Non-Hermitian Physics-Inspired Voltage-Controlled Oscillators with Resistive Tuning

In quantum mechanics, an open system can be generally modeled as a gain (or loss) unit with a resonant frequency as shown in Fig. 1(a). Such systems are described by non-Hermitian Hamiltonians which preserve complex eigenvalues, i.e., $\omega_{1}-ig$ or $\omega_{2}+il$. However, non-Hermitian quantum systems built upon two coupled units, one with gain and the other one with loss as shown in Fig. 1(b), possess purely real eigenfrequencies in certain regimes as derived below. Note that such systems are also specifically termed parity-time-symmetric (PT-symmetric) systems [10]. The system dynamics in Fig. 1(b) is expressed as

where the subscript G (or L) refers to the gain (or loss) unit and $a_{\text{G,L}}$ is the field amplitude defined such that $|a_{\text{G,L}}|^{2}$ represents the energy stored in each unit. $g$ (or $l$) presents the gain (or loss) of the unit. $\kappa$ indicates the coupling strength between the two units and $\omega_{1,2}$ represents the resonant frequency of each unit. To find the eigenfrequencies, we let $a_{\text{G,L}}\propto{\exp^{i\omega t}}$ and obtain the characteristic equation as

For a balanced system where the gain is equal to the loss, i.e., $g=l$, the solutions are then given by the following expression:

Eq. (3) shows that when the coupling strength $\kappa$ is stronger than a threshold determined by the gain-loss contrast $g$, i.e., $\kappa>g=l$, the system has a pair of real eigenfrequencies. Particularly, these eigenfrequencies could evolve with gain-loss contrast in a wide range as long as $g=l\in(0,\kappa)$. This simple analysis suggests that the interplay between gain/loss and their coupling provides a new degree of tuning freedom, i.e., gain/loss tuning freedom, to modulate the behaviors of a system. In the next section, we discuss how this physical principle (i.e., PT-symmetric topology) can be applied to design VCOs.

Before introducing the proposed VCO topology, Fig. 2 first re-examines conventional single-core and multi-core VCOs. A single-core VCO can be simplified into an active LC resonator shown in Fig. 2(a). It consists of a gain ($-R$) and an LC core with an intrinsic loss ($R_{0}$). A multi-core VCO (e.g., dual-core) is built upon two coupled single-core VCOs with a fixed coupling strength $\kappa$, each of which is simplified into an active LC resonator. For both VCOs, at the start-up phase, the gain is set to be slightly higher than the inherent loss to produce an oscillation with exponentially-growing amplitude. As the amplitude grows, the gain saturates and equates the loss in the large-signal domain due to nonlinearity. The oscillation then becomes stable, and the frequency (i.e., $\omega_{\text{S}}/\omega_{\text{M}}$ in Fig. 2(a)/(b)) can only be adjusted via $\omega_{0}$ by tuning the core’s capacitance or inductance. In particular, the multi-core VCO has two frequency modes as shown by $\omega_{\text{M}}$ in Fig. 2(b).

This re-examination shows that in the conventional VCOs, the gain-loss distribution plays only a trivial role in the transient behavior of oscillators, i.e., the gain is used to compensate for the undesired loss to establish the start-up condition for the exponential growth of oscillation amplitude. Fortunately, based on the non-Hermitian quantum mechanics introduced before, the loss is useful if the gain-loss distribution in a system is properly manipulated. Inspired by this physical principle, our method explores the interplay between the gain/loss distribution and their coupling to enhance the frequency tuning bandwidth of VCOs. Fig. 2(c) exhibits the simplified topology of the proposed VCO. It is built upon two coupled LC cores, one active with a negative resistance $-g$ and the other one dissipative with an equal amount of loss $l$, and the two cores have the same capacitance and inductance. The proposed VCO exhibits two frequency modes as shown by $\omega_{\text{P}}$ in Fig. 2(c) and an extra resistive tuning freedom (i.e., $g$) that is orthogonal with the typical capacitive/inductive tuning freedoms of $\omega_{0}$. Fig. 2(d) numerically compares the frequency tuning of these three types of VCOs. Both the single-core VCO and multi-core VCO have only individual frequency points (i.e., blue circle and red asterisks), which are independent of $g$. However, the proposed VCO has a very wide FTR enabled by $g$. The comparison shows that the proposed VCO with the resistive tuning freedom can achieve a wider FTR than conventional VCOs given the same capacitive/inductive tuning ability preserved by $\omega_{0}$.

