Closed-Form Design Formulas for the Equivalent Circuit ... - IEEE Xplore

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 53, NO. 4, NOVEMBER 2011

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Closed-Form Design Formulas for the Equivalent Circuit Characterization of Ferrite Inductors Krishna Naishadham

Abstract—Miniature inductors, consisting of thin-wire solenoidal or toroidal coils wound on a high-permeability soft ferrite core, find wide application in the filtering of noise or electromagnetic interference at RF. An understanding of the high-frequency parasitic and packaging effects of the inductor, such as stray capacitance, magnetic losses, self-resonance, etc., can be gained from an equivalent circuit characterization of the inductor. In this paper, we present a rigorous experimental method to fully characterize the RF behavior of ferrite inductors. The equivalent circuit parameters of the inductor, namely, series inductance, loss resistance, and stray capacitance, as well as the effective permeability of the core, are extracted in closed form from an accurate measurement of the RF impedance, without recourse to cumbersome optimization procedures usually followed in equivalent circuit extraction from measured data. We derive design equations for the equivalent circuit elements based on the proposed measurement-based model, and rigorously validate the model using inductors wound on commercial toroidal and rod-type Ni–Zn ferrite cores. Index Terms—Electromagnetic compatibility (EMC), electromagnetic interference (EMI), equivalent circuit analysis, ferrite inductors, magnetic materials, permeability, RF inductors.

I. INTRODUCTION ERRITE inductors, consisting of an insulated copper wire wound on slug-type, toroidal, or brick-type magnetic cores, find useful application in filtering noise resulting from electromagnetic interference (EMI) at RF. Besides the insertion loss of the inductor, parasitic effects, such as interturn and interwinding capacitance, self-resonance, dielectric and magnetic losses of the core, play an important role in the design of such filters. The distributed capacitance across the coil acts like a lumped capacitor in shunt with inductor, and results in occurrence of self-resonance at some frequency, above which the impedance becomes predominantly capacitive [1], [2]. The useful operating frequency range of the inductor is thus limited by its selfresonant frequency (SRF). In automotive electronics, broadband wireless communications and cable television systems, it is now required to design inductors with resonant frequency extending into the RF regime (100 MHz to 1 GHz). Thus, there is a need to understand the full circuit characterization of the RF inductor,

F

Manuscript received July 8, 2009; revised February 25, 2010, July 26, 2010, and January 11, 2011; accepted January 15, 2011. Date of publication May 27, 2011; date of current version November 18, 2011. The author is with the Sensors and Electromagnetic Applications Laboratory, Georgia Institute of Technology, Atlanta, GA 30080 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TEMC.2011.2116795

including all the frequency-dependent circuit-parasitic, material, and packaging effects. At RF, direct measurement of the inductance, resistance, and stray capacitance of the inductor, as functions of frequency, is difficult. It thus becomes important to develop a technique for extracting these circuit parameters from measurable physical parameters, such as insertion loss, resonant frequency, and impedance. Prior investigations on equivalent circuit characterization of inductors have been limited to frequencies much lower than the self-resonance, where the circuit parameters can be assumed as constant. An electromagnetic formulation based on quasi-static Maxwell’s equations [3] is adequate in this case to describe the inductor. At high frequencies, the assumptions of quasi-static analysis are violated. Bartoli et al. [4] proposed a method for estimating the self-capacitance of an inductor by measuring its resonant frequency with the assumption that the inductance near resonance is unchanged from its low-frequency value. In ferrite inductors, this assumption is valid only if the SRF is smaller than the intrinsic “rolloff” frequency at which the real part of the permeability begins to roll off from a constant low-frequency value, and where the imaginary part peaks [5]. Typically, the inductance remains constant below this rolloff frequency, and decreases significantly in its vicinity and above it due to increasing core loss. Therefore, the upper frequency bound of the inductor is limited by the intrinsic loss peak. For RF inductors, with SRF in the 100 MHz to 1 GHz range, which is significantly higher than the material rolloff frequency for typical Mn–Zn or Ni–Zn cores, it is inaccurate to assume that the inductance remains at its low-frequency limit corresponding to the initial permeability of the core. The permeability varies around the rolloff frequency, and effectively, the equivalent circuit elements of the inductor become frequency-dependent. Liu [6] described a method for extracting the self-capacitance of the inductor by neglecting the influence of the core. Thus, it is assumed that the capacitance with a ferrite core is unchanged from its value for a winding with an air core (also see [16]). For ferrite cores, this assumption is grossly violated because both Mn–Zn and Ni–Zn cores have dielectric constants ranging from 10 to 18 [7], [8]. Additionally, the high resistivity of these materials has to be taken into account in evaluating the shunt (leakage) resistance. Massarini and Kazimierczuk [9] derived expressions for estimating the capacitance of multilayer inductors by considering the approximate field distribution in the vicinity of the core. The effects of the dielectric core on the stray capacitance are not considered in their equations. Due to the difficulty in accurately estimating the actual path of electric flux or the surface area of the distributed capacitor, it is found that the expressions given in [9] underestimate the capacitance by as

