Stability analysis of GPS carrier tracking loops by phase margin approach

Abstract

Stability, which is significantly related to the loop parameters, is an important factor in the traditional GPS tracking loop design. Through the analysis of phase margin values in the discrete GPS PLL tacking loop, we are able to theoretically reveal the relationship between loop stability, equivalent noise bandwidth B _n , predetection integration time T, and loop parameters. We calculate the theoretical limitations for B _n T, that is, the product of equivalent noise bandwidth multiplied by predetection integration time, for second- and third-order phase-locked loop, respectively. The results are verified by actual data from GPS receivers.

Appendix: PM solution in third-order tracking loop

For the general cubic equation $ax^{3} + bx^{2} + cx + d = 0$, there are three roots,

$$ \left\{ {\begin{array}{*{20}c} {x_{1} = - \frac{b}{3a} - \frac{C}{3a} - \frac{{b^{2} - 3ac}}{3aC}} \hfill \\ {x_{2} = - \frac{b}{3a} + \frac{{C\left( {1 + i\sqrt 3 } \right)}}{6a} + \frac{{\left( {1 - i\sqrt 3 } \right)\left( {b^{2} - 3ac} \right)}}{6aC}} \hfill \\ {x_{3} = - \frac{b}{3a} + \frac{{C\left( {1 - i\sqrt 3 } \right)}}{6a} + \frac{{\left( {1 + i\sqrt 3 } \right)\left( {b^{2} - 3ac} \right)}}{6aC}} \hfill \\ \end{array} } \right. $$

(33)

Where $Q = \sqrt {\left( {2b^{3} - 9abc + 27a^{2} d} \right)^{2} - 4\left( {b^{2} - 3ac} \right)^{3} }$ and $C = \root{3} \of {{\frac{1}{2}\left( {Q + 2b^{3} - 9abc + 27a^{2} d} \right)}}$. Assuming x is the $\cos \omega_{p}$ in cubic equation (25), the coefficients a, b, c, d can be represented by

$$ \left\{ {\begin{array}{*{20}c} {a = 8} \hfill \\ {b = 4G_{1} \left( {G_{1} + G_{2} + G_{3} } \right)T^{2} - 24} \hfill \\ {c = - 2\left( {2G_{1} + G_{2} + G_{3} } \right)\left( {2G_{1} + G_{2} } \right)T^{2} + 24} \hfill \\ {d = \left( {\left( {G_{2} + G_{3} } \right)^{2} + \left( {2G_{1} + G_{2} } \right)^{2} } \right)T^{2} - 8} \hfill \\ \end{array} } \right. $$

(34)

Because the phase margin $(\cos \omega )$ is a real number, only real root x ₁ can be used when solving cubic equation (25). Thus, we get

$$ \cos \omega_{p} = - \frac{b}{3a} - \frac{C}{3a} - \frac{{b^{2} - 3ac}}{3aC} $$

(35)

and

$$ \omega_{p} = {\text{arc}}{ \cos }\left( { - \frac{b}{3a} - \frac{C}{3a} - \frac{{b^{2} - 3ac}}{3aC}} \right) $$

(36)

Substituting loop coefficients G ₁, G ₂, G ₃ and prediction integration time T into (24), (34) and (36), the phase margin figure of third-order loop can be drawn as Fig. 7.