**1. Introduction**

Recently, the demand for radar sensors has been rapidly increasing with the development of the Internet of Things (IoT) industry and the autonomous vehicle industry. The complementary metal-oxide semiconductor (CMOS) radar is characterized by various operating methods such as doppler, frequency-modulated continuous wave (FMCW), and (continuous-wave) CW. In the doppler radar, a low frequency to millimeter-wave (mm-Wave) must be used to acquire a two-dimensional image through synthetic aperture radar (SAR). Bandwidths of 500 MHz or more are used to obtain high-resolution images [1,2]. In addition, wideband performance is very important in frequency-modulated continuous wave (FMCW) radars because wideband chirp is directly related to the high-resolution distance information [3]. Therefore, the wideband performance of the signal generator, the core of the sensor, is required [4,5].

Generally, the performance of the phase-locked loop (PLL) in the signal generator must be concerned to obtain low noise mm-Wave signals. Figure 1 shows the block diagram of conventional PLL structure that consists of a phase-frequency detector (PFD), charge pump (CP), low-pass filter (LPF), voltage-controlled oscillator (VCO) and frequency divider. The key blocks that determine the specification of the PLL in the mm-Wave band are VCO [6,7] and frequency divider [8]. The mm-Wave frequency divider should operate at high speed and should have a wide operating range for applying the wideband sensor applications.

Frequency dividers are designed as the current mode logic (CML) divider, regenerative divider, and LC oscillator-based injection-locked frequency divider (ILFD). The CML divider is a combination of two flip-flops that perform simple logical operations [9–12]. Generally, CML dividers have a wide operating range and occupy a small chip area with no inductor design. However, CML dividers suffer from large power consumption, limited maximum operation frequency, and process, voltage, and temperature (PVT) variation at the mm-Wave. To address these shortcomings, tunable self-resonant circuit [9], dynamic latches with load modulation [10,11], and additional calibration circuits [12] have been studied. However, these still consume large powers of 4.8 [11], and 6.2 mW [12], respectively. The regenerative divider and ILFD are also popular frequency dividers. These two types of frequency dividers are LC oscillator-based circuits and both of them are quite similar. The regenerative divider comprises an LC-based band pass filter (BPF) and active-type mixer [13–15]. The active type of mixer consumes power and takes over the role of -g m core. Conversely, the ILFD comprises an LC-based BPF, -g m core, and passivetype mixer that does not consume power. Therefore, regenerative dividers consume more power than ILFDs and are not generally used for mm-Wave applications because of the influence of many parasitic capacitors of the active-type mixer such as the Gilbert cell. The even-harmonic mixer [14] and digital-assisted circuit [15] are employed to widen the locking ranges of the regenerative divider. Their locking ranges are 33% and 57.4%, respectively. However, the highest input frequencies are limited to 18.4 and 14.8 GHz, consuming 10.8 and 12 mW power, respectively.

**Figure 1.** Conventional phase-locked loop with mm-Wave frequency divider.

The most attractive mm-Wave frequency divider is the LC oscillator-based ILFD.The reasons for its high popularity are as follows. First, the ILFD self-oscillates when there is no input signal applied. It is possible to obtain a large output signal with a small input signal using the oscillator-based operation. Second, because of the LC resonator, the ILFD is advantageous for operation at the mm-Wave band. Finally, because the ILFD uses a passive type of mixer, it consumes less power than regenerative and CML dividers.However, the disadvantage is that the locking range is narrow because of a high-quality factor (Q) LC resonator. Several studies are being conducted to widen the locking range of ILFD [16–18]. The forward-body-bias techniques [16,17] are some of the effective ways of increasing the gain of the mixer and extending the locking range. Although the ILFD with the forward-body-bias techniques have a wide locking range of 90% in [17], there are several reasons why this technique is impractical in mm-Wave synthesizers. First, if a positive bias is applied to the body of an n-channel metal-oxide-semiconductor fieldeffect transistor (MOSFET), the leakage current cannot be ignored, and the possibility of a large diffusion current flow because of forward-bias increases. Second, the power of the harmonic signal increases because of non-linearity in devices. Applying an injection signal with an edge frequency in the locked range can make it difficult to distinguish the power difference between the output and harmonic signals. Finally, an additional circuit may be required to control the harmonic power, which can increase the circuit complexity and power consumption. The dual-resonance resonator is also considered as a suitable technique [18]. This ILFD has a locking range of 71.46%; however, it requires external bias control and has a small output power. Moreover, when a −3 dBm injection power is applied, an unlocking part occurs in the locking range.

