Engineering metamaterial-based devices such as terahertz bandpass filters (BPFs) play a definitive role in advancement of terahertz technology. In this article, we propose a design procedure to obtain a considerably broadband terahertz BPF at a normal incidence; it shows promising filtering characteristics, including a wide passband of ∼1.34 THz at a central frequency of 1.17 THz, a flat top in a broad band, and high transmission, compared to previous reports. Then, exploiting the voltage-dependent carrier density control in an AlGaN/GaN heterostructure with a Schottky gate configuration, we investigate the tuning of the transmission properties in a narrow-band terahertz filter. A combination of the ultra-wide, flat-top BPF in series with the tunable, narrow band filter designed in the current study offers the ability to tune the desired resonance frequency along with high out-of-band rejection and the suppression of unwanted resonances in a large spectral range. The proposed structure exhibits a frequency tunability of 103 GHz for a voltage change between -8 and 2 V, and a transmission amplitude change of ∼0.51. This scheme may open up a route for the improved design of terahertz filters and modulators.