Frequency Tunable Magnetostatic Wave Filters With Zero Static Power Magnetic Biasing Circuitry

A single tunable filter simplifies complexity, reduces insertion loss, and minimizes size compared to frequency switchable filter banks commonly used for radio frequency (RF) band selection. Magnetostatic wave (MSW) filters stand out for their wide, continuous frequency tuning and high-quality factor. However, MSW filters employing electromagnets for tuning consume excessive power and space, unsuitable for consumer wireless applications. Here, we demonstrate miniature and high selectivity MSW tunable filters with zero static power consumption, occupying less than 2 cc. The center frequency is continuously tunable from 3.4 GHz to 11.1 GHz via current pulses of sub-millisecond duration applied to a small and nonvolatile magnetic bias assembly. This assembly is limited in the area over which it can achieve a large and uniform magnetic field, necessitating filters realized from small resonant cavities micromachined in thin films of Yttrium Iron Garnet. Filter insertion loss of 3.2 dB to 5.1 dB and out-of-band third order input intercept point greater than 41 dBm are achieved. The filter's broad frequency range, compact size, low insertion loss, high out-of-band linearity, and zero static power consumption are essential for protecting RF transceivers and antennas from interference, thus facilitating their use in mobile applications like IoT and 6G networks.

The growth of multi-band and high frequency communication systems has resulted in single bandpass filter technologies being unable to satisfy the filtering requirements for all bands due to the crowded radio-frequency (RF) spectrum. 1,2,3 This is especially problematic as frequencies are scaled beyond the spectrum allocated for 5G (3)(4)(5)(6), where RF silicon-on-insulator switches exhibit unacceptably high loss when utilized in switched filter banks. 4,5 For example, the FR3 band from 7.125 to 24.25 GHz under exploration for 6G networks will require extensive innovations in both RF switch 4,6 and acoustic filter 7,8,9,10 technologies if it is to adopt the massively parallel switched acoustic filter banks utilized in 4G and 5G networks. As compared to the traditional implementation of a switched filter-bank, a single tunable filter has great potential to reduce the system cost, size, complexity, and remove entirely the additional switch paths loss. 2,11,12,13 Tunable bandpass filters are needed in applications beyond 6G wireless such as cognitive radios 14 , frequency hopped receivers 15 , satellite communications 16 , base stations 17 , and multiband radar 18 .
For the frequency spanning from S band to X band (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12), numerous tunable filters have been developed to eliminate out-of-band noise while preserving the in-band signal. However, most of these filters have a limited frequency tuning range. This is also elaborated on in Supplementary Note 1. Mechanically tunable filters offer high quality factor (Q) but require external circuitry and motors for tuning and have a limited center frequency range with a maximum center frequency tuning ratio of 1.2:1 19,20,21 . RF MEMS enabled tunable electromagnetic cavity filters achieve a center frequency tuning ratio up to 2:1 and high-power handling capabilities but are relatively large and sensitive to shock and vibration 22,23,24 . Varactor based tunable filters are small, have fast tuning speeds, and moderate center frequency tuning ratio of 2:1, but suffer from low Q values 25,26,27,28 . To the best of our knowledge, the highest reported S12 quality factor (Q-factor) is about 46 at L band and is heavily dependent on frequency 26 . This limits the minimum filter bandwidth and the steepness of the filter skirts. In addition, while a tunable filter attenuates out-of-band interferer signals, the intermodulation distortion of the filter could permit out-of-band signals a path to mix into the filter passband which can deteriorate receiver signal-tonoise ratio. Although much higher linearity can be achieved for mechanical and RF MEMS based tunable filters, the out-of-band (OOB) 3 rd order input referred intermodulation intercept point (IIP3) of other tunable filters is usually in the range of 23.5 dBm 29 to 27 dBm 30 .