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much as 40% when compared with an experimental method to extract the stray capacitance from RF measurements [10]. The equivalent circuit parameters have been also obtained from EM simulation methods, such as the quasi-static finite element [11] and the finite difference [12] methods. For toroidal inductors, the intrinsic permeability in the frequency range from 10 kHz to 10 MHz has been computed using approximate solution of steady-state-field equations to determine electric and magnetic energies stored inside the core [13], [14]. Kazimierczuk et al. [15] evaluate integrals for core losses and the total inductance of a solenoidal core inductor in terms of Bessel functions derived from a solution to the Helmholtz equation in cylindrical coordinates. Due to nature of the low-frequency approximations used, the solution in [15] is valid at frequencies up to only a few MHz. In summary, the analyses in [11]–[15] are specific to particular geometries of the inductor, involve timeconsuming numerical analysis, and are not applicable at RF due to the quasi-static assumption. Recently, there has been an interest to characterize RF inductors using measurements on impedance and network analyzers. Yu et al. [16] developed a method, based on network analyzer measurements, to estimate the self-capacitance of inductors by measuring the change in resonant frequency when a known (external) capacitor is shunted with the coil. The inductance and resistance are determined from the capacitance and the frequency shift. The inductance is assumed to vary slowly with frequency, a conjecture strictly valid for frequencies below the rolloff frequency of the core [10]. Due to the need to have closely spaced external capacitance values for determining the frequency dependence of the inductance, the experimental method in [16] also suffers from limited frequency resolution. However, the results appear to be vastly improved over previous attempts [9] to measure the stray capacitance. Using the well-known transmission line analysis of scattering parameters measured on coaxial samples with a vector network analyzer, Shenhui and Quanxing [17] extract the complex RF permeability of ferrite cores up to 1 GHz. The cavity resonances of the sample and effect of air gaps corrupt their results at the higher frequencies. In this paper, we present a closed-form measurement-based model, which comprehensively characterizes the RF inductor in terms of its equivalent circuit representation. The author developed a methodology for extracting in closed form the frequency-dependent equivalent circuit parameters of surface mount device (SMD) inductors [18], based on measured S-parameters of inductors mounted on calibrated test boards. This method has been extended to the equivalent circuit characterization of other packaged RF components for printed circuit applications, such as low-loss interconnects and SMD capacitors [19]. The nonideal behavior associated with board layout effects, device packaging, and component parasitic effects, has been considered in deriving these models, which are valid over a wide frequency band. Although not as versatile as commercial CAD packages in the sense that the measurement-based models in [18] and [19] are component-specific, they are derived without recourse to cumbersome optimization procedures normally followed in RF circuit synthesis. Because these component models are developed in-situ in the same configuration

Fig. 1. Measurement setup for HP 4291 Impedance and Material Analyzer (left) with a test station (right top) for mounting four sample fixtures (right) used to measure discrete components and magnetic materials [20].

as their actual board layout, they can be easily incorporated in circuit design, as we have demonstrated successfully for a coplanar waveguide filter [19]. In this paper, we apply the modeling methodology developed in [18] and [19] to extract in closed form, the frequency-dependent RLC equivalent circuit parameters of rod and toroidal ferrite inductors, from impedance measurements. Expressions for complex intrinsic and rod permeability of toroidal and slug-type ferrite cores, respectively, are also derived. In the sequel, we briefly describe the measurement procedure in Section II, followed by formulation of the closed-form equivalent circuit model in Section III. Simple expressions are presented to calculate the effective permeability in complex form for both rod and toroidal core inductors. Sample results on the equivalent circuit parameters and the effective permeability of inductors mounted on ferrite rods and toroidal cores are presented in Section IV, and validated with well-established low-frequency models or manufacturer’s specifications where applicable. Important conclusions are summarized in Section V. II. MEASUREMENT PROCEDURE The measurement setup, shown schematically in Fig. 1, entails direct measurement of the RF impedance on the HP 4291 Material and Impedance Analyzer (MIA) [20]. The MIA provides accurate impedance measurements from 1 MHz to 1.8 GHz with a high dynamic range (from about a tenth of an ohm to few hundred kiloohms). Separate test heads are employed for measuring high and low impedances. An axial lead fixture is connected to the test head for the impedance measurement of toroidal or solenoidal inductors. Likewise, one uses with the test head either a magnetic material fixture or a dielectric material fixture, for the measurement of permeability or permittivity, respectively. All these fixtures, shown in the photograph in Fig. 1 [20], are internally compensated for electrical length and parasitic impedance variations by calibrating against known impedance standards. The relative permeability of toroidal cores is calculated internally by the instrument from

NAISHADHAM: CLOSED-FORM DESIGN FORMULAS FOR THE EQUIVALENT CIRCUIT CHARACTERIZATION OF FERRITE INDUCTORS

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TABLE I MODEL PARAMETERS AT RESONANCE COMPUTED USING THE RESONANT FREQUENCIES FOR THE TWO CASES DESCRIBED

Fig. 2.

