Frequency Response.pdf

The article discusses the investigation of the frequency response characteristic of a software phase-locked loop (PLL) device. The goal is to experimentally measure the frequency response of the PLL transfer function using frequency-modulated signals. The PLL device consists of three main components: a phase detector, a digital filter, and a voltage-controlled oscillator. The device is implemented on a STM32F4 Discovery board, which includes a 32-bit microcontroller, 12-bit analog-to-digital converters, and digital-to-analog converters.

The transfer function of the PLL is described by the equation:

H(s) = (K0 \ KFD \ HCF(s)) / (s + K0 \ KFD \ HCF(s))

where KFD and K0 are the gains of the phase detector and voltage-controlled oscillator, respectively, and HCF(s) is the transfer function of the digital filter.

The digital filter is a proportional-integral filter with a transfer function:

HCF(s) = (1 + s \ Tm) / (1 + s \ Tcf)

where Tm is the time constant of the filter and Tcf is the cutoff frequency.

The frequency response of the PLL is measured by applying a frequency-modulated signal to the input and measuring the amplitude of the signal at the output of the digital filter. The measurement method is based on the formula:

|H(jω)| = (Ucf \ Δω) / (K \ Ω)

where Ucf is the amplitude of the signal at the output of the digital filter, Δω is the frequency deviation, and Ω is the modulation frequency.

The results of the measurement are compared with theoretical calculations, and the conclusion is made that the measurement method provides a high-quality assessment of the PLL frequency response with sufficient accuracy.

Key parameters of the PLL, such as the resonance frequency and damping factor, can be estimated from the frequency response characteristic. The resonance frequency is related to the parameters of the digital filter, and the damping factor is related to the gain of the phase detector and voltage-controlled oscillator.

The article also discusses the hardware and software implementation of the PLL device, including the use of a microcontroller, analog-to-digital converters, and digital-to-analog converters. The experimental setup is described, and the results of the measurement are presented in graphical form.