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## 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...

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##### Topics of Interest

2.5 Definition of Loop Order Recall the form of the phase-locked loop's transfer function, . The order of the PLL is defined as the highest order of s in the denominator of the loop transfer...

2.4 Differential and Integral Equations Problem 2.14 Solve the following differential equation: where f ( t) is plotted in Figure 2.7. Figure 2.7: The function...

2.2. Laplace Transform Let us first state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of...

4.3 SOLUTION OF DIFFERENTIAL AND INTEGRO-DIFFERENTIAL EQUATIONS The time-differentiation property of the Laplace transform has set the stage for solving linear differential (or integro-differential)...

OVERVIEW The Laplace transform provides a useful technique for solving linear differential equations. The basic idea is to first rewrite the equation in the transformed space, where the original...

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