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Charge pump with reduced current mismatch for reference spur minimization in PLLs

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Abstract

In this article, a charge pump circuit featuring minimal mismatch between its up and down currents is proposed. In conventional charge pumps, where error amplifiers are used in feedback, the factor that hinders exact current matching is the offset voltage of the error amplifier. In the proposed design, the input offset voltage is computed and additional current delivering/comsuming branch is implemented to supplement for the error current. The new charge pump requires a few error amplifiers and a dynamic comparator for its operation. Simulations considering process variations show current mismatch of less than 20 nA even at the worst case event. The proposed charge pump has been utilized in PLL circuits to reduce reference spurs and simulations of these PLLs show remarkable spur reduction.

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Correspondence to Debdut Biswas.

Appendices

Appendix A: Details of equations in deriving the magnitude of spurs

The waveform of \(V_C(t)\) is divided into three regions—(1) when \(I_{\mathrm {UP}}\) flows, (2) when \((I_{\mathrm {DN}} - I_{\mathrm {UP}})\) flows, and (3) when no current flows into the loop filter. The durations of the intervals when \(I_{\mathrm {UP}}\) and \((I_{\mathrm {DN}} - I_{\mathrm {UP}})\) flow are assumed low compared to \(T_{\mathrm {ref}}\). The equations for \(V_C(t)\) when \(I_{\mathrm {UP}}\) and \((I_{\mathrm {DN}} - I_{\mathrm {UP}})\) flow are thus linear. The duration during which no current flows is comparable to \(T_{\mathrm {ref}}\). A general expression of \(V_C(t)\) in this region is expressed in Eq. (5). It is derived as follows.

With some voltage \(V_3\) initially present across the capacitor \(C_2\), the control voltage \(V_C(s)\) is expressed as,

$$\begin{aligned} V_C(s) = \frac{V_3}{s} \frac{s + \frac{1}{R_1 C_1}}{s + \frac{1}{R_1 C_{eq}}} \end{aligned}$$
(27)

Assuming \(C_1 \gg C_2\), the expression of \(V_C(t)\) when no CP current flows is expressed by Eq. (5).

The time instants \(t_2\) and \(t_3\) are given by,

$$\begin{aligned} t_2&= t_{\mathrm {PFD}}\left( \frac{I_{\mathrm {DN}}}{I_\mathrm {UP}} - 1 \right) \end{aligned}$$
(28)
$$\begin{aligned} t_3&= t_{\mathrm {PFD}}\frac{I_{\mathrm {DN}}}{I_{\mathrm {UP}}} \end{aligned}$$
(29)

In the Eq. (4), if \(V_C(t)\) is equated to 0, then we get the expression for \(V_1\).

$$\begin{aligned} V_1 = t_2 \frac{I_{\mathrm {UP}}}{C_2} \end{aligned}$$
(30)

\(V_3\) can simply be obtained by using the Eq. (5) with the assumption \(T_{\mathrm {ref}} \gg t_{\mathrm {PFD}}\).

$$\begin{aligned} V_1 = V_3 \frac{C_1}{C_1 + C_2} \exp \left( -\frac{T_{\mathrm {ref}}}{R_1 C_{eq}} \right) \end{aligned}$$
(31)
Fig. 22
figure 22

Rectangular approximation of the PLL control voltage

The expression of \(V_2\) in the rectangular approximation of \(V_C(t)\) is realized by equating the areas under the waveforms as shown in Fig. 22. From the figure,

$$\begin{aligned} a_1 + a_2&= b_1 + b_2 +b_3 \nonumber \\ \frac{1}{2}V_1t_2 + \frac{1}{2}V_3 (t_3 - t_2)&= (V_1 - V_2)t_2 + (V_1 - V_2)(t_3 - t_2) \nonumber \\&\quad + (V_3 - V_1)(t_3 - t_2) \end{aligned}$$
(32)

