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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39266620
72383
10.1038/s41598-024-72383-2
Article
Synthetic rotational Doppler shift on transmission lines and it’s microwave applications
Seyedrezaei Zohreh
Rejaei Behzad
Memarian Mohammad mmemarian@sharif.edu

https://ror.org/024c2fq17 grid.412553.4 0000 0001 0740 9747 Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran
12 9 2024
12 9 2024
2024
14 2130313 7 2024
6 9 2024
© The Author(s) 2024
2024
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Rotational Doppler shift of a circularly polarized wave impinging normally on a rotating anisotropic surface, causes scattered waves with frequency shift equals twice the surface rotation frequency. We show that virtual rotational Doppler shift can be realized in transmission line platforms through a time-varying junction. In a system consisting of a pair of decoupled but identical transmission lines, voltage waves with a 90-degree phase difference between the two lines mimic a circularly polarized wave. A junction, comprising three time-varying capacitors and a static two-port network, connects the two lines and acts as a synthetically rotating anisotropic surface. As a result, the reflected and transmitted voltage (or current) waves undergo a frequency shift equal to twice the synthetic rotation frequency. Utilizing this effect, a full frequency converter is then proposed by augmenting the synthetically rotating capacitive junction with a dispersive phase shifter, followed by a short circuit. The system efficiently converts the incident tone into a single down- or up-converted tone, with amplification observed in the case of up-conversion. The frequency converter is subsequently employed to design a magnetic-free isolator. Circuit simulations with both ideal and switch-based time-varying capacitors match theoretical predictions.

Subject terms

Electrical and electronic engineering
Applied optics
Optical materials and structures
Metamaterials
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pmcIntroduction

Introducing time variations to the electromagnetic properties of materials (e.g. time varying permittivity) and circuits elements, has presented a distinctive opportunity for a diverse range of applications including but not limited to frequency conversion, parametric amplification, non-reciprocal transmission and broadband camouflage1–4. The so-called time refraction leads to frequency shift of a wave at a constant linear momentum, thus enabling frequency converters5–8. Additionally, time variations can break time-reversal symmetry, allowing for non-reciprocal behaviour. This alternative to conventional methods, alleviates the need to utilize nonreciprocal material such as bulky magnetized ferrites, and allows for the development of magnet-free, integratable isolators, circulators, and nonreciprocal antennas9–12.

In addition to investigations on time-varying media, numerous studies have employed time-varying circuit components in transmission line platforms. Their applications span a wide range, including parametric conversion, amplification, isolators, circulators, and enhancement of current circuit performance13–27.13–15 investigates a transmission line loaded with shunt varactors modulated with a travelling wave. A signal wave aligned in the direction of the modulation undergoes gradual conversion to frequency-shifted sidebands, while a signal in the opposite direction remains unchanged. This configuration is suitable for circulator applications and can be optimized to function as a Low-Noise Amplifier (LNA) when the modulation frequency exceeds the signal frequency14. In an alternative approach, effective electronic spin is induced in coupled resonators or filters by spatiotemporal modulation of their resonant frequencies16–19, resulting in significant nonreciprocal transmission and isolation. Recent developments have demonstrated the potential of N-path filter-based nonreciprocity for creating linear, miniaturized, and lossless isolating devices. Phase nonreciprocity is achieved by introducing phase shifts in the commutation of input and output switches within a 2-port N-path filter20–26. This concept has enabled the implementation of magnetless circulators across a broad frequency range from MHz to GHz by several research groups.

The complexity of designing and understanding time-varying and time-periodic systems is compounded by the broad spectrum of harmonics they generate. The rotational Doppler effect (RDE), inherently related to the vector nature of electromagnetic fields, offers a solution with minimal harmonics. Circularly polarized waves normally incident on a rotating anisotropic surface exhibit frequency shift (up or down), depending on the relative handedness of the wave w.r.t the sense of rotation of the surface. RDE -the rotational counterpart to the translational Doppler effect- has been explored in various studies28–32, where frequency shifts in the Hz-kHz range is achieved using mechanical rotation. In our previous work33, an electronic virtual rotation of axes of an anisotropic metasurface was proposed to yield arbitrary levels of RDE and in the MHz range for a CP wave.

It is also worth noting that electromagnetic waves carrying orbital angular momentum (OAM) encountering a rotating object will undergo the rotational Doppler effect34–36. This phenomenon has found applications in various fields, including remote sensing and angular velocimetry37–40, manipulation of rotational motion through rotational Doppler cooling and heating40, OAM detection for wireless communication37,41, nonreciprocity42, and frequency conversion33.

In33 the solution of scattered waves from a virtually rotating metasurface (VRM) was analytically expressed in closed form, simplifying the design without the need of time-consuming numerical simulations. The VRM was then used to design a a full frequency converter for radiating circularly polarized electromagnetic waves.

Transferring the concept of the rotational Doppler effect (RDE) to guided wave devices is crucial due to their potential immense range of applications. This paper extends the RDE (a concept for plane waves) to the realm of guided waves and transmission lines. By correlating x and y polarizations with a pair of decoupled transmission lines, and introducing a time-varying junction with only three time dependent capacitors, synthetic rotation and RDE are achieved (analogous to the Virtually Rotating Metasurface (VRM) in a free-space scenario). We validate the analytical findings by realizing the time-varying capacitors using varactor diodes and an alternative approach with switches and a capacitor bank. It is demonstrated that engineering the phase delay of generated frequency harmonics allows for ideal frequency conversion and amplification at microwave frequencies. Additionally, the frequency converter can be used to construct a non-magnetic isolator, achieving significant isolation (over 33 dB in the example).

