Nonlinear Transmission Line Performance as a Combined Pulse Forming Line and High-Power Microwave Source as a Function of Line Impedance

Abstract

Nonlinear transmission lines (NLTLs) offer compact, low-cost, all solid-state high-power microwave (HPM) generation. This article experimentally investigates the RF output power for composite-based 10, 25, and 50 Ω NLTLs used as a combined pulse forming line and HPM source. We manufactured coaxial NLTLs containing 10% barium strontium titanate and 15% nickel zinc ferrite encased in polydimethylsiloxane. The output voltage and power in the time and frequency domains, respectively, showed that the 10 Ω NLTL generated the greatest RF output. The 25 Ω NLTL generated greater output power from 500–1100 MHz than the 50 Ω NLTL. This occurs because reducing the NLTL impedance induces a larger transient current for a given charging voltage. This transient current corresponds to a stronger transient magnetic field, which facilitates magnetic moment alignment to allow for coherent magnetic moment rotation to occur. This setup eliminates the separate pulse forming network and magnetic field bias that typically occurs in other NLTL systems, which provides additional flexibility in tuning the NLTL impedance and reducing device footprint.

1. Introduction

Over the past several decades, high-power microwave (HPM) devices have transitioned from large scale laboratory systems to more practical, commercially available devices [1]. This progress has motivated the development of compact, solid-state equipment capable of generating HPM [1,2,3]. One such solid-state device under investigation for generating HPM is the nonlinear transmission line (NLTL) [2], which can achieve repetition rates above 1 kHz with consistent radiofrequency (RF) output [4,5]. In tandem with their HPM generation capabilities, NLTL systems can also sharpen input pulses by using materials whose permittivity and/or permeability varies with electric field and/or magnetic field, respectively. For ferroelectric and ferromagnetic materials, the inverse relation between phase velocity and a medium’s electromagnetic properties causes the crest of an electromagnetic pulse to propagate faster than the base, sharpening the pulse. HPM generation is realized by different means depending on topology.

Gyromagnetic NLTLs (GNLTLs) use gyromagnetic precession to modulate incident pulses into RF waveforms as wideband sources [6]. One recent study used nonlinear coaxial transmission lines with magnesium manganese ferrite rings with a rectangular hysteresis loop to achieve a wide frequency range (~5.5–12.7 GHz) and peak microwave oscillations up to 350 kV [7]. This approach may potentially eliminate the need for an external biasing field and may achieve oscillation frequencies up to 20 GHz [7]. Another recent GNLTL system attained a maximum peak power of 110 MW at a central frequency of 1.3 GHz by periodically placing nickel zinc ferrite (NZF) saturated by an axial magnetic field produced by permanent magnets inside the NLTL [8]. More compact GNLTLs (60 cm) can generate a broad frequency spectrum from 200 MHz to S-band with RF conversion efficiency above 10% [9].

Alternatively, one may use nonlinear capacitance to modulate an input pulse and induce RF formation. While ceramic capacitors can be used, the resulting system frequencies are restricted to ~100 MHz at high power due to parasitic inductances [10]. This has motivated the use of silicon carbide Schottky diodes to achieve frequencies up to 200 MHz [10]. Another modality leveraging nonlinear permittivity uses ferroelectric materials, such as barium titanate (BT) or barium strontium titanate (BST) [11]. An applied pulse moves an atom in the crystalline structure to another stable position, which rotates the atom’s polarization vector [12,13]. The frequencies for ferroelectric NLTLs are generally on the order of tens of MHz, compared to GHz for ferromagnetic NLTLs, because the inertia of the atom in a ferroelectric line exceeds the inertia of the electron in a ferromagnetic line [14].

More recent studies have focused on hybrid NLTLs, which use both nonlinear capacitance and inductance [1]. One hybrid NLTL constructed using commercial off-the-shelf components produced RF between 55 MHz and 80 MHz for 5 kV and 8 kV pulse voltages, respectively, with 600 ns pulse durations [15]. A lumped element hybrid NLTL composed of BT dielectric nonlinear capacitors and ferrite bead inductors generated ten RF cycles with frequency ~33 MHz for each input pulse [16], producing higher voltage modulation depth compared to NLTLs with a single nonlinear component [17]. In addition to nonlinear capacitors in the lumped element topology, other studies have assessed nonlinear resistances (as a function of voltage) with gas gap switches [18] or periodically placed high voltage silicon carbide Schottky diodes as switches [19].

