text: Vacuum-tube feedback oscillators became the basis of radio transmission by 1920.  However, the triode vacuum tube oscillator performed poorly above 300 MHz because of interelectrode capacitance.  To reach higher frequencies, new "transit time" (velocity modulation) vacuum tubes were developed, in which electrons traveled in "bunches" through the tube.  The first of these was the Barkhausen–Kurz oscillator (1920), the first tube to produce power in the UHF range.  The most important and widely used were the klystron (R. and S. Varian, 1937) and the cavity magnetron (J. Randall and H. Boot, 1940).
  Mathematical conditions for feedback oscillations, now called the Barkhausen criterion, were derived by Heinrich Georg Barkhausen in 1921.  The first analysis of a nonlinear electronic oscillator model, the Van der Pol oscillator, was done by Balthasar van der Pol in 1927. He showed that the stability of the oscillations (limit cycles) in actual oscillators was due to the nonlinearity of the amplifying device.  He originated the term "relaxation oscillation" and was first to distinguish between linear and relaxation oscillators.  Further advances in mathematical analysis of oscillation were made by Hendrik Wade Bode and Harry Nyquist in the 1930s.  In 1969 Kaneyuki Kurokawa derived necessary and sufficient conditions for oscillation in negative-resistance circuits, which form the basis of modern microwave oscillator design.