Tunable filters realized using magnetostatic wave resonators (MSWR) are a promising technology to fulfill the demands of broad and continuous frequency tuning range with high quality factor >1000 31 . Magnetostatic waves (MSW), also known as dipolar spin waves, are long wavelength spin waves, where the magnetic dipolar interactions dominate both electric and exchange interactions. 32 Because the MSW group velocities are slower than that of electromagnetic waves and are variable with applied bias magnetic field, magnetic field tunable magnetostatic wave filters (MSWF) with a wide frequency tuning range are possible. 32,33 Micrometer thick, single crystal yttrium iron garnet (YIG) thin films exhibit the lowest damping for MSW and thus the smallest propagation loss as compared to other common ferromagnetic materials. 34  Despite the small size and high Q of YIG MSWRs, YIG tunable MSWFs still suffer from large size, high power consumption, and slow tuning speed from the use of bulky and energy intensive electromagnets to supply the necessary magnetic bias field for MSWFs. 36 SI Note 1 shows a size comparison between this study and two leading commercial YIG-based tunable filters. These filters, incorporating electromagnet drivers for YIG sphere resonator frequency tuning, result in sizes exceeding 23 cm 3 and power consumption surpassing 2 W. These limitations hinder their applicability in Internet of Things (IoT) and mobile phone technologies.
Filters based on YIG sphere resonators have demonstrated low loss across a broad tuning frequency range (9:1).
YIG sphere resonators, however, are too large to fit within the miniature magnetic bias circuit reported here. 37,38 Previously reported planar geometry MSWR formed through standard microfabrication processes have a form factor compatible with the reported small, tunable, magnetic bias circuits. Previous filters realized from MSWR, however, exhibited 20 to 32 dB insertion loss when operating with a wide frequency tuning range between 2 to 12 GHz. 39 This is mainly due to the difficulty of obtaining large coupling and a well matched impedance across a broad frequency range. Low insertion loss planar YIG tunable filters were only demonstrated over a limited frequency range: ~5.3 dB loss with a center frequency tuning ratio of 1.5:1 in X band 40 and 5.8-6.4 dB loss with a tuning ratio of 1.5:1 in X and Ku bands 41 . In order to meet the requirements of insertion loss and out-of-band suppression, a planar MSWR with a large area of 4 mm × 10 mm 42 or a five layer stack MSWR with dimensions of 2 mm × 2 mm × 0.62 mm were needed 41 .
In this study, we demonstrate miniature, narrowband, frequency tunable filters (3.3:1) with zero static power consumption and exceptional out-of-band linearity. Fig. 1 a-c depicts the device assembly and shows images of the tunable filter assembly with a total volume of only 1.68 cm 3 . To tune the cavity center frequency, current pulses were applied to AlNiCo pieces in the magnetic bias assembly to alter their nonvolatile magnetic remanence. Using this approach, the tunable magnetic bias circuit only consumes transient power to tune the magnetic field and filter frequency and enables the frequency tunable filter to operate without any steady state power consumption.
Magnetostatic surface waves were utilized, where an in-plane magnetic bias field is established in the YIG perpendicular to the direction of magnetostatic wave propagation. The MSWF is based on cavities microfabricated in a YIG thin film with straight edge reflectors. By using planar microfabricated YIG cavities to form MSWFs, the size remains small such that the filters can fit within in the small, tunable bias assembly, which can only produce large and uniform magnetic fields over a small area. To optimize for low insertion loss, aluminum input and output transducers were placed directly on the YIG film to efficiently excite and collect the magnetostatic waves with high coupling. The geometrical parameters were also optimized based on the equivalent circuit shown in Fig. 1d to enable impedance matching to 50 Ω over the broad 3.3:1 frequency tuning range. These innovations enabled low filter insertion loss of 3.2-5.1 dB across the entire 3.4-11.1 GHz frequency tuning range with a YIG filter occupying only 200x70 µm 2 of area.