An alternative approach to the lumped element topology is to construct a coaxial line made of a composite with both nonlinear dielectric and magnetic inclusions. Adjusting the volume loading of these inclusions provides tunability in both linear [20,21,22] and nonlinear [23] electromagnetic properties for adjusting NLTL output waveforms. Knowing this, we developed a compact system that eliminated the need for a separate pulse forming network (PFN) and used the composite-based NLTL simultaneously as a pulse forming line (PFL) and HPM source [24]. For a 10 Ω NLTL comprised of 15% BST/10% NZF, we achieved 160 kW with a 15 kV charging voltage without a biasing magnetic field. Because the NLTL in this combined PFL/HPM format does not require an external PFN, the user may, in principle, select any impedance for the NLTL without needing to match the output of the PFN. Since standard NLTLs are 50 Ω to match the remainder of the system, this raises the question of the importance of impedance on NLTL output for this design.

This paper shows that increasing the NLTL impedance reduces the RF output because it reduces the current through the NLTL, concomitantly reducing the generated magnetic field and induced shock wave. Section 2 summarizes the experimental setup and NLTL configurations to achieve 10, 25, and 50 Ω NLTLs composed of 15% NZF/10% BST. Section 3 provides the results and discussion. We make concluding remarks in Section 4.

2. Materials and Methods

Composites containing 10% BST (TPL Inc. HBS-8000) and 15% NZF (Powder Processing & Technology FP350) inclusions encased in polydimethylsiloxane (Sylgard^TM 184, PDMS, Midland MI, USA) were manufactured. X-ray diffraction and scanning electron microscopy were used to assess composition and inclusion aspect ratio. The compositions of NZF and BST were ${\mathrm{Ni}}_{0.5}{\mathrm{Zn}}_{0.5}{\mathrm{Fe}}_2{\mathrm{O}}_4$ and ${\mathrm{Ba}}_{0.45}{\mathrm{Sr}}_{0.55}{\mathrm{TiO}}_3$, respectively. SEM showed that BST aspect ratios tended towards unity while the NZF particles were slightly greater than unity. Additionally, BST particles were on average 500–700 nm in diameter while NZF particles were 10–20 µm in diameter. Due to the small BST inclusion size, smaller samples (50 mm in length) were first made to evaluate the efficacy of the manufacturing procedure by using 3D-X-ray microscopy (XRM) to assess composite homogeneity.

Composites were made by first weighing out a base of the two-part silicone mixture. The necessary mass of both BST and NZF inclusions for the desired volume fraction could then be calculated. The BST and NZF inclusions were then added to the base, and stirred using a glass rod for 5 min. The mixture was then inserted into a planetary centrifuge (Thinky Mixer AR-100) for 5 min. The planetary centrifuge further homogenizes the mixture and removes any air bubbles introduced by hand stirring. We next placed the mixture in an ultrasonic bath (Crest Ultrasonics CP200HT) for 4 h to break up any conglomerations. We allowed the mixture to cool before hand stirring in the appropriate amount of curing agent (1/10th PDMS mass). The cooling step is crucial since the material’s Pot life decreases drastically with temperature. The material was then outgassed for 30 min at 0.1 MPa below atmosphere to allow for any air to vacate the mixture.

A custom coaxial mold was fabricated out of aluminum to cast the sample in. The mold had a center conductor with a 3 mm diameter and an outer diameter of 7 mm. Before injecting the mold with the composite mixture, a silicone mold release was applied and allowed to dry for 30 min to facilitate sample removal. The mixture was then injected into the mold from the bottom up. We then outgassed the mold for 5 min to remove any air bubbles introduced during the injection process. The samples were then placed in an oven (Thermo Scientific Herathem OGS180) for 2 h at 100 °C to allow for the samples to fully cure. We then performed XRD on the samples to assess the effectiveness of the manufacturing procedure.

While some small conglomerations of BST (<1 mm in width) were present, the mixtures were generally well dispersed with no evidence of voids or air bubbles. Figure 1 shows X-ray microscopy images for a representative composite containing 10% BST and 15% NZF.