This paper is structured as follows. “Rotational Doppler effect in a pair of transmission lines” describes the foundational analysis of the RDE in transmission lines using synthetic polarization. “A capacitive junction for realizing synthetic rotation” details the realization of the capacitance matrix C¯¯(t) with time-dependent capacitors to induce the RDE, validated by varactor diodes simulation, and a capacitor bank with modulated switches. “Applications of synthetic rotational Doppler shift in transmission lines” presents applications of the synthetic RDE at microwave frequencies, including a frequency converter and a magnet-free microwave isolator. “Discussion and conclusion” concludes. Methods covers the transmission matrix of a rotating junction.

Rotational Doppler effect in a pair of transmission lines

In this section we demonstrate how the rotational Doppler effect by an anisotropic, virtually rotating surface33 can be mimicked using transmission lines and time-varying circuit elements. To achieve this, two independent eigenstates, such as the x- and y-polarized states, are first mapped to two independent (decoupled) transmission lines to achieve synthetic polarization.

The time domain Maxwell equations for plane waves propagating along z direction in free space are1 ∂∂zE→(z,t)=-μ0∂∂tH→(z,t),∂∂zH→(z,t)=-ϵ0∂∂tE→(z,t),

where2 E→(z,t)=Ex(z,t)Ey(z,t),H→(z,t)=Hy(z,t)-Hx(z,t).

A +z direction propagating wave can be presented with two linear polarization, Ex, Hy and Ey, -Hx. For each electric field polarization and their associated magnetic fields, we consider a pair of voltage and current associated to a TL like that in Fig. 1b, V1, I1 are associated with the Ex and Hy polarized wave, and Ey and -Hx are related to V2 and -I2. The time-domain Telegrapher’s equations governing voltages and currents on the transmission line pair (denoted by 1 and 2) areFig. 1 (a) Scattering of a circularly polarized plane wave by a virtually rotating metasurface in free space33. (b) Two transmission lines along the z direction and a time varying junction to emulate virtual rotation leading to synthetic RDE. (c) Scatering of a LCP wave with the frequency ω~ and amplitude AL+ which travels in the +z direction from the time varying junction.

3 ∂∂zV→(z,t)=-L0∂∂tI→(z,t),∂∂zI→(z,t)=-C0∂∂tV→(z,t),

where4 V→(z,t)=V1(z,t)V2(z,t),I→(z,t)=I1(z,t)I2(z,t).

The parameters L0 and C0 represent the per unit length series inductance and shunt capacitance of the two transmission lines. These equations are identical to Maxwell’s equations for plane waves propagating along z in free space, if one identifies V1,V2 with Ex,Ey and I1,I2 with Hy,-Hx, and replaces μ0,ϵ0 by L0,C0. We can now create a model for the propagation of a plane wave with arbitrary polarization through these two transmission lines, by driving them with the necessary phase difference. For instance, a circularly polarized signal is obtained when the 1 and 2 transmission lines are driven with a 90∘ phase difference. Specifically, a right-hand circularly polarized (RHCP) signal emerges when the second transmission line is phase retarded by a quarter period relative to the first transmission line.

In analogy with the scattering of a plane wave by a VRM in free space shown in (Fig. 1a)33, we next investigate the reflection and transmission of waves travelling on the two transmission lines by a time-varying junction at z=0, connected between the two lines as shown in Fig. 1b. Anticipating practical realizations, we restrict ourselves to a network of time-dependent capacitors. Denoting the 2×2 capacitance matrix of this junction by C¯¯(t), the voltages and currents must satisfy the boundary conditions5 V→(0-,t)=V→(0+,t),I→(0-,t)-I→(0+,t)=∂∂tC¯¯(t)V→(0-,t).

Following our previous work33, we require that the capacitive junction is anisotropic, and without loss of generality assume it undergoes a right-hand sense synthetic rotation, i.e.6 C¯¯(t)=U¯¯(t)·C¯¯s·U¯¯T(t).

7 U¯¯(t)=cos(Ωt)-sin(Ωt)sin(Ωt)cos(Ωt),C¯¯s=c100c2,

and Ω is the angular frequency of rotation. The particular form of the capacitance matrix (6) allows us to map the problem onto a time-invariant one by applying the transformation8 V→~(z,t)=U¯¯T(t)V→(z,t),I→~(z,t)=U¯¯T(t)I→(z,t),

which results in the modified TL equations9 ∂∂zV→~=-L0∂∂tI→~+L0ΩΣ¯¯I→~,∂∂zI→~=-C0∂∂tV→~+C0ΩΣ¯¯V→~,

and the boundary conditions10 V→~(0-,t)=V→~(0+,t),I→~(0-,t)-I→~(0+,t)=C¯¯s∂∂tV→~(0-,t)-ΩΣ¯¯C¯¯sV→~(0-,t),Σ¯¯=01-10.

Despite being more complicated, the transformed boundary conditions are independent of time corresponding to a static junction. The resulting differential equations will therefore have time-independent coefficients, allowing simple solutions.

