Effect of Higher Harmonics on the Accurate Calculation of Dynamic Phase-Locked-Loop Characteristics Using the Quasi-Harmonic Method

Abstract—The effect of higher harmonics on the accurate calculation of the dynamic characteristics of piecewise-linear phase-locked-loop frequency control using the quasi-harmonic method has been demonstrated at exactly known values of the lock-in range and the frequency of rotational motions.

Rigorous Methods of Investigation of Piecewise-Linear PLL Systems

Let us transform Eq. (2) into a system of equations that does not contain derivatives of function $\Phi (\varphi )$

$$\dot {\vec {y}} = A\vec {y} + \vec {c}\Phi (\varphi ),$$

((A.1))

where $\vec {y} = {{({{y}_{1}}, \ldots ,~{{y}_{n}})}^{T}}~$ (T is the transposition symbol),

$$\begin{gathered} ~{{y}_{1}} = \varphi ,\,\,\,\,~{{{\dot {y}}}_{i}} = {{y}_{{i + 1}}},\,\,\,\,~i = 0,~1, \ldots ,\,\,\,~n - m - 1, \\ {{{\dot {y}}}_{i}} = {{y}_{{i + 1}}} + {{c}_{i}}\Phi ({\varphi }),\,\,\,\,~~i = n - m, \ldots ,~n - 1, \\ {{c}_{{n - m}}} = - \frac{{{{b}_{m}}}}{{{{a}_{{n - 1}}}}},\,\,\,\,~~{{c}_{{n - k}}} = - \frac{{{{b}_{k}}}}{{{{a}_{{n - 1}}}}} \\ - \,\,\sum\limits_{i = n - m + k}^{n - 1} {\frac{{{{a}_{{n - m + k - i}}}}}{{{{a}_{{n - 1}}}}}} {{c}_{{i - k}}},~\,\,\,\,~k = 1,~ \ldots ,~m - 1, \\ \end{gathered} $$

((A.2))

$$\begin{gathered} \vec {c} = {{(0,~ \ldots ,~0,~{{c}_{{n - m}}},~\, \ldots ,~{{c}_{n}})}^{T}}, \\ {\mathbf{A}} = \left( {\begin{array}{*{20}{c}} 0&1&0& \ldots &0 \\ 0&0&1& \ldots &0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0&{{{ - {{a}_{0}}} \mathord{\left/ {\vphantom {{ - {{a}_{0}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}}&{{{ - {{a}_{0}}} \mathord{\left/ {\vphantom {{ - {{a}_{0}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}}& \ldots &{{{ - {{a}_{0}}} \mathord{\left/ {\vphantom {{ - {{a}_{0}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}} \end{array}} \right). \\ \end{gathered} $$

((A.3))

Separatrix Loop in Piecewise-Linear Systems

Let us consider the system of differential equations (A.1) with piecewise-linear approximation (5) of function $F(x).$ In this case, the system can be transformed into form

$$\dot {\vec {y}} = {{{\mathbf{A}}}_{i}}\vec {y} + {{s}_{i}}\vec {g},\,\,\,\,~~i = 1,~2,$$

((A.4))

where

$$\begin{gathered} {{{\mathbf{A}}}_{i}} = \left( {\begin{array}{*{20}{c}} {{{g}_{1}}{{k}_{i}}}&1&0& \ldots &0 \\ {{{g}_{2}}{{k}_{i}}}&0&1& \ldots &0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {{{g}_{n}}{{k}_{i}}}&{{{ - {{a}_{0}}} \mathord{\left/ {\vphantom {{ - {{a}_{0}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}}&{{{ - {{a}_{0}}} \mathord{\left/ {\vphantom {{ - {{a}_{0}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}}& \ldots &{{{ - {{a}_{{n - 2}}}} \mathord{\left/ {\vphantom {{ - {{a}_{{n - 2}}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}} \end{array}} \right), \\ {{k}_{1}} = {{c}^{{ - 1}}},~\,\,\,\,{{k}_{2}} = - {{(\pi - c)}^{{ - 1}}},~\,\,\,\,{{s}_{1}} = - \beta ,\,\,\,\, \\ ~{{s}_{2}} = \pi {{(\pi - c)}^{{ - 1}}} - \beta , \\ \vec {g} = {{({{g}_{1}},~ \ldots ,~{{g}_{n}})}^{T}},~\,\,\,\,{{g}_{1}} = \ldots = {{g}_{{n - m - 1}}} = 0, \\ \end{gathered} $$

and ${{g}_{{n - j}}}$ are calculated according to (A.2), and $j = 0,~ \ldots ,m - 1.$ The roots of the characteristic equation in regions I and II are denoted using ${{p}_{i}}$ and $p_{i}^{'}~~\left( {i = 1,~ \ldots ,~n} \right);$${{\vec {f}}_{i}}$ and $\vec {f}_{i}^{'}~$ are the eigenvectors corresponding to eigenvalues ${{p}_{i}}$ and $p_{i}^{'}.$

