The atmospheric electric fields in thunderclouds have been shown to significantly modify the intensity and polarization patterns of the radio footprint of cosmic-ray-induced extensive air showers. Simulations indicated a very nonlinear dependence of the signal strength in the frequency window of 30–80 MHz on the magnitude of the atmospheric electric field. In this work we present an explanation of this dependence based on Monte Carlo simulations, supported by arguments based on electron dynamics in air showers and expressed in terms of a simplified model. We show that by extending the frequency window to lower frequencies, additional sensitivity to the atmospheric electric field is obtained.