The behaviour of a first-order phase-locked loop is critically examined under conditions where it is tracking a carrier corrupted by discontinuous (square-wave) amplitude modulation. The transient response of the system is first derived for the simple case of a PLL without time delay, and a comparison is made with a series of practical results obtained from a first-order loop. The agreement between the predicted response and that observed in practice is found to be extremely good. The theoretical behaviour of the loop is further analysed for the case of a system incorporating time delay. It is shown that such delay is detrimental to the tracking performance of the loop, when it is necessary to track a carrier modulated in the manner outlined above.