Solutions of the transformed TL equations may be written in a form which mimics circularly polarized (CP) waves with the angular frequency ω~,11 V→~R±(z,t)=AR±cos(ω~t∓β~Rz),sin(ω~t∓β~Rz),I→~R±(z,t)=±1Z0V→~R±(z,t),

12 V→~L±(z,t)=AL±cos(ω~t∓β~Lz)-sin(ω~t∓β~Lz),I→~L±(z,t)=±1Z0V→~L±(z,t).

The subscripts R and L designate solutions where the phase difference between V1 and V2 is +90∘ and -90∘, respectively. Following the standard terminology of plane waves, we shall call these two cases RCP and LCP waves, respectively. Note, however, that unlike common practice, our definition of the sense of rotation is independent of the direction of propagation of the waves, and is defined w.r.t. to the +z axis, as similarly done in33. The constants AR±,AL± denote the wave amplitudes (the superscript ± denotes the direction of propagation) and Z0=L0/C0 is the characteristic impedance of the transmission lines. The propagation constants depend on the sense of polarization and are given by13 β~R=(ω~+Ω)/v0,β~L=(ω~-Ω)/v0,

where v0=1/L0C0 is the phase velocity on the lines. Consider next, in the transformed system, an LCP wave with the frequency ω~ and amplitude AL+ which travels in the +z direction,14 V→~i(z,t)=AL+cos(ω~t-β~Lz)-sin(ω~t-β~Lz).

As depicted in Fig. 1c, this wave is scattered by the junction and the reflected and transmitted waves will contain both LCP and RCP components at the same frequency ω~ since the junction is anisotropic and static in the transformed system. The reflected wave which propagates in the -z direction, may therefore be expressed as15 V→~r(z,t)=AL+|rLL|cos(ω~t+β~Lz-ϕLL)-sin(ω~t+β~Lz-ϕLL)+AL+|rRL|cos(ω~t+β~Rz-ϕRL)sin(ω~t+β~Rz-ϕRL),

where |rLL|, |rRL| and ϕLL, ϕRL denote the magnitudes and phases of the corresponding reflection coefficients, respectively. From the point of view of the original system, however, where the junction is varying in time, the incident and reflected fields are found by using the inverse transformation of (8),16 V→i(z,t)=AL+cosω(t-z/v0)-sinω(t-z/v0),

17 V→r(z,t)=AL+|rLL|cosω(t+z/v0)-ϕLL-sinω(t+z/v0)-ϕLL+AL+|rRL|cos(ω+2Ω)(t+z/v0)-ϕRLsin(ω+2Ω)(t+z/v0)-ϕRL,

where ω=ω~-Ω. Therefore, from the viewpoint of the original system, an incident LCP wave with the frequency ω is reflected partially as another LCP wave with the same frequency, and partially as an RCP wave with the frequency ω+2Ω. This is also true for the transmitted waves. A similar analysis shows that for an incident RCP wave, the reflected and transmitted waves contain RCP components of the same frequency, and LCP waves with the frequency ω-2Ω. Therefore, conversion of polarization is accompanied by a shift of ±2Ω in frequency, depending on the relative phase of the voltages on the two transmission lines. No other harmonics are generated, which is uncommon in typical time varying systems. This phenomenon is analogous to rotational Doppler shift of circularly polarized plane waves that are scattered by rotating objects. Note that the sign of Ω depends on the sense of synthetic rotation of the junction defined by the rotation matrix U¯¯(t) in Eq. (7).

The derivation of the reflection and transmission coefficients of the synthetically rotating capacitive junction is very similar to the calculation presented in33 and is summarized in the Method section. Here, we report the final results. For an incident LCP wave, the reflection parameters are18 rLL=|rLL|e-jϕLL=1+j(ω+2Ω)cpZ02ξ(ω+Ω)-1

19 rRL=|rRL|e-jϕRL=-j(ω+2Ω)cmZ02ξ(ω+Ω),

where20 cp=12(c1+c2),cm=12(c1-c2),

21 ξω~=1+j(ω~-Ω)2cpZ0×1+j(ω~+Ω)2cpZ0+ω~2-Ω24cm2Z02.

The complex transmission coefficients for the transmitted LCP (frequency ω) and RCP (frequency ω+2Ω) waves are given by tLL=rLL+1,tRL=rRL. Similarly, an incident RCP wave is partially reflected as an RCP wave (frequency ω) and a LCP wave (frequency ω-2Ω) with the reflection coefficients22 rRR=|rRR|e-jϕRR=1+j(ω-2Ω)cpZ02ξ(ω-Ω)-1,

23 rLR=|rLR|e-jϕLR=-j(ω-2Ω)cmZ02ξ(ω-Ω).

The transmission coefficients are given by tRR=rRR+1,tLR=rLR. Note that if c1=c2 then the junction capacitance matrix given by (7),(6) becomes independent of time. No frequency harmonics will then be produced as can be seen from the expressions for the reflection coefficients rRL,rLR which will become zero when cm=(c1-c2)/2=0. Finally, it must be mentioned that the results are identical for waves that are incident on the junction from the right.

The synthetic rotational Doppler shift discussed above may be used to implement transmission-line-based frequency converters and isolators as we shall see later. This effect is due to the particular form of the junction capacitance matrix given by (7),(6). In the next section we describe how the capacitance matrix C¯¯(t) may be realized by a network of time-dependent capacitors.