Magnetostatic Wave Resonator (MSWR):
Straight edge MSWR consisting of a ferrimagnetic resonant cavity made of a 3.3 m film of YIG grown on top of a Gadolinium Gallium Garnet (GGG) substrate were patterned into a rectangular shape by wet etching. The transducers made of 2 m thick aluminum (Al) microstrips were fabricated on top of the YIG. The transducers width is approximately 7 m unless stated otherwise. Fig. 1c shows one typical fabricated device, where the width (W) is defined as the coupling length of the Al electromagnetic transducer on the MSW, whereas the length (L) is defined as the MSW cavity length in the direction of MSSW propagation. The fabricated MSWR was first measured under the magnetic probe station where two electromagnets were used to generate the magnetic bias field, as further illustrated in Supplementary Notes 2 and 3.
Inside the YIG cavity, the MSW is stimulated by inductive antennas, inducing oscillating magnetic fields through RF currents, as shown in Fig. 1d. The structure of the YIG cavity consists of two parallel reflecting interfaces formed by wet-etched YIG edges. As a result, spin waves entering the YIG cavity circulate coherently with minimal damping. By placing a single Al transducer on top of the YIG, the YIG cavity is configured as a MSWR.
Alternatively, the MSWR employing two Al transducers produces a filter-like bandpass frequency response with high out-of-band rejection. The cross-section view depicts the unidirectional propagation of the MSW, which exclusively propagate along the surface of the YIG and are reflected onto the other surface. 43, 44 As a result, the propagation path is not reciprocal between the two ports. This MSWR response can be represented by an equivalent circuit model of a parallel RLC circuit in series with an ohmic series resistance (Rs) and a selfinductance (Ls). 31 The resonance tank within the MSWR exhibits maximum impedance at the resonance frequency.
Consequently, the return loss displays a dip at the resonance frequency. Two-port MSWF can be modeled by connecting the resonance tank with a coupling inductor Lc, which considers the direct electromagnetic coupling between the two ports where the series inductance satisfies the relationship = ′ + .
A single RLC tank circuit with magnetostatic resistor Rm, magnetostatic inductor Lm, magnetostatic capacitor Cm only captures the response at one frequency. The complete impedance response can be modeled by the multi-mode circuit model in Fig. 2a with the number of modes, p. Fig. 2b compares the modeled and measured MSWR input impedance of a MSWR with W = 200 m and L = 70 m with an Al transducer width of 4 m. The impedance of the resonance frequency, fs, is typically evident without circuit modeling. However, the single-mode circuit model plays a crucial role in accurately determining the precise frequency and magnitude of the antiresonance frequency, fp. At this point, the impedance of the fundamental mode becomes extremely small, and the influence of other modes can impede its observation. Moreover, the multi-mode circuit successfully predicts the impedance response across a wide frequency range. Supplementary Note 4 details the circuit modeling procedure.
The circuit modeling and analysis were performed directly on the measured data without any de-embedding process.
The single-mode circuit model has proven to be effective in accurately predicting bandwidth, magnitude, and phase of the impedance around the peak frequency of the MSWR. A MSWF's performance can be optimized by increasing the coupling coefficient (K 2 ), Q, and figure of merit (FoM = K 2 Q) of the MSWR mode. Thus, it has been utilized to study these important performance parameters and the magnetostatic resistance, Rm. Fig. 3 compares the influence of the width of the MSWR on these parameters, with a fixed MSWR length of 70 m.
Increasing the width of the YIG cavity results in a higher K 2 , FOM, and magnetostatic resistance, Rm,, as the coupling distance increases. A maximum K 2 = 2.4 % is achieved at a width of 600 m. The Q-factor does not show a significant change with the width of the YIG cavity. However, the Q-factor generally increases with frequency. This helps to achieve an almost constant filter bandwidth with center frequency tuning, which is one of the advantages of MSSW filters to achieve constant data rates at various frequencies. The maximum Q-factor measured is 1313, which is for the MSWR with width of 150 m at a frequency of 11.6 GHz. Future studies could explore the use of thicker aluminum transducers or new layout designs, which could increase the coupling and quality factor of the devices.
To design a MSWR with better FoM, the width effect of the Al transducers and the length effect of MSWR are also discussed in Supplementary Notes 5 and 6. Impedance matching plays a crucial role in the design of a low loss filter, as further illustrated in Supplementary Note 7. Overall, W= 150-200 m MSWR are better matched to the 50  source impedance at high frequencies whereas the W = 600 m MSWR are matched to 50  at low frequency. Although higher K 2 and FoM were achieved in the wide MSWR, the impedance mismatch causes the insertion loss for W = 600 m to be higher than that of W= 100 or 200 m when taken across the tunable frequency range.

