How does an anode voltage affect electron flow in a thermionic diode?

In a thermionic diode, the anode voltage plays a critical role in regulating the flow of electrons emitted from the cathode. When a cathode is heated, it releases electrons that form a cloud in the space between the electrodes. This cloud creates a space-charge effect, where the negative charges of the electrons repel one another and hinder further emission.12

Applying a positive voltage to the anode creates an electric field that attracts these electrons away from the cathode, effectively overcoming the space-charge limitation. As the anode voltage increases, the flow of electrons to the anode also increases. This continues until the device reaches a state known as saturation, where the anode voltage is high enough to collect all the electrons being emitted by the cathode.1

In the space-charge-limited region, the relationship between the current density and the anode voltage is described by the Langmuir-Child law, which accounts for the voltage and the distance between the electrodes.1