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The vector fitting (VF) algorithm is applied to the equivalent circuit modelling of an antenna over a broad frequency band. To ensure that the resulting circuit model is realisable, the passivity constraint must be imposed during the modelling process. Three methods are investigated to ensure the passivity, each based on VF: (i) the use of quadratic programming proposed earlier by Gustavsen and Semlyen, (ii) a technique based on constraining a second-order rational function and (iii) a search method using a particle swarm optimisation with constraints. These techniques are evaluated through three sample antennas: a commercial broadband horn, an ultra-wideband planar monopole with a notch-band and an active integrated microstrip antenna. The results are compared in terms of the modelling error against the model order and the computation time required.