Fig. 3 shows the proposed VCO circuit. The gain side has a tunable gain rate generated by cross-coupled differential pairs (XDPs) and an inherent loss rate $R_{\text{G0}}$, leading to the total gain of $-R_{\text{G}}=(-1/G_{m})\verb||||R_{\text{G0}}$; $G_{m}={(g_{\text{mn}}+g_{\text{mp}})/{2}}$, where $g_{\text{mn}}$ and $g_{\text{mp}}$ are the small signal transconductance of NMOS and PMOS XDP. The loss side has an intrinsic loss rate $R_{\text{L0}}$ and a variable loss rate $R_{\text{L1}}$, giving the total loss of $R_{\text{L}}=R_{\text{L0}}\verb||||R_{\text{L1}}$. To make loss adjustable, a variable resistor based on stacked transistors is parallelly connected to the loss side. The subset in Fig. 3 shows the schematic of the variable resistor $R_{\text{L1}}$. All the gates of MOSFETs are connected together. By tuning the gate voltage $V_{\text{BIASL}}$, the resistance can be continuously adjusted in a wide range. The capacitor $C_{\text{G}}$ ($C_{\text{L}}$) in each LC core is composed of a parasitic capacitance $C_{\text{G0}}$ ($C_{\text{L0}}$), a fixed Metal-Insulator-Metal (MIM) capacitor $C_{\text{G1}}$ ($C_{\text{L1}}$) with high-quality factor (high-Q) and an adjustable varactor $C_{\text{G2}}$ ($C_{\text{L2}}$). The varactor takes up a small proportion of the total capacitance and is mainly used to compensate for the fabrication mismatch of the fixed MIM capacitors on each side. The coupling capacitance ($C_{\text{C}}$) is realized by two equal MIM capacitors ($C_{\text{C1}}$ and $C_{\text{C2}}$) which are serially connected through an on-chip switch (SW). The inductance ($L_{\text{G}}/L_{\text{L}}$) in both cores comes from the high-Q symmetrical parallel inductor (symindp) of the technology. A center tap connection is provided such that both cores share the same common mode voltage by connecting the center taps of the inductors. The balanced condition is satisfied by setting $R_{\text{G}}\approx R_{\text{L}}=R$, $L_{\text{G}}\approx L_{\text{L}}=L$, and $C_{\text{G}}\approx C_{\text{L}}=C$.

We perform a qualitative analysis on the phase noise (PN) of the proposed VCO. It is well-established in the classic VCO theory that a multi-core VCO built upon a coupled structure can lead to PN reduction as compared to a single-core VCO. By taking a two-core VCO as an example, the improved PN can be intuitively understood as that the equivalent current noise of each LC core experiences twice the capacitance, and therefore its PN contribution is reduced by $6$ dB. Two noise contributions from the two cores are uncorrelated and can be summed up, ideally leading to a $3$ dB reduction of the total PN. Generally, for an $N$-core VCO built upon a coupled structure, its PN is lower than a single-core VCO by $10\log_{10}N$ dB. Thanks to the coupled structure, the proposed VCO topology also inherits the PN advantage of conventional multi-core VCOs. On the other hand, the gain-loss tuning physically realized by active devices, although it does not generate more inherent losses, does contribute additional noise to the system. However, this is not a big issue as the unique gain-loss tuning also increases the carrier amplitude which suppresses the effect of noise. It can be shown as follows. For the proposed VCO, the gain not only compensates the inherent loss in the active core but also offsets the tunable loss in the coupled lossy core. Assuming the ratio between the tunable gain $-G_{m}$ generated by XDPs and the inherent loss $R_{0}$ is $\beta$ ($\beta>1$), the PN of the active core in our proposed VCO can be obtained based on the well-known PN model of single-core VCOs as below:

where $k$ is the Boltzmann constant; $T$ is the absolute temperature; $R_{0}$ is the inherent resonator loss; $m$ ($m>1$) is a constant noise factor of active elements; $V_{\text{osc}}$ is the amplitude of the carrier; $Q_{S}$ is the quality factor of the LC core; $\mathcal{L}_{\text{S}}(\triangle\omega)$ is PN of a conventional single-core VCO. Compared to the PN of conventional single-core VCOs, both the noise factor $m$ of active devices and the amplitude $V_{\text{osc}}$ of carrier increase in the PN formula of the active core of the proposed VCO. But the carrier amplitude increases to $\beta^{2}\times$ of the conventional one because the current flowing into the core is quadratically proportional to the gain when XDPs operate in the saturation region. Therefore, the proposed VCO topology shows better PN performance than conventional single-core VCOs.