Federated Learning (FL) is an emerging field of research that contributes to collaboratively training machine learning models by leveraging idle computing resources and sensitive data scattered among massive IoT devices in a privacy-preserving manner without raw data exchange. The majority of existing research has concentrated on developing efficient learning algorithms that demonstrate superior learning performance. Despite the extraordinary advancement, FL encounters three challenges that need to be resolved jointly, specifically, (1) how to properly measure the client reliability and contribution that serve as the basis for compensation allocation, (2) how to reasonably activate reliable clients, whose dataset grows gradually, to avoid over-learning, and (3) how to efficiently formulate the optimal local training decisions to improve model performance and energy efficiency of battery-constricted devices. To address the above challenges, this paper proposes a Reputation based Triple-step Incentive mechanism of Federated learning (RTIFed), which (1) introduces reputation as the metric to measure client reliability and contribution while employing blockchain to accomplish decentralized and tamper-resistant reputation management, (2) activates clients with high reputation and informative data pursuant by a Richness-of-Information Activation Strategy (RIAS), (3) determines training epochs for each client based on a Stackelberg Game according to an Optimal Training Decision Strategy (OTDS). Numerical results clearly show that the proposed RTIFed effectively motivates high-quality clients and improves learning accuracy while reducing the energy cost to meet the low resource consumption need of battery-constricted devices in smart city FL scenarios.