MICROWAVE AND OPTICAL TECHNOLOGY LETTERS 1

Optimization of a Broadband VHFLumped-Element Ferrite CirculatorJacob R. Smith*, Hang Dong*, Jeffrey L. Young* and Brandon Aldecoa**

Abstract—A multistage optimization procedure is provided tomaximize the operating bandwidth of a VHF circulator. We showthrough hardware validation that bandwidths on the order of 52.3MHz to 80.9 MHz are achievable using a 15 dB isolation specifica-tion. Simulation studies indicate that bandwidths on the order of100% are possible, but such wideband performance has not beenachieved in hardware to date. The optimization procedure usesboth genetic algorithms and orthogonal sweep methods to find thebest crossover geometry and ferrite properties, but the outcomeof both methods is roughly the same. Matching networks are usedto tune the response and to widen the operating frequency band.These networks are optimized using Pareto tradeoff charts in thecontext of power metrics like return loss, isolation and insertionloss.

Index Terms—Circulators, bandwidth, ferrites, VHF, optimiza-tion

I. INTRODUCTION

THE concept of a lumped-element, crossover circulator foruse in the VHF-UHF range was pioneered by Konishi

[1] in 1965. Since that time there have been numerous con-tributions to its underlying theory of operation and methodsof fabrication [2] - [5]. However, there are still questionsabout maximum wideband operation for a given isolationspecification. In the VHF range, marketed circulators tend tobe narrowband, but this performance is not driven by physicallimitations. The emergence of computational electromagneticsoftware in recent years has dramatically improved the waycirculators can be designed. And, through the use of new, so-phisticated methods of optimization and fabrication describedherein, it is now possible to close the gap between the upperbound of theoretical performance and the performance ofphysical devices.

As experimentally verified with a fabricated VHF circulator,bandwidths on the order of 54.7% are physically possibleusing a novel design process. Simulation studies indicatethat bandwidths can be as large as 100%, but parasitics notaccounted for in simulation tend to limit actual hardwareperformance.

To achieve wideband performance a multistage optimizationmethodology is utilized that exploits data obtained from bothsimulation and experiment. Due to the overwhelmingly largenumber of free design parameters (e.g., geometry values,material values, biasing value, matching network values) it

*Authors are with the Department of Electrical and Computer En-gineering, University of Idaho, Moscow, ID, 83844-1024 USA e-mail:jyoung@uidaho.edu.

**B.Aldecoa is with First RF, Boulder CO, USA.This work was supported by the Office of Naval Research under grant

number N00014-08-1-0286.

Fig. 1. A non-reciprocal, three-port network connected to a matching network.

is neither practical nor insightful to attempt an optimizationover the entire design space. Instead, we first focus on findingthe optimal geometry and materials for the crossover network,then the optimal biasing field and finally the optimal matchingnetwork. This methodology hinges on the notion of the circu-lation impedance and the knowledge it provides in estimatingbandwidth – prior to matching network design – as describednext.

By definition the circulation impedance Zc

is the loadimpedance connected to a linear, non-reciprocal, three-portnetwork that results in perfect isolation. It is also theimpedance associated with perfect return loss and insertionloss if the network is lossless. Thus, the network, if losslessand non-reciprocal, is a perfect circulator when all three ofits ports are loaded with Z

c

. From standard network analysisit can be shown that the circulation impedance is purely afunction of the network’s impedance parameters [6]:

Zc

= Rc

+ jXc

⌘ Z2

21

Z31

� Z11

. (1)

We may view the concept of the circulation impedance interms of a non-reciprocal ferrite network loaded by matchingnetworks. See Figure 1 for a network depiction and for thevarious impedance definitions used in this discussion. The roleof the matching networks is to realize as closely as possiblethe frequency response of Z

c

over some band of frequenciesas viewed by the non-reciprocal network.

Consider Figure 2, which shows typical Zc

data for a ferritecrossover network without tuning or matching. By placingC� across the ports we tune the response to a particular

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS 2

Fig. 2. Typical circulation impedance data without Co

and Cg

.

