A Wide-Band Divide-by-2 Injection-Locked Frequency Divider Based on Distributed Dual-Resonance Tank

Abstract

A wide-band divide-by-2 injection-locked frequency divider (ILFD) based on a distributed dual-resonance high-order tank is presented. The ILFD employs a distributed LC network as the dual resonance tank and achieves an ultra-wide locking range. Fabricated in a 65 nm 1P7M LP-CMOS process, the divide-by-2 ILFD consumes 7 mW from a 0.7 V power supply and realizes a locking range of 87.0%, from 13 GHz to 33 GHz. The core circuit occupies an area of 0.22 mm × 0.5 mm.

1. Introduction

In modern RF phase-locked loops, the frequency divider is one of the critical buildings. Among various dividers, the injection-locked frequency divider (ILFD) draws most of the attention, as it can work at a higher frequency and consume less power.

An original ILFD injects the input signal indirectly from the tail transistor of an oscillator [1]. However, this indirect injection topology entails two issues. Firstly, the injected current may flow to the ground through the parasitic capacitance at the common node and cause a loss of injected current. Secondly, the injected current is steered to the tank through two switch transistors, and the current gain from the injection current to the tank is 2/π, less than 1. Thus, the indirect injection topology is of low current injection efficiency. Reference [2] proposed a direct injection topology, where the input signal is directly injected into the tank. Though coming to a larger locking range as compared to its indirect injection counterpart, the direct injection ILFD is still limited to a relatively narrow locking range. In Reference [3], a multi-order LC oscillator is used to expand the locking range to 25.9%. In References [4,5], a wide-band ILFD was realized by a transformer-based fourth-order resonator. The work in [4] was further investigated in [6], and the locking range was expanded to 62.9%. In this case, however, a transformer with a low coupling factor was needed. The transformer should be carefully designed and occupy a large layout area due to the low coupling factor.

This paper proposes a wide-band divide-by-2 ILFD implemented in the 65 nm 1P7M LP-CMOS process. The ILFD uses a distributed dual-resonance resonator to expand the locking range. The analysis of the dual-resonance tank is presented in Section 2. Implementations and measurement results are shown in Section 3. Section 4 investigates the electromagnetic (EM) coupling of the inductors. Finally, the conclusion is given in the last section.

2. Analysis of the Proposed Dual-Resonance Tank

An ILFD functions as a free-running oscillator and has an oscillation frequency f_osc, if no signal is injected in. When the injected signal is large enough, the ILFD will oscillate at a new frequency f_new determined by the injection signal. At the free-running frequency f_osc, the phase response of the tank impedance is zero, while at the new frequency f_new, a phase shift is induced by the tank. The injected current should be able to compensate for this phase shift to sustain the ILFD oscillating at the new frequency f_new.

The total current I_tank flowing into the tank consists of the oscillating current I_osc and the injection current I_inj, that is:

$$\overrightarrow{I\mathrm{tank}}=\overrightarrow{I\mathrm{osc}}+\overrightarrow{I\mathrm{inj}}$$

(1)

A phase shift occurs between the current I_tank and the voltage V_tank at the tank, if the operating frequency deviates from the resonating frequency of the tank. In order to sustain the waveform in the oscillator, I_osc and V_tank should be in phase with each other. Refer to Figure 1a,b, the maximum θ_max is obtained when I_inj is perpendicular to I_osc:

$$\sin \theta \max =\left|I\mathrm{inj}/I\mathrm{osc}\right|$$

(2)

Thus, the phase shift caused by the tank should be less than θ_max:

$$\left|\angle Z\mathrm{tank}(\omega )\right|<\arcsin (\left|I\mathrm{inj}/I\mathrm{osc}\right|)$$

(3)

With a given I_inj and I_osc, a large locking range can be obtained if the phase shift caused by the tank is kept near zero across a large frequency range.

