Circulators are essential elements in high-power microwave systems and antenna networks in which energy must be directed and isolated. Circulator design involves the use of well-studied models based on classic equations presented by Bosma.¹ Although computers gain in processing speed, and computer-aided-engineering (CAE) programs grow in sophistication, these two equations nonetheless provide the background for high-frequency circulator development and modeling. Using these two equations as a reference, for example, the author developed several basic approaches to circulator design and analysis which are suitable for modeling in modern CAE programs.

The Y-junction circulator^{1-3, 8, 9} is a three-port junction employing a resonating nonreciprocal element.^10-15 The three ports connect the junction to the external world. A signal arriving at port 1 will circulate to port 2 but not to port 3; port 3 is said to be isolated from port 1. Similarly, a signal arriving at port 2 will circulate to port 3 but port 1 will be isolated from port 3 (Fig. 1). A ferrite or garnet disk or cylinder within a vertical static magnetic field usually serves as the nonreciprocal element. For signal circulation to occur, the two classic equations must be satisfied simultaneously:

The two equations are derived from the electromagnetic (EM) field solution for perfect circulation given by Bosma in ref. 1. Parameters P, M, and N are defined as:

where ψ is half the coupling angle at any one of the three ports and

where the J_n represent Bessel functions of the first kind with order n and

is the wave propagation constant where

ε_r = the relative dielectric constant,
f = the frequency,
c = the speed of light, and

is the effective permeability for the given Polder tensor parameters µ and κ.

Parameter Q is defined as

where the parameters Z_d and Z_eff are the external intrinsic impedance at the circulator ports and the intrinsic impedance of the magnetized ferrite, respectively. To help engineers better understand circulator design, this report will present graphic solutions for the two circulation equations at different levels of complexity.

Figures 2 and 3 represent classic solutions to the Eqs. 1 and 2 under ideal conditions and when retaining the first three terms of N and M (n = 0...2). These graphs are presented numerous times in earlier publications.^1,3 The graph in Fig. 2 gives the roots to Eq. 1 by presenting values of the electric radius, sR, as a function of |κ/µ| for various coupling angles. Figure 3 gives the roots to Eq. 2 by presenting the wave impedance ratio Z_eff/Z_d as a function of |κ/µ| for various coupling angles (half the coupling angle is denoted as ψ (Fig. 1).

There are two possible algorithms for solving Eqs. 1 and 2 for the variables sR and Z_eff/Z_d as a function of |κ/µ|. One way is to solve Eq. 1 for sR, use the obtained values of N and M to calculate Q, and then solve Eq. 2a for Z_eff/Z_d. This is an adequate way to solve the two equations but it may result in higher-order results. To avoid this, a limit is required for the acceptable range of solutions for Z_eff/Z_d. A preferred approach is to solve both equations simultaneously.

The simple solution proposed here involves solving Eq. 1 for sR under perfect conditions, i.e., no loss and for the first three terms (n = 0, 1, 2) as given in the literature for the classic solution. Each solution for sR also provides values for parameters M and N, which are then used in Eq. 2 to calculate the value of Z_eff/Z_d. A solution to Eq. 1 is within a given range, i.e., 0 < sR ≤ 3.5 with values of Z_eff/Z_d only acceptable within the range 0 ≤ (Z_eff/Z_d) ≤ 2.2.

Figures 4 and 5 show these solutions graphically, with curves identical to those in Figs. 2 and 3. The missing points are due to solutions with large error-function values (where no solutions were found, i.e., any circumstance where an error-function value exceeded 0.01). The lack of some points may also be due to differences in computational tools, i.e., personal computer (PC) versus workstation computer.

When the two equations are solved simultaneously, keeping the two variables within a given range i.e. 0 < sR ≤ 3.5 and 0 ≤ (Z_eff/Z_d) ≤ 2.2, the graphs of Figs. 6 and 7 result for the conditions used to generate Figs. 4 and 5. Other than the curves for ψ = 0.2 in Figs. 6 and 7, the other values of ψ are in close agreement with the values of Figs. 2 and 3.

Applying the DC vertical magnetic field, H_i, instigates the magnetic resonance frequency, f₀. The field intensity is chosen as the minimum required to ensure saturation (just saturated), i.e., H_i = H_m. At saturation, there is a cutoff frequency, designated f_m. The two frequencies are defined in Eqs. 3 and 4, respectively:

where:

γ_e = the gyromagnetic ratio and

4πM_s = the magnetization saturation.

Using the definition δ = f₀/f and assuming the ideal (lossless) case, the Polder parameter µ can be defined as

For each value of |κ/µ|, it is possible to directly find µ, µ_eff, and s. For the second variable, R and sR will be calculated as well. Since s is proportional to f/c and R is proportional to c/f, the product sR (the root of Eq. 1) is frequency independent. On the other hand, the solution to Eq. 2 seems to be frequency dependent at ψ = 0.2. Although there are some differences in the calculated curves for µ, µ_eff, and s (Figs. 8 to 11), the general trend is identical to Fig. 3. Figure 9 shows the results at close proximity to the cutoff frequency. Figure 10 shows the results at a higher frequency, and Fig. 11 shows the results at an even higher frequency.

The circulator's ferrite resonant material is not ideal, yielding a resonant width, ΔH, rather than a single-frequency resonance. Equation 5 relates the loss due to this resonance width:

where:

f = the frequency at which the width is measured.

At this point, it is necessary to define the parameters of the Polder Tensor. Common practice is to solve Gilbert's equation⁷:

where:

H = the DC magnetic field vector,
M = the magnetization vector,
t = the time variable,
γ_e = the gyromagnetic ratio, and
α_h = the damping magnitude coefficient.

Equation 6 is a good approximation of the Landau and Lifshitz equation⁵:

Equations 8 and 9 provide The Polder Tensor parameters:

When derived from the Gilbert's equation (Eq. 6), and using the definition p_s = f_m/f , the terms for the above equations are as follows:

For the derivation of the original Landau and Lifshitz equation (Eq. 7), the terms are different and are as follows: