Complementary BAW oscillator for ultra-low power consumption and low phase noise

Abstract

A complementary cross coupled BAW parallel resonance oscillator offering ultra-low power consumption and a good phase noise performance is presented. The power consumption in this structure is 50 % less than the classical NMOS based structure without any penalty in the phase noise performance. Rather, this structure serves to reduce the noise contribution of the biasing transistors at the output leading to a marginal improvement in thermal noise performance as compared to the NMOS based structure. Furthermore, the flicker noise upconversion of this complementary structure can be minimized by proper design considerations. The power consumption in case of such a complementary structure based oscillator (designed in 180nm CMOS process) employing a 2.497 GHz BAW resonator is around 675 μW for an amplitude of 300 mV with a phase noise of −140 dBc/Hz at 1 MHz offset.

Appendix

1.1 Calculation of the impedance seen by the noise current sources

The impedance seen by a noise current flowing across the output can be calculated from the small signal equivalent circuit of Fig. 2b. Writing the Kirchoff’s laws at various nodes in the circuit, this small signal circuit can be reduced to the one shown in Fig. 11. Y_c in this figure represents the admittance of the transistors seen at the output and it approximately equals −2G_m. Computing the equivalent impedance of the parallel combination of C_L and Y_c,

$$ {\rm Z}_{{\rm eq}1}=\frac{2}{{\rm Y}_{{\rm c}}}\left|\right|\frac{1}{{\rm sC}_{\rm L}} =\frac{2}{{\rm Y}_{\rm c}+2{\rm sC}_{\rm L}} $$

(18a)

$$ {\rm Z}_{{\rm eq}1}=\frac{2}{{\rm Y}_{\rm c}+2{\rm j}\omega {\rm C}_{\rm L}} $$

(18b)

$$ {\rm Z}_{{\rm eq}1}=\frac{2\left({\rm Y}_{\rm c}-2{\rm j}\omega {\rm C}_{\rm L}\right)}{{\rm Y}_{\rm c}^{2}+4\omega^{2} {\rm C}_{\rm L}^{2}}\approx \frac{2{\rm Y}_{\rm c}}{4\omega^{2} {\rm C}_{\rm L}^{2}}+\frac{1}{{\rm sC}_{\rm L}} $$

(18c)

This equivalent circuit is depicted in Fig. 12, where ${\rm R}^{\prime}$ represents ℜ[Z_eq1] ≈ G_m/C ²_L ω². This real impedance compensates the motional resistance R_m of the resonator at the working frequency,i.e. $-{\rm R}_{\rm m}+{\rm R}^{\prime}=0$. The impedance of the motional branch is given by ${\rm Z}_{\rm m}={\rm R}_{\rm m}+{\rm sL}_{\rm m}+\frac{1}{{\rm sC}_{\rm m}}$. The total impedance seen by the noise current source is

$$ {\rm Z}_{{\rm eq}}={\rm Z}_{\rm m}\left|\right|{\rm Z}_{{\rm eq}1}. $$

(19)

Simplifying the above equation, substituting ${\rm R}_{\rm m} = -{\rm R}^{\prime}$ and noting that the oscillation frequency is given by $\omega_0=\omega_{\rm m}\sqrt{1+\frac{{\rm C}_{\rm m}}{{\rm C}_{\rm L}}}$, replacing for ω₀ and L_mC_m = 1/ω ²_m , Z_eq becomes,

$$ {\rm Z}_{{\rm eq}}=-\frac{{\rm R}_{\rm m}\left(\omega^{2}+\omega_0^{2}-2\omega_{\rm m}^{2}\right)}{\omega^{2}-\omega_0^{2}}-{\rm j}\frac{1+{\rm C}_{\rm L}{\rm C}_{\rm m}{\rm R}_{\rm m}^{2}\omega^{2}-\frac{\omega^{2}}{\omega_{\rm m}^{2}}}{\omega\left({\rm C}_{\rm L}+{\rm C}_{\rm m}-\frac{{\rm C}_{\rm L}\omega^2}{\omega_{\rm m}^{2}}\right)} $$

(20)