Equivalent circuit of an inductor.

measurement of the impedance on a one-turn “coil” (the conducting path for the current is provided by a shorted coaxial fixture). For impedance measurements, we have used the HP 16194 Axial Lead Impedance Fixture with a high impedance test head. For toroidal core characterization, HP 16454 Magnetic Material Fixture and the low impedance test head have been used. DC biasing and dynamic magnetization curves are not considered in this study. Due to a large leakage flux path, the rod permeability of solenoidal cores cannot be measured on the impedance analyzer. In addition to the complex impedance as a function of frequency, the resonant frequency of the inductor was measured by looking for the maximum of the resistive part. The quality (Q) factor at resonance was measured from the resonant impedance and the 3 dB bandwidth. The equivalent circuit characterization and evaluation of other physical parameters, such as effective permeability of the core, are then accomplished from these three measured values—impedance, resonant frequency, and Q. III. EQUIVALENT CIRCUIT REPRESENTATION A. Closed-Form Circuit Model At RF, the inductor is represented by the lumped-parameter equivalent circuit shown in Fig. 2, where R, L, and C are the equivalent series resistance, inductance, and stray capacitance, respectively. R is mainly caused by the resistive (skin effect) losses in the winding, and the magnetic losses in the ferrite core. The latter are predominantly due to imaginary (or resistive) part of the permeability, and the core resistivity. At RF, these core losses dominate the winding losses. Other core losses, such as eddy current loss, hysterisis loss, and dissipation due to residual magnetism or magnetic domain polarization, are neglected in comparison with the resistive core losses. The lumped parasitic capacitance C includes the turn-to-turn distributed capacitance, turn-to-core capacitance, and for multilayer coils, also the layer-to-layer capacitance. In fact, for a long coil, the capacitance is distributed, but if the operating frequency and the circuit size are such that only the dominant resonant mode is excited, then the premise of an equivalent lumped capacitance is justified. For the RF coils that we measured, this requirement was fulfilled. Thus, the equivalent circuit derived in this paper is valid only for the dominant resonant mode. Transmission line concepts will have to be used to extract the high-frequency distributed equivalent circuit components of ferrite inductors, which support multiple resonant modes. We used

connecting leads less than half an inch long. Therefore, the lead inductance and the parasitic lead capacitance were ignored. Denoting the measured complex impedance as Z(ω) = Zr (ω) + jZi (ω), we obtain from Fig. 2 Z(ω) =

R + jωL 1 − ω 2 LC + jωRC

(1)

where ω = 2πf is the radian frequency. The impedance Z(ω) in (1) presents two (real) equations at each frequency to determine the three variables, L, R, and C. However, as explained shortly, the capacitance C, assumed to be frequency invariant, can be explicitly computed from the resonant characteristics, thus leaving two degrees of freedom for the calculation of L and R from the two independent equations in (1). Z(ω) resonates at the frequency ω0 with a quality factor Q, both of which can be obtained explicitly from the measured data. The resonant frequency is determined from measured data as the frequency at which Re(Z) is maximum, which is close to the frequency defined by the circuit theory criterion Im(Z) = 0 (see Table I and relevant discussion) and Q is computed as Q = ω0 /Δω, where Δω is the measured 3 dB bandwidth of the impedance. Using circuit theory, the resonant frequency and the quality factor for Z(ω) may also be deduced from Fig. 2 as ω0 = √ Q=

1 L0 C

Q2 1 + Q2

1/2

ω0 L0 R0

(2) (3)

where L0 and R0 denote inductance and resistance, respectively, at the resonant frequency. The impedance Z(ω) in (1) may then be rewritten in terms of resonant frequency as follows: Z(ω) =

R + jωL2 C(ω02 − ω 2 ) . (1 − ω 2 LC)2 + (ωRC)2

(4)

It is emphasized that at RF, both resistance R and inductance L are, in general, frequency-dependent, because the permeability and resistivity of the ferrite-core material vary significantly with frequency around resonance. The ferrite permittivity, however, is not a strong function of frequency. Hence, we assume that the stray capacitance C, which is affected only by the permittivity of the ferrite core, remains constant with frequency. At the resonant frequency ω0 , let Z0 ≡ Z(ω0 ) (a real number) be the impedance. Using (2) and (3) in (4), with ω = ω0

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substituted, we obtain at resonance R0 =

Z0 1 + Q2

(5)

L0 =

QR0 . ω0

(6)

The self-capacitance, assumed to be constant about the first resonant frequency, is then calculated using (2) and (6): Δ

C = C(ω0 ) =

1 ω02 L0

Q2 . 1 + Q2

(7)

Therefore, knowledge of the measured resonant frequency, quality factor, and the impedance at resonance, facilitates unique determination of one of the three equivalent circuit parameters for the RF inductor, namely, the frequency-invariant series capacitance C. After some algebra, the remaining two variables, the frequency-dependent inductance and resistance, can also be obtained in closed form in terms of the measured impedance (see Appendix) Zi (ω) + ωC |Z(ω)|2 [1 + ωCZi (ω)]2 + [ωCZr (ω)]2

(8)

Zr (ω) . [1 + ωCZi (ω)]2 + [ωCZr (ω)]2

(9)

