## 2 Materials and Methods
### Synchronous Filtering Concept
First, in case LPF core filters are missing (i.e., bypassed), Equation (7) is valid, resulting in restoration of the input signal at the QDR output by a gain \(V_{r}\),2, which depends neither on the DC nor AC form of \(V_{\textit{SIG}}\):  
Footnote 2: The \(V_{\textit{SIG}}\) is defined as the sum of the \(V_{\textit{SIG}}\) over all \(V_{\textit{SIG}}\). \[V_{\textit{SF}}=V_{\textit{SIG}}V_{\textit{r}}^{2}(\sin^{2}(\omega\text{t})+ \cos^{2}(\omega\text{t}))=V_{\textit{SIG}}V_{\textit{r}}^{2} \tag{7}\]  
Otherwise, in case LPF core filters are present and the input signal is a pure DC offset, then it is converted by the first mixers into sine and cosine waves with a frequency \(f_{\textit{rpf}}\). The LPL core filters reduce output amplitudes (\(V_{x}\)) and introduce a phase lag of 90\({}^{\circ}\). Further, in the second mixers, LPF outputs are again multiplied by sine and cosine waves, and their sum produces zero QDR output (\(V_{\textit{SF}}\) in Figure 2b), given the basic trigonometric identity in (8):  
\[V_{\textit{SF}}=V_{\textit{SIG}}V_{x}V_{\textit{r}}\Big{(}\sin \Big{(}\omega\text{t}-\frac{\pi}{2}\Big{)}\sin(\omega\text{t})+\cos\Big{(} \omega\text{t}-\frac{\pi}{2}\Big{)}\cos(\omega\text{t})\Big{)}=0 \tag{8}\]  
Therefore, it is important that LPFs have a minimum phase response, thus introducing a phase lag of 90\({}^{\circ}\) in their stop bands. An important effect of the QDR operation is the suppressed input DC offset. In a similar manner, according to the basic trigonometric identity:  
\[V_{SIG}=V_{s}\sin(\omega\mathrm{t}) \tag{1}\]0  
the second harmonic, produced by the useful differential signal \(V_{\textit{SIG}}\) from Equation (1) does not generate a third harmonic but can produce only a fundamental harmonic. While QDR is connected in the negative feedback, its band-pass response is subtracted from the input signal and is converted into a band-stop response for the forward path, thus rejecting the PLI. Such a closed-loop synchronous filter for PLI extraction with QDR in the negative feedback is shown in Figure 3a. QDR estimates the PLI error appearing in the demodulator output signals and automatically changes the amplitude and phase of the synthesized PLI in order to minimize the estimated error. The operation of the closed-loop SF concept from Figure 3a is illustrated in Figure 3b. It can be seen that after settling of the in-phase and quadrature control signals for the remodulator, namely, I and Q, the PLI is canceled. The longest transient processes of I and Q (settling over less than 0.2 s) is normally observed after turning on the filter, hereafter referred to as the start-up time. The closed-loop architecture must ensure automatic zeroing of the steady-state residual PLI. Therefore, the loop filter must have infinite gain for low frequencies and DC, and low gain for high frequencies, which can ensure a stable response. The digital integrator is the main gain block in servo control. The loop filter must contain at least one integrator. The transfer function of the loop filters \(LF(z)\) used in [37] is given by (10):  
\[V_{SIG}=V_{s}\sin(\omega\mathrm{t}) \tag{1}\]1Equation (10) includes the coefficient \(k_{i}\), two cascaded averagers (one PLI period), and a digital integrator. We can further consider the stability of the closed-loop system in detail by using (10) as an example for \(LF(z)\). The two averagers in (10) are optional, and, if omitted, the stability constraints are simplified. Note that the digital integrator behaves as an ideal integrator whose frequency response has an infinite DC gain and roll-off of -20 dB/dec. Thus, its infinite DC gain minimizes the steady-state error within \(\pm\)1 LSB. The sampling frequency is set up to \(F_{s}\) = 2 kHz and corresponds to an integration time constant \(\tau_{i}\) = 1/\(F_{s}\) = 0.5 ms or to an integrator unity-gain frequency, UGF = 1/(2\(\pi\tau_{i}\)) = 318 Hz.