CMOS SUBHARMONIC INJECTION-LOCKED FREQUENCY TRIPLERS
2.1 THEORETICAL MODEL FOR INJECTION-LOCKED FREQUENCY TRIPLER
Based upon the locking mechanism for a small injection signal [26] and the simple ILO model [67], a physical representation of the proposed ILFT with a frequency pre-generator to generate the third-order harmonic signal connected to an ILO is shown in Fig. 2.1. In the ILO model, H(jω) is the transfer function of the band pass LC-tank filter used to eliminate undesired frequencies generated by the frequency pre-generator. The active devices of the ILO are modeled as the linear constant transconductance stage Gm. The frequency pre-generator is modeled as the nonlinear characteristic function f(vI). Both the Gm and H(jω) with a feedback path form the ILO. Without any input signal, the ILO has a steady output signal if the Barkhausen criterion is satisfied in the close-loop structure. An incident signal vI(t) with input frequency ωI is injected into the oscillator via a frequency pre-generator.
The output frequency ωO is the function of input frequency ωI while the oscillator is under the locked situation.
If the ILFT is under the locked condition, the following apply:
vI(t)=Vicos(ωIt+θ) (2.1)
vO(t)=Vocos(ωOt) (2.2)
vI,ILO(t)= f(vI(t))= f(Vicos(ωIt+θ)) (2.3)
where vI(t) is the incident signal with input frequency ωI, amplitude Vi, and phase θ;
vO(t) is the output signal with frequency ωO = 3ωI and amplitude Vo; and vI,ILO(t) is the output signal of the frequency pre-generator.
２７
From [68], vI,ILO(t) can be expressed as a polynomial series:
∑
∞ than three. The coefficient a3 is proportional to the conversion gain of the third-order harmonic frequency generator. The output current of the transconductance stage Gmcan been written as
iOUT =Gm[f(vI(t))+vO(t)]
=Gm
[
vI,ILO(t)+vO(t)]
. (2.5)By substituting (1)–(3) into (5),
iOUT =Gm[vI,ILO(t)+vO(t)]
=Gm[f(Vicos(ωIt+θ))+Vocos(ωot)]. (2.6) By neglecting the O((vI(t))4) term in (2.4), by substituting other terms in (2.4) into (2.6), by assuming that any frequency not close to ωO is filtered out by the frequency selective load H(jω), and by rearranging the terms, (2.6) can be rewritten as
２８
and a3 is the coefficient of cubic term in the nonlinear characteristic function of the frequency pre-generator.
The approximate transfer function of the band pass LC-tank filter H(jω) can be written as
where ωr and Q are the resonant frequency and quality factor of the LC-tank, respectively. H0 is the impedance of the LC-tank at resonant frequency.
If the Barkhausen criterion is satisfied in the close-loop, the phase shift of the close-loop should be zero. Thus
2
( ) ( )
0Combing (2.9) and (2.11), gives
( )
By rearranging (2.12) and finding the solution for θ, the following is derived
２９
The output voltage amplitude can be written as
Vo = iOUT H(jω) amplitude can be rewritten as
(
0)
In general, the locking range is limited by failure of either the phase condition (2.15) or the gain condition (2.16) [67]. From (2.15), it can be seen that the locking
３０
range increases with an increase in either the conversion gain of the frequency pre-generator or the incident amplitude Vi. The degradation of the LC-tank quality factor Q can also improve the locking range. However, the latter causes a decrease in the impedance of the LC-tank H0 and, thus, the output voltage amplitude also decreases (2.17). This result is consistent with the results in [26]. According to the proposed ILFT model, the design principle can be developed. It can be seen from (2.17) that the quality factor of the LC-tank can be maximized in order to obtain increased output amplitude. The resulting degradation of the locking range can be improved by increasing of the conversion gain of the frequency pre-generator (2.15).
The overall ILO output phase noise is characterized by the noise contributions of all blocks in an ILO [69]. The simplified noise source model of the proposed ILFT is shown in Fig. 2.2 where the conversion gain of the third-order harmonic signal in the frequency pre-generator is simplified to be a constant value AFPG and vI,ILO3ω is the signal with frequency 3ωI. The noise contribution from the frequency pre-generator and the ILO are modeled as nFPG(t) and nILO(t), respectively. The linear phase-domain model [70] is adopted to calculate the output phase noise.
The simplified noise source model of the proposed ILFT, as shown in Fig. 2.2, can be divided into two parts. One part is the noise calculation of the frequency pre-generator and the other is the noise analysis of the ILO. First, the noise characteristic between vI and vI,ILO3ω is considered. The phase noise spectral density SIN,ILO(ωm) at vI,ILO3ω node can be expressed as [69]
SIN,ILO
( )
ωm =32⋅SINJ( )
ωm +SFPG( )
ωm (2.18) where SINJ(ωm) and SFPG(ωm) are phase noise spectral densities of the injection signal and frequency pre-generator, respectively. ωm is the offset frequency from output３１
where the corner frequency of the ILFT noise transfer function ωp can be written as
η
In the above equations, SOUT(ωm) and SFreeRun(ωm) are phase noise spectral densities of output and internal circuits, respectively; AFPG is the conversion gain of the third-order harmonic signal in the frequency pre-generator; ωr and Q are the resonant frequency and the quality factor of LC-tank in the band pass filter, respectively; H0 is the impedance of the LC-tank at resonant frequency; and Vi and Vo indicate the amplitudes of input and output, respectively.
The combination of (2.18) and (2.19) results in the following:
)
As may be seen from the first and the second terms in (2.22), the noise from the input signal and frequency pre-generator are passed through the low-pass filter so that their noise transfer functions have low-pass transfer characteristics. Thus, the output
３２
phase noise is dominated by these two noise sources at small offset frequency ωm. If the noise contribution from the frequency pre-generator is negligible, the output phase noise is 9.5 dB [=10log (32)] higher than that from the input signal with a small offset frequency. The noise from internal circuits as given in the third term of (2.22) has a high-pass transfer characteristic. At large offset frequency ωm, the output phase noise is dominated by this noise and has a high-pass shape. To minimize the output phase noise, the corner frequency ωp can be increased to filter out the internal noise. As may be seen from (2.20) and (2.21), ωp can be increased by either degradation of the LC-tank quality factor Q or the high incident amplitude Vi.
A summary of the proposed ILFT can be developed from (2.6)–(2.11). The quality factor Q of the LC-tank is maximized for a large output voltage swing and for low-power consumption. The degradation of the locking range and the output phase noise from the increase in quality factor Q can be compensated for by increasing the conversion gain of the frequency pre-generator.
If the frequency pre-generator is removed from ILFT, the nonlinear characteristic function is performed by ILO. Thus, the locking range can be derived as
( )
Whereas the output amplitude is represented as
３３
It can be seen from (2.23) that the locking range can be increased by increasing
|a3|. In general, the value of a3 is negative, and an |a3| which is too large would degrade the output amplitude of the ILFT in (2.24) significantly. Obviously, if an ILFT works without the frequency pre-generator, the extra power consumption is required for both a large locking range and large output amplitude.