With the sample homogeneity satisfactory for our application, we next measured the linear and nonlinear permeability [21,23] and linear and nonlinear permittivity [21,25]. These parameters are critical for NLTLs because they are used to calculate the line impedance. The saturation impedance of a lossless coaxial structure with nonlinear loading is given by

$$Z_{sat}=\frac{\ln \left(b/a\right)}{2\pi }\sqrt{\frac{{\mu }_{sat}{\mu }_o}{{ϵ}_{sat}{ϵ}_o}}$$

(1)

where $b$ is the diameter of the outer conductor, $a$ is the diameter of the inner conductor, ${\mu }_{sat}$ and ${ϵ}_{sat}$ are the relative saturation permeability and permittivity, respectively, and ${ϵ}_o$ and ${\mu }_o$ are the permittivity and permeability of free space, respectively. From previous studies [23,24], a 15% NZF/10% BST composite had ${\mu }_{sat}=1.15$ and ${ϵ}_{sat}=7.2$.

We extended the manufacturing procedure for producing the smaller samples to full size NLTLs by replacing the aluminum mold with the NLTLs outer and inner coaxial electrodes. This eliminated the requirement for mold release since the composite did not need to be removed and the composite could be form fitted to different sized center conductors. An additional outgassing step was added after filling the NLTL to ensure no air bubbles were present in the line. Aluminum caps were used to fix the center and outer conductors in place while the composite cured. The impedance of the manufactured NLTLs was altered for a fixed composite loading by varying the diameter of the center conductor with the inner diameter of the outer conductor fixed at 25.4 mm. The diameters of the center conductor for the 50 Ω, 25 Ω, and 10 Ω NLTLs were 3.7 mm, 9.5 mm, and 14.7 mm, respectively. Using (1), these parameters give saturation impedances of 13.1 Ω, 23.5 Ω, and 46.1 Ω.

Figure 2 shows a photograph and a schematic of the experimental setup. While our previous experiments focused on various compositions of BST and NZF [22,23], this study used a mixture of 10% BST and 15% NZF inclusions in polydimethylsiloxane (Sylgard^TM 184, PDMS), which was chosen to match the highest NZF loading in our previous study [22]. The reasoning for selecting this volume fraction is further discussed in the Discussion section but the primary motivation was that NZF exhibits greater nonlinearity relative to BST at the field strengths presented here.

We charged the NLTLs with DC voltages ranging from 6 kV to 20 kV using a high voltage supply (Glassman model EJ40P15) by attaching the output of the HV supply to the center conductor of the NLTL. The opposing end of the center conductor was connected to a spark gap switch (SGS), which was designed and constructed in-house using a Formlabs3 + 3D printer and Formlabs durable resin. The switch was pressurized using dry nitrogen (Indiana Oxygen, Lafayette, IN USA). The spark gap switch utilized round electrodes which were fabricated out of brass. Gap spacing was fixed at 3 mm while the pressure was adjusted between 10 and 65 psi to achieve the desired standoff voltage. The lines were terminated with the appropriate resistive load, made in-house using HVR RT series non-inductive resistors (HVR Advanced Power Components, Cheektowaga, NY, USA), to match the impedance of the NLTL. Due to power handling concerns, the loads were made by making four branches of four resistors in series. Each branch was then connected in parallel to achieve the appropriate load termination. Figure 2 shows an example of this setup with four 40 Ω branches, each consisting of four 10 Ω resistors in series, connected in parallel to form the load for the 10 Ω NLTL. Additionally, the output waveforms for each line were measured using a voltage divider, which was attached to the output of the spark gap switch. The voltage divider consisted of HVR RT series resistors with a 30 dB attenuator. The output waveform was viewed using a 4 GHz oscilloscope (Tektronix MSO64, Beaverton, OR, USA).

