A 0.6 V 2.7 mW 94.3% locking range injection ‐ locked frequency divider using modified varactor ‐ less Colpitts oscillator topology

This study presents a 4.6–12.8 GHz injection ‐ locked frequency divider (ILFD) based on modified varactor ‐ less Colpitts oscillator topology. A modified Colpitts ILFD is proposed to improve the LR and reduces power consumption, simultaneously. Meanwhile, the forward body bias technique is utilized to further decrease power consumption. Based on the 55 nm CMOS technology, a modified varactor ‐ less Colpitts ILFD with a free ‐ running frequency of 4.05 GHz is implemented. The proposed modified varactor ‐ less Colpitts ILFD exhibits 94.3% LR from 4.6 to 12.8 GHz with an injection power of 0 dBm. It consumes 2.7 mW power from 0.6 V supply voltage and the core area is 0.38 mm 2 .


| INTRODUCTION
With the rise of multistandard multiband communication systems, broadband building blocks have attracted great attention [1][2][3][4]. Phase-locked loops (PLLs) are one of the most important parts in modern wireless communication systems [5,6]. LC-based ILFDs are excellent candidates for the first-stage frequency divider in PLL because of their high operating frequency and low power consumption [7,8]. The LC ILFDs based on cross-coupled oscillator topology are widely studied because they are easy to implement and have low power consumption. However, the cross-coupled ILFDs generally suffer from a narrow LR. In order to meet multiband applications and enhance the robustness, a sufficiently large LR is required. Boosting the injection efficiency [9] and using high-order resonators [10] are common methods to extend the LR. In addition, other oscillator topologies can be employed to find out a wide LR ILFD [11][12][13][14][15][16].
Recently, Colpitts oscillator has been extensively researched because of its high performance [12,13]. It is meaningful to explore the application of Colpitts oscillator in ILFD. An 8-12 GHz Colpitts ILFD was reported in [14].
Although the LR is wide, it required a complex calibration system to adjust the varactors and the phase noise may be ruined by the varactors with low-quality factor. In addition, a 3-D helical transformer [15] was employed in Colpitts ILFD to extend the LR by decreasing the quality factor of the tank, which achieved a 34% LR. In [16], a 101.4% locking range Colpitts ILFD was implemented, but the power consumption reached 9.75 mW. Although previous works made good performances, the LR can be further improved and the power consumption can be lowered to satisfy the multistandard multiband applications.
A 94.3% LR varactor-less Colpitts ILFD with low supply voltage and low power consumption is presented. By utilizing the modified Colpitts structure, the LR can be enhanced significantly and the negative transconductance also is boosted to reduce the power consumption. Furthermore, forward body bias scheme is applied to further decrease power consumption. This article is organized as follows: Section 2 comparing conventional cross-coupled ILFD and Colpitts ILFD. Section 3 gives an analysis of the proposed modified Colpitts ILFD. Simulation results are demonstrated in Section 4. Finally, Section 5 draws a conclusion.