ωL(ω) =

R(ω) =

To the best of our knowledge, (7)–(9) are the most accurate expressions available to calculate the equivalent circuit parameters of an RF inductor. These simple equations are valid at frequencies beyond the SRF, and correctly predict the RF equivalent circuit parameters for any inductor that can be characterized by the equivalent circuit in Fig. 2. As shown in Section IV, the model produces less than 1% error in comparison with the measured impedance for both toroidal and solenoidal inductors. A similar agreement has been observed in [19] for SMD inductors. As a CAD tool, this equivalent circuit model is expected to be invaluable to RF engineers for the design and characterization of EMI suppression components, as it simplifies the analysis considerably without compromising accuracy. A discussion on the capacitance calculation is in order. Previous investigations [4], [6], [16] have estimated the selfcapacitance to be inversely proportional to the low-frequency L instead of the inductance at resonance. However, the inductance calculated from (8) for RF inductors yields a value at resonance that is larger than its low-frequency value. Hence, it appears that the previous studies overestimate the capacitance in comparison with the analytical formula (7), which uses the inductance at resonance instead of the quasi-static value. B. Calculation of Effective Permeability The rod permeability of a slug, defined as the relative permeability of a hypothetical magnetic circuit, which provides the same reluctance as the magnetic core [5], is difficult to measure accurately because of the large magnetic flux leakage path for cylindrical cores. Next, we will describe an approximate method to deduce the effective permeability of slugs as well as toroids, which yields consistent results and corroborates well

with available manufacturer’s specifications. As demonstrated by the results in Section IV, this method provides the most accuracy for RF inductors on toroidal cores, in which the intrinsic permeability can be measured accurately because of a closed magnetic flux path confined to the core. In order to estimate the effective permeability of open cores (such as ferrite slugs), which is much lower than the intrinsic permeability for a toriodal core of the same material, we first measure the impedance of a ferrite-core coil and calculate the inductance and resistance from (8) and (9), respectively. Next, we measure the impedance of an identical air-core coil (obtained by removing the ferrite core from the coil) and calculate the inductance La and the resistance Ra , as described shortly. The air-core coils have Q’s and resonant frequencies much higher than those of the corresponding ferrite-core inductors. For example, a 5-mm diameter, 20-mm long slug-type ferrite inductor of 12 turns has a resonant frequency of 137 MHz and a Q of 10, whereas the corresponding coreless inductor resonates at 450 MHz and has a Q of 779. This means that the capacitance of the air-core coil can be neglected, and the coil can be represented accurately by a series RL equivalent circuit. Therefore, if we measure the complex impedance Zair = Rair + jXair of the air-core coil, we can approximate Ra ≈ Rair and La ≈ Xair /ω within the desired frequency range. The effective relative permeability μe = μe − jμe is then calculated using

μe (ω) =

L(ω) La

(10)

μe (ω) =

R(ω) − Ra ωLa

(11)

where the inductance L and the resistance R for the ferritecore inductor are computed using (8) and (9), respectively. The permeability model in (10) and (11) assumes that the coil winding resistance is unchanged by the presence of the ferrite core, i.e., if we write the resistance of the ferrite-core inductor as R = Rc + Rw , where Rc accounts for core losses and Rw for winding losses, then (11) assumes that the winding resistance is only slightly perturbed from the value, Rw ≈ Ra . This is true only if the current distribution in the coil is unchanged by the additional magnetic flux introduced by the ferrite core. The winding losses at high frequencies are dominated by the skin effect, which depends strongly on the current distribution. An investigation of the nonuniformity in current distribution introduced by the ferrite core is needed in order to determine the upper frequency limit of this assumption. The validity of this assumption in (11) will be qualitatively established in Section IV using measured data. In the case of toroidal inductors, we have corroborated the measured air-core inductance with the well-known static formula available in standard references (cf., [2]). We have also observed good agreement (within 10%) between the permeability computed for a toroidal ferrite core using (10) and (11), and the measured intrinsic permeability (see Fig. 6).


Fig. 4. Fig. 3. Measured complex relative permeability of 4B1 toroidal core. (a) Real part μ r . (b) Imaginary part μ r . Permeability measurements conducted using HP 4291 Material Analyzer and HP 16454 Magnetic Material Fixture.

IV. MEASURED RESULTS We have measured the impedance of toroidal and solenoidal inductors, and calculated their equivalent circuit parameters, as well as the effective permeability of the core, using the expressions derived in the previous section. In this section, we present the results on the following inductors: 1) A 6-turn coil (AWG 20.5) wound on a ferrite toroid (Philips 4B1 nickel-zinc) of outer diameter 14 mm, inner diameter 9 mm, and height 5 mm. This will be referred to as Inductor 1. 2) A 12-turn coil (AWG 16) wound on a ferrite slug (Philips 4B1 nickel-zinc) of diameter 5 mm and length 20 mm. This will be referred to as Inductor 2. 3) An 11-turn coil (AWG 18) wound on a ferrite slug (FairRite 43 nickel-zinc) of diameter 5 mm and length 15 mm. This will be referred to as Inductor 3. A. Toroidal Inductors Fig. 3 displays the intrinsic complex relative permeability of the toroidal 4B1 ferrite core from 1 MHz to 1.8 GHz, measured on the HP 4291A Material Analyzer with the HP 16454 Magnetic Material Coaxial Fixture. The permeability of a rod core cannot be measured directly because of the large leakage path of the magnetic flux. Later in this section, we present results on the effective rod permeability calculated from impedance measurements for helical coils wound on slugs. The permeability at 1 MHz (see Fig. 3) is about 229 and relates to the low-frequency initial permeability of 250 specified in the 1998 Philips Catalog (No. MA01) for Soft Ferrites [21]. The real part increases to about 265 at 9 MHz, and then falls steadily with increasing frequency, with a value of about 32 at 100 MHz. The rolloff frequency corresponding to the magnetic loss peak occurs at 25 MHz, where the imaginary part is maximum. The traditional low-frequency models of ferrite inductors (cf., [6]) are valid below this frequency. An approximate value of saturation flux density Bsat in the core can be calculated using Snoek’s law given by 4πγBsat = μi ωres , where μi is the initial relative permeability, γ is the gyromagnetic ratio (a constant ∼1.76 × 1011

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Measured impedance of the 4B1 toroidal inductor.

