The approaching threshold in the design and optimisation of rotating components in gas-turbine engines has drawn attention to the inter-spool duct (ISD) due to its potential to enhance the aerodynamic efficiency of compression systems. The ISD is a crucial passage to deliver the required quantity and quality of the flow between low-pressure and high-pressure spools in a turbofan engine. In this study, an inter-spool duct is designed using a cubic Bezier curve matching the exit dimensions of a transonic compressor stage. In the first part of this study, the performance characteristics of the stand-alone compressor stage are validated at 80% and 90% of the design speed with experimental data. The second part analyses the integrated performance of the compressor stage and ISD at 100% of the speed using steady and unsteady numerical simulations. The performance characteristics of the integrated system show a reduction of 5.8% in the choke margin, but a 4.8% increase in the stall margin. The thick boundary layers on the duct walls have reduced the overall mass flow in the system by 1.6%. In addition, a component-wise loss decomposition through entropy generation is provided for the case with and without ISD. Signal processing of unsteady data is also performed, where the fast Fourier transformation (FFT) of the unsteady pressure signals acquired in the upper and lower walls of the duct inlet showed the frequency of the rotor stall ($0.024\times BPF$) and the other two frequencies ($0.4\times BPF$ and $0.61\times BPF$) related to stator instabilities.