| COMPARISON OF CROSS-COUPLED ILFDS AND COLPITTS ILFDS
A conventional cross-coupled ILFD is depicted in Figure 1a [9,10] which has an injection device M inj . C db , C gb , C gd , and C gs are parasitic capacitances from cross-coupled pair and C ind is the parasitic capacitance of the inductor. Under these simplifications, the input LR of conventional cross-coupled ILFD is expressed as [17].
where g d and C M,inj are the equivalent conductance and parasitic capacitance of injection device, respectively. g m is the effective gain of the cross-coupled pair, g max is the equivalent transconductance of injection device, R eff is the impedance seen from injection device, R is the loss of the inductor, and r 0 is the on-resistance of cross-coupled pair. It is clearly seen from Equation (1) that the LR can be enhanced by widening the size of the injection device to improve g max . However, the increase in parasitics will increase power consumption. Lowering g m and R eff is another way to F I G U R E 1 Schematic of (a) the cross-coupled, (b) the Colpitts ILFD F I G U R E 2 Simplified half-circuit model of the Colpitts ILFD F I G U R E 3 Theoretical and simulated LR of the conventional Colpitts ILFD F I G U R E 4 Proposed modified varactor-less Colpitts ILFD improve LR but it may cause a risk of stopping oscillation. Furthermore, decreasing the capacitance C p,con to extend the LR is feasible which will be discussed later.
The Colpitts ILFD is shown in Figure 1b, which consists of a Colpitts topology [18] and injection device (M inj ). M 3 and M 4 make up the switching current source which reducing current consumption by almost twice. Feedback capacitors C 1 and C 2 generate a positive feedback to maintain oscillation. M inj injects the input signal into the LC tank and LC tank selects the desired signal and filters the clutter. C db and C gb are the parasitic capacitances of M 1 and M 2 .
In order to obtain the LR of the Colpitts ILFD, the simplified equivalent half-circuit model of the Colpitts ILFD is illustrated in Figure 2, where R is the loss from L and C p,col is the total parasitic capacitance at output. C E and −G E represent the parasitic capacitance and negative conductance of the switching current source, respectively. The transfer function of M 1 can be expressed as From Equations (4) and (5), the oscillation frequency of the Colpitts ILFD can be approximated by ω 0 ¼ 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi By substituting Equation (6) into Equation (1), the LR can be given as where G m represents the equivalent negative transconductance of the Colpitts ILFD and R eff1 is the effective impedance of the output node. It can be known from Equation (7) that the LR of the Colpitts ILFD is related to the feedback capacitance C 1 and C 2 , but not to the parasitic capacitance of the output node. The LR simulated by Cadence Virtuoso and our theoretical results are shown in Figure 3. As seen, the results obtained by the Cadence Virtuoso simulator are close to the proposed model. For general cases, the value of g m R eff in Equation (1) should exceed unity to satisfy the gain conditions [17]. In the design of ILFD, its value should be as small as possible to obtain a maximum LR [19,20], so the value in conventional cross-coupled ILFD is unity. In the same way, in order to satisfy the gain conditions of the Colpitts ILFD while obtaining a maximum LR, the value of G m R eff1 in Equation (7) should be equal to 4 [13]. When the operating frequency of the conventional cross-coupled ILFD is high, the size of the crosscoupled pair is small, and the quality factor of LC tank is high. Therefore, the parasitic capacitance at output is small and the cross-coupled ILFD has a wide LR. But when the operating frequency is low, the quality factor of the LC tank becomes poor, and a large-sized cross-coupled pair is required to generate a larger negative transconductance to maintain the oscillation. Therefore, the parasitic capacitances of the output node will be large, which will cause the LR of the crosscoupled ILFD to decrease sharply. But in Colpitts ILFD, the parasitic capacitance of the output node does not affect the LR. Only a reasonable design of the ratio of C 1 and C 2 is needed to make the ILFD meet the feedback, and the value of C 1 and C 2 can be designed small enough to obtain a large LR in medium and lower frequencies.
From above analysis, it can be concluded that the following inequality holds in the middle and low frequency bands Therefore, Colpitts ILFD has a larger LR than crosscoupled ILFD in medium and lower frequencies. Another advantage of Colpitts ILFD is that under the same supply voltage, the Colpitts structure has a greater output power. However, the Colpitts structure often has a large power consumption. In order to solve this problem, this article proposes a modified Colpitts structure that not only enhances LR but also reduces power consumption.

| PROPOSED MODIFIED VARACTOR-LESS COLPITTS ILFD
To further improve the LR and reduce power consumption to meet multistandard multiband applications, the modified Colpitts structure is proposed. Two inductors (L 1 ) are inserted between Colpitts ILFD core and the inductors (L) to overcome these difficulties as depicted in Figure 4. In this way, not only the LR is improved but the equivalent negative transconductance is increased, resulting in lower power consumption. The design considerations for the proposed modified Colpitts ILFD are presented as follows.

| Analysis of the LR
The LR of ILFD is greatly limited by the capacitance of output node. Therefore, capacitance at output node should be minimized to extend the LR. To this end, two inductors (L 1 ) are introduced to separate the feedback capacitor from the output node. From the simplified equivalent half-circuit model of the modified Colpitts ILFD shown in Figure 5, the transfer function of M 1 can be written as By solving Equations (9) and (10) correctly, the oscillation frequency can be derived as ω 0 ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi At the same time, the quality factor can be expressed as By substituting Equations (11) and (12) into Equation (1) and obtains where G m,c and R eff2 are the equivalent negative transconductance and the effective impedance of the output node of the modified Colpitts ILFD, respectively. In this design, L 1 < L, so we can get 1/(1 − (L 1 /L)) > 1. Therefore, the LR is improved using the modified Colpitts structure. The LR of theoretical and simulation results is depicted in Figure 6, and Figure 7 shows the relationship between the LR and the inserted inductor L 1 at an injection power of 0 dBm. By using L 1 , the LR has been significantly improved. Table 1 lists the simulation parameters in Spectre.
To demonstrate the effectiveness of the modified Colpitts structure, a pre-simulation using Cadence Virtuoso is performed. The simulated input sensitivity curve of the conventional and modified Colpitts ILFD is shown in Figure 8. This plot suggests that the modified Colpitts ILFD has a larger LR than the conventional Colpitts ILFD at the same supply voltage and DC current.

| Analysis of the power consumption
The conventional Colpitts structures usually consume a large power consumption to meet the start-up conditions, which limit its application in multistandard multiband systems. To reduce power consumption, the equivalent negative transconductance should be increased so that it can consume less current to satisfy gain conditions. The modified Colpitts ILFD can effectively increase the equivalent negative transconductance.
To analyse the impact of inserting inductors (L 1 ), simplified schematics and equivalent half circuit of conventional and F I G U R E 9 Simplified schematic and equivalent half circuit model of (a) and (c) conventional Colpitts ILFD, (b) and (d) modified Colpitts ILFD LI ET AL.
modified Colpitts ILFD are depicted in Figure 9a-d. According to Figure 9c, the negative conductance of the conventional Colpitts ILFD can be derived as The imaginary part of the admittance can be written as where G s is the equivalent parallel negative transconductance of Colpitts ILFD. The circuit of Figure 9d can be performed similar calculations, and it yields (see Equations (16) and (17)) For a typical design, R 1 (C p,col + C E ) 2 ω 2 << G S , R 1 G S << 1 and L 1 2 G S 2 << 2L 1 (C p,col + C E ). Therefore, the G m,c can be simplified as F I G U R E 1 0 Simulated (a) equivalent negative transconductance, (b) imaginary part of the input admittance