2.1. Zero and Pole Analysis

Figure 2a shows the conceptual diagram of the proposed divide-by-2 ILFD, and Figure 2b is a simplified AC equivalent circuit of the tank in the ILFD. The input impedance of the dual-resonance tank can be written as:

$$Z_{\mathrm{in}}(s)=\frac{1}{s{C}_{\mathrm{s}}}\Vert (s{L}_{\mathrm{s}}+(s{L}_{\mathrm{p}})\Vert \frac{1}{s{C}_{\mathrm{p}}})$$

(4)

Replacing s with jω, we can obtain:

$$Z_{\mathrm{in}}(\mathrm{j}\omega )=\frac{\mathrm{j}\omega (L_{\mathrm{p}}+L_{\mathrm{s}}-{\omega }^2{L}_{\mathrm{p}}{L}_{\mathrm{s}}{C}_{\mathrm{p}})}{1-{\omega }^2(L_{\mathrm{p}}{C}_{\mathrm{p}}+L_{\mathrm{p}}{C}_{\mathrm{s}}+L_{\mathrm{s}}{C}_{\mathrm{s}})+{\omega }^4{L}_{\mathrm{p}}{L}_{\mathrm{s}}{C}_{\mathrm{p}}{C}_{\mathrm{s}}}$$

(5)

From (5), the tank owns two pole frequencies and one zero frequency. Defined as:

$${\omega }_1=\frac{1}{\sqrt{L_{\mathrm{p}}{C}_{\mathrm{p}}}},{\omega }_2=\frac{1}{\sqrt{L_{\mathrm{s}}{C}_{\mathrm{s}}}},b=\frac{{\omega }_1^2}{{\omega }_2^2}, \mathrm{and} a=\frac{C_{\mathrm{s}}}{C_{\mathrm{p}}}$$

(6)

Thus, the zero frequency ω_z and the two pole frequencies ω_p1 and ω_p2 are given as:

$${\omega }_{\mathrm{z}}^2=(b+a){\omega }_2^2$$

(7)

$${\omega }_{\mathrm{p}1}^2=\frac{{\omega }_2^2}{2}(b+a+1+\sqrt{{(b+a+1)}^2-4b})$$

(8)

$${\omega }_{\mathrm{p}2}^2=\frac{{\omega }_2^2}{2}(b+a+1-\sqrt{{(b+a+1)}^2-4b})$$

(9)

If the zero frequency ω_z is set to be equal to ω₂, that is, b + a=1, then:

$${\omega }_{\mathrm{p}1}^2={\omega }_2^2(1-\sqrt{a})$$

(10)

$${\omega }_{\mathrm{p}2}^2={\omega }_2^2(1+\sqrt{a})$$

(11)

The two pole frequencies ω_p1 and ω_p2 are located at the left side and right side of the zero frequency ω_z, respectively. Thus, following the method proposed in [4], if the losses of the inductor and capacitor are considered, the phase variety caused by the zero frequency may be compensated by the two pole frequencies. This compensation may result in a zero-phase-shift (ZPS) plateau in the phase response of the tank around the zero frequency ω_z, and lead to a larger locking range.

2.2. Phase Response

Taking the loss of the inductor L_p into consideration, the tank impedance in (4) may be re-written as:

$$Z_{\mathrm{in}}(s)=\frac{1}{s{C}_{\mathrm{s}}}\Vert (s{L}_{\mathrm{s}}+(s{L}_{\mathrm{s}})\Vert \frac{1}{s{C}_{\mathrm{p}}}||R_{\mathrm{p}})$$

(12)

$$Z_{\mathrm{in}}(\mathrm{j}\omega )=\frac{-L_{\mathrm{p}}{L}_{\mathrm{s}}{\omega }^2+\mathrm{j}\omega R_{\mathrm{p}}(L_{\mathrm{p}}+L_{\mathrm{s}}-L_{\mathrm{p}}{L}_{\mathrm{s}}{C}_{\mathrm{p}}{\omega }^2)}{R_{\mathrm{p}}[1-{\omega }^2(L_{\mathrm{p}}{C}_{\mathrm{p}}+L_{\mathrm{p}}{C}_{\mathrm{s}}+L_{\mathrm{s}}{C}_{\mathrm{s}})+{\omega }^4{L}_{\mathrm{p}}{L}_{\mathrm{s}}{C}_{\mathrm{p}}{C}_{\mathrm{s}}]+\mathrm{j}\omega L_{\mathrm{p}}(1-L_{\mathrm{s}}{C}_{\mathrm{s}}{\omega }^2)}=\frac{A{e}^{\mathrm{j}\delta }}{B{e}^{\mathrm{j}\beta }}$$

