The sheet beam klystron is a kind of novel powerful microwave and millimeter-wave vacuum electron device, in which used is a thin rectangular sheet beam with high aspect ratio in order to improve beam-wave interaction efficiency by improving space-charge-limiting current of electron beam and obtaining big electric current, and it has many actual and potential applications. Based on the motion of the single electron under the small signal condition, the transit-time effect of electron beam in 2π-mode standing wave electric field in a multiple-cavity resonator is investigated, the expression of electron load conductance in a multiple-cavity resonator is presented, and the influence of the cavity number N on transit-time effect in a multiple-cavity resonator is discussed. The high frequency characteristics of the three-gap extended cavity are studied. The abilities for the single-gap cavity and three-gap cavity to modulate the sheet beam are compared by 3D PIC simulation. The simulation result shows that the three-gap extended interaction cavity operating at 2πmode is better than the single-gap cavity. The electron load conductance is derived and corrected based on the theory of relativity, by which a more accurate relation of electron load conductance to transmit angle can be obtained. In order to improve the output power and electron efficiency, the three-gap extended output cavity is used in the relativistic klystron to replace the single gap output cavity. By using the electromagnetic simulation software and 3D PIC code, a Ka-band sheet beam relativistic extended interaction klystron amplifie is designed. A sheet electron beam with a width-to-height ratio of 30 is adopted to reduce the space charge effect. In the PIC simulation, when the beam voltage is 500 kV and current is 1 kA, the device can generate a 190 MW output power at 40 GHz with an efficiency of 38% and a gain of 69 dB. The 3 dB bandwidth of the EISBK is about 150 MHz. Meanwhile, the output microwave is without the clutter jamming, which makes the contribution avoid the shrinkage of output microwave impulse. This study is of great importance for the physical design and process in engineering of the Ka-band sheet beam extended interaction relativistic klystron amplifier.