A capacitive junction for realizing synthetic rotation

Theory

The components of the virtually rotating capacitance matrix (Eq. 6) are24 C¯¯(t)=c1cos2(Ωt)+c2sin2(Ωt)(c1-c2)sin(Ωt)cos(Ωt)(c1-c2)sin(Ωt)cos(Ωt)c1sin2(Ωt)+c2cos2(Ωt).

It may seem, at first sight, straightforward to realize such a junction by a Π-network of three time-dependent capacitors connected between a pair of transmission lines. A possible realization using two microstrip lines is illustrated in Fig.  2a. The resulting capacitance matrix of the Π-network of three time-dependent capacitors is25 C¯¯N(t)=cx(t)+ca(t)-ca(t)-ca(t)cy(t)+ca(t).

Comparison with Eq. (24) however, shows that for C¯¯ and C¯¯N to be equal, ca(t) (and even cx(t),cy(t), depending on the choice of c1,c2) must become negative at certain times which is impossible. On the other hand, we know that, in the frequency domain, an inductor behaves like a negative capacitor from an impedance point of view. Could we thus supplement a Π-network of time-varying capacitors by other, possibly inductive, elements to realize the capacitance matrix (Eq. (24))?

Let us consider the shunt connection of the Π-network of capacitors with another two-port which is assumed to be static as shown in Fig. 2a with blue components. Working in time-domain, the relationship between the port currents and voltages on the two ports of the junction is26 I→J(t)=-ddtC¯¯N(t)V→J(t)+∫-∞ty¯¯(t-t′)V→J(t′)dt′,

where I→J(t)=I→(0-,t)-I→(0+,t) is the vector of port currents entering the junction and V→J(t)=V→(0-,t) is the vector of transmission line voltages at the junction at z=0. The integral containing the matrix kernel y¯¯ describes the currents entering the added two-port network in response to the port voltages. This term is written as a convolution integral as the additional two-port is assumed to be time-invariant. Suppose that we design the static two port such that27 ∫-∞ty¯¯(t-t′)V→J(t′)dt′=D¯¯dV→J(t)dt,D¯¯=-dγdαdα-dγ,

where dα,dγ are positive constants. Then, substitution in (26) leads to a junction capacitance matrix C¯¯N(t)+D¯¯ (see Eq. (5)). Equating this matrix with Eq. (24) results in28 cx(t)=cp+cm2sin2Ωt+π4+dγ-dα,cy(t)=cp+cm2sin2Ωt-π4+dγ-dα,ca(t)=dα-cmsin2Ωt,cp=12(c1+c2),cm=12(c1-c2).

The (positive) constants dα,dγ will be chosen such that the time varying capacitors comprising the junction remain positive at all times.

To see how a static two-port network can satisfy (Eq. (27)), take note that if, in the end, we succeed in implementing the rotating junction, then only three frequencies will be involved. Any incident wave with the base frequency ω propagating on the two lines will be a combination of LCP and RCP waves. When scattered by the junction, only the frequencies ω±2Ω will be produced in case of polarization swap. Even if the rotating junction is part of a larger system in which the produced harmonics are reflected again and travel back to the junction, a simple analysis shows that no other frequencies are generated, provided that the rest of the system is static. This means that the junction voltage vector V→J(t) only contains sinusoidal components with the frequencies ω,ω±2Ω and may be written using phasors29 V→J(t)=ℜV→0ejωt+V→-ej(ω-2Ω)t+V→+ej(ω+2Ω)t.

For (27) to be valid, it is then sufficient to require30 Y¯¯(ν)=-jνD¯¯,ν=ω,ω±2Ω,

where31 Y¯¯(ν)=∫0∞y¯¯(τ)e-jντdτ,

is the 2×2 admittance matrix of the static two-port network. For simplicity, let us assume that dα=dγ. As a result the admittance matrix (30) is implemented as a single admittance32 Ya(ν)=-jνdα,ν=ω,ω±2Ω,

in shunt connection with the capacitance ca(t) in Fig. 2a. For capacitor values in (28) to be always positive, we require cp>2cm,dα>cm. The admittance Ya(ν) is purely imaginary and decreases with increasing frequency. This, however, is prohibited by Foster’s theorem and thus resonating structures are required for its implementation. For the frequency converter discussed in the next section only two frequencies will be involved (either ω,ω+2Ω, or ω,ω-2Ω) so that we may require (Eq. (32)) to be valid for two frequencies instead of three. Two possible realizations are shown in sub-figures in Fig. 2a.Fig. 2 (a) Configuration of the proposed time varying junction to emulate synthetic rotation in transmission lines concept, placed at z=0, as well as wo possible resonating structures for implementing needed admittance in Eq. (32) are depicted in sub figure. (b) Schematic of the circuit simulation of the ideal time-varying junction, where c1=37.55 pF, c2=19.45 pF, and dα=25 pF. The static two-port circuit parameters are C0=45.3 pF, L1=2 nH, and L2=8.75 nH.

The first step in the design of a single synthetically rotating junction is based on the frequency of the incident signal (wave) and the synthetic rotation frequency. Depending on the application (frequency converter or any other application), the scattering parameters required for the junction is determined. Using the scattering requirements, and with the aid of Eqs. (22) and (23) the capacitance values c1 and c2 are determined. It is also crucial to ensure that dα>cm, to maintain a positive ca(t). Finally, the parameters of the accompanying static two-port network are determined using (32), depending on which circuit topology is chosen from subfigures in Fig. 2a. This summarizes both the design procedure and the process to exactly determine all circuit values for design of a single synthetically rotating junction.

