## 1 Introduction  
Strip steels with a dual-phase (DP) microstructure are widely used in the automotive industry. The microstructure of DP steels consists of ferrite with 20%-80% dispersion of second phase (e.g., martensite, tempered martensite, and/or bainite) islands. The microstructure is produced either by controlling the transformation of austenite after hot rolling or by heat treatment after cold rolling. The amount and type of any second phase plays an important role in determining its mechanical properties. In order to achieve accurate control of the steel quality, in particular of the mechanical properties, during manufacturing, it is important to be able to monitor non-destructively the volume fraction and spatial distribution of the phases, as there is a direct link between microstructure and mechanical properties. Previously, multi-frequency electromagnetic (EM) sensors have been shown to be able to measure austenite-ferrite fraction from 0 to 100% in model (HIPped austenitic-ferrite stainless steel powder) alloys [1; 2] and ferrite-pearlite steels with uniform second phase distribution in C-Mn steels with different carbon concentrations [3].  
When considering the effective electrical or magnetic property of a material, which has two components with contrasting properties, a finite element microstructure model can be used to predict the relative permeability [4; 5; 6] based on actual microstructures. The microstructural phases were considered as constituents with different relative permeability values. To date, FE microstructure modelling to predict magnetic relative permeability has only been carried out in two dimensions (2D), whereas the electromagnetic field generated by the EM sensor [7; 8; 9; 10; 11; 12] interacts with the microstructure in three dimensions, so it is important to study how the permeability-ferrite-fraction relationship changes when modelling is carried out in 3D.  
Previously, we have proposed a strategy that can directly link the microstructure with the sensor's output [2; 13; 14]. The EM sensor output has been verified to be an important response symbol of the microstructure. However, the sensor's geometry can also affect the output. Therefore, results obtained from one EM sensor setup cannot be applied to another. In addition, different measurement instrument constructions (ferrite-cored or air-cored probe) and complicated microstructures would make it very challenging to directly link the sensor output with the microstructure. Fortunately, an alternative technique--the full transduction chain--has been proposed, which can be split into in two procedures [15]. The first is consists in linking microstructures to a magnetic property, i.e., permeability, and the second consists in linking the permeability to the sensor output signal. This technique enables us to build general models without considering sensor geometry.  
In this paper, the modelling of the effect of the 2 nd phase on the effective magnetic relative permeability has been extended to 3D microstructures, which are generated using a Voronoi modelling method based on the random close packing of spheres (RCPS-VM). This also allows for the precise control of phase fraction.