Pseudo-LTI Magnetless Circulators

• Pseudo-LTI magnetless circulators are passive three-port RF devices that achieve nonreciprocal signal routing using spatiotemporal modulation instead of magnetic bias.
• They employ synthetic angular momentum bias and noncommuting frequency conversion to deliver externally time-invariant S-parameters free from in-band intermodulation.
• Implementations across CMOS, MEMS, and SAW platforms demonstrate low insertion loss, high isolation, and compelling performance for full-duplex wireless and quantum applications.

Pseudo-Linear Time-Invariant Magnetless Circulators are passive, three-port microwave or RF components that achieve nonreciprocal signal routing—typified by directional TX→ANT→RX flow—without requiring magnetic materials, ferrites, or external DC magnetic bias. Instead, nonreciprocity is engineered via temporally and spatially modulated reactive networks (capacitors, inductors, or delay elements), imparting a synthetic angular-momentum bias or exploiting noncommuting frequency conversion operations. Critically, these networks exhibit "pseudo-linear time-invariant" (pseudo-LTI) behavior: while the network is fundamentally periodic in time (LPTV), the external S-parameters for the fundamental frequency response are invariant in time and free of in-band intermodulation products, emulating true LTI operation over a finite bandwidth. Pseudo-LTI magnetless circulators have now been demonstrated from MHz to mm-wave frequencies in IC, MEMS, SAW, and hybrid platforms, with performance approaching—and in some metrics surpassing—classic magnetic devices.

1. Fundamental Principles of Nonreciprocity and Pseudo-LTI Behavior

The essential mechanism in pseudo-LTI magnetless circulators is the deliberate breaking of Lorentz reciprocity with time- and/or space-dependent modulation. In conventional ferrite circulators, a static magnetic field induces nonreciprocal Faraday rotation. In pseudo-LTI devices, the role of the magnetic bias is replaced by:

• Spatiotemporal modulation of network parameters: e.g., varactor-modulated resonant frequencies or switched transmission-line impedances, phase-shifted by 120° between ports (Kord et al., 2017, Kord et al., 2017, Kord et al., 2020).
• Noncommuting frequency conversion and delay operations: Circulators based on sequential mixing (frequency shift) and time-delay exploit the noncommutativity of these operations to accumulate nonreciprocal phase (Rosenthal et al., 2017).
• Synthetic angular-momentum biasing: Spatial phase progression in the modulation waveform breaks the degeneracy of the clockwise and counterclockwise eigenmodes of a cyclically symmetric network (Kord et al., 2020).

A defining feature of "pseudo-LTI" is that, under cyclic symmetry, careful phasing, and appropriate spectral separation of spurious harmonic products, the external S-parameters at the RF ports are independent of time and free of intermodulation. This emulates LTI circulator behavior in-band, while avoiding the drawbacks of both ferrite-based and active transistor-based nonreciprocal devices (Kord et al., 2017, Kord et al., 2020, Reiskarimian et al., 2018).

2. Core Architectures and Implementation Modalities

Pseudo-LTI magnetless circulators can be classified by their underlying nonreciprocal elements and circuit topologies:

• STM-Resonant Junctions: Three resonant LC tanks (bandstop/Δ or bandpass/wye) connected in a ring with each branch's resonance modulated as $C_n(t) = C_0 + \Delta C \cos(\omega_m t + \phi_n)$, $\phi_n = (n-1)2\pi/3$ (Kord et al., 2017, Kord et al., 2017). Both single-ended and differential architectures are realized; differential arrangements with 180° anti-phase modulation cancel all intermodulation (IM) products.
• Switched Transmission Lines: RF delay lines whose electrical length or impedance is periodically modulated by gating segments (via CMOS switches) with clocks phase-shifted by 120°, yielding broadband pseudo-LTI port-to-port transmission (Nagulu et al., 2018).
• Conductivity Modulation and N-Path Filtering: A synthetic traveling-wave conductivity profile ($$\sigma(x, t) = \sigma_0 + \Delta \sigma \cos(k_m x - \omega_m t)$$) is synthesized by commutating banks of capacitors (N-path filters) or delay lines; the nonreciprocal phase shift accumulates across the ring (Reiskarimian et al., 2018).
• Temporal Nonreciprocal Phase Shifters: Linear arrays or loops with dynamically modulated capacitance/inductance networks acting as time-dependent phase shifters. Harmonic cancellation is accomplished by paired temporal loops and destructive interference (Taravati et al., 2021).
• Chiral Edge-State and Topological Platforms: Coupling between chiral quantum anomalous Hall edge magnetoplasmons and conventional LC resonators in a non-Hermitian manner, leading to strong nonreciprocity and pseudo-LTI behavior without classical modulation (Martinez et al., 12 May 2025).
• SAW Filters and Acoustically Modulated Networks: Parametric SAW (surface acoustic wave) filters in a ring, each modulated with phase-shifted AC bias to realize circulator action at lower modulation frequencies than electrical counterparts (Yu et al., 2019).