In this paper, a low power and wide locking range LC oscillator-based current-reuse (CR) ILFD using a fourth-order resonator with the distributed inductor is proposed. The CR technique is employed to reduce power consumption. This paper is organized as follows. Section 2 presents an analysis of the ILFD locking range. The limitations of the maximum locking range and harmonic issues are also presented. Section 3 presents the circuit design of the proposed CR-ILFD including the modeling of the transformer and

design flow chart. The measurement results are shown in Section 4. Finally, conclusions are organized in Section 5.

#### **2. Locking Range Analysis of ILFD**

Figure 2a shows a schematic of the conventional cross-coupled pair ILFD with a second-order resonator. This ILFD consists of an N-channel metal-oxide semiconductor (NMOS) cross-coupled pair (M1, M2), injection switch (M3) and LC resonator. The ILFD self-oscillates if there is no injection signal at the gate of M3. Biasing the injection signal of Vinj,2w at the gate of M3, the ILFD outputs V+out,w and V<sup>−</sup>out,w. When the frequency of the output signal is exactly half the frequency of the injection signal, it is referred to as "locking". To easily understand the locking operation, the current is classified into three types, namely, *Iso*, *Iinj*, and *Iout*. *Iso* represents the self-oscillation current flowing through the core when the ILFD self-oscillates without an injection signal. *Iinj* is the injection current flowing through M3 when an injection signal is applied. *Iout* is the output current, which is the sum of *Iso* and *Iinj*. Figure 2b shows the phasor diagram for the three current types. The phasor rotates clockwise. Point "a" shows that the phase has changed from *Iso* by *φ*. Point "b" shows the phase when the ILFD self-oscillates without an injection signal. The relational expression of the current vectors is as follows.

$$I\_{out} = I\_{so} + I\_{inj}.\tag{1}$$

**Figure 2.** (**a**) Schematic of the conventional cross-coupled pair ILFD with second-order resonator and (**b**) phasor diagram for the basic principle of the conventional ILFD.

Two waves are shown in Figure 2b, one is the self-oscillation signal of the ILFD and the other is the injection-locked signal. Point "b" of the self-oscillation signal is moved to point "a" by the injection signal. Therefore, the phase at 180◦ of the injection-locked signal is point "a" of the self-oscillation signal. Injection is instantaneously performed every half period, and the range of *φ* can be derived using the following equations.

$$
\phi = \angle \text{I}\_{\text{out}} = \angle \left( I\_{\text{so}} + I\_{\text{inj}} \right), \tag{2}
$$

$$V\_{\rm out} = Z\_L \cdot I\_{\rm out} \tag{3}$$

$$
\angle I\_{\text{out}} = \angle V\_{\text{out}} - \angle Z\_{\text{L}} \tag{4}
$$

where *Vout* is the output voltage signal when the ILFD is locked, and ZL represents the load impedance of the LC resonator. Equation (4) can be derived using the phasor in (3).

To replace *Vout* with the self-oscillation and injection signals, the following equations are derived as

$$V\_{\rm out} = V\_{\rm so} + V\_{\rm inj}.\tag{5}$$

where *Vso* is the output voltage signal when the ILFD self-oscillates and *Vinj* is the injection voltage signal generated from M3. It should be noted that *Vinj* is different from the input voltage signal, *Vinj,*<sup>2</sup>*w*. According to Equations (4) and (5), the *φ* is calculated as

$$
\phi = \angle \left( V\_{\text{so}} \pm V\_{\text{inj}} \right) - \angle Z\_{\text{L}}.\tag{6}
$$

The sign of *Vinj* is determined based on the value of the locked frequency relative to the self-oscillation frequency. When the ILFD self-oscillates with no injection signal, (6) is calculated as follows.

$$
\phi|\_{V\_{\text{inj}}=0} = \angle V\_{\text{so}} - \angle Z\_{\text{L}}.\tag{7}
$$

*Vinj* is zero, and *Vso* is expressed as the product of Iso and ZL. Because ZL is canceled out, the following equation is satisfied:

$$
\phi|\_{V\_{inj}=0} = \angle I\_{\text{sq}}.\tag{8}
$$

Meanwhile, *φmax* is derived when the following condition is satisfied:

$$I\_{\rm out} \perp I\_{\rm inv}.\tag{9}$$

The largest angle between *Iso* and *Iout* can be realized by considering the phasor as shown in Figure 2b. This is the condition of (9) where *Iout* and *Iinj* are vertical. Using the trigonometric function,

$$
\sin \phi\_{\text{max}} = \pm \frac{|I\_{\text{inj}}|}{|I\_{\text{so}}|},
\tag{10}
$$

$$\phi\_{\text{max}} = \pm \arcsin\left(\frac{|\mathcal{g}\_{inj} \cdot V\_{inj}|}{|\mathcal{g}\_m \cdot V\_{so}|}\right),\tag{11}$$

where *gm* and *ginj* represent the transconductance of the cross-coupled pair and injection switch, respectively.

