Date of Award

Spring 1976

Document Type


Degree Name

Doctor of Engineering Science in Electrical Engineering


Electrical Engineering

First Advisor

Jacob Klapper

Second Advisor

Stanley Habib

Third Advisor

Joseph Frank

Fourth Advisor

Sol Rosenstark


In this dissertation, a second-order phase-locked loop (PLL) in which the loop filter contains complex zeros is investigated in both its linear and nonlinear modes of operation; prior designs used a filter containing a simple zero on the negative real axis. This "generalized" second-order PLL had been heretofore essentially unexplored.

The basic characteristics of the generalized second-order PLL operating in the linear mode including the open and closed-loop responses and the corresponding root locus were generated and compared against those of the conventional second-order PLL. As in the conventional case, the generalized second-order PLL is found to be unconditionally stable. The closed-loop response of the generalized second-order PLL indicates a noise bandwidth which is theoretically infinite, thus making the predetection filter critical to the performance of this PLL. Such is not the case in the conventional PLL.

A method is presented for achieving an optimum design for the generalized second-order PLL for a number of useful modulation types including a single-channel FM speech signal, FDM-FM and FDM-PM. This optimum design is in terms of threshold performance and theoretically predicts that superior performance is possible over the conventional second-order PLL.

Using the Continuous System Modeling Program (CSMP), a nonlinear model of the generalized second-order PLL was simulated for the test-tone modulation case, both in the absence of noise and with the signal corrupted by bandpass additive Gaussian noise. In addition, preliminary simulation results were obtained for the case of a single-channel FM speech signal. Using simulation techniques, a measure of the mean-square phase error at threshold for the generalized second-order PLL was obtained. This parameter is useful in the optimum design procedure.

Additional insight into the operation of the generalized second-order PLL was obtained through investigation of its acquisition and tracking behavior. This was accomplished using phase plane techniques to study the nonlinear differential equation which governs the loop operation. The results indicate that two distinct types of behavior are theoretically possible depending upon the loop parameters. In one case the behavior is not unlike that of the conventional second-order PLL. In the second case, however, additional singularities are introduced into the phase plane and the behavior is seen to change markedly.