C/kg in MKS units), and ωres = 2πfres denotes the radian frequency at the loss peak [5]. Since ferrite cores used in low-power applications (such as cable television) are concerned with magnetic parameters below this frequency, rarely does the ferrite manufacturer publish data for complex permeability above this frequency. However, higher frequency data are essential when specifying ferrite cores used in the suppression of RFI, and core saturation effects take on paramount importance if high currents are generated by the noise source (such as permanent magnet DC motors in automotive applications). Fig. 4 depicts the measured impedance of (toroidal) Inductor 1 over a frequency range from 1 to 500 MHz. The resonant frequency (where the reactance is almost zero) is measured as 102.16 MHz and the Q at resonance as 0.4352. The peak resistance is about 906 Ω. It is clear from Fig. 4 that the reactance is predominantly inductive below the resonant frequency and capacitive above that frequency. The equivalent series inductance and resistance of the toroidal inductor, computed from the measured impedance using (8) and (9), respectively, are plotted in Fig. 5 as functions of frequency. For comparison, two curves are plotted: 1) equivalent circuit parameters computed using the resonant frequency definition corresponding to maximum Zr (ω), the real part of the measured impedance; and 2) those computed using the resonant frequency at which Zi (ω) 0, where Zi (ω) is the imaginary part of the measured impedance. For convenience, we refer to the former as Case B, and the latter as Case C. We infer from Fig. 4 that Case B gives a resonant frequency of 115.77 MHz, while Case C yields 102.16 MHz, a discrepancy of 13%. The discrepancy is due to the low Q of the inductor, and not an error in the measured result. For high Q circuits, these two definitions of the resonant frequency yield the same result. Despite this discrepancy in resonant frequencies, as summarized in Table I, the deduced model parameters at resonance for these two cases, such as equivalent inductance L0 from (6) and capacitance C0 from (7), are all very close, suggesting that Case B definition of resonant frequency is acceptable, and provides an accurate model. This validation is very useful, because in practice, it is easier to measure the peak of Zr (ω) than the zero or minimum of Zi (ω). Henceforth, we will employ Case B definition. The low-frequency inductance in Fig. 5 is 3.6 μH at 1 MHz and decreases to 0.5 μH at coil resonance. This is a large swing

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Fig. 5. Computed (a) resistance and (b) inductance of the 4B1 toroidal inductor.

Fig. 6. Computed relative permeability of the six-turn toroidal inductor compared with measured intrinsic permeability of a 14/9/5 4B1 ferrite toroid. (a) Inductive permeability. (b) Resistive permeability.

in contrast to helical coils, which exhibit much smaller change in inductance at resonance from its low-frequency value (see Fig. 9). The reason is perhaps the complete trapping of the magnetic flux inside the toroidal core in contrast to a large leakage flux outside the rod cores. This conjecture is supported by an order of two magnitude reduction in the effective permeability of the rod core in comparison with the material’s intrinsic permeability, as will be demonstrated shortly. Equivalent capacitance of the toroidal inductor, computed from (7) is 0.69 pF over the entire band. Fig. 6 displays the effective (in this case, intrinsic) permeability of the 4B1 toroidal core, computed from the extracted equivalent resistance and inductance of Inductor 1, as described in Section III. The nominal low-frequency inductance La of the corresponding air core is about 0.2 μH. The winding resistance Ra remained less than 2.5 Ω between 10 and 500 MHz, which is much smaller than the peak resistance of 906 Ω for the ferrite inductor. In order to validate the computed permeability in Fig. 6,

measured values of the permeability are replotted from Fig. 3 at a few frequencies. Excellent corroboration is observed between model and measurement for the inductive permeability, whereas about 12% discrepancy occurs in the resistive permeability at frequencies near the rolloff frequency. This may be due to neglecting the effect of ferrite core on the winding resistance (see (11) and the discussion thereof). For further verification of permeability calculations, we plot in Fig. 6(a) the permeability obtained using the series inductance calculated from measured impedance as L = Zi (ω)/ω, the number of turns, and the geometric core constant C1 = i i /Ai , where i is the effective path length and Ai is the area of cross section for a uniform (ith) segment of the toroidal core. This quasi-static model neglects frequency-dependent effects. The core constant is calculated as C1 = 2.84 mm−1 from the toroid’s physical dimensions according to IEC Standards Document 205, as outlined in [21]. The formula for the relative


Fig. 7.

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Measured impedance of the 4B1 solenoidal inductor (Inductor 2).