Implementation of synthetic rotation using varactors

To verify theoretical results, a harmonic balance circuit simulation was performed using Agilent ADS software. The simulation included three ideal time-dependent varactors with linear voltage-dependent capacitance, following the formulation in Eq. (28). The varactors simulated a time-varying junction synthetically rotating at a frequency of Ω=350 MHz in a right-handed sense relative to the +z direction.

We represent the tensor C¯¯s as a diagonal matrix, where c1=37.55 pF and c2=19.45 pF. Additionally, we assume dα=25 pF to ensure ca(t)>0. As shown in Fig. 2b, the upper static two-port circuit illustrated in subfigure of Fig. 2a is used in the schematic of the circuit simulation. The incident signal is assumed to be an LHCP signal with an amplitude of 1V and a frequency of 200 MHz. Thus, the microstrip lines are associated with 1V sinusoidal voltage sources, with the first line retarded by 90∘ relative to the second line. The microstrip lines are designed with a 50Ω line impedance, using a 20 mil FR4 substrate. Immediately after the junction, the lines are terminated with 50Ω resistive loads. Figure 3a shows the analytical and simulation results of the transmitted voltage magnitudes at the load terminations. The transmitted wave shows only two frequency components at 200 MHz (the drive frequency) and 900 MHz (the up-converted frequency), as expected analytically. Simulation results also confirm the RCP and LCP nature of the transmitted voltages at ωi=200 MHz and ωi+2Ω=900 MHz, respectively.Fig. 3 Analytical and circuit simulation results illustrate the Fast Fourier Transform (FFT) amplitude of the transmitted voltage magnitude from a right handed synthetically rotating junction. The synthetic rotational frequency of the junction is 350 MHz, with capacitance values set at c1=37.55 pF, c2=19.45 pF, and dα=25 pF. The parameters of static two-port network are C0=45.3 pF, L1=2 nH, and L2=8.75 nH. (a) The drive signal is a LCP signal with frequency of 200 MHz. (b) The drive signal is a RCP signal with frequency of 900 MHz. (c) Approximated modulated time-varying capacitors with 5 discrete states for cx,y,a(t), with c1=14.95 pF, c2=5.05 pF, and dα=9 pF and a modulation frequency of 40MHz. (d) Schematic of the switch-based circuit for implementing a piece-wise constant time-varying capacitor.

In Fig. 3b, the transmitted voltages for a down conversion case are depicted. The rotating junction remains consistent with the configuration in Fig. 3a, but the driving signal is a RHCP signal at 900 MHz. The transmitted voltage spectrum reveals the presence of only the injected tone and a single down-converted signal at ωi-2Ω=2π(200) MHz. Both analytical and simulation results exhibit excellent agreement, confirming that the transmitted voltage at the down-converted tone has a reversed sense of polarization compared to the injected signal.

The results obtained from ideal time-varying varactor simulations perfectly validate the concept of synthetic rotation in the context of transmission lines and time-dependent junction. The selection of the time modulation order and the magnitude of time-varying varactors is determined by reference examples from various time-varying systems that have been implemented using actual varactor diodes8,14,15,43,44.

In addition to varactor diodes, an alternative approach for realizing a time-varying capacitor involves implementing a piecewise constant time-varying capacitor using an array of switches and a capacitor bank. By approximating the necessary time-varying capacitances with predefined discrete levels, we can change the total capacitance through an accumulative approach, implemented by switches. When the modulation period (ΠΩ) of the time-varying capacitors is divided into N intervals (assuming N is even), the number of discrete levels is N2+1. Within each interval, the capacitor values remain constant and are equal to the value of cx,y,a(t) evaluated at the midpoint of the interval. Figure 3c illustrates the necessary temporal evolution of cx,y,a(t) for continuous modulated capacitors with c1=14.95pF, c2=5.05pF, and dα=9pF, as well as when they are discretized into 8 time intervals. Given that the minimum phase difference resolution required for realizing cx,y,a(t) is π4, the smallest number of discretization intervals would be 8. The schematic for implementing a piecewise constant time-varying capacitor is depicted in Fig. 3d.

Now we investigate the synthetically rotating junction designed with the switch-based c(t). cx,y(t) are assumed to be c(t)=10+7sin(2Ωt±π4)pF, with a modulation frequency of 40MHz, respectively, and the signal frequency is 400MHz. Assuming dα=9pF, we can determine ca(t). The magnitude of the Fourier transform of transmitted voltages at load terminations for a time-varying junction realized by switches, with an LCP and RCP incident signal, is presented in Fig. 4a, b, respectively. It can be observed that in the discretized case, some unwanted harmonics may occur, although their amplitudes are negligible. Additionally, the phase error between the first and second voltage components is less than ±1∘ in up-conversion case and less than -1.5∘ in down-conversion scenario. This indicates that the polarization of the transmitted signal aligns with expectations. However, the axial ratio in the upconversion case is 0.98 at 400 MHz and 0.89 at 440 MHz, which are the incident and upconverted frequencies, respectively. This indicates that the axial ratio for the upconverted tone deviates slightly from 1. Furthermore, the magnitude differences at 400 MHz and 440 MHz compared to the theoretical results are 3% and 15%, respectively. Similar calculations for Fig. 4b determine that the axial ratio error is respectively less than 1% and 2% at 400MHz and 440MHz, respectively. These circuit simulations suggest that practical state-switching systems can also achieve effective performance in implementing rotational Doppler effect on a microstrip line platform.Fig. 4 Simulation results depicting the magnitude of transmitted voltages at load terminations after encountering a right handed synthetically rotating junction implemented by switches. The junction is characterized by c1=14.95 pF and c2=5.05 pF, and dα=9 pF, with a rotational angular frequency of 20MHz. The static two-port circuit parameters are C0=1 nF, L1=15.9 nH, and L2=145 pH. (a) Drive signal is an LCP signal with 1V amplitude and frequency of 400 MHz. (b) Drive signal is an RCP signal with 1V amplitude and frequency of 440 MHz.

