  The application of Carr-Purcell-Meiboom-Gill (CPMG) $\pi-$trains for
dynamically decoupling a system from its environment has been extensively
studied in a variety of physical systems. When applied to dipolar solids,
recent experiments have demonstrated that CPMG pulse trains can generate
long-lived spin echoes. While there still remains some controversy as to the
origins of these long-lived spin echoes under the CPMG sequence, there is a
general agreement that pulse errors during the $\pi-$pulses are a necessary
requirement. In this work, we develop a theory to describe the spin dynamics in
dipolar coupled spin-1/2 system under a CPMG($\phi_{1},\phi_{2}$) pulse train,
where $\phi_{1}$ and $\phi_{2}$ are the phases of the $\pi-$pulses. From our
theoretical framework, the propagator for the CPMG($\phi_{1},\phi_{2}$) pulse
train is equivalent to an effective ``pulsed'' spin-locking of single-quantum
coherences with phase $\pm\frac{\phi_{2}-3\phi_{1}}{2}$, which generates a
periodic quasiequilibrium that corresponds to the long-lived echoes. Numerical
simulations, along with experiments on both magnetically dilute, random spin
networks found in C$_{60}$ and C$_{70}$ and in non-dilute spin systems found in
adamantane and ferrocene, were performed and confirm the predictions from the
proposed theory.