Fig. 8. Measured impedance of the Fair-Rite 43 (ATR) solenoidal inductor (Inductor 3).

permeability is given by μe (ω) =

L × C1 1.257N 2

(12)

where N is the number of turns, C1 is in mm−1 , and L is in nanohenry. This inductive permeability, calculated using the measured series inductance for L, is plotted in Fig. 6(a). It is observed that the empirical formula (12) corroborates well with measurements up to the rolloff frequency occurring at 25 MHz (less than 5% error). At higher frequencies, this formula deviates significantly from measurements, because it is derived from a static magnetic flux distribution, which does not consider core saturation, core losses, and frequency-dependent magnetic domain effects that become important above the rolloff frequency. The accuracy of static formulas thus becomes questionable at the higher frequencies, and the measured results presented in Fig. 3 should then be used for the inductive permeability of toroidal cores. Alternatively, the frequency-dependent permeability extracted from the circuit model in (10) and (11), and plotted in Fig. 6, may also be used. This conclusion emphasizes the necessity for engineers using ferrite cores for RFI suppression to experimentally characterize the intrinsic permeability of core materials at RF. The core manufacturer does not supply this information, as it typically involves expensive RF instrumentation. Instead, the manufacturer specifies the low-frequency permeability computed in (12).

Fig. 9. Computed (a) resistance and (b) inductance of the 4B1 and ATR soleniodal inductors.

B. Solenoidal (Helical) Inductors Fig. 7 depicts the measured impedance of Philips 4B1 helical inductor (no. 2) over a frequency range from 10 to 500 MHz. The resonant frequency and the Q at resonance have been measured as 137.4 MHz and 10.5, respectively, and the peak resistance is about 12 kΩ. The inductor is designed for a nominal inductance of 2 μH. It is again evident that the reactance is predominantly inductive below the resonant frequency and capacitive above that frequency. We plot in Fig. 8 the measured impedance of the 11-turn ATR inductor (no. 3) that employs Fair-Rite 43 ferrite material, and is designed for a nominal inductance of 1.4 μH. The resonant frequency is measured as 150.9 MHz and the Q at resonance as 10. The peak resistance is about 8.2 kΩ. Next, we evaluate the equivalent circuit parameters for both of these inductors, computed from the measured impedance as described in Section III. The equivalent series inductance and resistance, obtained from (8) and (9), respectively, are depicted in Fig. 9 as functions

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of frequency. In contrast to a toroidal inductor, resistance of the helical inductor increases monotonically with frequency. For Inductor 2, the resistance reaches 250 Ω at the resonant frequency of 137.4 MHz, and a peak value of 7.3 kΩ at 387.3 MHz. The small size of the solenoidal core causes the resistance peak at the latter frequency, which appears to be pertinent to loss peak in the magnetic core subject to leakage flux. Inductor 3 exhibits a resistance of 210 Ω at the resonant frequency of 150.9 MHz, and a peak value of 6 kΩ at 370.4 MHz corresponding to loss peak in permeability. The inductance stays relatively constant well beyond the first resonance (from 10 to 300 MHz), with a value between 1.98 and 2.15 μH for Inductor 2, and between 1.35 and 1.55 μH for Inductor 3. These values compare very well with the nominal value of 2.0 and 1.4 μH, respectively, for Inductors 2 and 3, measured at 20 kHz on a low-frequency LCR bridge. Past the frequency at which resistance peaks, inductance falls rapidly and becomes negative around 380 MHz, causing the closed-form model in (8) to become invalid. Therefore, inductance in Fig. 9 is not plotted beyond 400 MHz. The frequency-independent stray capacitance, calculated from (7), is 0.66 and 0.8 pF, respectively, for Inductors 2 and 3. We may then conclude that the equivalent circuit behavior with frequency is consistent between these two inductors. It is worth mentioning that the nominal capacitance of 0.8 pF for Inductor 3 is more reasonable than the 4.188 pF that has been estimated using the experimental method in [11]. In this paper [11], the capacitance appears to be grossly overestimated for reasons cited following (9) at the end of Section III-A. Fig. 10 shows the effective permeability of the Philips 4B1 ferrite rod core (Inductor 2) and the Fair-Rite 43 ferrite rod core (Inductor 3), computed using the equivalent series resistance and inductance from Fig. 9, as described in Section III. The striking similarity between the frequency dependence of the toroidal core (intrinsic) permeability in Fig. 6 and the effective rod permeability in Fig. 10 is noted. We observe that the rolloff around 400 MHz in the real part of permeability [see Fig. 10(a)] is associated with the corresponding loss peak in imaginary part of the permeability [see Fig. 10(b)]. This indicates that the permeability rolloff in Fig. 10 indeed corresponds to magnetic material behavior near material resonance, akin to the permeability rolloff in Fig. 6. This important observation needs further research for validation. The low-frequency inductive rod permeability (at 10 MHz) is about 9.5 for the 4B1 core, which is two orders of magnitude smaller than the intrinsic permeability (∼230) from Fig. 6, and can be justified physically by the large leakage magnetic flux of the rod core. The inductive permeability varies only slightly, between 8.5 and 9.5 for Inductor 2, or 6 and 7 for Inductor 3, over a wide frequency range spanning 10 to 300 MHz. This shows that our model is very stable in spite of the large leakage flux and its effect on inductance. The resistive permeability increases with frequency till the material loss peaks around 400 MHz for both inductors, and then falls to small values, akin to the resistive permeability in Fig. 6 for the toroidal core. Over a span of 300 MHz, resistive permeability also varies slightly, between 1 and 3.4 for Inductor 1, or between 0.2 and 2.5 for Inductor 2.