Applications of synthetic rotational Doppler shift in transmission lines

In this section we present two important applications of synthetic rotational Doppler shift on transmission lines at microwave frequencies. The first example is a full frequency converter applicable in telecommunication receivers and transmitters. This design is similar to that of a free-space frequency converter for electromagnetic plane waves presented in33. The full frequency converter will then be used to propose a magnet-free microwave isolator as our second example.

Full frequency converter

Consider a LCP wave with the frequency ω that is incident on the dynamic capacitive junction from the left. The reflected and transmitted waves contain both LCP (frequency ω) and RCP (frequency ω+2Ω) components. Let the two transmission lines on the right be terminated by two identical lossless networks, as shown in Fig. 5a. As seen from z=0 this network will reflect the LCP and RCP waves without flipping their sense of polarization (this is because the two transmission lines are terminated by identical networks). Besides, the reflected waves will have the same amplitudes as the incident waves since the terminating networks are assumed to be lossless. However, the reflection phase may be different for the LCP and RCP waves since they differ in frequency. The LCP and RCP waves that are reflected by the termination will travel towards the junction from the right and will partially undergo polarization flip accompanied by frequency conversion. However, since the frequency of the RCP wave is ω+2Ω, reversal of its polarization results in LCP waves with the frequency ω. Therefore, only two tones will be present in the end: ω and ω+2Ω.Fig. 5 (a) Full frequency conversion circuit based on virtually rotating junction, augmented with a disperssive phase shifter.(b,c) Analytic and circuit simulation results showing FFT amplitude of the reflected voltages magnitude from the right handed synthetically rotating junction/shorted microstrip line combination. Rotational frequency of the junction is 350 MHz, c1=37.55pF, c2=19.45pF, d=10 cm. The parameters of static two-port network are C0=45.3 pF, L1=2 nH, and L2=8.75 nH. Drive signal is an LCP signal with frequency of 200 MHz in case (a), and drive signal is an RCP signal with frequency of 900 MHz in case (b).

Taking into account all waves incident on the junction, one has33 BL+=tLLAL++rLLBL-+rLRBR-,

34 BR+=tRLAL++rRLBL-+rRRBR-,

35 AL-=rLLAL++tLLBL-+tLRBR-,

36 AR-=tRLAL++tRLBL-+tRRBR-,

where37 BL-=e-jθLBL+,BR-=e-jθRBR+,

in which θL and θR are the reflection phases of the lines as seen from the junction at frequencies ω and ω+2Ω, respectively. Using the reflection and transmission coefficients reported in “Rotational Doppler effect in a pair of transmission lines”, and noting that these coefficients must be calculated for the RCP wave at the frequency ω+2Ω instead of ω, one then obtains the overall reflection coefficients38 RLL=AL-AL+=2Δ1+j(ω+2Ω)2cpZ0+jtanθR2-1,

39 RRL=AR-AL+=-j(ω+2Ω)ΔcmZ0,

where40 Δ=1+j(ω+2Ω)2cpZ0+jtanθR2×1+jω2cpZ0+jtanθL2+ω(ω+2Ω)4cm2Z02.

Similar to33, simple algebra shows that if the conditions41 tanθL2=-12ωcpZ0±14ω(ω+2Ω)cm2Z02-11/2,

42 tanθR2=-12(ω+2Ω)cpZ0±14ω(ω+2Ω)cm2Z02-11/2,

are met, then43 |RLL|=0,|RRL|=ω+2Ωω1/2,

so that there is no reflected LCP wave. The incident LCP wave is thus reflected as a RCP wave with the frequency ω+2Ω and an amplitude that is larger than that of the incident wave. Full frequency up-conversion is achieved together with amplification. In a similar way it can be shown that an incident RCP wave with the frequency ω is fully down-converted with attenuation to a LCP wave with the frequency ω-2Ω if the phases θL,θR satisfy (41),(42) with ω replaced by ω-2Ω. The corresponding reflection coefficients then satisfy44 |RRR|=0,|RLR|=ω-2Ωω1/2.

The reflection phases θL, θR of the LCP, respectively, RCP waves given by (41),(42) may be realized by using dispersive phase-shifting networks since these waves have different frequencies (Fig. 5a). Although these phase shifters may be easily designed for any given combination of cp,cm,Z0, in particular cases (see Ref45) it is sufficient to use a finite length (D) of two short-circuited transmission lines which results in the phase shifts θL=π+2ωD/v0, θR=π+2(ω+2Ω)D/v0.

To summarize the design of a full-frequency converter using a synthetically rotating junction, it is important to note that the parameters c1 and c2 cannot be chosen arbitrarily. According to (41) and (42), cm2 must be greater than a minimum value to ensure the expression under the square root remains positive. Additionally, if the phase values in (41) and (42) are to be implemented using transmission lines (as done here), the phase shift introduced by the TL should be considered, and these equations should be solved accordingly for c1 and c2.

