## 3 Stability Analysis for Upper Limit
### Root Loci  
As is well known in control theory, the stability of linear time-invariant systems in the continuous-time domain can be determined by determining the location of the roots of the characteristic equation of the system. For bounded-input-bounded-output stability, the roots of the characteristic equation (i.e., the poles of the system) must all lie in the left half of the \(s\)-plane. Therefore, the stability of the system can be determined by checking the root locus diagram, which is a pictorial representation of the poles of the closed-loop transfer function as a function of the loop gain. The root locus diagrams for discrete-data systems are constructed in the \(z\)-plane using essentially the same properties as those of the continuous-data systems in the \(s\)-plane, except that the relationship between the root locations and stability must be made concerning the unit circle \(|z|=1\) in the \(z\)-plane [10]. Here, the loci of roots are defined when only \(BT\) varies instead of the loop gain \(K\) (unity gain is assumed for the discriminator gain \(A\)) to test the stability of a DPLL as a function of \(BT\). The closed-loop transfer function of the DPLL in (7) is then rewritten as:  
\[H_{\rm D}(z)=\frac{(z-z_{m})(z-z_{m-1})\cdot\cdots\cdot\cdot(z-z_{0})}{(z-p_{ n})(z-p_{n-1})\cdot\cdots\cdot\cdot(z-p_{0})}=\frac{\prod_{i=0}^{m}(z-z_{j})}{ \prod_{i=0}^{n}(z-p_{i})} \tag{9}\]  
where \(p_{i}\) and \(z_{j}\) represent the poles and zeroes of the system, respectively. It is obvious that as \(B\) increases, the roots of the closed-loop transfer functions in the \(s\)-domain deviate from the origin, but still stay in the left half of the \(s\)-plane for all values (i.e., the system is unconditionally stable). However, for the second- or higher-order systems, such pole deviation causes oscillation of their step responses. The left-side figures of Figure 3 illustrate the same effect in the \(z\)-plane for the first-/second-/third-order closed loops as described above but with different numerical integration methods for \(N(z)\) and \(F(z)\), and with the assumption of the zero computational delays. The trajectory for all cases begins at \(z=1\) when \(BT=0\) and then moves to the inside or outside of the unit circle as \(BT\) increases. The presence of roots at \(\left|z\right|=1\) causes the step response of the system to oscillate with a constant amplitude, and the system becomes marginally stable. Thus, the DPLL will be unstable for \(\left|z\right|>1\), even if its counterpart in the continuous-time domain is unconditionally stable. Therefore, the value of \(BT\) that makes the system marginally stable is the upper limit of available \(BT\) values for the stable DPLL. The limit is defined as \(BT_{\text{osc}}\) which indicates the \(BT\) when \(\left|z\right|=1\) (i.e., \(BT\) at the crossing point between the root trajectory and the unit circle) and is the point that makes the system marginally stable, so the step response begins to oscillate at this point. Figure 3: Root location of the DPLL closed-loop transfer functions with three different integration methods for \(N(z)\) and \(F(z)\) (\(t_{\text{D}}=0\)). (**a**) First-order root loci. (**b**) First-order root magnitude. (**c**) Second-order root loci. (**d**) Second-order root magnitude. (**e**) Third-order root loci. (**f**) Third-order root magnitude. The right-side figures of Figure 3 show \(\mid\!\!z\!\mid\) for the same case as discussed above with respect to \(BT\). As mentioned earlier, \(\mid\!\!z\!\mid\) should be less than 1 for a stable system. For the first-order DPLL in Figure 3b, the root loci for SI are outside the unit circle in the \(z\)-domain for \(BT\) > 0.5 (i.e., \(BT_{\text{osc}}\) = 0.5), whereas those for BL and II converge to values on the unit circle (\(\mid\!\!z\!\mid\) = 1) and the origin (\(\mid\!\!z\!\mid\) = 0), respectively, as \(BT\) increases. A similar phenomenon is observed in the second- and third-order cases.