3. Modal Analysis and Scattering Matrix Characterization

The small-signal, periodic nature of pseudo-LTI magnetless circulators allows rigorous analytical modeling in terms of modal decomposition and frequency-domain (Floquet) expansions:

• Cyclic Symmetry and Modal Decomposition: The 3-port network supports three orthogonal modes—one in-phase and two counter-rotating. Perfect cyclical symmetry ensures only the rotating modes are accessed under STM biasing; the modulation lifts degeneracy and imparts nonreciprocal phase accumulation to the modes (Kord et al., 2020).
• Floquet Scattering Matrix: The periodically time-varying circuit admittances or impedances can be analyzed by truncating to the fundamental harmonic under suitable $f_m/f_0 \ll 1$ ratios, yielding an effective (pseudo-)LTI S-matrix seen by external ports (Kord et al., 2017, Kord et al., 2019).
• Harmonic Cancellation in Differential Architectures: In fully differential designs, the two halves' spurious sidebands appear with opposite phase and sum to zero at each port, eliminating all even and odd order intermodulation products (Kord et al., 2017, Kord et al., 2020).
• S-Parameter Optimization: Analytical conditions for perfect circulation (e.g., 30° mode split, specific $\alpha = \tan^{-1}(2 Q_0 \Delta C / C_0)$ modulation depths) provide guidelines for achieving ideal transfer functions $S_{21}$, $S_{31}$, and return losses (Kord et al., 2017, Kord et al., 2018).

4. Quantitative Performance Metrics, Bandwidth Bounds, and Trade-Offs

Key technical metrics for pseudo-LTI magnetless circulators include:

Metric	Typical Range	Determinants
Insertion Loss (IL)	0.6–7 dB	$Q_0$, modulation depth, switch $R_{on}$
Isolation (IX)	20–60 dB	Modulation phase balance, symmetry
20 dB Isolation BW	2–15% (RF)	Filter order, $f_m$, $\phi_n = (n-1)2\pi/3$0
Noise Figure (NF)	≈ IL + 0.5 dB	$\phi_n = (n-1)2\pi/3$1, architecture, IC parasitics
P₁dB, IIP₃	+28 to +45 dBm	Switch design, varactor/inductor choice
Area	0.2–25 mm²	IC process, passive/discrete

• Global Bandwidth Bounds: The cyclic-symmetric, passive, and matched pseudo-LTI circulator with fundamental-only operation is subject to a bandwidth upper bound:

$\phi_n = (n-1)2\pi/3$2

where $\phi_n = (n-1)2\pi/3$3 is the loaded quality factor and $\phi_n = (n-1)2\pi/3$4 the maximum in-band reflection (Kord et al., 2018).

• Power Handling and Linearity: Multi-watt P₁dB performance and IIP₃ >+28 dBm have been measured in PCB and IC prototypes (Kord et al., 2017, Kord et al., 2017, Taravati et al., 2021). Switch $\phi_n = (n-1)2\pi/3$5, varactor stacking, and differential symmetry are pivotal.
• Ant Interface Efficiency ($\phi_n = (n-1)2\pi/3$6): A unified figure of merit quantifies system-level efficiency, factoring in IL, NF, P₁dB, and DC power; modern pseudo-LTI designs reach $\phi_n = (n-1)2\pi/3$7 values of >23% (Reiskarimian et al., 2018).
• Bandwidth-Insertion Loss-Isolation Tradeoff: IL and IX are competing figures depending on $\phi_n = (n-1)2\pi/3$8 (modulation amplitude), switch performance, and filter design; practical optimal regimes are identified via small-signal simulation and confirmed experimentally (Kord et al., 2017, Kord et al., 2018, Kord et al., 2020).