Let us assume that roots ${{p}_{i}}$ and $p_{i}^{'}$ in regions I and II satisfy the following conditions:

$${{p}_{i}} \ne {{p}_{j}},\,\,\,\,~p_{i}^{'} \ne p_{j}^{'}~\,\,\,\,~{\text{at}}\,\,\,\,~i \ne j,\,\,\,\,\operatorname{Im} ~p_{1}^{'} = 0,$$

$${\text{Re}}~p_{1}^{'} > 0,\,\,\,\,~{\text{Re}}~p_{k}^{'} < 0,~\,\,\,\,k = 2,~ \ldots ,~n,~\,\,\,{\text{Re }}{{p}_{i}} < 0.$$

Let ${\mathbf{W}}\left( {{{p}_{1}},~ \ldots ,~{{p}_{n}}} \right)$ denote the matrix whose columns are the eigenvectors corresponding to eigenvalues ${{p}_{i}}.$ Let ${{w}_{i}} = w({{p}_{1}},~ \ldots ~,~\left. {{{p}_{n}}} \right|a)$ be the determinant from $\det {\mathbf{W}}\left( {{{p}_{1}},~ \ldots ,~{{p}_{n}}} \right)$ by replacing the ith column with vector column a; ${{\vec {e}}_{1}} = {{(1,~0,~ \ldots ~,~0)}^{T}},~$$\mu = {\pi \mathord{\left/ {\vphantom {\pi {c - 1}}} \right. \kern-0em} {c - 1}}.$

In this case, the theorem stated in [13] is valid.

Theorem.For $a$ separatrix loop in system(A.4), which increases (decreases) periodic coordinate${{y}_{1}} = \varphi$by$2\pi$, to exist, we must have$\beta = {{\beta }_{{kc}}}~\left( {\gamma = - {{\beta }_{{kc}}}} \right),$where

$$\begin{gathered} ~~{{\beta }_{{kc}}} \\ = \,\,{{\,\left( {w - \sum\limits_{i = 1}^n {{{\omega }_{i}}} \exp ~\left( {{{p}_{i}}\tau } \right)} \right)} \mathord{\left/ {\vphantom {{\,\left( {w - \sum\limits_{i = 1}^n {{{\omega }_{i}}} \exp ~\left( {{{p}_{i}}\tau } \right)} \right)} {\left( {w + \sum\limits_{i = 1}^n {{{\omega }_{i}}~\exp } ~\left( {{{p}_{i}}\tau } \right)} \right)}}} \right. \kern-0em} {\left( {w + \sum\limits_{i = 1}^n {{{\omega }_{i}}~\exp } ~\left( {{{p}_{i}}\tau } \right)} \right)}}, \\ \end{gathered} $$

((A.5))

and τ is the root of equation

$$\sum\limits_{i = 1}^n {{{\omega }_{i}}~{{v}_{{i1}}}\exp ~\left( {{{p}_{i}}t} \right)} = 0,$$

$${{\omega }_{i}} = \mu {{w}_{i}}\left( {{{p}_{1}},~ \ldots ,~\left. {{{p}_{n}}} \right|f_{1}^{'}} \right) - \left( {\mu + 1} \right){{w}_{i}}\left( {{{p}_{1}},~ \ldots ,~\left. {{{p}_{n}}} \right|{{e}_{1}}} \right),$$

$${{v}_{{ij}}} = {{w}_{j}}\left( {p_{1}^{'}, \ldots ,\left. {p_{n}^{'}} \right|{{f}_{i}}} \right) - \left( {\mu + 1} \right){{w}_{j}}\left( {p_{1}^{'},~ \ldots ~\left. {p_{n}^{'}} \right|{{e}_{1}}} \right),$$

$$i,~j = 1,~ \ldots ~,~n,~\,\,\,\,~w = \det {\mathbf{W}}\left( {{{p}_{1}},~ \ldots ,~{{p}_{n}}} \right).$$

Limiting φ-Cycles in Piecewise-Linear Phase Systems

In triangular approximation of the nonlinearity for phase system (A.4), for φ-cycles to exist, the following conditions must be satisfied:

((A.6))

Here ${\mathbf{P}}(\tau ) = {\mathbf{W\Lambda }}(\tau ){{{\mathbf{W}}}^{{ - 1}}},$${\mathbf{Q}}(\theta ) = ~{\mathbf{V\Lambda }}{{'}}(\theta ){{{\mathbf{V}}}^{{ - 1}}},~$${\mathbf{\Lambda }}(t) = {\text{diag }}(\exp \left( {{{p}_{1}}t} \right), \ldots ,\exp \left( {{{p}_{n}}t} \right)),$$~{\mathbf{\Lambda }}'(t) = {\text{diag}}~(\exp \left( {p_{1}^{'}t} \right), \ldots ,\exp \left( {p_{n}^{'}t} \right))$ are the diagonal matrices; τ, θ is the time of travel of the point over regions I and II, respectively; W and V are the matrices whose respective columns are eigenvectors $\overrightarrow {{{f}_{i}}}$ and $\vec {f}_{i}^{'}~$. Let us assume that is the cycle frequency. In this case, relations (A.6) make it possible to obtain the oscillation characteristic in form