Fig. 10. Computed permeability of the 4B1 and ATR solenoidal inductors: (a) inductive permeability and (b) resistive permeability.

C. Model Error Since the RF equivalent circuit model for the inductor is based on closed-form derivation of frequency-dependent circuit parameters from measured data, the error between model and measurement is expected to be very small. This should be contrasted with model parameters determined using numerical optimization, in which case, the error is small only in selective regions of validity (e.g., just around local minima of the penalty function). We calculate the relative model error as |(y − yˆ)/y| × 100, where y is real (imaginary) part of the measured impedance, and yˆ is real (imaginary) part of the model impedance. The latter is calculated by substituting in (1), the equivalent circuit parameters determined from the measured data using (7)–(9). As an example, Fig. 11 depicts the relative model error for the 4B1 toroidal inductor. The maximum error is less than 1% for both real and imaginary parts of the impedance, thereby demonstrating accuracy of the equivalent circuit


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It is emphasized that R and L are explicit functions of frequency; the subscript on C in (13) simply indicates that the capacitance C0 is constant with frequency, as given by (7). R may be expressed in terms of L using the real part in (13) 1 ± 1 − (2Yr ωL)2 , 2Yr ωL < 1. (14) R= 2Yr Note that R is positive for both signs before the radical in (14). It is not necessary to choose one sign or the other, as clarified in (17). A squaring operation removes the sign ambiguity. The imaginary part in (13) may be written as follows: Fig. 11. Percentage relative error between measurements and the equivalent circuit model for the real and imaginary parts of the impedance of the 4B1 toroidal inductor. The model is calculated from (1) by substituting the equivalent circuit parameters from (7)–(9). The model is in excellent agreement with the measured data in Fig. 4, with maximum relative error less than 1%.

parameters, and validating the assumptions used in the model. Similar corroboration has been observed for other inductors. V. CONCLUSION We have presented a rigorous experimental method to fully characterize the RF behavior of ferrite-core inductors over a wide frequency band, by considering all the parasitic effects, such as flux leakage, stray capacitance, core losses, etc. It is believed that the proposed characterization method will be useful to engineers designing RF noise suppression circuits in automotive and wireless communications applications, where maximum noise suppression is desired in the frequency range from 100 MHz to 1 GHz, and no single reliable technique exists to characterize inductors at these frequencies. The equivalent circuit parameters of the inductor, as well as effective permeability of the core material, have been extracted in closed form from measurements of the RF impedance, performed using an impedance analyzer. Sample results measured on several ferrite inductors, both toroidal and solenoidal, yield consistent data, which corroborate well with manufacturer’s specifications where available, and with independent permeability measurements on the HP 4291 Impedance/Material Analyzer for the toroidal cores. The equivalent circuit model described in this paper provides rigorously derived formulas to aid the design of ferrite inductors. The proposed model is expected to shorten the length of design cycles considerably by mitigating repetitive measurements to characterize a wide array of inductors. APPENDIX We now derive the equivalent circuit model parameters in (8) and (9) for the frequency-dependent inductance and resistance, respectively. To derive the inductance L(ω) in (8), we begin with the admittance in complex form, given by reciprocal of Z(ω) in (1) R L + jω C0 − 2 . Y = Yr + jYi = 2 R + ω 2 L2 R + ω 2 L2 (13)

ωLYr . (15) R Substituting for R from (14) in (15) yields an equation for L of the form Yr 1 and X = . X ± X 2 − 1 = A, where A = ωC0 − Yi 2ωLYr (16) Transposing X to the right side of (16), squaring both sides, and solving for X in terms of A, we obtain Yi = ωC0 −

1 + A2 (17) 2A irrespective of the sign in front of the radical. Back substituting for X and A from (16) and rearranging terms results in the sought-after expression for L(ω) X=

ωL(ω) =

1 . ωC0 + ((|Y |2 − ωC0 Yi )/ωC0 − Yi )

(18)

Realizing that |Y | = 1/ |Z| , Yr = Zr / |Z|2 , and Yi = −Zi / |Z|2 , (18) after simplification yields (8). Substitution of ωL from (18) into (15) results in the expression for the resistance Yr (19) R(ω) = 2 (ωC0 ) − 2ωC0 Yi + |Y |2 Once again, writing Yr , Yi , and |Y | in terms of Zr , Zi , and |Z|, respectively, we obtain R(ω)in (9). REFERENCES [1] C. R. Paul, Introduction to Electromagnetic Compatibility, 2nd ed. New York: John Wiley, 2006. [2] H. W. Ott, Noise Reduction Techniques in Electronic Systems, 2nd ed. New York: John Wiley, 1988. [3] R. W. P. King, Electromagnetic Engineering, Vol. I: Fundamentals. New York: McGraw-Hill, 1945, p. 463. [4] M. Bartoli, N. Noferi, A. Reatti, and M. K. Kazimierczuk, “Modeling winding losses in high-frequency power inductors,” J. Circuits, Syst., Comput., vol. 5, no. 4, pp. 607–626, 1995. [5] E. C. Snelling, Soft Ferrites, Properties and Applications, 2nd ed. Boston, MA: Butterworth, 1988. [6] Y. Liu, “Power system equipment modeling based on wide frequency external impedance measurements,” Ph.D. dissertation, The Ohio State Univ., Columbus, OH, 1989. [7] Soft Ferrite Components and Accessories, MMG Product Catalog, Issue 1A, 1995. [8] Fair-Rite Soft Ferrites Catalog, 13th ed., Fair-Rite Products Corp., Wallkill, NY, 1996. [9] A. Massarini and M. K. Kazimierczuk, “Self-capacitance of inductors,” IEEE Trans. Power Electron., vol. 12, no. 4, pp. 671–676, Jul. 1997.