3. Results

Oscillation formation correlated strongly to NLTL impedance in the combined PFL and HPM configuration. We observed weak modulation on the flat tops of the pulses generated using the 10 Ω and 25 Ω lines and no modulation on the pulses generated using the 50 Ω lines. In conjunction with oscillation formation, significant pulse sharpening occurred for all lines. When comparing the highest charging voltage to the lowest for each impedance, the rise time, taken from 10% to 90% of the peak amplitude on the rising edge, decreased by 2.4 ns, 2.7 ns, and 2.5 ns for the 10 Ω, 25 Ω, and 50 Ω lines, respectively. Observable oscillations formed on the 10 Ω line for charging voltages as low as 10 kV, while a charging voltage of 15 kV was required to induce noticeable oscillations for the 25 Ω line. Interestingly, strong oscillations form after the initial pulse on the 10 Ω line with an RF burst duration of ~5 ns.

Figure 3 details the progression of the output waveform for different charging voltages for the 10 Ω, 25 Ω, and 50 Ω NLTLs. While pulse sharpening occurs for all NLTL impedances, oscillation formation does not. No oscillations occur for the 50 Ω NLTL for any of the charging voltages considered. Reducing the NLTL impedance to 25 Ω resulted in observable oscillation formation on the flat top of the pulse. Further reducing the NLTL impedance to 10 Ω resulted in oscillations on the flat top of the pulse and after the pulse for charging voltages of 15 kV and higher. The waveform profile presented in Figure 3a is consistent with the results for other 10 Ω composite-based NLTLs studied in the combined PFL/HPM source configuration [24].

Figure 4 shows the power spectrum density analysis on the output waveforms of the 10, 25, and 50 Ω lines for a 20 kV charging voltage. For the 10 Ω line, lobes are observed primarily at 1.08 and 1.43 GHz with peak power of 48.1 dBW (64 kW) and 39.34 dBW (8.5 kW), respectively. The lobes have bandwidths, which are taken as −3 dB down, of 76 MHz and 102 MHz. The power spectra for the 25 Ω and 50 Ω lines are very similar with differences between 500 MHz and 1 GHz. For the 25 Ω line, lobes are present at 513 MHz, 762 MHz and 943 MHz that are not present for the 50 Ω NLTLs. This difference is attributed to the weak RF modulation in the 25 Ω line. Further supporting this argument is the period of the observed modulation for a 25 Ω line charged to 20 kV lies between 1.3 and 2 ns, making 500 MHz to 1 GHz the region of interest when looking for power spectrum differences between the 25 Ω and 50 Ω NLTLs.

Additionally, the 10 Ω NLTL yielded greater power output at frequencies between 1.5 and 2.3 GHz compared to the other NLTL impedances. For example, at 2 GHz, the output of the 10 Ω line is 17 dBW, while the outputs of the other lines are ~−40 dBW, a 57 dB difference.

4. Discussion

The NLTLs presented here utilize a combination of both ferroelectric and ferrimagnetic materials. As such, one naturally wonders the degree to which each nonlinear material is responsible for oscillation formation. The power spectrum density and the resulting frequencies of enhancement suggest that the ferrite in the line is primarily responsible for the observed oscillations because the low-GHz frequencies generated are similar to the frequencies generated by other gyromagnetic lines [26]. For ferrites, the damped gyromagnetic precession of the electrons’ magnetic moment modulates the incident pulse; however, for ferroelectrics, the motion of the induced dipole moment induces the modulation [21]. The magnetic moments may precess at much faster rates compared to dipole moments, generating higher frequencies. In this setup, we hypothesize that the output oscillations are directly tied to the impedance of the line because a larger impedance correlates to a lower transient current, which correlates to a lower transient magnetic field. This, combined with the mechanism of pulse formation, results in a more effective gyration of the material’s magnetic moments as the pulse propagates. The gyration of the material’s magnetic moment modulates the pulse and forms oscillations through gyromagnetic precession.

Lines that utilize ferroelectrics are typically limited to output frequencies of hundreds of MHz, making it unlikely that BST causes the low GHz lobes observed in the power spectrum of the 10 Ω line [2]. Thus, while BST is crucial for tuning impedance, minimizing loss tangents, and preventing breakdown within the line [21], it does not necessarily provide additional nonlinearity under these conditions [25]. Additionally, the volume-averaged electric field suggests that BST is not in the nonlinear regime at these charging voltages [27,28]. Thus, we conclude that the gyromagnetic precession of the ferrite’s moments is most likely responsible for the RF formation at these charging voltages. Future efforts will assess higher charging voltages that will push BST into a more nonlinear region so that the BST and NZF can harmoniously aid in RF generation to enhance NLTL performance [25]. We hypothesize that if both the NZF and BST can harmoniously modulate the pulse, a broader power spectrum will result. We are currently working on preventing flashovers at the spark gap of our system to experimentally test this hypothesis.

