## 2 Materials and Methods  
Composites containing 10% BST (TPL Inc. HBS-8000) and 15% NZF (Powder Processing & Technology FP350) inclusions encased in polydimethylsiloxane (SylgardTM 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 Ni\({}_{0.5}\)Zn\({}_{0.5}\)Fe\({}_{2}\)O\({}_{4}\) and Ba\({}_{0.45}\)Sr\({}_{0.55}\)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 \(\upmu\)m 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 CP200 HT) for 4 h to break up any conglomerations. We allowed the mixture to cool before hand stirring in the appropriate amount of curing agent (1/10 th 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 \({}^{\circ}\)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  
\[Z_{sat}=\frac{\ln(b/a)}{2\pi}\sqrt{\frac{\mu_{sat}\mu_{o}}{\epsilon_{sat} \epsilon_{o}}} \tag{1}\]  
Figure 1: Cross section of a 3D-XRM images of a composite containing 10% BST and 15% NZF. The composite contains some conglomerations (<1 mm in width) of BST, but the dispersion is good overall. where \(b\) is the diameter of the outer conductor, \(a\) is the diameter of the inner conductor, \(\mu_{sat}\) and \(\epsilon_{sat}\) are the relative saturation permeability and permittivity, respectively, and \(\epsilon_{o}\) 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 \(\epsilon_{sat}=7.2\).