5. Representative Implementations Across Technologies

Pseudo-LTI magnetless circulators have been realized in several IC, hybrid, and quantum platforms:

• CMOS ICs (180/65/45 nm): STM angular-momentum biasing with switched capacitors or transmission lines on silicon enable form factors of <1 mm², IL ≈ 5 dB, IX > 50 dB, and BW ≈ 2–18% in the GHz to mm-wave regime (Nagulu et al., 2018, Reiskarimian et al., 2018, Kord et al., 2019).
• Discrete PCB Prototypes: Bandstop/delta and bandpass/wye circuits with varactor stacks demonstrate IL = 0.8–3.3 dB, IX = 24–55 dB, P₁dB = +29 dBm, and competitive noise figures (Kord et al., 2017, Kord et al., 2017).
• SAW-Based Circulators: High-$\phi_n = (n-1)2\pi/3$9 SAW filters with spatiotemporal modulation achieve ultra-low power, high isolation, and low insertion loss at lower modulation frequencies (Ω reduced by factor ≥20 vs. electrical implementations) (Yu et al., 2019).
• Topological QAH Insulator Platforms: Asymmetric, chiral coupling between edge magnetoplasmon and LC resonators delivers $$\sigma(x, t) = \sigma_0 + \Delta \sigma \cos(k_m x - \omega_m t)$$0 dB isolation, 6 dB IL, and MHz bandwidth in a magnetless, time-invariant design, exploitable at cryogenic temperatures for quantum-classical integration (Martinez et al., 12 May 2025).
• Temporal Nonreciprocal Phase Shifters: Loop-based phase shifters cancel all time-harmonic conversion products, enabling BW ≈ 20%, high linearity (IIP₃ = +45 dBm), and IL < 10 dB in integrated architectures (Taravati et al., 2021).

6. Differential and Broadband Architectures

Advanced architectures employ differential signal processing, external filters, and phased modulation to enhance pseudo-LTI performance:

• Differential STM Circulators: Pairing two single-ended circulators with anti-phase modulation completely suppresses IM products at all orders, yielding S-parameters practically indistinguishable from those of an LTI network, improved power handling, and relaxed requirements on modulation amplitude (Kord et al., 2017, Kord et al., 2020).
• Broadbanded Cyclic-Symmetric Topologies: Addition of second-order or higher Chebyshev bandpass filters to the STM junction extends the 20 dB isolation bandwidth to >13.9% (measured at 1 GHz) (Kord et al., 2018). Theoretical bounds confirm that combined modulation and filtering determine overall performance.
• Scaling and Practical Limitations: To achieve mm-wave bandwidths, architectures based on switched transmission lines (scaling $$\sigma(x, t) = \sigma_0 + \Delta \sigma \cos(k_m x - \omega_m t)$$1) and simplified switch clocks have been demonstrated, with isolation $$\sigma(x, t) = \sigma_0 + \Delta \sigma \cos(k_m x - \omega_m t)$$2 dB and IL $$\sigma(x, t) = \sigma_0 + \Delta \sigma \cos(k_m x - \omega_m t)$$3 dB over >10% BW (Nagulu et al., 2018, Reiskarimian et al., 2018).

7. Comparative Evaluation and Applications

Pseudo-LTI magnetless circulators rival and often surpass ferrite or active counterparts in size, linearity, and integrability, albeit with BW and insertion loss subject to modulation and physical constraints:

Feature	Ferrite	Active LTI	Pseudo-LTI Magnetless
Magnet Required	Yes	No	No
Integrability (Si, MEMS, etc.)	Poor	Good	Good
Insertion Loss	Low	Low/gain	Medium (1–5 dB)
Isolation	20–30 dB	20–30 dB	30–60 dB
Bandwidth (%)	10–20	5–10	3–15
Power Handling	>+50 dBm	<+20 dBm	+28 to +45 dBm
Noise Figure	<1 dB	>10 dB	2–5 dB

Applications encompass full-duplex wireless, RF front-ends, mm-wave 5G, cryogenic quantum control/readout, and high-linearity, integrable antenna interfaces (Kord et al., 2020, Reiskarimian et al., 2018, Martinez et al., 12 May 2025). The pseudo-LTI concept enables scalable on-chip nonreciprocal elements without the size, weight, and bias demands of ferrites or the nonideality of active topologies.