## 5 Applied Magnetic Linkage Estimation Methods
### The Used Analogue Method  
The electronic diagram of the developed analogue integrating circuit is shown in Figure 13. The analogue integrator consists of six subcircuits:  
1. Input resistive voltage divider with an adjustable ratio in the following range: \[\theta_{\text{d\_IN}}=\frac{R_{\text{2}}}{R_{\text{1}}+R_{\text{2\_max}}}= \frac{(0\div 20)\text{ k}\Omega}{300\text{ k}\Omega+20\text{ k}\Omega}=(0\div 0.0625)\] (46) The divider provides an initial adjustment (reduction) of the input signal amplitude.  
2. A voltage buffer with a voltage gain equal to the following: \[G_{u1}=1\] (47) 3. High-pass filter with a cut-off frequency as follows: \[f_{\text{c\_1}}=\frac{1}{2\cdot\pi\cdot(R_{\text{f\_1}}\cdot C_{\text{f\_1}})}= \frac{1}{2\cdot\pi\cdot(330\text{ k}\Omega\cdot 330\text{ nF})}=1.46\text{ Hz}\] (48)  
This filter ensures the cut-off of the DC component that could appear in the input signal. However, the use of high-pass prefiltering causes deterioration of the parameters of the integrator in the electronic circuit.  
4. Non-inverting active amplifier with adjustable voltage gain in the following range: \[G_{u2}=1+\frac{R_{\text{f}}}{R}=1+\frac{(0\div 20)\text{ k}\Omega}{100\text{ k}\Omega}=(1\div 6)\] (49)
5. Inverting integrator (low-pass filter), with additional parallel connected resistance \(R_{\text{f\_INT}}\) in the feedback loop, with the following cut-off frequency: \[f_{\text{c\_INT}}=\frac{1}{2\cdot\pi\cdot(R_{\text{f\_INT}}\cdot C_{\text{f\_INT }})}=\frac{1}{2\cdot\pi\cdot(470\text{ k}\Omega\cdot 330\text{ nF})}=1.03\text{ Hz}\] (50)  
Figure 13: Schematic diagram of an electronic circuit performing analogue integration.  
And the blocking frequency:  
\[f_{0\_{\text{INT}}}=\frac{1}{2\cdot\pi\cdot(R_{\text{1\_INT}}\cdot C_{\text{f\_INT }})}=\frac{1}{2\cdot\pi\cdot(47\text{ k}\Omega\cdot 330\text{ nF})}=10.26\text{ Hz} \tag{51}\]  
The cut-off frequency \(f_{\text{c\_INT}}\) and the stop frequency \(f_{0\_{\text{INT}}}\) selected in this way for the expected input signal with a frequency of 25 Hz allows for relatively accurate integration of the waveform. The phase shift (delay) of the output signal relative to the input signal with a frequency of 25 Hz is \(-83.93^{\circ}\), and for a frequency of 50 Hz, it is \(-87.30^{\circ}\). This value takes into account the initial signal shift (lead) introduced by the high-pass filter (differentiator). An additional parallel resistor \(R_{\text{f\_INT}}\) in the feedback loop ensures gradual discharge of the capacitor \(C_{\text{f\_INT}}\), due to which the system suppresses the DC component in the output signal with a time constant equal to the following:  
\[f_{0\_{\text{INT}}}=\frac{1}{2\cdot\pi\cdot(R_{\text{1\_INT}}\cdot C_{\text{ f\_INT}})}=\frac{1}{2\cdot\pi\cdot(47\text{ k}\Omega\cdot 330\text{ nF})}=10.26\text{ Hz} \tag{52}\]  
6. Inverting active amplifier with adjustable voltage gain in the following range:  
\[G_{u3}=-\frac{R_{\text{f}}}{R}=\frac{(0\div 100)\text{ k}\Omega}{20\text{ k}\Omega}=(-5\div 0) \tag{53}\]  
The voltage signal generated by the Omicron CMC tester was fed to the input of the analogue integrator, while the output signal of the analogue integrator was finally transmitted to the input of the analogue voltage input system (IN_F input terminals without galvanic isolation).