## 5 Applied Magnetic Linkage Estimation Methods
### The Used Digital Method  
To implement the digital integration of the phase waveform of the secondary voltage, a numerical method known as the trapezoidal method (sometimes also called the modified Euler method, which can also be interpreted as a variant of the Heun method) was used. The trapezoidal method is an implicit second-order method. The basic implicit Forward Euler method with next-value prediction can be derived from the Formula (54), which is a formula for the derivative of a function. \[\frac{\text{d}y}{\text{d}t}=\lim_{\Delta t\to 0}\frac{y(t+\Delta t)-y(t)}{ \Delta t} \tag{54}\]  
The differential expression contained in formula (54) was converted into the form of a function:  
\[\frac{\text{d}y}{\text{d}t}=\text{f}(t,y) \tag{55}\]  
The time interval \(\Delta t\) contained in Formula (54) is replaced by a constant and finite step size \(h\):  
\[\Delta t=h \tag{56}\]  
Assuming a constant and finite step \(h\), the limit for \(\Delta t\) tending to zero was eliminated from Formula (54) and an approximation of the form (56) was obtained. The obtained result is more accurate in relation to the analytical solution, with the smaller the value of step \(h\). \[\text{f}(t,y)\approx\frac{y(t+h)-y(t)}{h} \tag{57}\]  
Formula (56) is written in discrete form:  
\[\text{f}(t_{n},y_{n})=\frac{y_{n+1}-y_{n}}{h} \tag{58}\]After transformation, Equation (58) was obtained, which expresses the implicit Euler method with the prediction of the next value:  
\[y_{n+1}=y_{n}+h\cdot\mathbf{f}(t_{n},y_{n}) \tag{59}\]  
For the continuous form of the input signal of the measured secondary voltage, we can write the following:  
\[\frac{\mathrm{d}y}{\mathrm{d}t}=\mathbf{f}(t,y)=u_{2}(t) \tag{60}\]  
For the discrete form of the input signal of the measured secondary voltage, we can write the following:  
\[\mathbf{f}(t_{n},y_{n})=u_{2}(n) \tag{61}\]  
Assuming that  
\[n=m-1 \tag{62}\]  
then, by substituting expression (62) into the indices in Formula (59), the prediction was shifted to the current step:  
\[y_{(m-1)+1}=y_{m-1}+h\cdot f(t_{m-1},y_{m-1}) \tag{63}\]  
By arranging Equation (59) and transforming it into the discrete form of the flux path, the following equation was obtained:  
\[\frac{\text{d}y}{\text{d}t}=\text{f}(t,y) \tag{55}\]0  
Similarly, the basic implicit Backward Euler Method can be derived from Formula (65) for the derivative of a function. This time, however, the derivative is determined from the current value and one step back (the earlier value):  
\[\frac{\text{d}y}{\text{d}t}=\text{f}(t,y) \tag{55}\]1  
Formula (65) was written for a discrete signal in the following form:  
\[\frac{\text{d}y}{\text{d}t}=\text{f}(t,y) \tag{55}\]2  
Formula (66) was transformed into the following form:  
\[\frac{\text{d}y}{\text{d}t}=\text{f}(t,y) \tag{55}\]3  
To move one step forward, Formula (67) is modified to the following:  
\[\frac{\text{d}y}{\text{d}t}=\text{f}(t,y) \tag{55}\]4  
Assuming that  
\[\frac{\text{d}y}{\text{d}t}=\text{f}(t,y) \tag{55}\]5  
then, by substituting expression (67) into the indices in Formula (69), the following equation was finally obtained for the flow course:  
\[\frac{\text{d}y}{\text{d}t}=\text{f}(t,y) \tag{55}\]6  
The modified Euler method used in the numerical integration algorithm is, in fact, a combination of the two basic Euler methods described earlier.