Consider, for instance, the synthetically rotating junction investigated in Fig. 3. With D=10 cm, both conditions (41),(42) are satisfied. The circuit simulation results, for frequency up-conversion of an incident LCP wave at 200MHz, and down-conversion with an incident RCP wave at 900MHz are shown in Fig. 5b, c. The simulation results validate the theoretical expectations. The Fourier transform of the reflected voltages is shown in Fig. 5b and it is clear that it only contains an up-converted tone at 900 MHz. Its magnitude agrees well with Eq. ((43)). Furthermore, the phase of the reflected V2 voltage is delayed by 90∘ with respect to that of the reflected V1, which means that the total reflected wave is an RCP wave as expected. For an incident RCP wave at 900MHz the Fourier transform of the reflected voltages is presented in Fig. 5c. The reflected wave contains a single down-converted tone at ω-2Ω=2π(200)MHz. The quadrature phase difference of x and y components of reflected voltages affirms their LCP character. The frequency conversion efficiency in this case is 47.14% which is in excellent agreement with those from theory in Eq. ((44)).

Magnetless isolator

The frequency converter presented above may be used to implement a non-magnetic isolator as described below. First, the two outputs of a 90∘ hybrid (ports 2,3) are connected to a pair of microstrip lines as shown in Fig. 6a. Excitation of either input ports 1 or 4 of the hybrid results in generation of RCP or LCP waves, respectively, on the two output lines where the phase difference between the voltages on the two lines designated by x and y is ±90∘. The microstrip lines are connected to a right-handed full frequency converter block via two band-pass filters with the central frequency of ω (the operation frequency of the isolator). The filters are designed such that signals with frequencies ω±2Ω are blocked. The frequency converter is designed to fully down-convert an incident RCP wave with the frequency ω to a reflected LCP wave with the frequency ω-2Ω. Two absorbing filters are placed in between the band-pass filters and the frequency converter such that signals with a frequency of ω-2Ω are completely absorbed. This configuration comprising a band-pass filter and a band-pass absorbing filter can be viewed as a diplexer, with one of its paths being connected to a matched load (Fig. 6b).

The case of port 1 excitation and the signal tracking is depicted in Fig. 6a. When port 1 is excited by a signal with the frequency ω, a RCP wave with the same frequency is produced on the two output ports (2,3) of the hybrid. The RCP wave passes through the band-pass filters and is left intact by the absorbing filters. It is reflected as a LCP wave from the frequency converter block, with frequency ω-2Ω (the reflected red-shifted waves are shown with red arrows in Fig. 6a). The reflected LCP wave travels back towards the absorbing filters and is completely absorbed, and does not reach ports 2,3 of the hybrid block. Since ports 1 and 4 of the hybrid are isolated by design, no output signal will be present on port 4 in the absence of reflected waves reaching ports 2,3. Hence, the signal fed to port 1 of the hybrid is totally absorbed and does not appear on port 4.

Excitation of port 4, is presented in Fig. 6b. When the same signal is fed to port 4 of the 90∘ hybrid, a LCP wave with the frequency ω appears on ports 2,3. The LCP wave will again pass through the band-pass/absorbing filters without any loss or reflection and reaches the frequency converter. Had the frequency converter been designed as an up-converter, the LCP wave would have been totally reflected as a RCP wave with the frequency ω+2Ω. However, due to the phase requirements (41) and (42), one cannot achieve full down- or up conversion of RCP or LCP waves, respectively, of the same frequency with the static network presented in Fig. 2a. As a result, the reflected wave will contain both LCP (frequency ω), RCP (frequency ω+2Ω) components, and higher frequencies. The LCP wave will not be absorbed and will pass through the band-pass filters to reach ports 2 and 3 of the hybrid circuit. The reflected RCP wave will not be absorbed either, but it cannot pass through the filters because its frequency lies outside their pass band. It will thus be reflected by the filters towards the frequency converter. Since the frequency of this wave is ω+2Ω and not ω as in the previous scenario, the frequency converter will return both RCP (at ω+2Ω) and LCP (at ω) components. The latter will pass through the filters while the former will again be reflected. This process of multiple reflections of LCP and RCP waves will continue indefinitely. In steady state, the superposition of all LCP waves escaping the band pass filter will reach ports 2 and 3 of the hybrid and will exit entirely through port 1.Fig. 6 (a,b) Magnet-free isolator design based on synthetically rotating junction. The case of port 1 excitation and the signal tracking is presented in (a). The case of port 4 excitation and the signal tracking is presented in (b).(c) Schematic of the diplexer designed for 3 GHz and 4 GHz. (d) The S-parameters of the diplexer designed for 3 GHz and 4 GHz. (e,f) Spectrum of recieved signals at port 1 and port 2 in dB, when one port is driven with a signal at 4 GHz with amplitude of 1 V and the other port is matched.The junction is characterized by c1=25 pF and c2=2.3 pF, and dα=12.6 pF, with a rotational angular frequency of 500 MHz. The static two-port circuit parameters are C0=162.45 pF, L1=0.16 nH, and L2=13.6 pH. Port 1 is driven in (e), and Port 2 is driven in (f).