According to (6), the conditions for extending the locking range of the ILFD can be determined qualitatively. First, the magnitude of the self-oscillation signal *Vso* is decreased by reducing the sizes of M1 and M2 to decrease the transconductance of the cross-coupled pair. However, when the transconductance of the cross-coupled pair is too small, it can make failure in the self-oscillation, causing the ILFD to act as a harmonic buffer. Second, to increase the amplitude of *Vinj* generated by M3, the size of M3 can be increased or the injection signal *Vinj,*<sup>2</sup>*w* can be amplified. However, the operation frequency may be limited by large parasitic capacitors. A pre-buffer, which consumes additional power, will be required to increase the amplitude of *Vinj,*<sup>2</sup>*w*. Finally, the phase of the load impedance can be changed. The phase of ZL can increase or decrease *φ*. However, the maximum and minimum values of the phase, ±*φ*max, limit the range of *φ*. Therefore, the phase of ZL should be close to zero value in the wide frequency range. In conclusion, the maximum and minimum values of *φ* are determined by (11), and the method of extending the range of *φ* is consistent with the equation in (6).

The power of the output signal should be greater than that of the input signal. Two graphs of the load impedance magnitude against the angular frequency are shown in Figure 3, which presents two cases. The first case is the normal case where the power of the input signal is significantly smaller than that of the output signal as shown in Figure 3a. The range from *w*1 to *w*2 is the operation frequency band obtained by dividing by two, and the range from 2*w*1 to 2*w*2 is the injection frequency band. The operation and injection frequency bands do not overlap in the normal case because 2*w*1 is larger than w2. Therefore, the input signal does not exceed the start-up condition and is not amplified more than

the output signal. The second case is the abnormal case where the power of the input signal can be larger than that of the output signal, as in Figure 3b. Here, the operation and injection frequency bands overlap because 2*w*1 is smaller than *w*2. The injection frequency band contains the parts that exceed the start-up conditions, which are determined by the following "Barkhausen formula".

$$
\lg\_m \cdot |Z\_L| \ge 1.\tag{12}
$$

**Figure 3.** Graphs of magnitude of load impedance against angular frequency; (**a**) normal case, (**b**) abnormal case.

In the abnormal case, the ILFD cannot be used in mm-Wave applications, because the input and output signals are amplified together in the frequency band used.

This problem can be solved by increasing the division ratio of the ILFD. However, to operate at high division ratio, a harmonic signal with a small magnitude should be used, which results in a narrow locking range of the ILFD [19,20]. Additionally, the injection mixer for the high division ratio creates larger parasitic capacitance than the injection switch of the divide-by-two ILFD. Consequently, an ILFD that operates at a high division ratio greater than two is disadvantageous for application in the mm-Wave band. Therefore, a divide-by-two ILFD optimized to have a wide locking range without including the abnormal case would be most suited as a mm-Wave frequency divider. The following equation is used to calculate the locking range of the ILFD.

$$LR = \frac{w\_2 - w\_1}{w\_1 + (w\_2 - w\_1)/2} \cdot 100 \text{ (\%)}.\tag{13}$$

Under the normal case condition, *w*2 < 2*w*1, the maximum locking range of the divideby-two ILFD can be obtained when w2 is equal to 2*w*1. Therefore, the maximum locking range is

$$LR\_{\text{max}}|\_{w\_2=2w\_1} = 66.7\%,\tag{14}$$

where *LR* is the locking range. If the locking range of the divide-by-two ILFD exceeds 66.7%, the power of the input signal may be greater than that of the output signal. In conclusion, the locking range of the ILFD should be designed to be less than 66.7%.