Fig. 11

Impedance seen by a noise current source—I

Fig. 12

Impedance seen by a noise current source—II

The frequency ω is taken to be offset from the oscillation frequency by $\Updelta\omega$ i.e. $\omega=\omega_0+\Updelta\omega$, the real part of the impedance seen by the noise current source is

$$ \Re[{\rm Z}_{{\rm eq}}]\approx-\frac{2{\rm R}_{\rm m}\left(\omega_0^{2}-\omega_{\rm m}^{2}\right)}{\omega^{2}-\omega_0^{2}}\approx-\frac{{\rm R}_{\rm m}\left(\omega_0^2-\omega_{\rm m}^2\right)}{\delta\omega\omega_0}. $$

(21)

Inserting $\frac{{\rm C}_{\rm m}}{{\rm C}_{\rm L}}=\frac{\omega_0^2}{\omega_{\rm m}^2}-1$, the above equation becomes

$$ \Re[{\rm Z}_{{\rm eq}}]\approx\frac{1}{{\rm Q}_{\rm m}{{\rm C}_{\rm L}}\Updelta\omega}. $$

(22)

In the imaginary part of the impedance in (21), the C_LC_mR_mω² term is negligible. Therefore, the imaginary term becomes

$$ \Im[{\rm Z}_{{\rm eq}}]\approx\frac{1-\frac{\omega^{2}}{\omega_{\rm m}^{2}}}{\omega {\rm C}_{\rm L}\left(1+\frac{{\rm C}_{\rm m}}{{\rm C}_{\rm L}}-\frac{\omega^{2}}{\omega_{\rm m}^{2}}\right)} $$

(23)

Simplifying yields

$$ \Im[{\rm Z}_{{\rm eq}}]\approx\frac{1-\frac{\omega_0^{2}}{\omega_{\rm m}^{2}}}{\omega {\rm C}_{\rm L}\left(\frac{\omega_0^{2}}{\omega_{\rm m}^{2}}-\frac{\omega^{2}}{\omega_{\rm m}^{2}}\right)} $$

(24)

Replacing $\omega=\omega_0+\Updelta\omega$,

$$ \Im[{\rm Z}_{{\rm eq}}]\approx\frac{\frac{{\rm C}_{\rm m}}{{\rm C}_{\rm L}}}{\omega {\rm C}_{\rm L}\left(\frac{\omega_0^{2}}{\omega_{\rm m}^{2}}-\frac{\omega^{2}}{\omega_{\rm m}^{2}}\right)} $$

(25a)

$$ \Rightarrow\Im[{\rm Z}_{{\rm eq}}]\approx\frac{1}{\omega {\rm C}_{\rm L}}\frac{\frac{{\rm C}_{\rm m}}{{\rm C}_{\rm L}}}{\frac{2\Updelta\omega\omega_0}{\omega_{\rm m}^{2}}} $$

(25b)

$$ \Rightarrow\Im[{\rm Z}_{{\rm eq}}]\approx\frac{1}{\omega {\rm C}_{\rm L}}\frac{\omega_{\rm m}{\rm C}_{\rm m}}{2\Updelta\omega {\rm C}_{\rm L}} $$

(25c)

The total impedance seen by the current source is therefore,

$$ {\rm Z}_{{\rm eq}}\approx\frac{1}{{\rm Q}_{\rm m}{\rm C}_{\rm L}\Updelta\omega}+\frac{1}{{\rm j}\omega {\rm C}_{\rm L}}\frac{\omega_{\rm m}{\rm C}_{\rm m}}{2\Updelta\omega {\rm C}_{\rm L}} $$

(26)

For calculating the noise PSD, only the magnitude of the impedance is taken into account. Since the absolute value of the real part in the above equation is much less than its imaginary counterpart,it can be neglected leaving

$$ \left|{\rm Z}_{{\rm eq}}\right|^2=\frac{1}{\omega^2 {\rm C}_{\rm L}^2}\frac{\omega_{\rm m}^2 {\rm C}_{\rm m}^2}{4\Updelta\omega^2 {\rm C}_{\rm L}^2}. $$

(27)