Fig. 3. Typical circulation impedance data with Co

and Cg

.

frequency and then use the C1

and L1

series network to matchthe positive slope of X

c

between the two zero slope points,as depicted in Figure 3. The resonant frequency of the LCnetwork !

1

, where !1

= 1/pL1

C1

, is set to the resonantfrequency for which X

c

= 0. The slope of the LC networkreactance through resonance is 2L

1

and is set to roughly equalthe slope of X

c

through resonance. These two conditions, i.e.resonance and slope through resonance, give a rough estimatesfor L

1

and C1

from the data of Xc

near resonance. As for theload resistance R

L

, it is set from the Rc

data near resonance;we initially let R

L

⇡ 2Rmax

c

/3, which corresponds to a returnloss of 14 dB. In addition to C�, C

1

and L1

tuning it iswell known that a capacitor C

g

tied between ground and thecommon returns of the ports also enhance bandwidth. Theproof of this assertion is based on an eigenvalue analysis ofthe impedance matrix Z by showing that C�, C

1

, and L1

onlyaffect two of the three eigenvalues and C

g

only affects theremaining eigenvalue [5].

As provided in the subsequent discussion, optimal valuesof L

1

, C1

, Cg

and RL

are determined from an optimizationprocedure that uses power metrics associated with the returnloss, insertion loss and isolation of the combined crossoverdevice and matching network; the previously discussed valuesare regarded as seed values to that optimization procedure. The

operational bandwidth of the combined crossover network andmatching network is estimated – prior to complete matchingnetwork design – from the two zero slope points of X

c

,as depicted in Figure 3. That insight allows us to optimizethe geometry and materials of the crossover network withoutsimultaneously optimizing the matching networks. It thusmakes sense to define a percent bandwidth metric BW ofthe optimizer in terms of these slope points:

BW ⌘ 100fb

� fa

fa

, (2)

where fa

and fb

correspond to the zero slope frequenciesof X

c

. Equally stated, they are the maximum and minimumvalues of X

c

on either side of resonance. The bandwidthmetric is normalized by f

a

so that 100% corresponds to oneoctave [7]. In essence, bandwidth is maximized by adjustingthe geometrical and material parameter space such that theslope of X

c

through resonance is minimal.From this previous discussion the multistage optimization

procedure can now be articulated:1) Use simulation tools and optimization search techniques

to maximize BW , as defined by Eqn. (2), and determinethe optimal crossover geometry and materials by includ-ing in the search optimal values of C�, C

g

and biasingfield strength. Discard the determined values of C�, C

g

and biasing field strength.2) Fabricate the crossover network and measure the

impedance parameters for a wide variety of biasing fieldvalues. For each field value design a matching networkand select the field value that results in the widestbandwidth.

3) Fabricate the combined crossover and matching networkon a single substrate and measure the final performance.

Each of these steps will be considered in more detail in theensuing discussions. However, it should be pointed out thatin step one the goal is only to find the best geometrical andmaterial parameters; C� and C

g

are only invoked for tuningpurposes; their values are eventually discarded since they aredetermined more accurately in step two. Step two is essentialgiven the difficulties in obtaining precise simulation data ofan actual ferrite device.

II. FORMULATION

There are seven design parameters to be considered whendesigning the crossover network shown in Figure 4 (left). Theseven parameters, which are defined in Figure 4 (right), are

x1

= Ferrite Height (mm),x2

= Ferrite Radius (mm),x3

= PCB Height (mm),x4

= Ferrite 4⇡Ms

(G),x5

= Trace Width (mm),x6

= Gap Width (mm),x7

= Gap Radius (mm).

