The recent results of Hirano et al. (Supercond. Sci. Technol. 33, 085002 (2020)) reported a high trapped field of 1.61 T in a composite MgB₂ ring comprising copper plates and and a soft iron yoke magnetised by pulsed field magnetisation (PFM). Inspired by these results, an investigation using systematic modelling methods was conducted to investigate the key parameters leading to the success of Hirano et al. Our results indicate that composite structures of MgB₂ rings present a viable method of trapping high magnetic fields when magnetised with PFM. Leveraging a finite element method modelling framework with a commercial software package (COMSOL Multiphysics), we have successfully modelled the experimental data with excellent agreement. We have paid careful attention to the assumptions regarding the thermal physics, which enabled the successful and accurate modelling of the experiment. Exploiting the flexibility of computational modelling, we extend our studies to investigate the influence of the constituent elements of the composite bulk on the electromagnetic and thermal behaviour, and discuss in detail how each can enhance the trapped field performance of the bulk. Aided by the models, it is shown how the number of copper layers influences the elongation of the applied pulse, reducing the field penetration and the maximum temperature rise of the bulk. The addition of the iron yoke significantly increases the trapped field, by concentrating flux during and after the pulse.