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[10] K. Naishadham, “A rigorous experimental characterization of ferrite inductors for RF noise suppression,” IEEE Radio Wireless Conf., Denver, CO, pp. 271–274, Aug. 1999. [11] Q. Yu and T. W. Holmes, “A study on stray capacitance modeling of inductors by using the finite element method,” IEEE Trans. Electromagn. Compat., vol. 43, no. 1, pp. 88–93, Feb. 2001. [12] R. F. Huang, D. M. Zhang, and K.-J. Tseng, “An efficient finite-differencebased Newton_Raphson method to determine intrinsic complex permeabilities and permittivities for Mn–Zn ferrites,” IEEE Trans. Magn., vol. 42, no. 6, pp. 1655–1660, Jun. 2006. [13] D. Zhang and C. F. Foo, “A practical method to determine intrinsic complex permeabilities and permittivities for Mn–Zn ferrites,” IEEE Trans. Magn., vol. 41, no. 4, pp. 1226–1232, Apr. 2005. [14] R. F. Huang, D. M. Zhang, and K. -J. Tseng, “Determination of dimensionindependent magnetic and dielectric properties for Mn–Zn ferrite cores and its EMI applications,” IEEE Trans. Electromagn. Compat., vol. 50, no. 3, pp. 597–602, Aug. 2008. [15] M. K. Kazimierczuk, G. Sancinento, G. Grandi, U. Reggiani, and A. Massarini, “High-frequency small-signal model of ferrite core inductors,” IEEE Trans. Magn., vol. 35, no. 5, pp. 4185–4191, Sep. 1999. [16] Q. Yu, T. W. Holmes, and K. Naishadham, “RF equivalent circuit modeling of ferrite core inductors and characterization of core materials,” IEEE Trans. Electromagn. Compat., vol. 44, no. 1, pp. 258–262, Feb. 2002. [17] J. Shenhui and J. Quanxing, “An alternative method to determine the initial permeability of ferrite core using network analyzer,” IEEE Trans. Electromagn. Compat., vol. 47, no. 3, pp. 651–657, Aug. 2005. [18] K. Naishadham, “Extrinsic equivalent circuit modeling of SMD inductors for printed circuit applications,” IEEE Trans. Electromagn. Compat., vol. 43, no. 4, pp. 557–565, Nov. 2001. [19] K. Naishadham and T. Durak, “Measurement-based closed-form modeling of surface-mounted RF components,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2276–2286, Oct. 2002. [20] Measurement of Impedance using the HP 4291 Impedance Analyzer, HP Product Note 4291-1, Hewlett-Packard, Palo Alto, CA, 1994. [21] Philips Soft Ferrites Data Handbook, MA-01, Philips, Amsterdam, The Netherlands, 1998.

Krishna Naishadham received the M.S. degree from Syracuse University, Syracuse, NY, and the Ph.D. degree from the University of Mississippi, Oxford, MS, both in electrical engineering, in 1982 and 1987, respectively. He was engaged with the faculty of electrical engineering for 15 years at the University of Kentucky, Wright State University as Tenured Professor, and Syracuse University as an Adjunct Professor. He taught courses in electromagnetics, microwave engineering and antennas, and performed sponsored and unsponsored research on a variety of applied electromagnetics (EM) topics, graduating three Ph.D. students and several M.S. students. In 2002, he joined Lincoln Laboratory, Massachusetts Institute of Technology, Cambridge, MA as a Research Scientist and contributed innovative asymptotic techniques and spectral estimation methods for the electromagnetic signature analysis of large objects containing sections with small features. In 2008, he joined Georgia Institute of Technology, Atlanta, GA, where he holds a joint appointment with the Georgia Tech Research Institute as a Principal Research Scientist, and the ECE Department as a Research Professor. He is currently leading research projects on novel multifunctional antenna design for aerial platforms, and carbon-nanotubebased chemical sensors. He served as an Associate Editor for the Applied Computational Electromagnetics Society Journal. He is currently an Associate Editor of the International Journal of Microwave Science and Technology. He has authored or coauthored four book chapters and more than 150 papers in professional journals and conference proceedings on topics related to computational EM, high-frequency asymptotic methods, antenna design, EMC, materials characterization, and wave-oriented signal processing. Dr. Naishadham is currently the Chair of the Joint IEEE AP/MTT Chapter at Atlanta and serves on the Technical Program Committee for the International Microwave Symposium.