To ensure that the optimal parameters are found, two op-timization methods are invoked. The first method uses agenetic algorithm (GA) for optimization and the other uses

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS 3

Fig. 4. Typical crossover network and its design parameters.

an orthogonal search. Both programs use the seven-dimensionsearch space defined as

0.3 mm x1

1.2 mm,8 mm x

2

12 mm,0.254 mm x

3

0.457 mm,1600 G x

4

2000 G,2.5 mm x

5

8 mm,1 mm x

6

6 mm,8 mm x

7

15 mm,

(3)

Ansys’ HFSS is the kernel of both optimization programs tosimulate the response of the crossover network models. Thethree-port Z-parameter simulation data are exported to Mat-Lab, which calculates the circulation impedance and estimatesthe bandwidth for each simulation result while searching C�and C

g

in the context of a specified operating frequency. Thedesign with the largest bandwidth is retained. This design fixesthe values of x

i

, but the values of C� and Cg

are discarded. Inthe following example we seek to find the largest bandwidth,with 80 MHz being the upper limit of the frequency range.

A. GA and Orthogonal Search MethodsThe Genetic Algorithm (GA) optimization program, as artic-

ulated more fully in [8], defines the seven design parameters asthe chromosome genes. The fitness function is the bandwidthdefined by Eqn. (2). A population of 20 chromosomes is usedin the present discussion. Each chromosome is broken intotheir 7 genes and 8 alleles are assigned per gene, such thateach chromosome appears as follows:

Chromo1

= 00100110| {z }gene1

10110000| {z }gene2

. . . 01000000| {z }gene7

.

Each gene’s binary string is encoded into a numerical value foruse in HFSS and the chromosomes are ranked by their fitness,i.e., bandwidth performance. Natural selection is evoked toeliminate 12 chromosomes from the population. Roulettewheel selection is used for the mate pairing and the followingequation provides the probability of chromosome selectionbased on its rank n:

Pn

=N

pool

� n+ 1PN

pool

n=1

n. (4)

Here Npool

is the mating pool size. Uniform crossover isused for offspring generation, which creates a new generation

Fig. 5. Learning curve of the GA for 50 generations.

of 16 chromosomes; migration is used to restore the newgeneration’s population to 20. Migration is implemented bycreating four new random chromosomes. A 10% mutationrate is impressed on each generation out of a total of 50generations.

From this GA process and HFSS simulation data, the bestperforming design has a bandwidth of 90.5% over the range48 MHz to 80 MHz when

x1

= 0.775 mm,x2

= 14.809 mm,x3

= 0.3588 mm,x4

= 1972 G,x5

= 5.765 mm,x6

= 5.219 mm,x7

= 13.52 mm.

(5)

The learning curve of this GA across 50 generations is shownin Figure 5. This graphic shows that only five generations areneeded to reach near optimality; the next 45 generations onlyslightly improve bandwidth performance.

A modified form of Powell’s method conducts orthogonalparameter sweeps until it converges on a local maximum. Astarting point P� is chosen at random and a sweep of x

1

values are conducted to seek out the widest bandwidth valuefor the next sweep point P

1

. From this new point, the x2

values are similarly swept and the best performing designis assigned as the starting point for the x

3

sweep, which islabeled P

2

. This process is repeated until the local maximumis discovered. Figure 6 shows the results of one full iteration ofvalue sweeps followed by the maximum value found, which is

Point x1 x2 x3 x4 x5 x6 x7 BWP� 0.75 10 0.305 1800 4.2 3 8 73.9%P1 0.6 10 0.305 1800 4.2 3 8 83.7%P2 0.6 12.8 0.305 1800 4.2 3 8 90.5%P3 0.6 12.8 0.305 1800 4.2 3 8 90.5%P4 0.6 12.8 0.305 1875 4.2 3 8 90.5%P5 0.6 12.8 0.305 1875 4.2 3 8 90.5%P6 0.6 12.8 0.305 1875 6 3.5 8 90.5%P7 0.6 12.8 0.305 1875 6 3.5 11.5 100%P0� 0.58 12 0.356 1850 5 2.5 11.5 105%

Fig. 6. Results from orthogonal parameter sweeps.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS 4

Fig. 7. Simulation data for an optimized VHF circulator; 39-80 MHz; BW=105%. IL is insertion loss, RL is return loss and I is isolation.

labeled P0�. As seen in Figure 6 the modified form of Powell’s

method outperformed the genetic algorithm by about 15%. Thehighest bandwidth discovered in this search was 105% with afrequency range from 39 to 80 MHz. But more importantly,both techniques, which use entirely different search methods,indicate that near 100% bandwidths are possible for VHFoperation. Thus, this value represents the goal for hardwareperformance. When this crossover network of Figure 6 (lastline) is tuned with an idealized matching network, the corre-sponding response as determined from simulation is seen inFigure 7.

