One of the other benefits of using the charge pump is the fact that we can select $I_{\text {pump }}$, that is,

$$
K_{P D I}=\frac{I_{\text {pump }}}{2 \pi}
$$


Again, let's set the lock range to 20 MHz . Using Eq. with $\zeta=1$ gives, again, $\omega_n=10 \times 10^6 \mathrm{radians} / \mathrm{V} \cdot \mathrm{s}$. From Eq. we get

$$
R C_1=200 \mathrm{~ns} \text { so let's use } R=20 \mathrm{k}, C_1=10 \mathrm{pF} \text {, and } C_2=1 \mathrm{pF}
$$

remembering that we generally set $C_2$ to one-tenth of $C_1$. Using Eq. we can now find the value of $I_{p u m p}$

$$
10 \times 10^6=\sqrt{\frac{I_{p u m p} \cdot 1.57 \times 10^9}{2 \pi \cdot 2 \cdot 10 \times 10^{-12}}} \rightarrow I_{p u m p}=8 \mu A
$$


Because this value isn't that critical, we'll round it up to $10 \mu \mathrm{~A}$. Figure shows the simulation results. Notice the $\zeta=1$ (or maybe a little less because the voltage does overshoot its final value by a little bit) shape of the VCO's control voltage. Let's modify the loop filter to show what $V_{\text {invco }}$ would look like if $\zeta=$ 0.1 . All we have to do, from Eq., is drop $R$ by a factor of 10 . The result is seen in Fig. The control voltage is oscillating and the loop isn't locking. Of course, we can also increase the damping factor. The loop now behaves sluggishly and does not respond to fast changes in the input signal. For general design, where the process and temperature vary, it's better to center the design on a larger damping factor to avoid instability.