(13)

wherein A and B are real values greater than 0, and:

$$\tan (\delta )=\frac{\omega R_{\mathrm{p}}(L_{\mathrm{p}}+L_{\mathrm{s}}-{\omega }^2{L}_{\mathrm{p}}{L}_{\mathrm{s}}{C}_{\mathrm{p}})}{-L_{\mathrm{p}}{L}_{\mathrm{s}}{\omega }^2}=R_{\mathrm{p}}{C}_{\mathrm{p}}\frac{{\omega }^2-{\omega }_{\mathrm{z}}^2}{\omega }$$

(14)

$$\tan (\beta )=\frac{\omega L_{\mathrm{p}}(1-L_{\mathrm{s}}{C}_{\mathrm{s}}{\omega }^2)}{R_{\mathrm{p}}[1-{\omega }^2(L_{\mathrm{p}}{C}_{\mathrm{p}}+L_{\mathrm{p}}{C}_{\mathrm{s}}+L_{\mathrm{s}}{C}_{\mathrm{s}})+{\omega }^4{L}_{\mathrm{p}}{L}_{\mathrm{s}}{C}_{\mathrm{p}}{C}_{\mathrm{s}}]}$$

(15)

$$\tan (\beta )=-\frac{\omega }{R_{\mathrm{p}}{C}_{\mathrm{p}}}\frac{{\omega }^2-{\omega }_2^2}{({\omega }^2-{\omega }_{\mathrm{p}1}^2)({\omega }^2-{\omega }_{\mathrm{p}2}^2)}$$

(16)

The phase shifts δ and β exhibit the same sign when ω is located between ω_p1 and ω_p2. The slopes of the two phase shifts versus frequency are determined by R_pC_p, and the two slopes show opposite dependences on R_pC_p. If R_pC_p is properly selected, the two phase shifts may cancel each other out across a wide bandwidth. Based on the tank shown in Figure 3a, Figure 4 presents the calculated δ and β from (14) and (15), where L_p, L_s, R_p, C_p, and C_s are 2 nH, 2 nH, 130 Ohms, 100 fF, and 50 fF, respectively. With the properly selected parameters, δ and β exhibit the same slopes from 11.8 GHz to 17.0 GHz, such that the resultant phase shift for Z_in(jω) within the frequency range is always 0.

The frequencies where the two phase shifts cancel each other out may be obtained by enabling δ = β:

$$R_{\mathrm{p}}{C}_{\mathrm{p}}\frac{1}{\omega }=-\frac{\omega }{R_{\mathrm{p}}{C}_{\mathrm{p}}}\frac{1}{({\omega }^2-{\omega }_{\mathrm{p}1}^2)({\omega }^2-{\omega }_{\mathrm{p}2}^2)}$$

(17)

Define Q_p to be the quality factor of L_p at ω₂, then:

$$\frac{1}{R_{\mathrm{p}}{C}_{\mathrm{p}}}=\frac{{\omega }_1^2{L}_{\mathrm{p}}}{R_{\mathrm{p}}}=\frac{b{\omega }_2}{Q_{\mathrm{p}}}$$

(18)

Thus, (17) may be simplified into:

$${\omega }^4-(2-b^2/Q_{\mathrm{p}}^2){\omega }_2^2{\omega }^2+b{\omega }_2^4=0$$

(19)

The real roots are present when the discriminant is larger than 0:

$$\Delta =\{{(2-b^2/Q_{\mathrm{p}}^2)}^2-4b\}{\omega }_2^4\ge 0$$

(20)

The real root ω₀ of (19) is:

$${\omega }_0^2=\frac{(2-b^2/Q_{\mathrm{p}}^2){\omega }_2^2\pm \sqrt{\Delta }}{2}\ge 0$$

(21)

Equations (20) and (21) hold only when:

$$Q_{\mathrm{p}}\ge \sqrt{\frac{b^2}{2(1-\sqrt{b})}}$$

(22)

The critical point of (22) may be expressed as:

$$Q_{\mathrm{p}}=\sqrt{\frac{b^2}{2(1-\sqrt{b})}}$$

(23)

For the tank impedance shown in (13), apart from ω₂, another frequency ω₀ where a zero phase shift is generated may be expressed as:

$${\omega }_0=\sqrt{\frac{2-b^2/Q_{\mathrm{p}}^2}{2}}{\omega }_2=\sqrt{\sqrt{b}}{\omega }_2$$

(24)

Referring to Figure 4, when (23) is satisfied, the phase shift from ω₀ to ω_z may be regarded as zero. Consequently, the zero-phase-shift bandwidth is evaluated by:

$$\frac{{\omega }_{\mathrm{bw}}}{{\omega }_{\mathrm{c}}}=\frac{{\omega }_2-{\omega }_0}{({\omega }_2+{\omega }_0)/2}=\frac{2(1-\sqrt{\sqrt{b}})}{1+\sqrt{\sqrt{b}}}$$

(25)

According to (6), we can obtain:

$$\frac{L_{\mathrm{s}}}{L_{\mathrm{p}}}=\frac{b}{a}=\frac{b}{1-b}$$

(26)

As is apparent from (25) and (26), a smaller L_s/L_p means a larger zero-phase-shift bandwidth.