Magnetostatic Wave Filter (MSWF)
Supplementary Note 8 illustrates the tunability of the MSWF via applied magnetic field. The relationship of the main resonance frequency with respect to the applied external magnetic bias field is linear with a slope of 2.9 MHz/Gauss. Fig. 4 shows the typical S12 frequency responses of the MSWF. All these MSSW filters exhibited less than 10 dB insertion loss with greater than 20 dB out-of-band isolation.
As was discussed in the previous section, the FoM increases with increasing width, resulting in a lower insertion loss for the wide MSWF at 3.4 GHz. However, the insertion loss of W = 600 m at 9.1 GHz is not lower than the W = 200 m filter. This is because the main resonant tank becomes over-coupled to the source impedance of 50 , While wider widths result in increased coupling to the MSW, they also lead to higher direct electromagnetic wave coupling between port one and port two. However, it is worth noting that the direct electromagnetic wave coupling can be mitigated to some extent by increasing the propagation distance of the MSW. This increased propagation distance helps to reduce the direct electromagnetic wave coupling between the ports, thereby improving the out- propagation in YIG. 48 When the input power is above P1dB and below approximately 8 dBm, the insertion loss is increasing, and the output power is saturating with increasing input power. When the input power is above 9 dBm, the direct inductive coupling between the input and output aluminum transducers dominates the insertion loss and the output once again increases linearly with input power. Such self-limiting behavior can be useful to protect receivers from damage under large in-band interference.
Supplementary Note 11 shows the in-band and out-of-band IIP3 measurements for W = 150 m and L = 70 m. Future studies with an improved IIP3 test setup or improved linearity in the aluminum transducers could achieve an increase in the out-of-band IIP3 of the tunable filters.

Magnetic Biasing Circuit
As shown in Fig. 1, the magnetic bias circuit comprises two neodymium-iron-boron (NdFeB) permanent magnets, two AlNiCo magnets wrapped with copper coils and two nickel-iron-molybdenum (NiFeMo) magnetic yokes.
The NiFeMo magnetic yokes provide a low reluctance path for magnetic flux due to low coercivity and high permeability. The NdFeB permanent magnets and the coil-wound AlNiCo magnets serve as a constant magnetic flux source and a tunable magnetic flux source, respectively. Compared with NdFeB material, AlNiCo material has a lower coercivity 49 . Therefore, the AlNiCo magnets can be magnetized and demagnetized by applying a pulse of current though the coils surrounding the AlNiCo material. Also, due to the high magnetic remanence of AlNiCo, the AlNiCo magnets can still retain magnetism and provide magnetic flux for the circuit after the end of the current pulse.
To study the tuning range of the magnetic field, the magnetic flux density in the middle of the two yokes where the YIG chip sits was measured using a Gaussmeter. Supplementary Note 13 describes the process of capacitor charging and current flow through the coil to magnetize the AlNiCo magnets. By charging the capacitor to a specific voltage and discharging it through the coil, a current pulse is generated for magnetization. The remanent flux of the AlNiCo magnets can be adjusted by controlling the capacitor charging voltage and the resulting current pulse amplitude, allowing for precise tuning of the magnetic field. Fig. 5a shows the measured magnetic flux density along the vertical direction at different charging voltages.
The origin of the vertical position is where the magnetic flux density reaches the maximum value, which is around the middle of the yoke. Due to the 2 mm thickness of the magnetic yoke, the magnetic field does not change from -1 mm to 1 mm. This establishes a uniform magnetic bias field for the YIG filter cavity which is required for proper filter operation. Supplementary Note 14 describes a separate device in which a 0.5 mm thick yoke was utilized, resulting in a non-uniform magnetic field. This non-uniformity introduces additional losses in the integrated filter. With increasing capacitor charging voltage, the current for magnetizing the AlNiCo magnets increased, and thus the magnetic field generated by the AlNiCo magnets increased. The total amount of the magnetic flux in the middle of the two yokes was equal to the sum of the magnetic flux generated by the two NdFeB permanent magnets and the magnetic flux generated by the two AlNiCo magnets. For positive capacitor charging voltage, the field direction of the magnetized AlNiCo magnets was in the same direction as the NdFeB magnets. Due to the saturation of AlNiCo magnets, when the charging voltage was larger than 50 V, the effect of increasing the capacitor charging voltage on increasing the magnetic field became less. Finally, the tunable filter was realized by placing the MSWF chip in the center of the magnetic biasing circuit. About 0.7 J of energy is needed to switch from the minimum to the maximum bias field. The tunable filter assembly has a total dimension of 20 mm × 12 mm × 7 mm and occupies a volume of only 1.68 cm 3 .