NLTLs that utilize ferrites frequently use an external magnetic field as an auxiliary system to allow for coherent precession of the magnetic moments; however, we do not require one here. We have hypothesized that the observed oscillations result from magnetic moment precession; however, our experimental setup does not use an external field to allow for the moments to precess coherently. This provides the benefit of reduced system weight and complexity but warrants a discussion on how such a result is possible. We conjecture that this behavior arises due to the mechanism of PFL operation coupled with the absence of a biasing magnetic field.

In this setup, all magnetic moments in the line are randomly oriented initially and the line is charged to a voltage $V_c$. Once the spark gap triggers, a voltage of $V_c/2$ is applied to the load and a voltage step travels down the line toward the power source. Upon reaching the source, the voltage step is reflected due to the high impedance, which in our case is 5 MΩ, and propagates back up the line uninverted. This gives a pulse width of

$$P_w=\frac{2L}{c_m}$$

(2)

where $L$ is the NLTL length and $c_m$ is the speed of light in the NLTL medium. Additionally, when the line discharges, the current through the load is given by

$$I_{load}=\frac{V_c}{2Z_{NLTL}}.$$

(3)

Since the magnitude of this current translates to the transient magnetic field strength, NLTLs with lower impedance will have greater magnetic moment alignment during the initial pulse formation due to the higher transient magnetic field. This ultimately enhances RF formation on the subsequent reflected pulse through a more coherent precession of the magnetic moments as the pulse propagates back toward the load.

This hypothesis also aids in understanding why the stronger oscillations observed on the output waveform follow the pulse. This behavior was also observed for other NLTLs used in this configuration [24] and was supported by LTspice simulations [24]. While weaker modulations occur on the flat tops of these pulses, which is the common location for the RF burst to occur for gyromagnetic lines, much stronger oscillations form after the pulse. This succeeding RF burst is much higher in peak power compared to the oscillations on the top of the pulse. We hypothesize that it is necessary to achieve a threshold current, and a concomitant threshold transient magnetic field, to sufficiently align the magnetic moments for this setup. Thus, we suspect that stronger oscillations will occur for 50 Ω and 25 Ω NLTLs when magnetic field strengths become comparable to those generated in the 10 Ω NLTL for the voltages used here. This threshold would theoretically differ for other ferrites/composites since the hysteresis behavior of the ferrite depends on numerous factors, such as composition, volume loading, and temperature.

Although not explicitly discussed in this paper, the results presented here also suggest the importance of future material development. This study and our previous NLTL system studies [24,29] have focused on various loadings of BST and NZF. However, other nonlinear materials could be used or developed to adjust nonlinear permittivity and permeability to better tune system impedance or output frequency by changing the conditions of the ensuing shockwave. Additionally, NLTLs used in this topology will ultimately be limited by the breakdown strength of the composite located between the inner and outer electrode. Previous studies have shown that a composite, such as the one presented here, has a DC breakdown strength of 401.20 ± 150.73 kV/cm [21]. Thus, higher charging voltages are feasible if flashover can be prevented. The present setup has been limited by flashover between the high voltage side of the spark gap housing and the load. Future studies may address this issue by altering spark gap geometry and/or completely redesigning the switch to achieve higher charging voltages.