B. Biasing Field Optimization

Based on the final optimal values of the geometrical andmaterial parameters, the resulting optimal crossover networkis fabricated and measured for a range of biasing field values.A rigid aluminum case holds the PCB, ferrites and three 3.5mm APC connectors; see Figure 8 for an image of the HFSSmodel and its fabricated counterpart. The ferrites are Trans-Tech TTVG-1850 series and have a linewidth less than 10 Oe,dielectric constant of 14.8 ± 5%, and 4⇡M

s

of 1850 G ± 5%.To simplify the measurement process, three-port symmetry isassumed and a two-port network analyzer is used to measurethe S-parameters of the fabricated crossover network, whilethe third port is loaded with a 50 ⌦ termination. The biasingfield strength is swept in discrete steps from high to low. Foreach bias point the S-parameter data is recorded and stored.This data is then searched to determine which bias point

Fig. 8. HFSS model and the fabricated device.

Fig. 9. Pareto trade-off population chart.

Fig. 10. Measured crossover network performance with ideal matchingnetwork; 45-84 MHz; BW= 86.7%. IL is insertion loss, RL is return lossand I is isolation.

corresponds to the widest possible bandwidth. For the finalselected value, a full, three-port measurement is made at theoptimal field strength using the two-port method provided in[10]. From this data the ideal matching network componentsare found using return loss, isolation and insertion loss powermetrics in conjunction with Pareto trade-off charts and auniversal search method [9]. We arbitrarily chose a minimumisolation of 15 dB and a return loss of 15 dB. The Pareto chartfor this outcome is shown in Figure 9. The component valueoutcome is:

Ha

= 1615 Oe,C

g

= 626 pF,C� = 162 pF,C

1

= 80.4 pF,L1

= 84.3 nH,R

L

= 22.5 ⌦.

(6)

By mating the data of the ideal matching network withthe measured data in code, we can anticipate the measuredperformance of the final fabricated device. As shown in Figure10 the resulting bandwidth is 86.7% with a range from 45 to84 MHz. The bandwidth difference between the data of Figure7 and Figure 10 is about 18%, which is expected given that

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS 5

Fig. 11. Hardware layout.

Component Manufacturer Part Number ValueL1 Coilcraft 1515SQ82NGEB 82 nHC

g

Johanson Tech 251R15S151JV4E 150 pF(for 1 of 3 251R15S470JV4E 47 pFlocations) 251R15S120JV4E 12 pF

C� Johanson Tech 251R15S151JV4E 150 pF251R15S121JV4E 12 pF

C1 Johanson Tech 251R15S330JV4E 33 pF251R15S470JV4E 47pF

Fig. 12. Components used in Figure 11. Multiple capacitors are placed inparallel to better match the ideal component values of Eqn (6).

certain parasitics cannot be accounted for in simulation.

C. Realization of Matching NetworkThe matching network of Figure 1 is incorporated with

the crossover network on a single PCB using the designconfiguration shown in Figure 11. In this design the circularferrite housing serves as the common node for C� and C

g

.To establish complete three port symmetry, the total capaci-tance of C

g

is distributed into three equal values and placedrotationally 120� from each other. The connection from thethree C

g

’s to ground is made by a via to a circular copperring on the bottom of the PCB which is pressed againstthe aluminum case, i.e., ground. The board is fabricated andpopulated with components that correlate as nearly as possiblewith the values shown in Eqn. (6); see Figure 12 for theactual component values. Because purchased components donot exactly match the values of Eqn. (6), some adjustmentsare needed. Specifically, the biasing field is again swept usinga similar method as before with the exception that three-portsymmetry is no longer assumed. The field is again swept fromhigh to low in discrete steps. In between each iteration ofthe biasing field strength measurement the R