Integrated device
A MSWF with W = 200 m and L = 70 m with Al transducer width of 4 m was laser diced and placed in the center of the gap of the magnetic biasing circuit. The filter was first measured inside a magnetic probe station with electromagnetic coils to provide the bias magnetic field. The frequency response was compared before and after integration with the tunable magnetic bias circuit. As shown in Fig. 5b-d, the S12 frequency response and insertion loss remains unchanged between 3 GHz and 12 GHz in both the magnetic probe station and magnetic biasing circuit assemblies. The out-of-band rejection is greater than 25 dB and the insertion loss is less than 5.1 dB with an average across the tunable frequency range of 4 dB. The magnetic bias circuit measurements show a 7.7 GHz filter tuning range with a frequency tuning ratio of 3.3 achievable with an 80 V programming range.

Discussion
In conclusion, we have demonstrated miniature and narrowband tunable filters with zero static power consumption, exceptional out-of-band linearity, and a frequency tuning range from 3.4 to 11.

Magnetic circuit fabrication:
The three different components of the magnetic bias circuit were prepared separately and then assembled. To form the magnetic yokes, a 2 mm thick NiFeMo sheet was cut using a 532 nm green laser to the required shape as shown in Fig. 1. The main body and protrusion part of the yokes have a size of 20 mm × 3 mm and 2 mm × 1.1 mm, respectively. The NdFeB permanent magnets had a dimension of 3.175 mm × 3.175 mm × 3.17 5mm and were purchased from K&J magnetics. The AlNiCo magnets had a dimension of 12 mm × 3 mm × 2 mm and were cut from a bulk AlNiCo bar using an electric discharge machining (EDM) copper wire with a diameter of 0.2 mm was wound around each AlNiCo magnet manually to achieve a total number of turns of 50. After all the magnetic parts were prepared, the two NiFeMo yokes and two NdFeB magnets were assembled and fixed on an acrylic substrate using epoxy. Then the coil-wound AlNiCo magnets were placed on the yoke and fixed using epoxy.

Measurement setup:
The YIG sample was characterized using a magnetic probe station (MicroXact's MPS-1D-5kOe). The magnetic field was generated by electromagnets inside the magnetic probe station. A Gaussmeter (Model GM2, AlphaLab Inc) was used to calibrate the magnetic probe station and the filter frequency responses were measured using a vector network analyzer (Keysight, P9374A).

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request. In Supplementary Figure 1, a size comparison between this work and commercial YIG spheresbased tunable filters, namely Teledyne F series and Micro Lambda Wireless MLFR series, is illustrated. 5,15 The images are depicted at the same scale. The YIG sphere-based filters are notably larger due to the electromagnets required to adjust their resonance frequency. Supplementary