To verify the above design, a harmonic balance circuit simulation using Agilent ADS software was carried out. The full frequency converter is designed for frequencies of 3GHz and 4GHz, meaning the synthetic rotation frequency of the junction is Ω=500MHz. The design parameters for the full frequency converter are: c1=25 pF, c2=2.3 pF, dα=12.6 pF, d=6.6 cm, C0=162.44 pF, L1=160 pH, and L2=13.6 pH. The diplexer is designed using two 4th-order Chebyshev band-pass filters at 3GHz and 4GHz, each with a 200 MHz bandwidth and a 0.1 dB pass-band ripple. Since the filter frequencies are well separated, we can connect them through a Tee microstrip line as depicted in Fig. 6c, and with some optimization achieve matching at the common port of the diplexer at the two required frequencies. The S-parameters of the diplexer are depicted in Fig. 6d. Simulation results for the isolation and through cases, presented in Fig. 6e, f , show ideal transmission (as no loss was considered in the model) and isolation above 33 dB.

As demonstrated in this section, the nonreciprocal behavior of the synthetically rotating junction enables magnetic-free isolator design. By placing a 90∘ hybrid before the full frequency converter and employing a diplexer tuned to the converter’s signal frequencies, it is possible to achieve a magnetic-free isolator with excellent performance for microwave applications. Simulations validate the design’s ideal transmission and over 33 dB isolation.

Discussion and conclusion

In summary, a novel synthetically rotating junction, comprising only three time-dependent capacitors and a static two-port network, was used to realize rotational Doppler effect with transmission lines. By introducing a synthetic polarization through the modes of an uncoupled pair of microstrip lines, it becomes feasible to observe the frequency shift resulting from the interaction of a circularly polarized (CP) signal with a rotating junction. The observed frequency shift, akin to the conventional rotational Doppler effect, is determined by the polarization alignment or misalignment with the junction. Similar to the conventional rotational Doppler effect, the shifted frequency equals twice the virtual rotation frequency, and no other harmonics are generated in this system.

Moreover, the addition of a dispersive phase shifter after the junction enables the realization of a full-frequency converter in reflection mode. It has been demonstrated that in the case of up-conversion, a parametric amplification proportional to the up-converted frequency relative to the incident frequency is achievable. Furthermore, leveraging the nonreciprocal behavior of the synthetically rotating junction facilitates magnetic-free isolator design. By incorporating a 90∘ hybrid and a diplexer tuned to the converter’s signal frequencies, a high-performance magnetic-free isolator for microwave applications can be achieved.

Circuit simulations, utilizing both ideal modulated capacitors and switch-based time-varying capacitors to emulate time-varying capacitance, align well with theoretical expectations. Notably, unlike conventional parametric conversions where the pump frequency is required to be twice or higher than the signal frequency, the synthetic rotating junction allows for pump frequencies both lower and higher than the signal frequency. Additionally, simulations confirm the isolator design’s excellent performance, showing ideal transmission and over 33 dB isolation. The results of this paper have potential applications in achieving frequency conversion for transceivers, developing non-magnetic isolators for microwave frequencies. As it can be implemented on a printed circuit board (PCB), it offers the possibility of creating an integrated frequency converter and isolator.

Methods

Formulation of the transmission matrix of a synthetically rotating junction

To compute the transmission matrix for the synthetically rotating junction, we solve the scattering problem using the transformed voltage and current vectors V→~(z,t) and I→~(z,t) in Eq. (8). These quantities satisfy the time-invariant equations (Eq. (9)) and boundary conditions (Eq. (10)) which can be dealt with by using the common phasor-representation,45 V→~(z,t)=ℜV→(z)ejω~t,I→~(z,t)=ℜI→(z)ejω~t.

The transmission line equations for the voltage and current phasors are simply found by replacing ∂/∂t by jω~ in Eq. ((9)). For the scattering problem considered, the solution is the region z<0 is written as46 V→(z)=AR+e-jβ~Rz+AR-ejβ~Rz1-j+AL+e-jβ~LzAL-ejβ~Lz1j,

47 I→(z)=1Z0AR+e-jβ~Rz-AR-ejβ~Rz1-j+1Z0AL+e-jβ~Lz-AL-ejβ~Lz1j,

where βR,βL are given by (13). Here AR+ and AL+ denote RCP and LCP waves incident on the junction. The reflected wave amplitudes are AR- and AL-. For z>0 we have48 V→(z)=BR+e-jβ~Rz1-j+BL+e-jβ~Lz1j,

49 I→(z)=1Z0BR+e-jβ~Rz1-j+1Z0BL+e-jβ~Lz1j.

By imposing the boundary conditions on the phasors, one finds after some algebraic manipulations,50 BR+BL+=T¯¯cp(ω~)AR+AL+,

where the transmission matrix T¯¯cp(ω~) is51 T¯¯cp(ω~)=1¯¯+Q¯¯(ω~)·χ¯¯cp-1,

52 Q¯¯(ω~)=C¯¯-1·jω~1¯¯-ΩΣ¯¯·C¯¯,

53 χ¯¯cp=12Y0C¯¯-1·C¯¯s·C¯¯,

54 C¯¯=11-jj.

The reflection matrix is defined by55 AR-AL-=R¯¯cp(ω~)AR+AL+,

and is given by56 R¯¯cp(ω~)=T¯¯cp(ω~)-1¯¯.

Author contributions

Z.S., B.R. and M.M. equally contribute to this research.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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