#### **3. Circuit Design of Proposed CR-ILFD**

#### *3.1. Fourth-Order Resonator and CR Core*

As mentioned in the previous section, to extend the locking range of the ILFD, the phase plot of the load impedance should be flat in the range of ±*φ*max [8,21]. A fourth-order resonator with two poles is required to flatten the phase plot. Figure 4a shows a schematic of the conventional cross-coupled pair-based ILFD with a fourth-order resonator consisting of a resonator (*L*1, *C*1, *R*1, *L*2, *C*2, *R*2), cross-coupled pair (M1, M2) and injection switch (M3). The "k" is the coupling factor between *L*1 and *L*2. *ZL* is the load impedance of the resonator, which is calculated as

$$Z\_L = \frac{(1 - k^2)L\_1 L\_2 \mathbf{C}\_2 \mathbf{s}^3 + L\_1 \mathbf{s}}{(1 - k^2)L\_1 L\_2 \mathbf{C}\_1 \mathbf{C}\_2 \mathbf{s}^4 + (L\_1 \mathbf{C}\_1 + L\_2 \mathbf{C}\_2)\mathbf{s}^2 + 1}. \tag{15}$$

**Figure 4.** Schematic of (**a**) conventional cross-coupled pair ILFD with fourth-order resonator and (**b**) CR core-based ILFD.

*R*1 and *R*2 are resistors that affect the quality (Q) factor of the resonator and have been approximated in this calculation. Two poles that make the denominator zero are represented using the following equation [22],

$$w\_{R,L} = \sqrt{\frac{L\_1\mathbf{C}\_1 + L\_2\mathbf{C}\_2 \pm \sqrt{\left(L\_1\mathbf{C}\_1 + L\_2\mathbf{C}\_2\right)^2 - 4(1-k^2)L\_1L\_2\mathbf{C}\_1\mathbf{C}\_2}{2(1-k^2)L\_1L\_2\mathbf{C}\_1\mathbf{C}\_2}}}.\tag{16}$$

Assuming that *L*1 = *L*2 and *C*1 = *C*2,

$$w\_{L,R} = \frac{1}{\sqrt{(1 \pm k)L\mathcal{C}}}.\tag{17}$$

According to (17), the distance between the two poles increases as the value of k increases and the distance between the two poles decreases as the k value decreases. If k is zero, the pole value is obviously equal to that of the second-order resonator (18). Figure 4b shows a schematic of the conventional ILFD with the CR core. For the CR core, M2 of the cross-coupled pair ILFD in Figure 2a is replaced by P-channel metal-oxide semiconductor (PMOS) [23–27]. The oscillation of the CR core can be divided into two half periods. In the first half period, the current flows through M1 and M2, and in the second half period, no current flows through M1 and M2. Unlike the oscillation in the cross-coupled pair core, the oscillation of the CR core reduces the current by simultaneously turning the MOSFET on and off [24].

Figure 5 shows the magnitude and phase plots of the second- and fourth-order resonator-based ILFDs. The schematic of the second-order resonator-based ILFD is shown in Figure 2a. Figure 5a shows the graph of the load impedance magnitude against the input frequency. The second-order resonator-based ILFD has one pole, w0, that is expressed as follows.

16 20 24 28 32 36 40 0 100 200 300 400 500 600 700 800 Magnitude (Ohms ) Input Frequency (GHz) 2nd order 4th order 6WDUWXS FRQGLWLRQ =/1/gm) *Z /& Z5 N /& Z/ N /&* :LGHQWKHPDJQLWXGH /RFNLQJ 5DQJH I*PD[* I*PD[* 16 20 24 28 32 36 40 -60 -40 -20 0 20 40 60 Phase (degree ) Input Frequency (GHz) 2nd order 4th order (**a**) (**b**) 

$$w\_0 = \frac{1}{\sqrt{LC}}.\tag{18}$$

**Figure 5.** (**a**) Simulated magnitude plot and (**b**) phase plot of second-order resonator-based ILFD and fourth-order resonatorbased ILFD.

If the fourth-order rather than the second-order resonator-based ILFD is applied, the magnitude plot of the load impedance becomes wider even if the maximum magnitude value decreases. However, because a new minimum value occurs between the two poles, it is necessary to simulate whether locking is sufficiently achieved at this value. If the minimum value between the two poles is less than the start-up condition (12), the ILFD does not operate in that frequency range. Figure 5b shows the phase plot against the input frequency. According to (11), the ±*φ*max limits the locking range of the ILFD. Unlike the phase of the second-order resonator-based ILFD, that of the fourth-order resonator-based ILFD has a value approximately equal to zero over a wide frequency range because of the formation of a ripple. Consequently, the simulated locking range of the ILFD is increased by 22% from 26–32 GHz (21%) to 22–36 GHz (43%).