Supplementary Note 4: Circuit Modeling for Magnetostatic Wave Resonators (MSWR)
Following the fabrication and measurement of the two-port MSWF, using the magnetic probe station, the performance of the fabricated MSWR and MSWF were evaluated by studying the circuit model of the devices. In the measurement of the MSWF, the full S-parameters of the two ports were recorded. To determine the characteristics of the one-port MSWR device, we utilized the values obtained from either 11 or 22 of the MSWR, representing the impedance when the other port is open.
As outlined in Supplementary Note 6, the MSSW filters exhibit multiple modes owing to the highly dispersive nature of spin waves in a planar rectangular YIG cavity. Consequently, these different modes are closely clustered together in frequency. Unlike spurious-free single mode resonators, the frequency response of MSSW filters can display numerous local maxima and minima. The anticipated behavior of an ideal resonance, characterized by maximum impedance at the resonance frequency ( ) and minimum at anti-resonance frequency ( ), becomes challenging to observe. Consequently, the accurate extraction of circuit parameters for MSWR becomes a laborious task. This issue is less pronounced in devices with low coupling (K 2 ), primarily because the narrow frequency spacing between and results in fewer spurious modes within that range.
In a conventional acoustic resonator, a multi-mode Modified Butterworth-Van Dyke (MBVD) circuit can be established by introducing several parallel branches of the motional resistor , the motional inductor , and the motional capacitor to the main mode resonance tank. The circuit parameters derived from these additional parallel branches enable the calculation of the coupling coefficient and quality factor of the spurious mode. 16 Inspired by the multi-mode MBVD circuit, Figure 2 illustrates the circuit diagram of MSWR with a total of p modes. In each resonance tank (designated as the nth tank in the circuit), there is a parallel magnetostatic resistor , one magnetostatic inductance ,and one magnetostatic capacitor . These resonance tanks are connected in series, the impedance of the resonance tank in the MSWR reaches its maximum at resonance and returns to a minimum at frequencies away from the resonance. This behavior contrasts with traditional acoustic resonators, where additional branches would remain open when only the main branch is resonating. In MSSW resonators, the additional branches remain electrically shorted when only the main branch is resonating. This unique property of MSWR leads to the 11 parameter of both the MSWR and MSWF exhibiting a bandstop characteristic, indicating the 11 impedance reaches the maximum at resonance frequency. Additionally, the 12 parameter of the MSWF shows a bandpass characteristic, indicating the 12 impedance reaches the maximum at resonance frequency.
As for a single mode MSWR, the impedance of a resonator is the series combination of series resistor and series inductance and the resonance tank. It can be expressed as follows.
= + _ = + + 1 1 + 1 + (S1) e total impedance of an MS R's tan consisting of a total of modes can be expressed by combining the impedance of each individual resonance tank, denoted as: . Consequently, the total impedance of a MSWR, , can be described as follows: With the establishment of the multi-mode circuit model for MSW devices, the process flow employed to extract the circuit parameters is described in Supplementary Fig. 4. The determination of these parameters assumes a critical role in precisely characterizing the device's behavior and facilitating performance optimization.
The procedure commences by importing the measured impedance data, . Subsequently, the and are calculated from the imaginary and real parts of the off-resonance impedance, respectively. When the frequency significantly deviates from the resonance, the overall 11 impedance of MSWR can be expressed as, .
Subsequently, the impedance response is analyzed to identify the frequency peaks of interest. These peaks, usually corresponding to local maxima with the highest magnitude, are then ranked based on their impedance values. For instance, the first peak, representing the resonance with the largest impedance, is used to calculate the circuit parameters of the resonance tank associated with the largest. The second peak represents the second-largest resonance impedance, and so on. By examining the resonance peaks in descending order, it becomes possible to observe and analyze the impedance changes of smaller resonance tanks, as the significant impedance of larger resonance modes can conceal the impedance response of smaller modes. To accurately determine the fitting frequency range, a local minimum preceding the selected peak and a local minimum following it are identified. This frequency range is crucial as it captures the dominant influence of the corresponding resonance tank's impedance on the overall circuit behavior. By focusing on this range, a precise characterization of the specific resonance mode can be achieved.
After that, the initial guess parameter for the resonance tank circuit parameters: , , and can been estimated for the initialization of the multi-mode circuit model recursive fitting. The initial condition for is t e frequency response's maximum magnitude of t e impedance. The Q can be approximated as the ratio of over t is pea 's 3-dB bandwidth. The initial condition for and can be expressed as: After obtaining the initial values for , , and , the random search process is employed to explore the parameter space for these three circuit parameters. This process begins by generating a series of three sets of random numbers that deviate from the initial three parameters. These random numbers serve as the test parameters for the subsequent analysis. By incorporating these test parameters into the circuit model, the corresponding impedance response of can be computed as the combination of the impedance of the series components with the impedance of the previously fit resonance tank, if any, and the resonance tank of interest with random generated test parameters.
In order to compare between the calculated impedance response and the measured impedance data, an error function of ( , ) can be calculated from the magnitude of the fit data, ( ) and the measured data, ( ) and the phase of the fit data, ℎ ( ) and the measured data, ℎ ( ), which is defined as: The normalization function, denoted as ( ), serves to eliminate the absolute scale difference between phase and magnitude. It can be mathematically expressed as follows: In this equation, is the data of interest. The ( ) and ( ) are the average and standard deviation of the data.
The error function is utilized to evaluate all the test parameters generated during the random search process. The test parameters associated with the smallest error function value are saved, as they indicate a closer match between the modeled impedance response and the measured data. This process of random number generation and error function evaluation is repeated multiple times until the error function converges, indicating a satisfactory level of parameter optimization. Once the error convergence is achieved, the circuit extraction process concludes with key fitting results for , , and . These results represent the optimized circuit parameters that best align with the measured impedance data. Subsequently, the recursive fitting procedure moves on to the next peak of interest and performs the fitting for the resonance tank associated with the next mode. This iterative approach ensures comprehensive characterization of each resonance mode and the determination of their respective circuit parameters.  Another application for the circuit model is to understand the effect of series resistance and inductance, as the spin wave Q-factor can be calculated from the circuit model. The spin wave Qfactor is defined as:

