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In this paper, we suggest applying fundamental-mode operation to orthogonal fluxgates with tube cores. Excitation current in these fluxgates flows through a toroidal coil wound around the tube core, whereas in the orthogonal fluxgates with amorphous wires, it flows directly through the core. Having no excitation current inside the core reduces its heating and, hence, decreases the fluxgate thermal drift. Employing the toroidal coil also allows decreasing the excitation current by simply increasing the number of coil turns, while keeping the same intensity of the excitation field. Our experiments have shown a much higher efficiency of the new operating mode as compared with the second-harmonic mode. Adding a great enough dc bias to the ac excitation has caused a dramatic noise reduction. This effect is especially pronounced at relatively low frequencies, below 10 kHz. The fluxgate resolution in the fundamental mode, 10 pT/radicHz at 1 Hz , is by a factor of 30 better than in the second-harmonic mode. The sensitivity in the fundamental mode exceeds by a factor of 12.5 the sensitivity in the second-harmonic mode. We have also observed an about two times lower thermal drift of the fluxgate output. We have also found in this work that the phase noise of the excitation current is the main contributor to the fluxgate noise at low frequencies. It contributes about 67% to the fluxgate noise power density near the fundamental.