The stability of the found cycle can be determined by considering the matrix of the point transformation of plane ${{y}_{1}} = - c$ into plane ${{y}_{1}} = 2\pi - c\,:$

$${\mathbf{U}} = {\mathbf{P}}(\tau ){{{\mathbf{H}}}_{1}}{\mathbf{Q}}(\theta ){{{\mathbf{H}}}_{2}},$$

((A.7))

where

$$\begin{gathered} {{{\mathbf{H}}}_{1}} = \frac{\pi }{{c\left( {c - \pi } \right)\left( {{{g}_{1}}\left( {1 - \beta } \right) + y_{2}^{2}} \right)}} \\ \times \,\,{{(0,~{{g}_{2}}\left( {y_{2}^{2} - c} \right),~ \ldots ,~{{g}_{n}}(y_{n}^{2} - c))}^{T}}e_{1}^{T} + {\mathbf{E}}, \\ \end{gathered} $$

$$\begin{gathered} {{{\mathbf{H}}}_{2}} = \frac{\pi }{{c\left( {c - \pi } \right)\left( { - {{g}_{1}}\left( {1 + \beta } \right) + y_{2}^{1}} \right)}} \\ \times \,\,{{(0,~{{g}_{2}}\left( {2\pi - c - y_{2}^{1}} \right),~ \ldots ,~{{g}_{n}}(2\pi - c - y_{n}^{1}))}^{T}}e_{1}^{T} + {\mathbf{E}}. \\ \end{gathered} $$

Here $\beta ,~y_{2}^{1},~ \ldots ,~y_{n}^{1}$ are found from (A.6) and $y_{2}^{2},~ \ldots ,~y_{n}^{2}$ are determined using

$${{(c - \beta c,~y_{2}^{2}, \ldots ,~y_{n}^{2})}^{T}} = {\mathbf{W}}~{\mathbf{\Lambda }}(\tau ){{W}^{{ - 1}}}{{( - c - {\beta }c,~y_{2}^{1},~ \ldots ,~y_{n}^{1})}^{T}}~.$$

One of the roots of characteristic equation $\det ({\mathbf{U}} - \lambda {\mathbf{E}}) = 0$ is unity. If the other roots are within the unit circle, the point transformation is stable and, accordingly, the periodical mode under consideration is orbitally stable and nonasymptotically stable in the sense of Lyapunov.

In the saw-tooth approximation of nonlinearity, system (A.1) takes form

$$\dot {y} = {\mathbf{A}}y - \beta \vec {g},~$$

((A.8))

where

$${\mathbf{A}} = \left( {\begin{array}{*{20}{c}} {{{{{g}_{1}}} \mathord{\left/ {\vphantom {{{{g}_{1}}} \pi }} \right. \kern-0em} \pi }}&1&0& \ldots &0 \\ {{{{{g}_{2}}} \mathord{\left/ {\vphantom {{{{g}_{2}}} \pi }} \right. \kern-0em} \pi }}&0&1& \ldots &0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {{{{{g}_{n}}} \mathord{\left/ {\vphantom {{{{g}_{n}}} \pi }} \right. \kern-0em} \pi }}&{{{ - {{a}_{0}}} \mathord{\left/ {\vphantom {{ - {{a}_{0}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}}&{{{ - {{a}_{1}}} \mathord{\left/ {\vphantom {{ - {{a}_{1}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}}& \ldots &{{{ - {{a}_{{n - 2}}}} \mathord{\left/ {\vphantom {{ - {{a}_{{n - 2}}}} {{{a}_{{n - 1}}}}}} \right. \kern-0em} {{{a}_{{n - 1}}}}}} \end{array}} \right).$$

Similar to the previous case, on the assumption that ${\mathbf{Q}}(\theta ) = {\mathbf{W\Lambda }}(\theta ){{{\mathbf{W}}}^{{ - 1}}}$ and (the cycle frequency), the characteristic of oscillation can be found in form

((A.9))

where $\overline {{{a}_{{11}}}} = {{a}_{{11}}} + 2$ and ${{a}_{{ij}}}$ are the elements of matrix ${\mathbf{Q}} - {\mathbf{E}}.$

The matrix of the point transformation has form

$${\mathbf{U}} = {\mathbf{Q}}(\theta ){\mathbf{H}},$$

where

$${\mathbf{H}} = - \frac{2}{{{{g}_{1}}\left( {1 - \beta } \right) + y_{2}^{1}}}\vec {g}e_{1}^{T} + {\mathbf{E}}.$$