Supplementary
Supplementary Figure 6 illustrates the spin wave Q-factor of various MSWR with different widths. The spin wave Q-factor and device Q-factor follow similar trends with frequency. The maximum spin wave Q-factor of 1777 for W = 150 m, L = 70 m at 11.5 GHz can be achieved. Compared to device Q-factor, spin wave Q-factor removes the impact of the Al trace resistance on overall device performance and is about 30 % ~ 50% higher than the device Q. This indicates significant energy is dissipated through the series resistance.
Supplementary Figure 6. Comparison of MSWR spin wave Q-factor for different width of YIG cavity.

Supplementary Note 5: Width Effect of the Aluminum Transducers
In order to achieve a low insertion over a broad frequency tuning range, the effect of the width of the aluminum (Al) transducers has been studied. A 2-dimensional finite element simulation was performed in COMSOL 17 to simulate the total integration of the magnetic flux density in the YIG when a DC current is applied to the aluminum trace line. The aluminum transducer was simulated with a trapezoidal shape to better represent the wet etching profile. The change in series resistance of t e aluminum transducers wit Al widt is muc less t an 1 Ω w en t e widt of t e aluminum transducer decreases from 19 m to 4 m. Thus, a constant current assumption of 1 A was used for the simulation. Supplementary Figure 7 shows that a narrower Al transducer achieves higher magnetic flux, which is mainly due to the increase of the y component of the magnetic flux.
Supplementary Figure 7. COMSOL simulation results on the magnetic flux inside the YIG with respect to the width of the aluminum transducer.
As a result of the increased magnetic field inside the YIG cavity, the Rm and insertion significantly improve between 3.4 ~ 12.9 GHz, as shown in Supplementary Figure 8. The increased FOM does not translate into an improvement in insertion loss for frequencies above 12.9 GHz because the MSWR resonator is over coupled to t e 50 Ω source impedance.
Supplementary Figure 8. Effect of the aluminum transducer width on the (a) radiation impedance and (b) insertion loss. The MSWF is designed with constant W = 200 m and L = 70 m.
Supplementary Figure 9 shows a typical frequency response for different Al transducer widths. A narrower aluminum transducer slightly reduces the out-of-band attenuation of the tunable filters as the increase of the current density in the aluminum trace slightly enhances the direct inductive coupling between the two ports.
Supplementary  Figure 3 (d) illustrates that Rm increases with the external magnetic field. For MSWR with W = 150 m, Rm is greater than 50 Ω for frequencies above 7.6 GHz and increases to 77 Ω at the frequency of 11.5 GHz. For the MSWR with W = 600 m, the Rm is greater than 50 Ω for all frequencies with a minimum of 115 Ω at 3.4 GHz and the maximum of 363 Ω at 11.6 GHz. This difference in Rm causes different degrees of impedance mismatch for the MSWF vs. frequency.

Supplementary Note 7: Impedance Matching of the Magnetostatic Wave Filters (MSWF)
Supplementary Figure 11 shows the return loss of the MSWF at the frequency where the S12 reaches its peak. Due to the increase of the magnetostatic resistance of the MSWF with W = 600 m, this return loss decreases from 18 dB and 22 dB at 3.4 and 6.3 GHz, respectively, to 14 dB and 8 dB at 9 GHz and 11.6 GHz, respectively. As a contrast, the MSWF with W = 150 m shows a minimum return loss of 6 dB at 3.4 GHz and achieves its maximum of 13 dB at 10.2 GHz. For the MSWF with W = 150 m or W = 200 m, the maximum of S11 is at the same frequency with the maximum of S12, indicating the same fundamental mode. However, the large maximum Rm of the devices with wider YIG cavities caused the maximum return loss to occur at a higher order mode of the MSWF where the Rm of that higher order mode is closer to the 50  termination impedance. At the fundamental mode where the maximum impedance is achieved, most of the signal is reflected back to the source and thus the insertion loss of the fundamental mode is high.
Supplementary Figure 12 depicts the insertion loss of the MSWF w en terminated wit 50 Ω. Due to the difference in the degree of the impedance matching, the insertion loss of the filters with a narrower YIG cavity (W = 150 or 200 m) are more consistent across the frequency band whereas the insertion loss of the wide YIG cavity (W = 400 or 600 m) decreases with increasing frequency.
To reduce the loss of the W = 600 m devices across the filter band, an external series inductor and parallel capacitor can be used to achieve better matching to the 50  termination impedance, but this requires matching networks that are tunable with frequency. Therefore, for the purpose of achieving a broad frequency tuning range of the filters, the MSWR should not only achieve the maximum FOM but also maintain good impedance matching to the 50  source impedance across the tunable frequency range.

