#### What is in this article?:

- Ferrite Transformers Fuel Broadband Power Dividers
- Transformer Takeaways

Understanding the characteristics of ferrite materials for transformers can help when forming RF/microwave power dividers/combiners with those transformers.

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Many passive multiport devices—such as power dividers/combiners, baluns, hybrids, and impedance transformers—rely on electromagnetic (EM) coupling for their intended functions. In the microwave range, the coupling is typically achieved through a capacitive (electrical) interaction between closely spaced transmission lines. With this coupling mechanism the broadest available bandwidth is achieved when the length of the coupling section approaches quarter of the wavelength. Consequently, as the frequency drops below the microwave range, the size of devices utilizing this coupling mechanism grows prohibitively large.

A viable alternative to the capacitive coupling in the RF range is the inductive coupling realized through a mutual magnetic interaction of the wire loops coupled together through a common ferrite core. With a proper selection of ferrite material, core dimensions, and the number of loop turns, device functionality can be significantly expanded toward the lower frequencies. Eventually, the use of ferrite transformers and magnetic coupling results in much smaller broadband devices which manifest better electrical performance because of their reduced ohmic losses.

Some applications, however, call for even broader devices operating in a frequency range covering both RF and microwave bands. But there is a fundamental conceptual difference in designing devices intended for use at RF and at microwave bands. The “lumped-element” approach is a prevalent design ideology at RF, while “distributed elements” and “transmission lines” dominate design approaches at microwave frequencies. There are certain limits as to how far each of these approaches can be stretched into the neighboring band.

More promising is the synthetic approach which combines both concepts into a single device. This approach was particularly successful in developing the ultrabroadband ferrite power dividers and couplers which combine capacitive coupled transmission lines, such as twisted bifilar wires and coaxial lines, with ferrite transformers to add magnetic coupling into the mix.^{1} Such a hybrid approach could potentially accomodate almost four decades’ worth of bandwidth in a single device.

To assess the suitability of ferrite material for transformer core applications, one should understand the response of nonmagnetized ferrite to an external AC magnetic field. In the absence of DC magnetic bias, the dynamic magnetic response of soft ferrite materials to an AC field is almost completely determined by the effective field of magnetocrystalline anisotropy, H_{A}, which is present in all types of ferrites. This internal field enforces preferential alignment of magnetic moments along some particular crystallographic directions. The random orientation of grains in polycrystalline structures, such as ceramic ferrites, averages this effect and leads to isotropic complex dynamic permeability given by the following relationship:^{2}

μ′_{i} = 1/3 + (2/3){[(H_{A} + 4πM_{0} + jωG)^{2} - (ωγ)^{2}]/[(H_{A} + jωG)^{2} – (ω/γ)^{2}]} = μ′_{i} + jμ″_{i} (1)

where:

4πM_{0} = the magnetization of saturation, and

G = the dissipation factor which describes dynamic losses in ferrite material.

The real part of magnetic permeability, μ′_{i}, further referred to as the initial magnetic permeability, characterizes the capacity of ferrite core to carry a magnetic flux. The imaginary part, μ″_{i}, describes the magnitude of active losses in the wires which are magnetically coupled with ferrite core. Equation 1 describes the broadband frequency response including also the “natural magnetic resonance,” which takes place when the frequency of the AC field approaches the frequency of the spin precession in the internal field of anisotropy, ωA = γHA. **Figure 1** shows a typical dispersion of the real (solid line) and imaginary (dashed line) permeabilities resulting from Eq. 1.

Because of significant losses associated with a nonmagnetized ferrite material, the resonance is very broad. One can see that the initial permeability remains almost flat to the resonance. After the resonance, it changes into a rapidly descending slope. As for the imaginary part, the loss gradually grows to reach a maximum at resonance and then slowly decreases with frequency. One very important observation is that the magnetic loss extends far into the microwave range, whereas the initial permeability reduces to the vacuum permeability almost immediately above the resonance.

The very same magnetocrystalline anisotropy which causes the “natural magnetic resonance” has an adverse effect on the magnitude of initial permeability:

μ′_{i} = 2M_{S}/3H_{A }(2)

As a result, the ferrites with stronger anisotropy provide broader bandwidth but less permeability and vice versa. This correlation is clearly seen in **Fig. 2**, which shows a comparative permeability for a group of soft ferrites with various strengths of anisotropy field. The analytical relation which describes the correlation between the bandwidth and magnitude of permeability, known as Snoek’s law, is shown in the same graph.

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