On the performance of a single loop OEO using dynamic control BPF in the face of noisy angle modulation

Abstract

Dynamically controlling the resonance frequency of a tuned circuit by the modulating signal extracted from the synchronizing FM input is a wonderful technique. It not only increases the linear region but also reduces the output THD by judicious control of the feedback gain. This communication presents the study of lock range of OEO using dynamic control BPF under the influence of noisy angle modulated signal. The instantaneous amplitude and phase governing equation of the OEO in presence of noisy FM signal has been derived. We show the dependence of the noise variance on the input carrier-noise ratio. It has been observed that higher values of input carrier-noise ratio and lower values of the noise variance gives large values of lock range for a given value of the input carrier-noise ratio. The improvement in performance of the OEO using dynamic control BPF over static BPF have been presented. The steady state condition existing in the OEO using an approximate and exact analysis is given.

Appendix

1.1 Steady-state condition

A steady state condition for the oscillator is obtained in the absence of synchronizing signal. This also implies that $\frac{dV}{{dt}} = \frac{d\theta }{{dt}} = 0$ in Eqs. (5) and (6). Thus it is not difficult to obtain in noise-free environment

$$2J_{1} \left( {V_{0} } \right) \times \cos \left( {\omega_{0} \tau } \right) + E \times \cos \phi_{0} = V_{0}$$

and

$$2J_{1} \left( {V_{0} } \right) \times \sin \left( {\omega_{0} \tau } \right) = E \times \sin \phi_{0}$$

Squaring, adding and rearranging terms, one gets

$$V_{0}^{2} - 4V_{0} \times J_{1} \left( {V_{0} } \right) \times \cos \left( {\omega_{0} \tau } \right) + 4J_{1}^{2} \left( {V_{0} } \right) - E^{2} = 0$$

Clearly, the above equation is quadratic in the steady state output voltage and is a function of the fiber delay. Further insight can be obtained regarding the dependence of ‘$V_{0}$’ on the fiber delay by considering the steady state RF output voltage to be small. In that case, we have an approximate relation for the steady state output voltage which will be given by

$$\begin{gathered} 2V_{0}^{2} - 2V_{0}^{2} \times \cos \left( {\omega_{0} \tau } \right) = E^{2} \hfill \\ V_{0} = \pm \frac{E}{{2\sin \left( {\frac{{\omega_{0} \tau }}{2}} \right)}} \hfill \\ \end{gathered}$$

Using the approximate results, the variation of steady state output voltage for two values of injection amplitude is shown in Fig. 9(a). However a closed from expression for the quadratic expression is not possible because of the presence of the Bessel term, and thus a numerical solution (Fig. 9(b)) is obtained which justifies the nature of Fig. 10.

Fig. 9

Steady state RF output voltage with fiber delay in ns a approximate result b exact result

Fig. 10

Variation of normalized amplitude with fibre delay in ns

1.2 Dynamic control action of the BPF

The resonance frequency of a single tuned parallel RLC circuit can be changed by feeding a control signal to the bank of varactor diodes which comes in parallel to the fixed capacitance in the RLC branch. The capacitance of the varactor diode will now change in accordance to the amplitude of the control signal resulting in the shift of the centre frequency of the dynamic band-pass filter (Bishayee et al 2018). The purpose of such shift in the center frequency of the BPF is to provide a more linear region for the FM-AM conversion process and the system will be capable of handling a wideband FM signal. Moreover the output THD can be reduced by a judicious choice of the feedback gain$\left( \alpha \right)$. In Fig. 11, the shift in the centre frequency of the BPF for two given values of alpha is shown.

Fig. 11

Shift of centre frequency of the BPF for two values of feedback gain