Supplementary Note 9: Mode analysis of Magnetostatic Wave Resonators (MSWR)
Because the MSSW travels along the surface of the YIG film and is reflected to the other side of the YIG surface at the straight edges, the resonant conditions are when the following equation is met: 20 2 = 2 = 1,2,3, … ..
Where is the average wavenumber for the top and bottom surfaces. As the length increases, the spacing between the resonance modes decreases in wavenumber, due to the strong dispersion of the MSSW. Although a wider MSWR can provide higher FoM and lower insertion loss, it also contains additional spurious responses. The dispersion relation with the width modes can be expressed as: 21 (2 ) = + + ( 2 − 2 )( − ) − + ( 2 − 2 )( + ) Inside the film, we have 2 = k 2 = ( ) 2 1 + 2 (S10) And outside 2 = k 2 = ( ) 2 + 2 (S11) where 1 = 1 − Ω /(Ω 2 − Ω 2 ) , Ω = ω/ν4π , Ω = H/4π , ω is the angular frequency, ν is the gyromagnetic ratio of 2.8 MHz/Oe for YIG, and 4π Ms is the saturation magnetization of the YIG film which is 1780 Gauss, H is the external magnetic field applied, t is the distance between YIG film and ground plane which can be chosen as any arbitrary large number here.
The calculated results are shown in Supplementary Figure 14 and the measurements of the S12 frequency response for the MSWFs with widths of 200 m, 400 m, and 600 m are shown in Supplementary Figure 15. Because the width of the MSWF is much smaller than the electromagnetic wavelength at these frequencies and the current remains almost constant along the transducer, only odd order width modes can be excited. The frequency spacing between two adjacent width modes increases for narrower YIG cavities. The frequency response of W = 600 m and W = 150 m confirms this result. There is also a frequency shift for the main resonance mode with increases in width, as can be seen from the dispersion curve where a constant wavenumber corresponds to a higher frequency as the width increases. As both theoretical calculation and experimental measurements confirm, the MSSW becomes more dispersive at higher frequency for wider MSSW filters.
Supplementary Figure 16 and 17 illustrate the S11 frequency response of devices with varying widths and lengths. Consistent with the earlier findings, it can be observed that longer and wider MSWFs exhibit more spurious responses. Additionally, wider MSWFs display a lower minimum value of S11. However, the length of the MSWF does not significantly affect the minimum value of S11, as the increase in Rm is primarily associated with the width rather than the length. These observations further support the conclusions regarding the influence of width and length on the spurious responses and impedance characteristics of the MSWRs. Supplementary

Supplementary Note 13: Capacitor Voltage and Coil Current During Capacitor Discharging
To generate a pulse of current for magnetizing/demagnetizing the AlNiCo magnets, a capacitor was used. First, the capacitor was charged to a voltage chosen to produce the desired magnetic field. Subsequently, it was connected to the coil and discharged, which produced a current response according to a series RLC circuit. Supplementary Figure 27 shows the measured voltage across the capacitor and current flowing through the coil during the capacitor discharging. The capacitor used in the experiment had a capacitance of 270 F and was charged to 19 V. The peak current flowing through the coil was approximately 27 A. The amplitude of the current pulse is proportional to the capacitor charging voltage. By controlling the capacitor charging voltage and the current pulse amplitude, the remanent flux of the AlNiCo magnets can be adjusted. Therefore, tuning of the magnetic field at the YIG chip is realized.
Supplementary Figure 27. Capacitor voltage and coil current during capacitor discharging.
Supplementary Figure 29. Comparison of the frequency response of the MSWF inside the magnetic bias circuit and magnetic probe station where the applied magnetic field of the magnetic bias circuit is nonuniform. (a) S11 frequency response, (b) S12 frequency response of the width = 150 m, length = 70 m MSWF.