Applying the principles of engineering to Synthetic Biology relies on the development of robust and modular genetic components, as well as underlying quantitative dynamical models that closely predict their behaviour. This study looks at a simple positive feedback circuit built by placing filamentous phage secretin pIV under a phage shock promoter. A single-equation ordinary differential equation model is developed to closely replicate the behaviour of the circuit, and its response to inhibition by TetR. A stepwise approach is employed to fit the model's parameters to time-series data for the circuit. This approach allows the dissection of the role of different parameters and leads to the identification of dependencies and redundancies between parameters. The developed genetic circuit and associated model may be used as a building block for larger circuits with more complex dynamics, which require tight quantitative control or tuning.