Figure 5a presents the zero-phase-shift plateau under three different cases, where the three cases are of the same ω₀ but with different ratios of L_s/L_p. The parameters for the three cases are listed in

Table 1. Parameter list for the tank shown in Figure 3a.

	Lp/H	Ls/H	Cp/F	Cs/F	Rp/Ohms	Ls/Lp	a	b	ω1/Hz	ω2/Hz	ω0/Hz
Case 1	4n	2n	75f	50f	146	0.5	2/3	1/3	9.19 G	15.9 G	12.1 G
Case 2	2n	2n	100f	50f	130	1	1/2	1/2	11.3 G	15.9 G	13.4 G
Case 3	1n	2n	150f	50f	112	2	1/3	2/3	13.0 G	15.9 G	14.4 G

. Simulation results in Figure 5a show that (25) is a pretty conservative estimation of the zero-phase-shift bandwidth, and the simulated bandwidth is approximately 1.5 times of the calculated values.

2.3. Magnitude Response

To sustain the waveform in the oscillator, the gain condition should also be satisfied. The tank impedance should be high enough to be easily compensated for by the negative resistance of the cross-coupled pair in the oscillator. A higher tank impedance means a lower current consumption and a higher output voltage amplitude. When ω = ω_z,

$${\left|Z_{\mathrm{in}}\left(\mathrm{j}\omega \right)\right|}_{\omega ={\omega }_{\mathrm{z}}}=\frac{L_{\mathrm{p}}{L}_{\mathrm{s}}{\omega }_{\mathrm{z}}^2}{R_{\mathrm{p}}{L}_{\mathrm{p}}{L}_{\mathrm{s}}{C}_{\mathrm{p}}{C}_{\mathrm{s}}({\omega }_{\mathrm{z}}^2-{\omega }_{\mathrm{p}1}^2)({\omega }_{\mathrm{z}}^2-{\omega }_{\mathrm{p}2}^2)}=\frac{1}{a{R}_{\mathrm{p}}{C}_{\mathrm{p}}{C}_{\mathrm{s}}{\omega }_{\mathrm{z}}^2}=\frac{1}{R_{\mathrm{p}}{\omega }_{\mathrm{z}}^2{C}_{\mathrm{s}}^2}$$

(27)

From (27), the magnitude of the tank impedance is inversely proportional to C_s when ω_z is fixed. That is, relatively larger inductors and smaller capacitors are preferred to raise the tank impedance.

The zero-phase-shift plateau is present when the critical condition (23) is satisfied, and thus the magnitude at ω_z is:

$${\left|Z_{\mathrm{in}}(\mathrm{j}\omega )\right|}_{\omega ={\omega }_{\mathrm{z}}}=\frac{\sqrt{2(1-\sqrt{b})}}{(1-b)C_{\mathrm{s}}{\omega }_{\mathrm{z}}}=\frac{\sqrt{2}}{\sqrt{{(1+\sqrt{b})}^2(1-\sqrt{b})}{C}_{\mathrm{s}}{\omega }_{\mathrm{z}}}$$

(28)

If we define:

$$x=\sqrt{b}, \mathrm{and} f(x)={(1+x)}^2(1-x)$$

(29)

$$df(x)/dx=(1-3x)(1+x)\le 0, \mathrm{when} x\ge 1/3$$

(30)

Thus, the magnitude response of the tank at ω_z exhibits a positive dependence on b when b ≥ 1/9.

