2.4 First Order Loop Responses

The simple first-order loop of Figure 2.2 has a filter, f( t) = F( s) = K _f. Substituting this first-order filter into Equation 2-48 yields

In Equation 2-50, the gain of the phase-locked loop, , is the dominant characteristic of the loop transfer function. For a first-order PLL, the only variable available to the designer is the loop gain, . The error transfer function, H _e( s) for the first order loop is

The error output, ? _e( s), is obtained from H _e( s) by

Using Equation 2-52, we can recompute the three signal cases previously computed for the first order loop. We will compute all of these cases with the Laplace Transform technique and compare them to the solutions we obtained from the differential equations.

Case I. ?( t) = ??, where ?? is constant. The Laplace transform of this input is . From Equations 2-51 and 2-52,

The inverse Laplace transform of ? _e( s), using Equation 2-22, or transform tables in Appendix A, yields

This matches our result in Equation 2-14, which we obtained through direct solution of the differential equation. (Note that we have included the gain of the phase detector and input amplitude in Equation 2-54.)

Case II. ?( t) = 2 ? f _? t The Laplace transform of ?( t) is . Again using...