Alternatively, the setup also provides the ability to explore other switching technologies. While spark gaps offer ease of manufacturing and simplicity, using them comes at the cost of repetition rate, which causes concern for achievable average output powers. As such, solid state switches are an appealing replacement for the spark gap in our system. Due to the voltage standoff requirements of the system, insulated-gate bipolar transistors (IGBTs) and power MOSFETs are an attractive alternative. Combinations of IGBTs in series and parallel configurations have successfully been used in modulators with voltage levels ranging from 1–150 kV with peak current amplitudes ranging from 0–5000 A and pulse repetition frequencies greater than 40 kHz [30,31]. While such systems have been realized, ensuring that the load is shared equally among the switch configuration is paramount for ensuring device survivability [31]. Compared to closing switches, semiconductor opening switches (SOS) make the inductance in the circuit an active, rather than a passive (or stray) component [32]. Thus, using an opening circuit results in transferring energy to the load in a shorter time and with greater voltage amplitude compared to the closing circuit [32]. One study used SOS to take an input pulse with amplitude of 500 kV and pulse duration at the full-width half-maximum (FWHM) of 7 ns to a gyromagnetic NLTL to output 740 kV with a pulse duration ~2 ns into a 40 $\mathsf{\Omega }$ load [33]. A subsequent study used two gyromagnetic NLTLs as magnetic compressions lines to take a similar input pulse and output it at 1.1 MV with a pulse duration of 0.65 ns and an increase in peak power from 6 to 30 GW [34]. Thus, future system construction could take our system where an NLTL can act simultaneously as a PFL and HPM source and explore designs using semiconductor switches. However, it is important to note that these systems can be quite large [32,35]. Thus, applying semiconductor switching necessitates close attention to system weight since stacks of semiconductors will most likely be needed for power handling. Depending upon application, replacing the spark gap with semiconductor switching could realize a fully solid-state, high power system with our configuration providing tunability based on composite properties.

Moreover, the ability to tune the impedance, by eliminating the need to match the impedance of a given PFL, provides additional flexibility in system output that does not exist in standard devices. This may enable a user to modify NLTL electrical performance by altering the composite blend without needing to adjust the physical dimensions of the NLTL. This would allow an operator to simply change out an NLTL to achieve the desired waveform without making any other changes to the system. Future work will assess different materials, loadings, and impedances to assess how much flexibility this approach provides.

The reduction of system size is a crucial step towards achieving portable HPM systems. The reported topology offers significant weight and size reduction, making it appealing to applications where systems need to be robust and highly mobile. Such systems necessitate careful consideration of system size, length, and weight. System parameter space depends heavily on application and mission needs. Modern gyromagnetic NLTLs with coaxial geometry have ranged in length from ~10–100 cm [36,37,38]. The NLTLs presented here measure 20 cm in length. Other systems have also required pulse forming networks and biasing coils for proper function. While benchtop gyromagnetic NLTLs have been realized recently, output powers are less than those shown here [37]. Admittedly, removing the biasing field reduces system tunability since the output can only controlled by the parameters of the input pulse; however, it may provide alternative tunability by permitting the rapid changeout of NLTLs with different composite loadings to achieve desired output waveforms. This necessitates a comprehensive knowledge of the composite behavior and careful consideration of NLTL length and impedance.

5. Conclusions

This study characterized the impact of NLTL impedance on RF generation when used in the combined PFL/HPM topology. Such a system provides a compact form factor compared to traditional NLTL systems by eliminating the need for a separate PFN and biasing magnetic field. The reduction in system size and weight provides a promising step toward a man-portable HPM systems. To test our hypothesis that reducing the NLTL impedance increased RF generation when using the NLTL as a PFL, we manufactured NLTLs with a fixed volume loading of BST and NZF and varied impedance by modifying the ratio of the inner and outer diameters of the coaxial structure. Both visual inspections of the output waveforms and power spectrum density analysis suggest that oscillation formation depends on NLTL impedance in this format. The 10 Ω NLTL generated 64 kW at 1.08 GHz and 8.5 kW at 1.43 GHz for a 20 kV charging voltage. Lobes had bandwidths of 76 MHz and 102 MHz. The power spectrum for the 25 Ω and 50 Ω NLTLs show 47 kW and 15 kW peaks at 556 MHz and 513 MHz with bandwidths of 22 MHz and 41 MHz, respectively. We conjecture that this arises because of how the PFL forms the pulse and the mechanism by which ferrimagnetic materials generate the oscillation (coherent gyromagnetic precession). Moreover, while one of our major motivations for this approach was to reduce the system’s footprint by eliminating the PFN, it may be possible to further increase the output power by using a biasing magnetic field, albeit at the cost of system size. Furthermore, the flexibility in device design presented by the composite structure and NLTL impedance may permit output frequency tuning in future systems.