\omega\mathbf{m}\hat{I}(\omega)=0\). Hence  
\[\hat{j}(\omega)=-(\mathrm{i}\omega\mathbf{M}+\mathbf{R})^{-1}\mathbf{m}: \mathrm{i}\omega\hat{I}=-\hat{j}_{ISR}(\omega)\cdot\mathrm{i}\omega\hat{I}( \omega). \tag{5}\]  
The inverse Fourier transform is performed to Equation (5) to get (\(*\) denotes convolution)  
\[j(t)=-(j_{ISR}*\frac{\mathrm{d}I}{\mathrm{d}t})(t). \tag{6}\]  
Provided that the magnetic-field response to an inverse step current in Equation (3) is known, the time evolution of the eddy-current magnetic field transient \(B(t)\) caused by any pulse waveform \(I(t)\) can be computed by  
\[B(t)=-(B_{ISR}*\frac{\mathrm{d}I}{\mathrm{d}t})(t). \tag{7}\]  
Equation (7) is suitable to estimate transient magnetic field for the CC and \(B_{p}\) coil in the SR. The currents in the \(B_{p}\) coil and CC are \(I_{p}(t)\) and \(I_{c}(t)\), respectively. Suppose the coil have individual inductances \(\mathbf{m}_{p}\) and \(\mathbf{m}_{c}\) with the eddy-current patterns, leading to corresponding magnetic fields, \(B_{ISR}^{(p)}\) and \(B_{ISR}^{(c)}\).