Local learning algorithms, such as Equilibrium Propagation (EP), have emerged as alternatives to global learning methods like backpropagation for training neural networks. EP offers the potential for more energy-efficient hardware implementation by utilizing only local neuron information for weight updates. However, the practical implementation of EP using memristor-based circuits has significant challenges due to the immature fabrication processes of memristors, resulting in defects and variability issues. Previous implementations of EP with memristor crossbars use two separate circuits for the free and nudge phases. This approach can suffer differences in defects and variability between the two circuits, potentially leading to significant performance degradation. To overcome these limitations, in this paper, we propose a novel time-multiplexing technique that combines the free and nudge phases into a single memristor circuit. Our proposed scheme integrates the dynamic equations of the free and nudge phases into one circuit, allowing defects and variability compensation during the training. Simulations using the MNIST dataset demonstrate that our approach maintains a 92% recognition rate even with a 10% defect rate in memristors, compared to 33% for the previous scheme. Furthermore, the proposed circuit reduces area overhead for both the memristor circuit solving EP's algorithm and the weight-update control circuit.