The magnitude of the tank impedance at ω₀ should also be considered:

$${\left|Z_{\mathrm{in}}(\mathrm{j}\omega )\right|}_{\omega ={\omega }_0}=\frac{A\cos (\delta )}{B\cos (\beta )}=\frac{{\omega }_0^2}{R_{\mathrm{p}}{C}_{\mathrm{p}}{C}_{\mathrm{s}}({\omega }_0^2-{\omega }_{\mathrm{p}1}^2)({\omega }_0^2-{\omega }_{\mathrm{p}2}^2)}$$

(31)

Substitute (10), (11), (23), and (24), into (31),

$${\left|Z_{\mathrm{in}}(\mathrm{j}\omega )\right|}_{\omega ={\omega }_0}=\frac{1}{\sqrt{2(1-\sqrt{b})}{C}_{\mathrm{s}}{\omega }_{\mathrm{z}}}<{\left|Z_{\mathrm{in}}(\mathrm{j}\omega )\right|}_{\omega ={\omega }_{\mathrm{z}}}$$

(32)

It is apparent from (32) that a larger b will improve the magnitude response at ω₀, similar to the case at ω_z. The magnitude of the tank at ω_z and ω₀ are plotted in Figure 6a, based on (28) and (32). With the increase of b, the magnitude rises slowly until b reaches 0.7. For a typical b within [1/3, 2/3], the variation of the magnitude is less than two times.

The phase response and magnitude response for the case shown in Figure 4 are plotted in Figure 6b The peaks in the magnitude response are located at ω_p1 and ω_p2, deviating from the zero-phase-shift plateau in the phase response. Thus, a waste of power consumption occurs, as compared to an LC-tank.

2.4. Influence of R_p and R_s

Referring to the tank shown in Figure 3a, the phase and magnitude responses versus R_p are plotted in Figure 7, where L_p, L_s, C_p, and C_s are 2 nH, 2 nH, 100 fF, and 50 fF, respectively. A zero-phase-shift plateau is achieved when R_p is 130 Ohms. A smaller R_p enables a sharper decline in the phase response, and a larger R_p may expand the locking range while increasing the in-band phase ripple. For the magnitude response, a larger R_p exhibits a lower in-band tank impedance. From this properly selected R_p of 130 Ohms, the quality factor Q_p of the inductor L_p can be calculated. Assume the operating frequency to be 16 GHz, then:

$$Q_{\mathrm{p}}=\frac{R_{\mathrm{p}}}{2\pi f{L}_{\mathrm{p}}}=\frac{130}{2\pi \times 16\times 2}=0.65$$

(33)

This is quite a low-quality factor, such that a generally explicit resistor is needed instead of only exploiting the loss of the inductor L_p and capacitor C_p.

The loss of inductor L_s is modeled by a series resistor R_s. The phase and magnitude of Z_in are shown in Figure 8. The loss of L_s generates a negative phase shift, but imposes little impact on the magnitude response of the zero-phase-shift plateau region. The negative phase shift generated by R_s can be easily compensated by adjusting C_p.

According to the analyses above, the design flowchart of the proposed ILFD can be outlined as follows:

Step 1. Choose a zero frequency ω_z according to the frequency requirement;

Step 2. Determine L_p/L_s ratio according to the bandwidth requirement, and thus C_p/C_s is also determined;

Step 3. Determining the minimum value of the capacitor C_s according to the parasitic capacitance of the active devices (e.g., the cross-coupled NMOS pair and the injection transistor), and thus the values of C_p, L_p, and L_s are determined;

Step 4. Choose a proper value of the resistor R_p to obtain a wide-band zero-phase-shift plateau.

If a phase ripple may be tolerated, the resistor R_p may be chosen to be larger than the one in step 4.

3. Implementations and Measurement Results

Figure 9 shows the circuit implementation of the proposed divide-by-2 ILFD. The core circuit of the proposed ILFD is composed of three NMOS transistors M₁, M₂, and M_inj, and a dual-resonance tank. The cross-coupled transistors M₁ and M₂ form a negative resistance to compensate for the loss of the tank. The input signal is directly injected into the tank through the gate of the transistor M_inj.

A differential inductor L₁ is used as L_p, and a differential inductor with its center-tap split into two terminals is used as L_s. Though the inductors L_p and L_s are implemented as separate inductors for simplicity, the two inductors may be EM coupled to each other and form a transformer to save the area. A paralleled resistor R is used to increase the loss of the inductor L₁. Except for the core circuit, the divide-by-2 ILFD also includes an output buffer to drive the 50 Ohms load of the spectrum analyzer. The output buffer is realized by a pseudo-differential common-source amplifier using inductors as loads.