F I G U R E 1 3 Simulated voltage transients for the modified Colpitts ILFD
In this design, L 1 (C p,col + C E )ω 2 < 1 is established. Consequently, the equivalent negative transconductance is increased by using the modified Colpitts structure. In order to understand this formula more intuitively, the simulation results on equivalent negative transconductance are depicted in Figure 10. It is clearly seen from Figure 10a that the negative transconductance can be increased sharply from 12 mS to 1 S when the inductance (L 1 ) is 0.45 nH near 6 GHz. However, the sharp fluctuation of the imaginary part shown in Figure 10b will introduce an additional phase shift at this time, which may cause the circuit to stop oscillation if the positive feedback condition is not satisfied. Therefore, the inductance (L 1 ) in this design is 0.65 nH, which improves the robustness of the circuit while increasing the negative transconductance.
The modified Colpitts ILFD can satisfy gain conditions by consuming less current to generate a negative transconductance as large as the conventional Colpitts ILFD, which greatly reducing power consumption. For further reduction in power consumption, the forward body bias technique is applied. The DC current of the Colpitts ILFD can be represented by þ γ � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi where W is the channel width, L is the channel length, μ n and C OX are electron mobility and gate oxide capacitance, respectively. V TH0 represents the threshold voltage when V BD = 0 V and Φ f is the Fermi potential. It can be concluded that when the body bias voltage V BD increases, the modified Colpitts ILFD can generate the same DC current as the conventional Colpitts ILFD through a smaller gate-to-source bias voltage V GS . Therefore, by using forward body bias technology, the voltage margin required for normal operation of the transistors is reduced to make the ILFD work normally at low

| SIMULATION RESULTS AND DISCUSSIONS
The modified Colpitts ILFD is implemented in 55 nm CMOS technology provided by Semiconductor Manufacturing International Corporation. Figure 11 illustrates the layout of the designed ILFD with a core size of 620 � 610 μm 2 . The post-layout simulation is performed using Cadence SpectreRF. The ILFD consumes 4.5 mA current at 0.6 V supply voltage.
The input sensitivity curve is obtained by simulating LR with various injected power levels and plotted in Figure 12. The LR is 94.3% from 4.6 to 12.8 GHz with 0 dBm injection signal. Compared with the pre-simulation results, the LR is slightly reduced mainly due to the routing effects. The simulated voltage transients of the ac injection and the ac output signals are depicted in Figure 13. There is an output peak voltage between every two injection signal peak voltages, so the ILFD can work normally. Figure 14 shows the simulated spectrum of the freerunning and locked ILFD. It can be clearly seen that the locked ILFD has a good suppression of harmonics. Phase noise performances of ILFD are illustrated in Figure 15. The injection signal is generated by an external low phase noise VCO. The phase noise difference is also approximately 6 dB in 1 kHz to 10 MHz offset frequencies, which is in good agreement with the theoretical value.
The PVT analysis is performed to verify the robustness and performance of the modified Colpitts ILFD. Five available process corners (SS, FF, TT, SF, FS), −40°C to 85°C temperature range, and 10% supply voltage disturbance are selected as PVT variables. Table 2 tabulates the PVT analysis results of the modified Colpitts ILFD. As seen, LR varies slightly from the best situation to the worst situation.
The fluctuation of DC current caused by process and mismatch also has great influence on LR, so the robustness of the modified Colpitts ILFD is further verified by Monte Carlo simulation as shown in Figure 16. The simulation results show that the DC current is 2.248 mA, and the corresponding standard deviation is only 3.34 μA under 8 GHz injection signal. According to the results of Monte Carlo simulation, the modified Colpitts ILFD has low sensitivity. Table 3 summarizes the post-layout simulation results of the modified Colpitts ILFD and compares with previous works. The FoM is defined by [21] FoM ¼ LRð%Þ

Power Consumption ðmWÞ ð21Þ
It can be concluded that the modified Colpitts ILFD has a comparable merit over other works. Through an analysis of the LR of ILFDs, it is proposed that Colpitts ILFD has a wider LR than conventional cross-coupled ILFD in the medium and lower frequencies as the capacitance at the output node is smaller. A modified Colpitts ILFD is proposed to improve LR and reduce power consumption by boosting negative transconductance. In addition, the forward body bias technique is applied to reduce supply voltage and thus further decrease power consumption. An ILFD based on modified varactor-less Colpitts oscillator topology is implemented in 55 nm CMOS technology. The simulated LR can remain at 94.3% from 4.6 to 12.8 GHz at 0 dBm injection power. The ILFD consumes 2.7 mW power from 0.6 V supply voltage, which corresponds to FoM of 34.93%/mW.