L

value is alsooptimized in code. The final outcome is H

a

= 1639 Oe andR

L

= 22 Ohms. The final measured performance is shown inFigure 13. The operating frequency band is from 52.3 MHzto 80.9 MHz, i.e. 54.7% for a minimum isolation of 15 dB. Inthis band the return loss never exceeds 16 dB and the insertionloss never exceeds 0.9 dB. The insertion loss at the center ofthe band is 0.8 dB.

Fig. 13. Measured VHF circulator parameters; 52.3-80.9 MHz; BW= 54.7%.IL is insertion loss, RL is return loss and I is isolation.

III. CONCLUSION

This paper has articulated a method for determining themaximum bandwidth of a VHF circulator and has validatedthat method using hardware prototypes. The outcome is a mea-sured bandwidth response of 54.7%. Moreover, a vast numberof simulation studies suggest that bandwidths on the order of100% are possible, but to date we have been unable to achievethis result in the laboratory. The reasons are many. First,our simulation models are not exact; we do not account forevery geometrical detail, since such details create unnecessaryburdens on the simulation software in terms of computationaltime. Second, we assume that the ferrite has a uniform internalbiasing field, when in fact it does not due to shape effects.Although we could model non-uniformities, we do not doso for the sake of increasing the optimization throughput.Third, it is very difficult to model electrical parasitics inHFSS near the locations of the components, since componentsare not modeled but are assigned values at simulation ports.Finally, the hardware components are far from ideal. This isparticularly true for L

1

, which tends to store some charge (i.e.,a capacitive effect) at the frequencies of interest and tends tohave a fairly low quality factor on the order of 120.

ACKNOWLEDGEMENTS

This work was supported by the Office of Naval Researchunder grant number N00014-08-1-0286. The authors appreci-ate the many insightful comments and suggestions offered byDr. Jeffrey Allen of Navy SPAWAR, San Diego, CA.

REFERENCES

[1] Y. Konishi,“Lumped element Y circulator,” IEEE Trans. MTT, vol. 13,no. 6, pp. 852–864, Jun. 1965.

[2] L.K. Anderson,“An Analysis of broadband circulators with externaltuning elements,” IEEE Trans. MTT, vol. 15, no. 1, pp. 42–47, Jan.1967.

[3] T. Miura, M. Kobayashi and Y. Konishi, “Optimization of a lumpedelement circulator based on eigenvalue evaluation and structural im-provement,” IEEE Trans. MTT, vol. 44, no. 12, pp. 2648–2654, Dec.1996.

[4] J.O. Bergman, “Equivalent circuit for a lumped-element Y circulator,”IEEE Trans. MTT, vol. 16, no. 5, pp. 308–310, May 1968.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS 6

[5] R. H. Knerr, “An improved equivalent circuit for the thin-film lumpedelement circulator,” IEEE Trans. MTT, vol. 20, no. 7, pp. 446–452, Jul.1972.

[6] J. L. Young, R. S. Adams, B. O’Neil and C. M. Johnson, “Bandwidthoptimization of an integrated microstrip circulator and antenna assembly:Part 2,” IEEE Antennas and Propagation Magazine, vol. 49, no. 1, pp.82–91, Feb. 2007.

[7] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 1st Ed.,New York, John Wiley and Sons, p. 260, 1981.

[8] R. L. Haupt, and D. H. Werner. Genetic Algorithms in Electromagnetics,Hoboken, N.J., IEEE Press, 2007.

[9] J. C. Allen, H1 Engineering and Amplifier Optimization, Birkhauser,Boston, MA, 2004.

[10] F. Lenk and R. Doerner, “A new multiport measurement-method using atwo-port network analyzer,” 2005 IEEE MTT-S Int. Symp. Dig., pp 1663- 1666.