Equations 7 and 8 are the Fourier series expansions of the rectangular and the exponential approximations. The Fourier series coefficients for the rectangular-approximated wave is derived by integrating \(V_2\) along with \(\exp \left( -j\omega _{\mathrm {ref}}kt \right)\) in the duration 0 to \(t_3\). The coefficients for the exponential approximation are given by,

$$\begin{aligned} c_{k,\mathrm {exp}} = \frac{1}{T_{\mathrm {ref}}} \int _{t_3}^{T_{\mathrm {ref}}} V_3 \frac{C_1}{C_1 + C_2} e^{-\frac{t-t_3}{R_1C_{eq}}} e^{-j\omega _{\mathrm {ref}}kt}dt \end{aligned}$$
(33)

A summary for final form of the magnitude of spurs derived in Eq. (9) is given below. The terms that need simplification can be aggregated as,

$$\begin{aligned} V'_{C,k} (t)&= \frac{A}{jk} \left\{ e^{-jk\omega _{\mathrm {ref}}\frac{I_{\mathrm {UP}}}{I_{\mathrm {DN}}}t_{\mathrm {PFD}}} - 1 \right\} \nonumber \\&\quad + \frac{B}{jk\omega _{\mathrm {ref}}R_1C_{eq} + 1} \left\{ e^{-\frac{T_{\mathrm {ref}}}{R_1C_{eq}}} - e^{-jk\omega _{\mathrm {ref}}\frac{I_{\mathrm {DN}}}{I_{\mathrm {UP}}}t_{\mathrm {PFD}}} \right\} \end{aligned}$$
(34)

where A and B are constants pertaining to Eqs. (7) and (8) respectively. With \(I_{\mathrm {UP}} \approx I_{\mathrm {DN}}\) and \(\omega _{\mathrm {ref}} t_{\mathrm {PFD}} \ll 1\), the first term approximates to 0 and \(V'_C(t)\) reduces to,

$$\begin{aligned} \left| V'_{C,k} (t)\right| = \frac{B}{k\omega _{\mathrm {ref}}R_1C_{eq}} \left| 1 - e^{-\frac{T_{\mathrm {ref}}}{R_1C_{eq}}} \right| \end{aligned}$$
(35)

where \(\omega ^2_{\mathrm {ref}}R^2_1C^2_{eq} \gg 1\). The simplified Eq. (35) is used to find the magnitude of spurs shown in Eq. (9).

Appendix B: Details of equations in deriving the critical current \(I_c\)

The critical current \(I_c\) for the transistors \(\mathrm {M_1}\) and \(\mathrm {M_2}\) of the CP in Fig. 4 are expressed as,

$$\begin{aligned} I_c&= \frac{\beta _n}{2} \left( V_b - V_Y -V_{Tn} \right) ^2 \end{aligned}$$
(36a)
$$\begin{aligned} I_c&= \frac{\beta _p}{2} \left( V_X - V_{\mathrm {out0}} - A_{\mathrm {c}}\frac{V_{DD}}{2} - V_{Tp} \right) ^2 \end{aligned}$$
(36b)

Equating the above two equations, a relation between \(V_X\) and \(V_Y\) is realized.

$$\begin{aligned} \frac{V_X - V_Y - V_p + V_n}{V_X + V_Y - V_p - V_n} = \frac{\sqrt{\beta _n} - \sqrt{\beta _p}}{\sqrt{\beta _n} + \sqrt{\beta _p}} = \frac{1}{K} \end{aligned}$$
(37)

Now, \(V_X + V_Y = V_{DD}\) and \(V_X - V_Y = 2R I_R\), and,

$$\begin{aligned} I_c = \frac{V_{DD}}{2R_B} - \frac{R + R_B}{R_B} I_R \end{aligned}$$
(38)

Substituting these values in Eq. (37), the Eq. (12) in Sect. 3 is obtained.

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Biswas, D., Bhattacharyya, T.K. Charge pump with reduced current mismatch for reference spur minimization in PLLs. Analog Integr Circ Sig Process 95, 209–221 (2018). https://doi.org/10.1007/s10470-018-1163-z

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