The divide-by-2 ILFD is fabricated in the 65 nm 1P7M LP-CMOS process. The core area of the die photo shown in Figure 10a is 0.5 mm × 0.22 mm. The measurement setup is shown in Figure 10b. The input signal is generated from a commercial Analog Signal Generator, and a GSG probe is used for signal injection. The output is captured with a GSG probe and characterized using a Spectrum Analyzer. The insertion losses of the coaxial cables and adapters have been calibrated.

The core circuit consumes a DC power of 7 mW under a supply voltage of 0.7 V. The free-running frequency of the ILFD is first measured. When no signal is injected, the ILFD oscillates at 12.56 GHz. Figure 11a shows the simulated and measured input sensitivity curves of the proposed divide-by-2 ILFD. When the injection power is 0 dBm, the ILFD obtains a locking range of 87.0%, from 13 GHz to 33 GHz. When the injection power is lowered to −6 dBm, the proposed ILFD still sustains a locking range of 69.6%, from 15 GHz to 31 GHz. When the injection power is reduced to −24 dBm, the locking range decreases to only 3.9%, from 25 GHz to 26 GHz.

The phase noise performances of the input signal and output signal are also measured through the spectrum analyzer, and the results are shown in Figure 11b. The phase noise of the output signal is approximately 6 dB lower than that of the input signal, which coincides well with the theoretical value.

Table 2. Performance summary and comparison.

Reference	[6]	[7]	[8]	[9]	This Work
Process	65 nm CMOS	65 nm CMOS	40 nm CMOS	65 nm CMOS	65 nm LP-CMOS
Injection Power (dBm)	0	0	−4	−5	0
Supply Voltage (V)	1	0.8	0.9	1.2	0.7
Power Dissipation (mW)	5.8	2.9	5.8	3.7	7
Locking Range	27.9–3.5 GHz (62.9%)	53.4–79.4 GHz (39.2%)	28.8–91.9 GHz (104.5%)	58 GHz (18.3%)	13–33 GHz (87.0%)
FoM * (GHz/mW)	4.41	8.97	10.9	2.87	2.86
Core Area (mm2)	0.18	0.126	0.02	0.029	0.11

* FoM = locking range (GHz)/power dissipation (mW).

summaries the performance of the fabricated divde-by-2 ILFD, and compares it with some other works.

4. EM Coupling Discussion

The inductors L_p and L_s may be EM-coupled to save the area. Referring to Figure 12a, the coupling factor of the two inductors is k, and the AC current flowing through L_p and L_s are i_p and i_s, respectively. The equivalent model of Figure 12a is shown in Figure 12b, where:

$$L_{\mathrm{pe}}=L_{\mathrm{p}}+\frac{i_{\mathrm{s}}}{i_{\mathrm{p}}}M, L_{\mathrm{se}}=L_{\mathrm{s}}+\frac{i_{\mathrm{p}}}{i_{\mathrm{s}}}M$$

(34)

$$i_{\mathrm{s}}=i_{\mathrm{p}}+i_{\mathrm{p}}s{L}_{\mathrm{p}}/(\frac{1}{s{C}_{\mathrm{p}}})$$

(35)

$$L_{\mathrm{pe}}=L_{\mathrm{p}}+(1-{\omega }^2{L}_{\mathrm{p}}{C}_{\mathrm{p}})M, L_{\mathrm{se}}=L_{\mathrm{s}}+\frac{M}{1-{\omega }^2{L}_{\mathrm{p}}{C}_{\mathrm{p}}}$$

(36)

Thus, the EM coupling can be decoupled, and the equivalent inductors L_pe and L_se can be derived following (36). Figure 12c shows the magnitude and phase response of the tank based on Figure 12a, where L_p, L_s, C_p, C_s, R_p, and k are 1 nH, 1 nH, 105 fF, 40 fF, 70 Ohms, and 0.3. It can be concluded that the EM coupling is possible to gain the same wide zero-phase-shift plateau while occupying a smaller area.

5. Conclusions

A wide-band divide-by-2 ILFD based on a distributed dual-resonance tank is reported. Based on the dual-resonance high-order tank, when the injection power is 0 dBm, the proposed ILFD obtains an ultra-wide locking range of 87.0%, from 13 GHz to 33 GHz, while consuming 7 mW from a 0.7 V power supply. When the injection power is lowered to −6 dBm, the proposed ILFD still sustains a locking range of 69.6%, from 15 GHz to 31 GHz.